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OPTIONS, CUTUkES, AN: OTHtR DERIVATIVES Jehn C. llnn and Risk Management Mapk Finatlcial tzrpvProfessorpf Derates Josep L. RotmanScool ofManagement Universityof Toronto
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Contents
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Contents
c onten j J
apter 21
atl*ngYolatl-ll-tl-es n n1 c0aela:-0ns..............................'................................ 469 Iqstl':latl-llgvolatl-h-ty 49 he exponentiallyweighted movingaveragemodel 471
e2l 0Ph'0nS................................................................................................. a1 33.1 alll-tall-llvestrflellt apprn.l-s valllatl-onfrarnework 33.2 Ilxtensl-o11 Of tlle rl-sl:-llelltral llsti:fl o f r1-sk........................................................ arl:etIlr1'ce atil)g 3.3 tlle rn 3 Of va1ll ation a llllsilless :1 to tlle 33.21 Ilpll'catl-o ..................'...............................................
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Contents
BUSINEjSSjAPSHOTS
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TECHNICAL NOTES ,
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Availableon the Author's Website WWW.rotman.utoronto.ca/-hull
1. ConvexityAdjustmentsto EurodollarFutures 2. Propertiesof the LognormalDtribution 3. Wanant ValuationWhenValueof Equityplus Warrantsls Lognormal 4. ExactProdure for Valuing mericanCallson StocksPayinga SingleDividend 5.,Calcplationof the CumulativeProbabilityin a BivariateNormalDistribgtion 6. DiferentialEquationfor Pri f a Derivativeon a jtock Payinga Known DividendZeld 7. DiferentialEquationfor Priceof a Derivativeon p, FuturesPri 8. AnalyticApproximationfor ValuingAmericanOptions Procedure 9. GeneralizedTree-Building 1. The Cornish-FisherExpansionto EstimateVaR 11 Manipulationof CreditTransition Matrics 12. Calculationof CumulativeNoncentralChsquare Distribution 13. Ecient Produre for ValuingAmerican-styleLookbackWtions 14. The Hull-WhiteTwo-FactorModel Bondsin a One-FactorlnterestRate Model 15. ValuingUptions on Coupon-Bearing of 16. Construction an Interest Rate Tree with NonconstantTime Stepsand NonconstantParameters l7. The Processfor the Short Rate in an HJM Term StructureModel 18. Valuationof a CompoundingSwap 19. Valuationof an EquitySwap 2. A Generalizationof the llkk-NeutralValuationResult 21. Hermite Polynomials and Thir Usefor lntegration 22. Valuatipnof a VarianceSwap .
prefAe Itis sometimeshardforme to believethatthefirsteditionof tilisbookwas only 33 pages and 13chapters long!Thebook has grown nd beenadaptedto keepup
ke contracts, an electivecoupe in investmentsprior to taking coufse basedon thisbok. Optiotls,Fgllfre, atld Oter Derivativescan be used fpr a flrstcourse in derivativesor for a more advnced course. There are many diferent ways it can be used in the classroom.lnstructorsteachinga firstcourse in derivativesare likelyto want to spend mostclassroom timeon thefrst half qfth book.lnstOctors teachinga more advanced coursewill findthat many diferent combinations of chapters in the second half of the book can e used. 1findthat the mpterial in Chapter34 works well at the en of either an introductoryor ap advancedcourse. 'a
Ghat': New? Materialhas been updated and improvedthroughoutthe book. The changes in the seventheditionincludethe following: '
1. The lnternational edition contains a new chapter on derivativesmarkets in developingcountries (Chaptef14).Tls focuss particularly on China,India, and Brazil. xix
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2. The chapter on citdit deritives (Chapter23) haS been expanded. lt discusses h0w the mortgages weyesecursubprimemortgage lendingin thr United states, crdit also provies cris. resultant It itized,and the a more complete treatmentof the valuation of CD0s, the implementationof the Gaussiancopula model,and alternativesto the Gaussiancopula model. 3. Options on futurysare n0w coveredin a seppratechapter from options on indices .and currencies (Chaptersl5 and 16).Therehas also been some restructuring of material.Chapter 15, which covers options on indittk and currencies,fuit gives examplesof howindexand currencyoptions can be usedand then covers valuation issues.This makes the chapter E0wbetter.Chapter 16, which covers options on futures, provides more dtails on how Black'smdel is psed as an altrnatlve to Black-scholesfor valuing a wide rane of Europeanoptions. 4. Chapter 17, which covers Greekletters,has alsobein restructured. Ddta, gamma, theta,vega, etc-, are Erst explained in the context of an option on non-dividendc payingstock.Formulasfor theGreeklettersfor other typesof ptiohs are givenin a tabletowardthe end of the chaptef. 1findthisapproach works wdl when I teachthe
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5. Chapter 4 contains a more detaileddescriptionof liquidty preirence theory and howbnks lanage net Interestincome, 6. Issues associayedwith tailingthe hedgeare covred in Chajter 3. This resolves a minor inconsistencybetweenformulasin the book and thoseused in CFA exams. 7. Many new topicsare covered. For example: (a) The V1Xvolatility indexis explainedin Chapters13 nd 14. (b) Varianceswaps,Volatilityswaps,and theirvaluation are cogeredin Chapter)4. (c) Tmnsactionsinvolvingcredit indicesare explained in detailin Chapter23. (d) Theri is more material on volatilitysmilesin Chapter18.The appendix includes an explanation (witha numerical example)of howth probability distribution for an asset price at a futuretimecan be calculated fromimpliedvolatilities. (e) Lookbackoptions al.e covere in mor detailin Chapter24, with flxedlookbacksbeingdistinguishe fromEoatinglookbacks. (f) Futures-styleoptions are covemdin Chapter 16. the symbol4, whkh 8. One small change has been made to the ntation dnotes normal distribution.As is the usul practici, the second arpment of 4 is Standard deviationof the dtribution. now the variance rather than the 9. New end-of-chapter problems havebeen added. '
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Sqware Version1.52of the widselyacclaimedsoftware,DerivaGem,is includedwith this book. An installationroutine is providedwith thisversionof thesoftware.Thislads ftlesinto the correct folders,creates icons, and makes it easier for studets to start using the software.DerivaGm consists of tw0Excelapplicatios: theOptiozsCalculatorand the Applicatiozs Builder. The OptilmsCalculatorconsists of easy-tus softwarefor Applications Builder options. consists The of a numbrr of valuing a wide range of Excelfunctionsfromwhih users can buildtheir0wn applications. lt includesa number of sample applications and eables studentsto explore the properties of optipnsand
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numerkal produres more easily.It also allows more interestingassignments to be designed.The software is describedmorefullyat the end of the book.Updatesto the softwarecan be downloadedfromthe author's website: hull wwprotman.utronto.ca/ew.
Slldes several hundmd Powerpointslides can be downloadedfrom Pearson's Instructor
kesurceCepterof frn my own websiti. Instructorswho adopt the text are welcome to adapt the slides to meet their ownneeds. SolutionsManual As in the sixth edition, end-of-chapterproblems are dividedinto two groups: S'Questions and Problems'' and 'sAssignmentQuestions''. to the lestipns and solutions Problems are in Optioss, Futures, asd Otber Derivatives e, Ihtersatiozal Editioz: whih is published by Pearson and can be SolutiozsJ/'JFII/J/ (ISBN:9135026369)1 purchased by students.
Ihstrudors Manual and t'Questions and Probto ll' qstions (both''AssignmentQuestions'' tz: Derivatives f,fcrrwfprlu/ Futures, Editioz.. e, ) are in optiozj, oter lems' I'utructors vtwpw/ (IsBN: 1314589x), which is made available by pearson to adoptinginstructois.The Isstructors Mwll also icludes nptes op the teachingof each chapter, test bank qestions, some notes on course organization, and questions and problems that are not in the 1)ook The test bank questions are also available from the PearsonInstructor Resour Centef.
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Technkal Notes Technical ptes are used to elaborafe'on pointsmade in the text.They are referred to in the text and an be downloadedfrom my website. By not includingthe Notes'' in my book, I am able to streamline the presentation of material so that it is morqstudent friendly. 'drrezhnical
Acknowledgments Many people haveplayed a part in the production of thisbook.Academics,students, 4ndpractitiners who have made exllent and useful suggestions includeFarhng
Aslani, Jas Badyal,Emilio Barone, GiovanniBaron-/gi, .AlexBergkr, George Blazenko,LaltrenceBooth, PhelimBoyle,Peter Carr, Dn Chance,..J.-P.Chateau, Michel Crouhy,Empuel Derman, Ren-Raw Cen, Dan Cline,GeorgeConstantinides, Drabin,JeromeDuncan,steinar BrianDonaldson,DieterDorp, scott Ekern,David Richard Goldfarb, Dajiang Guc, Jrgen Forfar, David Fowler, Louis Gagnon, Hallbeck,Ian Hawkins,Michael Hemler,SteveHeston,BernieHildebrandt,Michelle Hull,Kiyoshi Kato, Kevin Kneafsy,IainMacDonald,BillMargrabe,EddieMizzi,Izzy Nelkin,NeilPearson,PaulPotvin,ShailendraPandit,EricReiher,Richard Rendleman, Gordon Roberts,Chri Robinson, CherylRosen,John Rumsey, Ani Sanyal,Klaus PietSercu,DuaneStock,EdwardThorpe, Michaelselby, Schurger,Eduardoschwartz,
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YisongTian, Alan Tucker,P.V. Viswanath,GeorgeWang,Jason Wei,B0b Whaley, 1anWliite,HailipngYang,VictorZak, an(j saf zemek. who read theoriginalmanuscript for I am particularly grateful to Eduardoschwartz, and tn the srstedition and mademanycommentsthat1edto signiscantimprovements, George Constantinides,Ramon Rabinovitch,and Rkhard Rendleman,who made specifksuggestionsthat led to ippmvtmentsin more rent editbns. The srstsix editions of this book were very popular w11 practitioefs and their commentsand suggestions have ld to many improvementsin th: book. I woul ' like to thank Dan Cline, DakidFodar, and Rkhard Gbldfarb.The MBA partiularly and MFinstudents at theUniversityofToronto avealsoplayed a signifkantrole in the evolutionof the book. I would partkularly liketo thank two MFin students, Derrik Knie and NelsonArruda, who provided exlknt research assistan for this edition. AlanWhite,a ctklleague at theUiversityofToronto,deservesa Special acknowledgment. Alan and l havebeencarrying0ut jointresearch in the area of derivativesfor over25 years. Duringthat timewe havesynt countlesshoursdijcussingdiferentissues clmrning derivatives.Manycf the new ideasin th.book, and mimy of the ntw ways usedto explain old ideas,areas much Alan'sas mine. Alpnread the originalversion of this book very carefullyand made many excellentsuggestionsfor improkement.Alan has also done most of the developmentWorkon the DerivaGemsoftware. Specialthanks are due to man people at Pearsonfor their enthusiasm, advi, qnd encouragemnt. l would parcularly like to thank Donna Batsta, m editor, and Mary-KateMurray,the Enanceassistat ediipr. I am alsogratefulto ScottBarr,Leah Jewell,PaulDonnelly,Maureen Riopelle,andDavidAlexander,who at diferenttimes have played keyroles in the developmentof the book. l welcome commentson the bookfromreaders. My e-mail addressis: '
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hullotm=.utoronto.ca
John HuB JosephL. RotmanSchoolof Management Universitygf Tomlto
Introductipn
In the last 30 years derivativeshavelco>xincreasingiy,ertyM,nan. Futures and options are now traded activelyon many exchangestluoughoutthe world. Many diferent typesof forwardcontracts, swa/s, optitms,and otherderivativesare regularly traded by fnancial institutions,fund managers, and corporate treasurefsin the overthe-cpuntermarket. Derivativesare Qded to bond sues, usedin executivecompensaplans, embedded c>pital in investmensopportunities,and so on, We have now tion reachedthe stage where anyone who works in fnan needs to understand how derivativeswork, howthey are used, and howtheyare priced. A deratiye can be dened as a financialinstrumentwhose value depens o (0r derivesfrom) the vales of Qther, more basic, underlying variables. Veryoften the variablesunderlying erivatives are the prices of traded assets. A stock option, f0r example, is a derivativewhose value is dependenton the price of a stock. However, derivatives can be dependentOn almost any variable, fromfhe price o) hogs to the amount Of snow fallingat a certain ski resort. Sincethe frst edition Of this book was published in 1988there havebeen many developmentsin derivativesmaykets. Thereis now active tradingin crrdit derivatives, electrkityderivatives,weather derivatives,and insuran derivatives.Many new types of interestrate, foreip exchange, and equity derivativeproducts havebeen created. Therehavr been many new ideasin risk management ad rk measurement. Analysts have also becomemore aware of the need to analyze what are knownas real optiolu. This edition of the book reqects a11thesedevelopments. In this openingchapter we take a ftrstlook at forward,futures,and optionsmarkets and provide an overviewof llowthey are used by hdgers, speculators, and arbitrageurs.Later chapters will give more detailsand elaborate pn many of the points made
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1.1
EXCHANGE-TRADED MARKETS A derivativesexchange is a markt where individualstrade standardized contracts that havebeendehnedby th exchange.Derivativesexchangeshaveexistedfor a longtime. The Cicago Boardof Trade(CBOT,www.cbot.com) was establhed in 1848to bring 1
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farmersand merchants together.Initiallyits main task wqsto standardize thequanlities ad qualities of the grain that were traded. Withina fewyears the srstfutures-type contractwas developed.It was knownas a to-arriyecoztract. Speculatprssoon became interestedin the contmct and fpundtmdingthe consract to be an attractivealternative k to tradingthe grain itself.A rivalfuttlresexhange, the Chicago ercantile Exchange (CME,www.cme.com), was establishedin 1919.N0wfuturesexchangesexist all over $e world. (Seetable at the end of tllebook.) The Chiago BoardOptionsExchange(CBOE,www.cboe.com) start:d tmdingcall optioncontracts on 16stocks in 1973.Ojtionshad tmdedprior to 1973,but theCBOE succeededin creatipg an orderly market with well-defned contracts. Put option contractsstarted tradingon the exchangein 1977.The CBOEnow trades optins on well over 1,000stoks and mny diferent stock indis. Like futures, options have provedto be vl.y popular contracts. Manyother exchangesthroughoutthe world now trade options. (Seetable at the ed of thebook.)The underlying assetsincludeforeig currenciesand futurescontract,sas well as stocks and stock indis.
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Traditionallyderivativesexchangeshaveused what is knownas tlle openoatcry system. This involvestradersphysicallymeting op the foor of the exchange, shoutihg, and usinga complicated set of hand signalsto indicat the tpdes they would like to carry out. Exhanjes are increasinglyreplcing the open outcrysystemby electrozictrading. 'fhis traders entering theifdesiredtradesat a keyboardand a computer being usedto matchbpyersand sellers.Theopen outcry systemhasits advocates,but, as time jasses, it is becominglessand lesscommonk ginvolves
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MARKETS OVER-THE-CQUNTER .: .
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Not a1l tralng is don 6n excanges. The oyer-te-counter arket is an important alternativeto exchanges and, measured in terms of the total volume of traing, has becope much largerthanthe exhange-traded market. It is a telephone-and computerlinkednetwork of dealers.Tradesarr done oveythephoneand are usually'between two injiiiutionand one of its clients (typically snancialinstitutionsof betweena a crporate treqsureror fundmanager).Financial institutionsoften act as market makers for the more commonly tradedinstrnments.This means that theyare alwaysprepared t quote both a bid price (aprice at which theyare prepared to buy)and an ofer price (a price at which they are prepared to sell). Telephoneconversations in the over-the-countermarket are usually taped.If thereis a disppteabout what was agreed, the tapes are rejlayed to resolve the issue.Tradesin the over-the-counter market are typicallymuch largerthan trades in the exchangetradedmarket. A keyadvantage of the over-the-countermarket is that the terms of a contractdo not haveto be yhosespecifed by an exchange.Marketparticipants are free to negotiate any mutually attractive deal.A disadvantageis that thereis usually some crezit risk in an over-the-countertrade (i.e.,thereis a smnll rijk that the contract will not be honored). As we shall see in the next chapter, exchanges have organized themselvesto eliminatevirtually a1lcredit risk. .
Market Size Both the over-the-counterand the exchangeutradedmarket for derivativesare huge. Althoughthe statistics that are collectedfor the twp markets are not exactly comparable, it is clear that the over-the-countermatket is much largerthan the exchangestartedecbllecting tradedmarket.The Bankfoylnternational settlements (www.bis.org) statisticson the markets in 1998.Figurel compares(a)the estimated total principal amountsunderlying transactionstht wre outstanding in the over-thecounter markets betweenJune 1998 and June 2007 and (b)the estimated totl value of the assets underlyingexchange-traded contfacts duringthe same period. Usingthesemeasures, wese that, bj June2($7,the ovepthe-countermarket had grown to $516.4trillionand the exchange-tradedmarket had grown to $96.7tllion. In interpretingthesenumbers, we should bear in mindthat the principal underlying transactionis not the sameas its value. An exampleof an over-thean over-the-cpunter Colmter Contfad iS 2l agrccment to buy 1($ il' llionUs dollarswith Britishpounds at a predeterminedexchange rate in 1 yeal'. The total principylamount underlying this ttansactin is $100miflion.However,the value of the contract might be only $1million. estimatesthe gross marketvalue of all ovtr-theThe Bankfor InternationalSettlements 1 countercontracts outstanding in June 2007 to be about $11.1trillion. .1
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FORWARDCONTRACTS A relativelysimplederivativtis zforward coztract. It is an agreementto buy or sell an assetat a certain future time f0r a rtain pri. It can be contrasted with a spot l A contract that is worth $1 million to one side and a grossmarket value of $1 million. having
-$1
million to the other side wold be counted as
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CHAPTER 1 Table 1
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USb/GBP excllange
the spotand forwardquotes for USD
rate, July2, 2007(GBP Britishpoupd; is numberof USb per GBP). =
Spot l-monthfomard 3-nionthfofward f-monthfomard
Bid 2.0558 2.0547 2.0526 2.0483
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Us dollar;qute
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Ojhr 2.0562 2.0552 2.0531 2.0489
which is n agreementto bu or sell ah asset today.A fomard contract is contract, tradedin the ver-the-cmnter market-usually betweh two hnanciarlinstitutionsor between a hnancialinstitutionand one of its clients.
One of the partiesto a fomard contract assllmesa Ing position and agreesto buy the underlying asseton a certain specised futuredatefor a certain specised pri. The other party assumes a sbort position and agrees to sell the asset on the jame date for the same price. Fomard contracts on foreignexchange re very popular. Mostlargebanks employ traders.Spot.tradep re trading a foreign both spot and fomard foreign-exchange almost dqlivety. F orwardtraders for immediate currency are tradingfor deliveryat a future time. Table 1.1 provides the quotes on the exchange rate betweenthe British pound (GBP) and the Us dollar (USD) that might be made by a largeinternational bank on July 2, 2007.The quote is for the number of USD per GBP. The srstrow indicatesthat the bank is prepared to buy GBP (alsoknwn as sterling) in the spt market (i.e.,for virtually immediatepelivery) at the rate of $2.0558 per GBP and sell. sterlingin the spdt market at $2.0562 per GBP. ne second, third, and fourth rows indkate that the bank is prepared to buy sterling in 1, 3, and 6 mnths at $2.0547, $2.0526,and $2.0483 pr QBP, respectively,and to sellsterlin in 1, 3, and 6 montlls pt respectiveiy. $2.0552,$2.05j1,and $2,0481 per GBP, Forward contracts can be used to hedgeforeip currency risk. that, on suppose Jly 2, 2007,the treasurerof a Us coporation knowsthat the corporation will pay f 1 million in 6 monthj (i.e.,on January20,2008)and wants to hedgeagainstexchange rate moves. Usingthe qnotes in Table 1.1, the treasurercan agree to buy f 1 million 6 months forward at an exchange rate of 2.0489.The corporaon then has a long forwardconttact on GBP. It hasagreedthat on January2, 2008,it will buyf 1 million million. Thebankhas a short forwardcontract on GBP. lt fromthe bank for $2.0489 million. Both has agreed that on January2, 2008,it will sell f 1 millionfor $2.0489 sideshavemade a bindingcommitment.
Payoffsfrom Forward Contracts Considerthe position of the corporation in the tradewe havejustdescribed.What are thepossible utcomes?The fomprd contract obligatesthe coporaion to buyf 1million lf the spot exchangetate rose to, say,2.1, at the end of the 6 months, for$2,048,900. to the the forWard contrad would be worth $51,100(= $2,100,000$2,048,900) million pounds purchased would enable 1 exchange be at an rate to coporation.lt -
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Figure 1 Payofs frm frward cntracts: (a)lng position, Deliverypri = K; price of assei at contract maturity= S1,. .2
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shrt position.
Payoff
Pamff
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0
K
yy
E
s'z
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ifthe spot exchnye rate fellto 1.9 at theend of 2.0489rathrr than z'.l. similarly, gf the6 npnths, thefrwafd contract woulp havea negativevalue to the corporatin of $148,900becauseit wuld lead to the corporation paying $148,900 more than the market price for the sterling. In general, tht payof from a long position in a forwardcontract n one unit f an asset is y .g
K is the deliveryplice and S is the pot prke qftltheasst at maturity of the cdntrct. Thisis becausetheholderof the cntract is bligatedto buy an asset worth S for.f. similarly, the paytd froma short position in a forwardcontract on one unit f an assd is
Lwhere
K Sy -
These payofli can be positive or negative.They are illustmtedin Figure1.2.Becauseit costsnothing to enter int a frward contract, the payos fromthe contract is also the trader s total gain r lss frm the contract. In the examplejustconsidered,K = 2.0489and tht cporation has a longcntract. WhenSy = 2.1, the payofis $0.0511 per t1;when Sz 1.9, it is per f 1. j
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Forward Prices and Spot Prices We shall be discussingin some detailthe relationship betweenspot and fomard jrices in Chapter5. For a quick previewof why the two are related,consider a stock that pays no dividendand is worth $60.You can borrowor lendmoney for 1 yearat 59:. What shouldthe l-yearforwardjriceof the stock be? The answtr is $6 grossed up at 5% for 1 ytar, or $63.If the fomard price is more than this,say $67,you could borrow$69,buyone share of the Stock, and sellit fomard
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for $67.After paylng ofl'theloan,yo wuld net a prost of $4in 1 yeaf. If the forward priceis lessthan $63,say $58,an investorownng the stock a! part of a portfolio would contrat to Imyit backfor $58in l year. sellthe stok f.or $6 ahd enter into a would proceeds beinvested of investment 'rhe at 5% to eanl $3.Theinvestorwould end up $5betterof than if the stock wefe keptin the portfolio for the year.
tomard
1.4
FUTURESCONTRACTS Likea forwardcontract, a futurescpntmd is an agreementbetweentwoparties to buyor sellan assetat a rtain timein thefqturefor a certain pri. Unlikefomard contracts, futurescontracts are normall? traded on an exchange. To make tradingpossible, the exchangespecises rtain standardizedfeaturesof te contract. Asthe two parties to the contractdo not ncessarily knoweach other,the exchange also provides a mechanism that gives the two partiesa guarantee that the contract will be honored. Thelargestexchangeson which futurescontracts are tradedare theChicagoBoard of Trade (CBOT)and the ChicagoMercantileExchange (CME). On these and other exchangesthroughoutthe worldj a very wide range of commoditiesand fnancial assets form the underlying assetsin the various contracts. The commoditks includepork bellies,livecattle, sugar,wool, lumber,copper, aluminum, gold, and tih. The snahcial assetsincludestock indis, currencies,and Treasurybonds.Futtyes pris are regularly reported in the snancial press. Supposethat, on September1, the December futurespri of gold is quoted at $680.nis is the price, exclusiveof commsigns, at whichtraderscan agreeto buy or sell gold for Dember delivety.lt is determinedon theCoorof the exchangein the samewayas otherpris (i.e.,bythelawsof supplyand demand).lf more traders want to go longthan to go short, the pri goes up; if the j reverseis true, then the pri goes own. Further detailson issuessuch as marginrequirements, dailysettlementprodures, deliveryprodures, bid-oflr spreads, and the role of the exchangeclearinghouse are givenin Chapter2. '
1.5
OPTIONS Options are traded both on exchangesand in the over-the-countermarket. There are twotypesof option. A call option givestheholderthe right to buythe underlying asset by a certain date for a rtain price. A pat optiongives the holderthe right to sell the lmderling asset by a rtain datefr a rtain pri. The price in the contract is known or strikeJrke; the datein the contract is knownas the expiration as the exerciseprice ztrlerktz optiozs cah be exercisedat any time up to the expiration date or matarity. date. Europeanoptions cap be exerdsed only on the expiration date itself.zMost of the options that are traded on exchangis are Aerican. ln the exchange-traded equity optionmarket, one contract is usually an agreementto buyor sell 1 shares. European options are generally easier to analyzethan Americanoptions, and some of the 2 Note that th termsAmericas and Zuropeaz do not refer to the locationof te option or the exchange. Some options tradingon North Amerkan exchangesare European.
projerties of an Americanojtion are frequentlydeducedfromthpse of its European coupterpart. that an pption givesthe holderthe right to d something. It should be The h.older doesnot haveto exepise th right. Thisis what distinguishesoptions iom forwardsand futures,where the holderis obligated to buy or sellthe underlying asset. Whereasit costs nothing to enter into a fomard or futurescontract, thereis a cost to acquiringan option. The largest exchange in the world for tradingstockoptions is the ChicagoBoard OptionsExchange(CBOE;www.cboe.com). Table 1.2 gives the midpoint of the bi2 symbol: a'nd ofer quotej for some of the American options tradlng on Intel (tickef INTC)on September12,26. Thequotes aa takenfromtheCBOEwebsite.TheIntel stockprice at the time of the quotes was $19.56.The option strike prices are $15., $ 2.00, $22.50,and $15.00.The maturitiesare October26, January 2007, $17.50, and April2007.The Octobr optionshavean expiration date of October21, 2006; the Januaryoptions havean expirationdate of January2, 20079theApriloptions havean expirationdate of April 21,)2007. Table 1.2illustratesa numet of properties of options. The pri of a call option decreasesas the strike pri increases;the *oriceof a put option inreakes as the strike pri increases.Both types of options tend to becomemore vajuable a! their tipr .to price shouldbe exercisedimmediately. maturityincreases.A put with a $25%trike That properties a11maturities. why price of options will These is is the same for the be discussedfurtherin Chapter9. Supposean investorinstructsa brokerto buy one Aprilcall option contract on Intel with a Strike pric of $20.. The broke:will relay theseinstructionsto a trader at the CBOE. Thistraderwillthen5ndanothertraderwho wants to sell one Aprilcll contzact strikeprice of $2.0C, an a pri will be agreed. For the purposes of our onIntelwith a example,we ignorethe bid-ofer spread and assunie that thepri is $1.65,as indicated in Table 1.2.Thisis the price for an option to buy one share. In the UnitedSyates,an option contract is a contract to buy or sell 10 shares. Thereforethe investormust arrangefo: $165to be remitted to the exchangethroughthe broker.The exchangewill then arrange for this amount to be passed on to the party on the other side of the transactipn. In our examplethe investorhas obtained at a cost of $165the rkht to buy 10 .epkasized
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CHAPTER 1 ----
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Sigllfe 1 from(a)purchasing a contractconsistingof 100Intel Net ProfhPer and (b)purchasinga contract consistingof Aprilcalloptions with a strikepri of $20.00 l Intel April put options wiyha strik pri of $17.50. Share
.3
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Intel shares $20.00each. The prty on the other side of the transactionhas receivedf 165 and has agreed to sell l Intel shares for $20.09per share if the investorchoosesto exercisethe optin. If the pri of Intel doesnot rise above $2. beforeApril 21, 2007,the option is not exercisedand the investorloses$165.But if the Intel share prke does well and th option is exercisedwhen it is $30,the investor is able to buy l sharesat $20.00per share when they are worth $3 per share. This . leadsto a gain of $1, or $835whenthe initial cost of the optionsare takeninto .for
account.
An alternativetrade f0r the investorwould be the purchaseof one Aprilput option coiract with a strike pri of $17.50.From Table 1.2 we See that tls would cost ; 1 x 0.725 or $72.50.The investorwould obtain the right sell l Intel shares f0r $17.50per share prior to April21,2007.If the Intelsharepri staysabove $17.50, the optionis not exercisedand the invstor loses$72.50.Butif the investorexerciseswhen the stock pri is $15,the investr makesa gain of $250by buying1 Intelsharesat T15and sellingthemfor $17.5. The net prtt afterthe cost of the options istakeninto accountis $177.50. 'fhe stock optionstrading on the CBOEare American.If we assume f0r simplicity that they are European, so that they can be exemisedonly at maturity,the investo'r's proft as a functionof the fnal stck pri for the tw0 tradesw haveconsideredis shownin Figure1.3. Furtherdetailsabout the operationof options marketsandhowpris such as those in Table 1.2 are determinedbytraderspregvenin later hapters.At this stagewenote that iherear four types of participants in optionsmarkets: '
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1. Buyersof calls of calls 2. sellrrs 3. Buyersof puts of puts 4- sellers Buyersare referred to as havingloq positiozs; sellrs ap referred to as havingshort positiou. Sellingan option is alsoknownas writingthe optiop.
Intyoduction ('
BuslnesjSnapshot1.1 Hedje Funds - '
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Hedgefunds liave become major users cf derivativesfor hedglng,speculation, and A edgeiknd i gmilr to a mutualfundin that it invsts fundson behalf arbitrage, unlike clients. hedgefunds are not retuired to rejister ktker mutual Hp of underUs fideral securitieslaw. Th is becausethey aept funds 9nly from stiplticated indikidulsanddo not publicly llbrtheir seurities. Mutual nancilly fuds are lubjet tp rgulaiionj kquiringthat sharesin the fqndsbe fairlypriced, that investpentpolicieslk disclosed;that that the sharts be reeemable at any time, that no shrt poitions ari taken,and o on. Htdge thr ust cf leviraje e tmitrd, fun ds r relatikelyffeeof thse regulations.ThisgivesShema great dealcf freedom proprietaryinyestmentstrategies. to 'develpp ophistictez, unconventional,and ' .. . feestharped hy hedgefund panagtrs rq depenpenton the fud s prformance and qrerelatively high-typially 1% tg 2% of the amountinvestedplus 1% of the have grw in populrity with over $1 trillinnbeinyinvsted pro s. Hedge throughokttheworld for clients. sFundsof funds'' havebp jet up tp investitl a of hedgefunds. Portfplio The investmeptstrategy follwed by a hedgefund manageroften inkolvesusing deriyAvej to sei up speculqtiveor arbitpge position. Oncethejtfategy h4s been a defljed, thehedgefupd managermus tvalktethe riqks to whic tli fundis ex/ose 'fuils
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2. Dide whichrisks art neraptablt and Fltich w111be hedged involving erivativei) to hdje the unacptable 3. Dkise strattgies (usually fisks ' .
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of te labelsued for hebgefundstogetherwiththi trding Hre g!'ejoli examplej .. ' . . . . . . , .. . ktrltidiisfolloFed: .
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arbitrage: Tke a lmg position in a convertiblebnd comdined with an Cotlvebtible atiyrly mapgd stprt position in the underling equity. . ijjueb by companies in cr close to bankruptcy. Distressed Ccprffk-r: uy sur lties ' ?j rgi4g mctca: lnvst in dibt and eqtty of companiesi devlopingor emerging countriesand in the dbt of the countries thilselves. Maro r Rloball Ude leverag and derivativesto speculateon interest rate and foreignechange rye movis. j; rkt nevtral: Purchasesicuritiesconsideredto be unrvalued and short securities to be overvaluedin sucha way thatthe exposureto theoyeralldirectio onsidered nf the marketis zero. .
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1.6 TYPESOF TRADERS Derivativesmarketshavebeenoutstandinglysuccessful.ne pain reason is that they haveattracted many diferent typesof tradersand havea great dealof liquidity.When an investorwants to take one side a contract, tbereis usually no problem in Ending someonethat is prepared to takethe other side. .cf
10
CHAPTER 1
Threebroad categories of traderscan be idpntihed:hedgers,speculatrs, and arbitrageurs.Hedgersuse derivativesto rcdu the risk that theyfa frompotential future movementsin a market variable. speculators pse themto bet on tlie futuredirectionof ofsetting positionsin two r more instrumentsto a market vayiable.Arbitragemstake 1.1, hedgefundshavebecomebig lockin a profh.As describedin Businesssnapshot . users of derivtives for a11threepurposes. - i few sections, we will conder the activities of each type of trader in In the nex more dCtail ,
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1.7
HEDGERS In this section we illustratehow heikerj can pduce their risks with fomard contracts an#
Options.
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Hdging Using Forward Contracts Supposethat it is July2, 27, and Imgortco, a company base/.in theUnitedstates, knowsthat it will have to pay f 10 mlllion on October2, 2007,for goods it has purchasedfrom a British supplier. The USD-GBPexchange rate quotes mde by a fmancialinstitutionare shown in Tahle1.1. Importco could hedgeits foreip exchange risk by buying pounds (GBP)fromthe Mancialinstitutionin te 3-monthfrward marketat 2.0531.Thiswould havethe llct of fxingthepri to be paid to theBfitish exporterat 520,531,000. Considernext another US company, kilich we will refer to as Exportco, that is exportinggoods to theUnitedKingdqmand, on July20,2007,knowsthatit willreceive f 30 million 3 months later. Exportco can hedgeits foreignexchange lisk by selling f 30 millin in the 3-monthforwardmarket at an exchangemte of 2,0526.Tiliqwould havethe ekct of lockingin theUs dollarsto be lealized forthe sterling at 561,578,000. .Notethat a company might do betterif it chooses not to hedgethan if it chooses to hedge.Altrnatively, it might do worse. ConsiderImportco. If the exchange rate is 1.9200on October20and thecompanyhasnot hedged,thef 1 millionthatit hast pay willcost $19,,, whiL islessthanS2,531,. 0n the other and, if the exhange tl company w.i1lwish that it rate is 2.1, thef l million will cst S21,,-and position of Exportcoifit doesnot hedgeisthe reverse.If the exchange had hedgd! The belessthan2.0526, the compay willwish thatit hadhedged;if in October rate provesto the rate is greater than 2.0526,it will be pleased that it has not done so. Tilis example illutmtes a keyaspect of hedging.Thepurposeof hedgingis to reduce risk. There is no guamntee that the outcome Fith hedgipgwill be better than the 011tC0mC Without hedging.
Hedging UsingOptions Optionscan also be used for hedging.Consideran investorFho in Mayof a particular Microsoftshares. The share pri is S28per shar. The investoris year owns 1, about oncerned a possible share price declinein the next 2 months and wants protection.The investorcould buy ten July put option contracts on Microsofton the Clcago Board OptionsExchangewith a strike pri of 527.50.This would give the
11
Intyoduction '
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FiSure 1.4 Value of Vicrosoft holdig in 2 months with and without hedging. '
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40,000 Valutcf
($) holdicg #
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investorthe right to sell a total of 1,0 shares for a price of $27.50.If the quoted optionprice is $1,then each option contract would cost 1 x $1 = $1 and the total cdstof the hedgingstrategy would be 1 x $1 = $1,. The strategycsts $1 bt guaranteestat tlie shares an be sld for at least$27.50 per share duringthelifeof the option. Ifthemarketpriceof Micrsoft fallsbelow$27.50, is forthe entire holding.Whenthe the options will be exerdsed, so that $27,500 realized of ptions the is $16,500. takeninto account, amount is If the market cost the the options are not exercisedand expireworthless.However,in pricestaysabove $27.50, value of thiscase the theholdingis alwaysabove$27,500 (orabove$26,500when the cost ofthe ptions istakeninto ccount).Fipre 1.4showsthe net valueof theportfolio (after takingthe cost of the options into account) as a functionof Microsoft'ssiockprice in 2 months. The dottedlin shows the value of the portfolio assumingno hedgng. 'realized
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A Comparison Thereis a fundamentaldiflkrencebetween4heuse of fomard contracts and options for hedging.Forward contracts are designedto neutralizerisk by flxipgthe prke fhat the hedgerwill pay or receive for the underlying asset. Optioncontracts, by contrast, provideinsuran. Theyofer a way for investorsto protect themselvesagainst adverse prke movements in the futurewhile still allowingthemto beneft fromfavorableprke movements.Unlikeforwards,options involvethe jayment of an up-frontfee.
1.8
SPECULATORS We now move on to consider how futurts and options markets can be used by speculators.Whereashedgerswant to avoidexposurefo adversemovementsin the price
12
CHAPTER 1 '
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f an asset, speculatorswishto take a positionlnthimarket.Eithertheyare bettingthat the price of the asset rillgo up or theyare bettingthat it will g? down.
Speculatien Using Futures '
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Consider> US,speculytrwho in Feruary thinksthatthe Btitislipound will strengthn, r' lativeto the US dollarover the next 2 months and ispreparedtobak that hunc to 0ne thingthe speculator can do is yurchase f 25, the tune of f 25,. in spot sold laterat a hlgherpri. (Thestetling once marketin thehopethatthe sterlingcan be purchgsedwould be keptin an interesbbearingatcount.)Another possibilityis to take E Aprilfuturescontracts on sterling. (Eachfuturesconttact a longpositionin four CM. of is for the purchase f 62,500.)Table 1.3 jummarizes the two altematives on the assumptionthat ihe current excange rate is 2.0470dollarsper pound and the April futurespriceis 2.0410dollarsper pound. lf the exchapgerate turns out to be 2.1 dollars per pouhd in April,the futurescontract alternativtenables the speculator to = 2.41) x 25, The spot market alternative pglize a prtt of (2.1 $14,750. unitk of an assetbeingpurchasedfor $2.047in Februaryand soldfor leadsto 25, + $13,250 2.0470)x 25, $2.1 in April,so that a prot of (2.1 is made. If the exchange rate fallsto zk dollarsper pound, thi futurescontractgivesrise to a (2.41 z.) x 25, = $10,250lss, whereas the spot market alternativegives rise to a loss of (2.04702.) x 250,000 $11,750.ne alternaves appear to give rise to slightlydiferent profts and losses.But these calculations do not reQectthe interestthat is earned or paid. As shown in Chaptrr 5, When the interestearned in sterlingand the interestforegoe on the dollarsysedto buythe sterlinz are takeninto account,the proft or lss fromth two alternatives is the same. What then is the diserencebetweenthe tw altematives? Theflrst alternative of buying sterling requires an up-front investmentof $5l1,750 (=25, x 2.47). In contrast, the second alternave requires only a spall amount of cash to be account''. The operation by the speculatorin what is termed a deposited of margin accounts is explained in Chppter 2. In Table 1.3, the initial margin in total, but in practice is assumed to be $5, per contract, or $2, requirement it might be even lessthan this. The futuresmarket allowsthe speculatotto obtain leverage.With a reltiyely smallinitial outlay, the investoris able to take a large speculativeposition. 'the
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Table 1
Speculationusing spot and futurescontracts. 0ne futurescontract is on f62,5. lnital marginon fur futurescontracts= $2,#. .3
Possibletrades
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Buy f250,000 Spotprice = 2.#7 . .
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Intodyctinn
speulatjnUsing options that it is Octoberand a speculator Optionscan also be used for speculatin, suppose valne stock considerstht a is likelyto increasein over th next 2 months.The stock option with call currently is a nd $20, pri a $22.50strikepricpis currently a z-month sellij for $1. Ta 1e 14 illustratestwo ppssiblealternatives,assuming that th speculatoris willing to invest$2,. Onealternativeis to purchasel shars; the othe: the purchaseof 2,0 cail options (i.e.,20 call option contracts). suppose that involves of price stock rises the and th to $27by December. thespeculator'shunchis correct The frst alternativeof buyingthe stockyklds a proft of .
l
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($27 $20) $7($ =
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However,the second alternativeistarnote proftable.A call option on thestockwith a strikeprice of $22.50givesa payof of $4.5, becauseit enablessomethingworth $27to be boughtfor $22.50.Thetotal payof fromthe 2, options that ari purchasedunder the second alternativeis 2, x $4.50 $9,000 =
sgbtractingthe original cost of the options yields a net proft of $9,000 $2,020 $7,000 =
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Theoptions strter is,therefol'e,l timesmoreprotable thandirectlybuyingthestock. Ojtions lso giverise to a peater potential loss.Supposethe stock pricefallsio $15 by December.The frst alternativeof buyingstock yields a loss of 1 x ($2 $15)= $500 -
Becausethe call options expirewithout beingexercised,the options strategywould lead to a lossof $2,-the original amount paid forthe optinj. Fipre 1.5showsthe proft or lossfromthe two strategies as a functionof the stock pri in 2 months. Optionslikefuturesprovide a form of leverage.For a given investment,the use of optionsmagnies the fnancialconsequens. Good outcomes becomeyzy good, while bad outcomes result in the whole initialinvestmentbeinglost.
A Cojarison Futures and optiohs are similarinjtrumentsfor speculators in that theyboth provide a way in which a type of leveragecan be obtined. However,there is an important Table 1
from two alternative Comparisonof profts tlosseasl using strategiesf0r $2,000to speculateon a stock worth $2 in October. .4
CHAPTER 1 Prtt or lossfrom two alternative strategiesfQr speculating on a stok currentlyworth $20. Fijufe 1
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10000 Prot ($) 2000 60
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25
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diflkrencibetweenthi two.When a speculptoruses futures,the potential lossas well as the potetial gain is very lrge. When options are used, no matter howbad thingsget, the speculator's lossis linted to the amount paid for the options.
1.9
ARBITRAGEURS Arbitrageursare a third imjortant group of participantsin futures,forward, and options mArkets. Arbitrageinvolveslockingin a risklss prot by simultaneously enteringinto transactionsin two or more markets. ln later chapters we will see how arbitrageis sometimespossiblewhen the futurespri of an asset gets out of linewith its spot price. Wewill also examinehowarbitrage can be used in options markets, This sectionillustratesthe concept of arbitrage with a very simple example. Ltt us consider a stock that is traded on both the New York stockExchange (www.nyse.co>)and the London stockExchange(www.stockex.co.uk). suppose that the stock price is $200in NewYork and f l in London at a time when the exchange rate is $2.0300per pound. An arbitrageur could simultaneously buy 1 shares o the stock in NewYork and sell themin London to obtain a risk-free
proft of
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or $300in the absen of transactionscosts. Transactionscosts would probably eliminatethe prost for a small investor.However,a largeinvestmeptbank fas very 1owtransactionscosts in both the stockmarket and the foreip exchangematket. lt would:nd the arbitrage opportunity very attractive and would try to take as much advantageof it as possible.
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BusinesjSnajshot 1.2 The BaringsBankDisaster
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Derifativesare keryversatik instrnenis. They can be used for hedging,speculation, andatbitrge. On of th risks fated Y a company that tratles dtrivgtivesis that an mploytewh has a mqndateto hedgeor to lpok fpr arbitrage may sprcultor, beconle a yickLesow.an employeeof BariugsBankin jhe jinga/oreoce in 1995,hd a madateto look fir rbitigi pjdrtunities btwen tht Nikkti225futureq n ylzeSing>poreextliangi d thse op theOsakaexchange.Ovr timeLeeln poed ftom bing pn >rbitrageutto beinga spetqlatpr without lnyonein the Barings Londn head oce fully understanding tht he had changedthe w&yhe waS using whkh he was able io llide.He then began to He beg n to inur lpsses deiivayives. ? take biggei sptculative positionsin : ap aitempt to reover the losses, htli only suceeded in mkitg the lossesworsz. Byth tim Leeson'sactivitieswere uncovered,te total losswas closeto 1 billion dollArj. As rsultj Baringsv: bgk that h>dbeenin existencefor 200 yearsMwas wipedt olt. 0ne of the lessonsftom Baringsis that it is njortant t desnt y unapl igups lijk limitsfor tfadersand then monitol carefujjy wjyat tjsy tjo to lke sure that thest lilnitsare adhered to.
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Arbitrage opportunities such as the 0ne just describedcannot llst for long. As arbitrageursbuy the stock in New York, th forcesof supply and demapdwill cause tLedollarpriceto rise. Similarly,as yheysell the stotk in Londoh,the sterlingpricewill be drivendown. Very quickly the two priceswill becom: equivalent at the cuyrent exchangerate. Indeed,the existencef profh-hungryarbitrageursmakesit unlikely that a major disparitybetweep4hesterlingprice and the dollarpricecould ever exist in the srst place. Generalizingfrom tls example,we can say that the very existence of only very smallaritrage opprtunities are observed arbitrageursmeans that in kractice quoted in the pricesthat are in most nancial markets. In this book most of the arjuments oncerning futurespris, fomard prices, and thc valuesof optlon contracts willbe based on the assumption that no arbitrage opportunities exist.
1.10 DANGERS ;
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Derivativesare very versatlleinstruments. s we have seen, they can be used for hedging,for speculation, and for arbitrage. It is this very versatility that can cause problems.Sometimestraders who have a mandate to hedge rkks or follow an arbitrage strategy become (cosciously or unconsciously) speculators. The results of by the activitiesof Nick Leeson can be disastrous.One example tls is provided 3 at BaringsBank (seeBusinessSnapshot1.2). To avpid the sort of problems Baringsencountered, it is very importantfor both fmancialand nonfnancial corporations to set up controls to ensure that derivativesare being used for their intendd purpose. Risklimitsshould be set and the activitiesof traders should be monitored dailyto elure Shattheserisk limitsare adheredto. 3 The movk Rogae Trader providesa good dramatizationof the failure of BaringsBank.
CHA/TERI
16 SUMMARY
0ne tf the excitingdeveloppentsi Enanceoverte last 30 years has beenthe gr-cwth of drivatives markets. In many situations, both hrdgersand speculators :nd it more attractiveto trpdea delivativeon an asset than to tradetheassd itself.some derivatives and are tradedon exchanges;othersare tradedbyEnancialinstitutions,fundmanagers, market,or added to new issuesof debtand equiyy coporationsin the over-the-counter securities.Much of thi bookis concernedwith the valuation of derivatives.The aimis to present a unifying frameworkwithin which al1 derivatives-not just options or futures=can be valued. . In this chapter we havetaken a srstlook at fomard, futves, and options contracts. A forwardor fuiurs contract involvesan obligation to buy or sell an assetat a ceftain timein the futurefor a certain price. Thereare two types of options: calls and puts. A call option givesthe holderthe right to buy an asstt by a rtaip date fcr a certain price.A put option gives the holderthe right to sell an asset by a certain date for a certainprice. Fomards, futures,and options tzadeon a wide range of diferent underlyingassets. Derivatives ave been very successfulinnovationsin capital markets.Three main typesof traderscan V identifkd:hedgers,spiculators, and arbitrageurs. Hedgersare in te position where they face risk associated with the price of an asset. use wishto bet on futuremovements derivativesto reduce or eliminatethis risk. Speculptors in the price of an asset. Theyuse derivaves to get exlra leverage.Arbitmgeursare in businessto take advantage of a discrepancybetweenprices in two diferentmarkets.If, for example, theysee te futurespri Of an assetgetting out of linewith the cash price, theywilltake ofsetting positions in the two markets to lockin proft. 'fhey
FURTHERREADING Chancellor E. Deyil Take te Hindmost-A History of Fiazcial Speculatioz.NewYork: Farra ! StrausGlroux,20().
Merton,R.C. 'EFihanceThery and Future Trends: The Shift to lntegratio,'' Risk, 12, (July
1999):48-51.
Miller, M. H. esFinancial lnnovation: Achievementsand Prospects,'' Joarnal of Applied Cpymrc/c Fizate, 4 (Winter 1992):Gl 1. azd te Fall p.f Barizgs 'tmk. New York: Haper Rawnsley, J. H. Total .Ryk.' Nick fzt?-yp?y Collins,199. Zhang, P. G. Barizgs Bazkruptcyand Fizazcial Derates. Sipgapore:World Scientifk,1995. :
and Problems(Answersin SolutionsManual) Questions 1.1. What is the diferencebetweena long forwardposition and a short fgrwardposition? l Explain carefully th diflrence betweenhedging,speculation, and arbitrage. 1 What is the diferencebetweenentering into a longforwrd contract when the forward price is $5 and taking a long positionin a call option with a strike price of $502 1.4. Explain carefully the diferencebetweensellihg a call option and buyinga put opticn. .2.
.3.
17
Introduction
1 5. An itkjtot
Brith pounds for US enyf into a short forwardcontract to sell l, pf dollap at an eqchange yate 1.9 US dollrs per pound. How much does the ipvestor gain 0f lose if the exchange rate at the end of the cntract is (a)1.8900and (b)
1.92?
1.6. A trader enters into a short ctton futures ontract when the futuresprice is 50 cents per pounds. How muh does the trader pound. The contract is for the deliveryof 5, gain f' le if the cotton pri at the end of the contract is (a)48.20 epts per pound and (b) 51.30cents ner pound? .
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1,7. Supposethat y0u write a put contract with a strikepri of $4 and an expiration date in 3 months. The current stockpri is $41and the contyact is on 1 shares.Whathav y0u commiqed yourself to? How much could you gain or lose? 1 8 What is the diference betweenthe over-tht-counter market and the exchange-traded mayket?What are the bid and ofer Quotes of a malet makerin the over-the-counter market? -
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1.9. You wonld like to speculateon a rise in the pri of a rtain stock. The current stock pri $29and a 3-month all with a strikepri of $3 costs $2.90.Youhave$5,800to invest.Identifytwo alternative investmentstrategies, one in the stock and the other in an option on the stock.What are the potential gains and lossesfrom each? 1.1'. Supposethat you own 5, sharesworth $25each. How can put options be used to provideyou
strikepri of $17.50, sell the 1 shares as soon as lntrl'slrice reaches $17.50.Discussthe a broker to and disdvantges of the t* strategks. advantages 1.29.A bondissuidby StandardOi1scce timt agc worked as fllows. Th: holderreceivedno At the bond's maturity the cpmpany promised to pay $1, plus an additional interest. amountbased on the price of oi1 at that time.The additional amountwas equal to the productof 17 and the excess(ifany) of the price of a barrelof oil at maturity over $25. additional amountpaid was $2,550(which cprresponds to a prke of $* The maximum combinationof regular bond, a longposition in call a perbarrel).Showthat thebondis a optionson oi1with a strike price.of $25,and a'short positiol in call opiionson oil with a strikeprke of $40. 1.3. Supposethat in the situation of Table 1.1 a corporate treasuref said: t$I will have f,1lillion to sell in 6 months. lf the exchangerate is lessthan 2.02, 1 want you to give m 2.02. If it is greater than 2.09, l will accept2.09. If the exchange rate i between2.02 and 2.09, l will sellthe Sterling for the exchangerate.'' How could you us options to satisfythe treasurer? 1.31.Describehowforeigncurrency options can be used forhedgingin the situation considered in Section1.7 so that (a)Importco i guarantee that its exchangerate will be lessthan 2.7, and (b)Exportcb is guaranteed that its exchangerate *i11 be at least2.4. Use DerivaGemto calculate the cost of setting up the hedgein each case assumingthat the exchangerate volatility is 12%, interestrates in the UnitedStatesare 5%, nd interest rates in Britainare 5.70/0,Assumethat the current exchangerate is the average of the bid and oflbr in Table 1.1. 1.32.A traderbuys a Europear call optionand sellsa Europeanput option. Theoptionshave the same underlying asset, strike price, and maturity. Describethe trader's position. Under what circumstances doesthe priceof the call equal the pri of the put?
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In Chapter 1 we xplained that both futuresand forwardcontracts are agreementsto buy or sell an asset at a futuretimefor a certainprice. Futurescontracts are tradedon an organized xchange, and the contract termsare standardized by that exchange. By contrast,forwardcontpcts are privaie agreementsbetweentwo fnancialinstitutionsor betwiena fmancialinstitutionand one of its clients. This chaptrr coversthedetailsof howfuturesmarkets work. Weexamineissuessuch asthe speccation of ontracts,the operation of mayginaounts, the organization of exchapges,the regulation of markets, the way in whkh quotes are made, and the treatmentof futurestmnsactionsfor accounng and tax purposes. We otnpap futures contractswith forwardcnttacts and explainthedifkren betweenthe payofs realized fromthem.
2.1
BACKGROUND As wesaw in Chapterl futurescontracts are now tradedactivly a11over the world. The Clcago Bord of Trade(CBOT,www.cbot.com) and theChicagoMercantileExchange (CME, www.cme.com) are the twolargestfuturesexchangesin the United States.(They fnalizedan agreement to merge in July2007.)Thetwo largestexchnges in Europe are wllich reached an agreement to merge with the New Euronext (www.euzonext.com), in 2006,and Eurex (www.euzexchuge.com), York Stock Exchange (www.nyse.com) and the SwissExchange.Otherlargeexchanges wlch is co-owned by Deutsche Brse B olsa in So Paulo,the Tokyo Mercadorias & Futuros de include (www.brnf,com.br) lnternationalFinancialFutures Exchange ('www.tiffe.or.jp), the Singaporelnternaand Exchange tional Monetary Ftltures Exchange (www.sgx.com),the sydney (www.sfe.com.au). A table at the epd of thisbook providesa more complete list. We examine h0w a futures coptract comes into existenceby considering the com futurescontract tradedon theChicagoBoard of Trade(CBOT).OnMarch5 a traderin New York might call a broker with instnlctiontto buy 5,000 bushelsof corn for deliveryin July of the same year. ne brokerwould immediatelyissueinstructionsto a trader to buy (i.e.,take a long position in) one July corn cotract. (Each col'n contract on CBOTis for the deliveryof exactly5,000bushels.)At about the sametime,another ,
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CHAPTER 2 '
Busiess snapjht 1.1 The UnAtiipAteb DelikeryJofqa FutuyesCot,iact '
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Thisstory (wch my:well e pocrphplllFas told to ihi agthor pf thisbook by a f a fnahciatinstiiition;It ntrns a newemjloyetQf thr fnrncial siioi execptive institutiqnkhohad iot prviolply jrkpd l) t1l., aiial sectorkOne ot th cliet j . ofthehpancialinstittion fejulrly e:ttred ipto a lonsflgyrs,glqfat on lir iqttie ' f9r hilgiqt jurpos apdisjzd ifytrcyins $4dpqi p4 tiqpositio o'tlkilaj4 d4y '
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... . . tpde pn. tlke hicgo .Vtntik (Livetqitleftuilohtictj f g antjeac cntrat i n k , 00 . pclurs E t4kl nei k iyloyEE T Acoqntr ) ; foi hling,the q ntibklity ( WIP k q y .: k. . dpsr jjut iotrpct th8 dt/lby ptid ihai Whtl. iheille ce to thetlkniw$ qtkjjt Ey qthr ekhajji &nd ijirqtied sll) ve zgntfact t traet t a lng n kstpiiuiipn'giikd aj nlstke jk f ihis qj: ytkf' :ni talt Clivi'attk w: th t $ saudrl tat. fuiiiq tttk tsi k psit mitky ( ii ctptr. qn tw ws jloited lg n C @ i mtrac 2) ( tradingin the hd tjajed. j ; : y j ( 'te Vgpcial institltip ,(Aoty1ht clkt) w j pqjj yj qypy ty Ay a, r starie to look hto theditils of the zliyiyitjiikints fi liye lile is rrjltj cotractswstpnetlii rl,t d iwyr dolp'btfr, dry t, teppspf thr ) fturx .c>tttecouldbt (klikerdqljy .y . tlkrparty sftjpsitiktypyyqnmberq , drili ,?fntfqd, ni,ted Stitllftty diflent locatiopsin th thk etwi it k7 sti pthig ) gri: with sllprt bt wjit,fr a 1?j, the:nanial institutipncmld p r' t/fnfrpfip: Kotic io pf/ tplr exhangnil) for ihrexaqge tp isse q ppsitn E ptit,t ingititin !. sip Nnantl thaty l y y yj to eyintuattyrrteived q . 'teexchotakfpdyiltt .. ' ' ' ( ( . . . . . ' . ngiicr fipi itld It rrtiivi tive a kytlxfoEnpkiikktd.'f sint 2,0 cattleat a loyipn ml es ;. rrrrinypyriwaj tlkiliayigtl to hapdlethipjs. It titp t'tlji il lptti pttle tip 49 bolgiiiE e' Tlkr positlp eV livtr ilt wa iiyTusday. prty kit ye pyt dliveir ihem.Unffyatety cattk cqitlr at 4heation nd tlien ptdiatilycilpn 'rlw until 'Ikeqdqy, eqplo resold nixt thr tllwiqg the attle not ke facedwith tlte prbtezj f nkig Arzqtjli j) rftk t tatdrt be ' . wasyhezefore i te nacial stof! ald fe for a weet.Thij Aa44 peat .stqrtto : nistjob htmsed
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traderin Kansasmightinstructa brokerto sell5, bushelspf corn for Julydelivery. Thisbrokerwould thenissueinstructionsto sell (i.e.,take a short potion in) onecorn contract. A prie would be dtermined and the deal wold be done. Under the traditionalopen outcrysystem,floortradersrepresentingeach party would physically meetto determinethe pri. Withelectronictrading,a computerwould mafch trades and.monitorprices. position in one The trader in New York who agreed to buy has a longfatares position in one contract;the trader in Kansas who agreedto sellhas a sbort fatares price foi July corn, say 300 cents contract.ne price agreedto is the cuaent fatures determined price,like otherprice,is bushel.This by the lawsof supplyand any per demand.lf, at a partklar time,mre traderswish to sell rather than buy July corn, the pricewill go down.Newbuyersthen enter the market so that a balancebetween buyersand sellers is maintained. If more traders wish to buy rather than sell July corn, the pri goes up. New sellersthen enter the market and a balan between buyersan sellersis maintained.
23
Mechana PJFutuet McrAel '
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Closing Out Positions
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The vast majority of futurescontracts do npt lead to delivery.The feason is ihatmost qaderj.chqoseto close out their pgsitions prior to the deliverypiriod specifed in.the contrci. Clsing ot position meal.ljeptering into the opposite tmde to the original # one. F0r exaplle, the New ork investorwho bught a July corn futurescontract on March 5 can lose ou4 the ppsition by sellig (i.e.,shofting)one July torn futures ontracton, say,April2. TheKansasinvestorwho sold (i.e.,shorted) a July contmct on March5 can close out tht psition by buyingone Julycmtract on, say, May25. In eachcase, the investor'stotal gain or lossis detrmined by the change in the futures yricebetweenMarch5 and the day when the contract is clostd out. Deliveryis so unusal that traderssmetimes forgethowthe deliverypross works 2.1). Nevertheless we will spendpart of this chptey reviewing (seeBusiness snapshot thedeliveryarrangements in futurescntracts. This is becauseit is the possibility of fnal deli ery tht tiesthe futurespriceto the spot price.l
SPECIFICXJIONOF A FUTURESCONTRACT
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Whendevelopinga new contract, the exchange m'ust specify in some detail the exact natureof the agrement betweenthe tWoparties.In particular, it must specifythe asset, how much of the assetwill be deliveredunder one contmct), ' the contract size (exactly whefedeliveyywiil be mad, and when deliwrywill be made. Sometimesalternativs are specifedfor the grade of the asset thatwillbedeliveredpr . forthe deliverylocations.As a generalrule, iyis the party with the short position (the : party that has agreedto sell the asset) that chooseswhat will happenwhen alternatives aw sptf ed the exchange. When the party with the short position is rea to it Elesa zotice of j/lfdrlffp?lto liverwiih the exchange. This notice indicates deliver, selectionsit has made with respect to the grade of asset that willbe deliveredand the deliverylocation. .
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Tbe Asset Whenthe asset is a commodity, theremay be quite a variation in the quality of what is availablein the marketplace.Whenthe assetisspecifkd,it istherefowimportantthatthe exchangestipulate the grade or grades of the commpdity that are aeptable. TheNew York Board of Trade(NYBOT)has spedfed the asset initsfrozenconcentrated orange juicefuturs contract as orange solids fromFloridaand/or Brazil that are US GradeA with Brix value of n0t lessthan 62.5degrees. For some commodities a range of gtades can be elivere, but the pri received dependson the grade chosen. F0r exnmple,in the ChicagoBoard of Tfadecorn futures contract,the standardgrade is $$No.2 Yellow'',but substitutions are allowedwith the pricebeingadjustedin a way stablished by the exchange.N0. 1Yellowis deliverable for 1.5 cents per bushel more than No. 2 Yellow.No. 3 Yellowis deliverablefor 1.5cents per bushellessthan No. 2 Yellow. The nancial assetsin futmescontracts are generallywell defnedand unambiguous. j
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As mentionep in Cliapter1, the spot pri is the price for almost immediate delivery.
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CHAPTER 2
Forexample,thereis no need to specif tllegrade of a Japaneseyen. However,thereare futures contracts someinrestingfeatures9f the Treasurybtmd and Trgsqry t yote underlpng asset ln the Trasury bond traded n the hicagoBord of Trade. bondthathas is long-terpUSTreasury a maturity of greatertan 15years contract any andis ot cllabl within 15yeafs. In theTreasurynote futurescontrat, the undeflying latgrity of no lessthan 6.5 years and no asset is any long-termTreasurynote with a delivery. Ip both fromthedateof l than cases, the exchangehas a formula yeap more reived accordingto the coupon and maturity dateof the bond for adjustingthe pri delivered.Thisis dcussed in Chapter6. 'fhe
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The contract Size The contracs size specifiesthe amountof the asset that has to be delivered under one ) contract.Thisis An imjrtant decisionforthe exhange. If the contract sizeis too large, many investorswho wish to hedge relativelysmall expures or who wish to take relativelyspall speculativepositionswill be unable to use the exchange. 0n the other hand, if the contract size tpo small,trading may be expensiveas there i a cost associatedwith each contract traded. 'l'hecorrect sizefor a contract clearlydependson thelikelyuser. Whereasthe value of whatis deliveredunder a futurescontract on an agrkultural productmight be $1,00 it is much igher for Fome fnancial futures. For example, under the to $2,, Trepsurybond futurescotract traded on the ChicagoBoard of Trad, instruments with a facevalue of $1,0 are delivered. ssminir' pntrts to attract smallerinvesIn som cases exchanges hpveintroduced CME'S exaiple, MiniNasdaql the contract is on 20 tiles the Nasdaq1 tors.For index,whereas the regular contract i! on l times the index. ' '
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'Theplace where deliverywill be made mujt be specihed by the exchange.This is pattkularlyimportantfor commoditiesthat involvesignicant transportationcosts. In concentrate orange jui contract, deliveryis to the cse of the NYBOT zchajezlicensed warehousesin Florida,NewJersey,or Delaware. When alternative deliverylocationsare specihed,the pre receivedbythe party with accordingto thelocationchosen bythat party. the short positionis sometimesqdjusted Thepricetendsto be higherfor deliverylocationsthat are relativelyfar fromthemain sonrcesof the commodity. 'frozen
DeliveryMonths A futprescontract ij refrre/ to by its deliverymonth.ne exchnge must specify the precise perio during the month when dilivery can be made, For many futures the deliveryperiodk the whole month. contracts, The deliverymonths vary fromcontract to contract and are cliosen by th exchnge to meet the needs of market participants.For example, corn futurestrazed on the ChicagoBoard of Tradehavedeliverymonths of March,May,July,September,and December. At any given timr, cotracts trade for the closest deliverymonth and a number of subsequent deliverymonths. The yxchange specihes when ttading in a particular month's contract will begin.The exchange also pecihes the last day on '
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Mechanics PJFutuyesMayets
whichtrading an take j lace a pivencoptmct. Trading generally ceasesa fewdays beforethe last day o whichdeliverycag be made. 'for
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Price Quotes The exchange dfnes how jces will be quoted. For example,crude oil pris on the NewYork MercantileEychangeare quoted in dollarsand cents. Treasury bond and Treasury note futureson the ChicagdBoard of Yrade are quoted in dollarsand thirtyof a dollar. seconds
Price Limitsand PositionLimits For most contracts, dpilyprice movement lipits are specifed by the exchange. If in a daythe prici moves downfrom the previous day's close by an amput equal to the dailyprice limit,the contract is said to be limitdowz.If it movesup by thelimit,it is said to be IimitgJ, A limitmoveis a move in either directionequal to the daily price limit.Normally,trading ases for the day once the contract is limitup or limitdown. However,in someinstancesthe exchangehas the authority to step in and cange tlie limits. The purpose of daily price limits is to prevent large price movementsfrom ouning becauseof speculative excesses.Howiver,limhs can becomean artifkial barrier to trading when the price of the underlying commodity is advancing or decliningrapidly. Whether prict limitsare, on balan, good for futuresmarkets is controvrsial. Positionlimits are the'maximum number vfcontpts that a speculatormay hold. The pumose of theselimitsis to prevent speculatorsfromexercisingundue ivuence on themarket.
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cosvEkcEscE
oF FUTURESpRlcE To 5PoT PRICE
As the deliveryperiod for a futurescontract is approached, the futurespri converges to the spot pri of the underlying asset. When the deliverypeliod is reached, the futurespri equals-or is very close to-the spot pri. T0 see why this is so, we frst supposethat the futurespri is above the spot price duringthe deliveryperiod. Traders thenhave.a clear arbitrage opportunity: '
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1. sell(i.e.,shorl) a futurescontract 2. Buy the asset 3. Makedelivery These steps are rtain to lead to a prot equal to the amountby which the futures pli exeds the spot prke. As tradersexploit this arbitrage opportunity, the futuresprice willfall.Supposenext that the futuresprice is belowthe sppt priceduringthedelivery period.Companiesinterestedin acquiring the asset will sndit attpctiveto enter into a longfuturescontract and thn wait for deliveryto be made.Astheydo so, the ftures pricewill tend to rise. The result is that the futurespriceis very close to the spot priceduringthe delivery period.Figure2.1 illustratesthe convergenceof the futurespri to the spot price. In
CHAPTE: 2
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Relationshipbetweenfuiufespric and spot priceas the deliverypefiodis approached: (a)Futurespiii ahve spotprics;(b)futufespricebelowspot price.
Figure 2. .1
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Figure 2.1(a) the futurespriceis above tlie sjot priceprior to the deliveryperiod.In Fipre 2.1(b)the futurespriceis btlowthe spot pliceplior to the deliverypefiod.The circumstancesundef which thesetwopattefnsafe observed afe discuksedin Chapter5.
2.4
AND MARGINS DAILYSETTLEMCNT If tw0 investorsget in tguchwith eachothef difectlyand agree to trade an asset in the future f0r a certain price,thefeare obvious risks. 0ne of the investorsmay regret the the investorsimplymay not havethe fmancial deal and try to back out. Alternatively, rotes of the eqchangt is to organize resoufcestp honor the agreement. 0ne pf the key trading so that contract dfaults are avoided. Tls is whefemargins come in.
The Operation of Margips To illustratehowmargins Fork,Fe considef an investorwo ontacts his of her broker on Thursday,June 5, to buy two Decembefgold futurescontracts on the COMEX division f the New York MercantileExchange (NYMEX).We suppose that the current futufesplice is $600per ounce. Bec>usethe contract sizeis 1 ounces, the investorhas contracted to b a total of 200ounces at thisprice.Thebrokerwillrequir theinvestoftp depositfundsin a margiz accoant.The amount that must bedepositedat the timethe contfact is entere into is knownas the initial margiz. We suppose this is $2,000pel' contfct, or $4,000in total. At the end of each trading day, the mal'gin accounti8adjusted to refkct the investor'sgain or loss.Thispracticeis referred t as markingto market the neztount. Suppose,fof example, that by the end of June5 the futufespricehas dfoppedfrom $600to $597.Theinvestorhas a loss of $600(=200x $3),becausethe 200 ounces of Decembergol, wlch theinvest contracted to buy at $600,can now be soldfor nly
27
Mechanicsof FutuyesMaykets
$597.Thebalapcein the pargin account wull thereforebe reduced by $600to $3,400. S1milarly, if the,rpriceof December gld rose to $603by the end f the rst day the 'balancein the margip aount would be' intreased y $6 to $46. A tradeis rst maf k:d to markei at the close of the day on which it takesjlscr.It is then marked to ,
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market at the close of tradingon each susequent day. Note tht marking to market is n0t merely an arrangement jjetween j)r ker and cltent.Whenthereis decreqsein the futuresprice so that the marginaount of an investorwith a long position is reduced by $6, the investor'sbrokerhas to pay the exchage $600>nd the exchangepasses thi money on to the brokerof ?.ninvestorwith a short position. Similarly,when thereis an increasein the futuresprice, brokersf0r partks with ort positions pay mney to the exchangeand brokersf0r parties with long positions receivemoney fromthe exchange.Later we will examinein moie (ktail the mechanism by which th happens. ' withdraw anybalan in the margin aount in exss ofthe ' The inkestor is entitled to initialmargin. T0 ensure thatthebalancein th marginaccountnever becomesnegative a maintezazce margiz, which is somewhat lowerthan the initialmargin, is set. If the balancein the margin account fa11sbelowthe maintenancemargin,theinvestorreceives a margin call and is expectedto t0p up themargin aount to theinitialmargin levelthe nextday.The extp fundsdepositedare knownas a varitioz margin. Iftheinvestordoes not provide the variation margin, the brokercloses 0ut the position. In the case of the investorconsidered earlier, closing ut the position would involveneutralizing the in Deeinber. existingcontract by slling 2 ouncesof gold f0r delivel-y Table2.1 illustratesthe operation of the margin account torone possible sequenceof futuresprices in the case of the investorconsideredearlkr. The maintenancemargin is assumedfor the purpose o theillustrationto be $1,5 per ontzact, or $3, in total. On June 13 the balancein the margin accountfalls$340belowthe maintenancemargin Table2.1 level.Thisdroptriggersa margin call fromthebrokerf0r an additional $1,340. assumesthat the investordgesin fact provide th margin by the close of tradingon June 16. On June 19the halancein ihemarn accountagain fallsbelowthe maintenance marginlvel, and a margin call for $1,260k sentout. Theinvestorprovidesthis margin by the close of trading on June 2. On June 26 the investordicidesto close 0ut the and the positionby selling tw0 contpcts. The futuresprice o!l that day is $592.30, investorhas a cumulative loss of $1,540.. Note that the invstor has excessmargin on 2.1 JuPC 16 23 24 and 25.Table assumesthat the eqgessis n0t withdrae. ,
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Further Details Many brokersallow an investorto earn intereston the balalkein a margin account. The balancein the aount does not, therefore,represent a tru cost, provided that the interestrate is competitivewith what could be earnedelsewhere.T0 satisfy the initial margin requirements (butn0t subsequent maigi calls), an investorcan sometimes depositsecurities withthe broker.Treasurybillsare usually acpted in lieu of cash at about 90% of theirfa.value. Sharesare alsosometimesaepted in lieu of carsh-but at about 5% of their market value. The eect of the marking to markit is that a futurescontrad is settleddailyl'ather than a11at the end of its life.At the end of each day,theinvestor'sgain (loss) is adyed value of from)the margin account,bringingthe the contmct back to tb (subtracted 0ut and rewritten at contract eflkctclosed is in A futures a new price eachday. zero.
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CHAPTER 2 '
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Onertkn f mamins for a lony position in two gold futures or $4,000in total, apd' cotracts.The initia)margn is $2,000per tontract, the maintenance mar/n is $1,590per cdtqact, or $3,000i total. The contract is etered into op June 5 at $600yndclojtd out on June 26 at $592.30.The numbers in the setqnd column?expt $e frst and the last, represeptthe futuresprices t the close of tmding. price
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pay June 5 June 6 June 9 June 1 June 11 June 12 ' June 13 Jun 16 June 17 Jupe 18
Minimumlevelsfor initial and maintenan margins are set by the exchange. lndividua1brokers lay rtquire greater tnar/ns fromtheir clients than those specifkd by the exchange. However, they tAnnot require lwer margins than those specifkd by the exchange.Margin levelsare determinedby the variabilityof the price of the underlyingasset. The higherthis variability,thehigherthemarginlevels.The maintnance margin is usually about 75% of the initialmar/n. Marginrequirementsmay pependon theobjectivesof thetmder.A bona:de hedger, such as a company that produces the compodity on which the tutures contmct is fequirements written,is oftensubjsctto lowermargin than a speculator.The reason is that thereis deemedto be lessrisk ofdefault.Daytmdesand spreadtransactionsoften giverise to lowermargin requirementsthan do hedgetmnsactions.ln a d?y tradt th traderannounces to thebrokeran intentto close out the position in the sameday.In a spread transactionthe trader-simultaneously buys (i.e.,takes a long position in) a contract on an asset for one maturity month and sells (i.e.,takes a short position in) a contract on the sameasset for another maturitymnth. Note that margin requirementjare the same on short futurespositions as theyare on
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longfuturespositions. 1$is justas easy to takea short futurespotion as it ikto take a longone. The spot niarket (loes pot haveyll symmetry.Tpkinga long position in the spotmarket involvesbuyingthe assetforimmediatedeliveryand presnts n? probleins.
Takinga short psition involvesselling an asset that you do not own.Tll more posslbti particular market. ip be ihat It transacpn may or may not a complex Chaiter discussd frther in 5.
The Clearinghouse @nd CleafingMargins A clearingoase acts as an intermediaryin futurestransactions.It the clearinghouse has a number of perfofmanceof the parties to eachtransction. members,who must post funds with the exchange. Brokerswho are not membefs main tpsk of the must channet their businessthfoug a member. themselves clearinghouseis to keeptrack of a11the transactionsthat take pla during a day, so tht it can calculate the net positionof each of its members. Jujt as an investoris required to mtintain a mar/n nf-nount with a brokr, thebroker is required to maintain a margin account Fith a clearingj ous messr aatj ts clearinghousemembtr required to mintain a margin aount with the learinghouse. The latter is known as a ckating margin. The margin aounts for clearinghouse membersare adjustedfor gains and lossesat the end of each tradingday in the same wayas are the margin atcountsof investors.Hpwever,in the caseof the clea/nghouse member,there is an originalmargin, but no maintenance margin. Everyday the account balance for each contract must be mpintained at an amount equal to the original margin times the mlmber of contracts outstanding. Thus, depeding on transactionsdufingthe dayandpricemovements,the clearinghousemember may have to add fundsto its margin acount at theend of theday.Alterntively,it may findit can remoyefunds from the aount at this time. Brokers who are not clearinghouse must maintain a margin aount with a clearin#puse member. members In determiningclearing mar/ns, the exchangeclearing11 ouse ca1culatesthe number ofcontracts outstanding on either a gross or a net bas. Whetl the gross basisis used, thenu'mber of contracts equalsthe snm of the long and short positions. When the net basis used, these are oflet against each other. suppose a clearinghousemember has position with other with a short position in in 29 contracts, the two clients: one a long 15 contracts. Gross margining would calclate the clearing pargin on the bas of 35 contracts; net margining would calculate the clearing maryin on the basis of 5 contracts.Most exchangescurrently use net margining. 'guarantees
'fhe
'fhe
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Credit Risk The whole purpose of the mar/ning systemis to eliminatethe rk that a trader Who makes a proht will not be paid. Overallthe system has been very successful.Traders enteringinto contracts at major exchangeshave always had their contracts honored. Futuresmarkets were testd on October19,1987,when the s&P500indexdelined by over20% and traderswith long positions in S&P599futuresfoundtheyhad negative marginbalances.Traderswho did not meet margin callswere closed out but stillowed did not pay and as a result somebrokerswent bankrppt their brokersmoney. some without clients' hecause, money, they were unable to meet margin calls on their contractsthey entered into on behalf of their clients. However,the exchanges had
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Mechanic OJFutuye
31
MJAAdS ' .
Collateralizatknsgniscanylyreuces the creit risk in oker-the-conter contracts an is discussedfurtherin Section22.(8.Collatemlkatim gmements were used by a: hedgefund, Long-TermCapitalManagemens(LTCM),in the Tey allowed LTCMtD be hkhly leveraged.The contracts did prpjde credit risk protection, but as descri ed in BusinessSnapshot2.2the lgh levrqe leftthe hege fundvulnerabletn other rks. .
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.19%s.
NEWSPAPERQUOTES
2.5
Many newspaperscarry futuresprices.Table2.2showjthe prkes2j or commojjyjo ( ay theyappepredin the WallStreetJpprwl of Tuesday,January9, 2007.The prices refer to the traing that took place on the previousday (i.e.,Mondy, January 8j,27). The pricesfor indexfutures,currency futures,anj intefestrate futures are given in Chapters3, 5, and 6, respectively. for contrats with relatively shnrt The Wall Street Jpprrxl only showj tuotes maturities.For most commodities, contracts trade with much lonyerpatufities than those lhown. However,trading volume tends to decmase as contract maturity .
j
nCrCasCs.
The asset underlying the futurescontract, the ekchange thpt the contrad is tr'aded on,the contract size,and how the prke is quoted are all shownat the top of each sectionin Table2.2. The srstasset is copper, traded on COMEX(a divisionof the ' New York MercantileExchange).The contract sizeis 25() lbsr and the price is r quotd in cents per lb. The maturity month of the coptrat is shpWnin the srst . .
,
Prices Thefirstthree numbers in ech row shnw the openingpri, the highestprke achieved in tradingduringthe day, and thelowestprice achlevedin tradingdnriy the day.The openingprice is representativeof lhe prices at which contracts were tradingimmeiatelyafter the opening beli. For Mamh2007copper on January8, 2007,the opening pricewas 253.50 nts per pou an, duringthe day,thepricetradedbetween247.00 and 258.95cents.
Settlement Price
'
The fourth number is the settlemeztprice. This is the prke used for calulating daily gainsand lgssesand marginiequirements.It is usuallycalculated as the price at which the contract ttadedmmidiately beforethebellsignalingthe end of tradingfor theday. The fth number is the change in the settlementpri fromthe preous day.For the March2007copper futurescohtract, the settlementpricewas 252.80cents on January8, 2007,down0.70 cents fromthe previous tradingday. In the case of the Mqrch2007futures,an investorwith a longpositio in one contract wouldfindhis or hermarginaount balancereducedby $175.00 (=25,000x 0.70cents) with position in one contmct would a short on January8, 2007.Similarly,an investor thisdate. findthat the margin balanceincreasedby $175.00 on
Sozrce Reprinted by yrmission of Dow Jones,Inc., via CopyrightClearan Center,Inc., ) 2907 DowJones& Company,Inc. A11RightsReservedWorldwide.
33
Mechanic; PJFutuyesMqrbts
()pen lntrest ''rhennalcolmn
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. . in Table2.1 showsthe opez izterest for eah cptract. Thisisth totai humberof contracts outstandinj. The open'rheinterestit thenllmbrr of lopgpositionsor, the number of short#ps itions. openinterestftr March2007Loppeiis equivalently, shownas 48,809contracts. Note tlkat tlie Qpen interestfor theJanuary2007contract ij muchloweibecausemost tradirsFho Vldlpngor shortpsitions in that contract have '
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alreadyclosed out , Sometimesthe volllmeof tradig ili a dayis greater thanthe open interestat theend of the day for a contratt. Th is'indicatie of a largenumber of daytrades. '
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Patterns of Futures prlces Puturespricestan showa numberof dibrent patterns.In Table2.2,gold,crude oil, and
naturalgaS a11haveSettlement pris thatincreasekith the laturity of te contract. This jj isknownas a zormal market. Table2.2shws that t e settlementpriqfor 7anuary and Marchorange jukefutureswere 201.90and 195.80cents, resp.ectively.Otherdata show that the May 2007 July 2007 September2007 and November2007contracts had settlementpricis of 193., $9,5, 187., ayd 181. nts, respectively,on January8, 2007.The orange juicefuturespri was thereforea decreasig funtion of maturity on January8, 2007.Thisis knownas an izvertedmarket. Figure2.2displaysthesettlement priceas functionof maturity for gold d orange juiceon January8, 2007, Fuyure pricescan showa mixture of normal and invertedmarkets. An exmple is by livecattle on January8, 20t7. s shoFl in Table2.2,the Aprilfutures Provided pricswas higheythan te Februaryfuturespfi. However,theJlpe futres pri'e (not shown) was lowerthan the Aprilfutus pri. i7or later maturities the futurespri decreasing,then incrased, thendecreasedagain as a functionof maturity. continued ..
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2.6
DELIVERY As mentioned earlier in this chapter, very fewof the futurescontracts that are entered into lead to deliveryof the underlying asset.Mostare closed out early. Neyertheless, it .
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Figure 2.2
Settlementfuturespri
2007,for (a)gold and
s a functionof contract maturity on January8, 21 Futumsprice(cents perlb) 200
630
190
620 60
''
(b)orange juice
650 Futuresprice($per oz) 64
610
-.' .
180 Contractm turitvmonth -
.
Jan-07 Apr-07 Jul-07 0ct-07 Jan-08
(a)00ld
t
170
ontractmaturitymonth .
Ja11-07 Mr-07 Vy-07 Jul-07
(b)
!ep-07Now07
ange juice
34
CHAPTER 2 '
(
ls the possibility of eventual deliverythat determipesthe futures price. An undercf deliveryprocedures is thereforemjoftant. standing The period during whkh deliverycan be made is defned by the exchange and varis frop contractto contract.Thedecisionon whrn to deliveiis madeby the party with the jhort posiyion,whom w shall refer t as investqrA, W.hen investorA decidesto deliver,investprA's brokerissuesa notice of intention to deliverto the exchageclearinghouse.This notice states how many contractswill be deliveredand, in the case of' ommodities,also specifks where dlivery will be made and what grade Wi11 b delivered.The exchange then ihopsesa paity with a long positionto aept
delivery. jupposethat the pgrty n theotherlideof investorA's futurescontzactwhen it was enteredinto was investor It ij importantto realize that ther is nb reason to expect thatit will be investorB who takeqddivery.InvestorB maywell haw closedout his pr herpositioh by trading with inyestorC, investorC may haveclosedout his or her positionby tradingwith investorD, apd so on. n usual rule chosen by the-echange of intep'tion to deliverm tp theparty with the oldestoutstanding is to pass the '
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'notice
deliverynotices. However,if th, long position. Partieswith longpositions must wcept r notis are transferable,long investorshave a short period of time, usually half an hour, to :nd another party with a long position thakis prpared to accept the notice from them. In the case of a commodity, takingdeliveryustlally means aceptipg a warehoue receiptin return for immdiate payment. I'hr party takingdeliveryis then responsible f?r a11warehousing costs. In thevcase0 f livestockfutgres,theremay bl costs associated with feedingand lookingafter the ahimals (seeBusinessSnapshotj.1). In the case of snanial futures,deliveryis usually made by wire transfef For a11contracts, the price paidis usually the most recent settlementprice. If specifed bytheexchange,thispriceis adjustedfor grade, locationof delivery,and so on. The whole deliveryprocedure frotn the issuanceof the notice of intentionto deliverto the deliveryitslf generally takes abput two to threedays. Thel'ear three critical daysfor a cotract, These are the frst notice day, the last noticeday,and thelpsttradingday.The/rlf zotice dayisthefirsydayon which a notice of intentionto make deliverycan b Submitted to the exchange.The Iast zotice dayis the last such day.Tht Iast tradihgdayis generallya fewdaytbtforethelast notice day. To avoid the rkk of hayingto take delivery,an investorwith a long position should closeout his or her contracts prior to the srstnotice day.
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Cash Settlement some snancial futures,such as those on stock indices,are settledin cashbecauseit is inconvenient or impossibleto diliverthe nderlying asset. In ihecase of the futures contract on the S&P5, for example, deliverinjthe underlying asset would involye deliveringa portfolio of 500stocks. Whena contractis settled in cash, a11outstanding contractsare declaredclosed on a predetermined day.Thefmalsettlementprice is set equal to the spot price of the underlying asset at either the opening or closeof trading on that day. For example, in the S&P500futurescontmcttrading on the Chicago MercantileExchange, the predetermineddayis the thkd Fridayof the deliverymonth 'and fnal setttement is at the openingprice. '
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Mfchans
p./
35
FutuyesMaykets
TYPESOF TRADERSANDTYPESOF OFDES
2.7
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Thereare twomain typesof tradersexeting trades:commissionbrokersand Iocals. brokersare followingtheinjtructidnsof theirclinyq and chargta comltnilCommissioz sionfor doing so) lqqls are .tradingon theirown acpunt. In (iividualstakt kp iitii , whether lcals or the cents of commissionbrokers can b8 categorized as hedgers,speculatorj, or rbitrageurs, as discussedi Chapter 1. specu.latorcan be classised as scalpers, day traders, or posin traders.Scalpers are watching for very shorbtrrm trendsand attemptto proft frm smatl change in the contractprice. Thev usuallv hold their positionsfor only a fewminutes. Day traders hold their positionsfor lessthan one tradingday.They re unwilling to take the risk that adversenews will our ovemight. Position tradersholdtheir positions for much longerperiodsof time.Theyhopeto make signifcantprofts frommajof movelents in the marke s.
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Orj ers The simplest type of order placedwith a broker a market order. It is a request that a at the best pce availablein the market. However, trade be carried out immed.iately orders. We will consider thosethat are more comnonly there are many other typesof usC d
A limitorder specihesa partiular price.The order can be executedonly at thisprice or at oe mori favorableto the investor.Thus,if the limitprice is Tj for an investor katzg io buy, the order will Ix executedonly at a pri of $3 or lesskThere of course,n guarantee that the orderwill be executed at all, bause the llmltprice may jwwr bg reached. orderalso specifes a particular price. The order is executed A stop order or at the bestavailablejrice oce a bid o? ofer is made t tht particular prke or a lessprice. suppose stop order to sell at $3 is sued when the market price favorable is $35.It becomesan prdr to sell when and if the pric fallsto $30.In efect, a stop orderbecomesa marketorder as soon as the specifed prke has ben hit. The purpose Of take a stop grder k usually to tloseout a pnsitionif unfarahle price Povements lace. It lints the lossthat can be incurred. P stoylimit order is a combinalionof a: stop order and a limit order. The rder A bcomes a limit order as sooh as a bid or ofer is made at a pri equal to or less favorablethanthe Stop prke. Twoprkes rwstbe specihedin a stop-limit order: the stop that at the timethe market price is $35,a stop-limit priceand the lilit price. suppose orderto buyis suedwith a stop price of $4 and a limitpriceof $41.As soon as there a bid or ofer at $40,the stop-limit becomesa limitortkr gt $41.lf the stoppriceandthe limitprice are the same,the order is smetimes calleda stop-and-limitorder. A market-ltoached '(M1T)-prr is executedat the bestavailableprice after a trd ocurs at a specihedprice or t a pri morefavotablethanthe specifkdpriie. In efect, an MIT becomesq market order once the specifedprice ha8beenhit.An MIT is also knwn as a board order. Consideran investoywho has a long posion in a futures contractand is sing instructionsthat Wouldlead to closicg out the'contract. A stop ' order is designedto placea limiton thelossthat can our in the event of unfavorable --
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CHA/TER 2
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price movements. By contfast, a arkebif-touchd order is designedt ensure that Profhs are takenif sucieitly fuvorablepri movementsour. order or market-nqt-eld order istrded as a parket order expt that A discretionary executin may be delayedat te broky's discrdipnin an atteppt to a better price.' Some rdersspecifytimeconditions. nlessothemke stated,an order is a day order and expifesat the end of the tradingday. A tim-of-dy oder specifes a particular periodof timedring th daywhen the order tanbe ixecuted.An opez order or a goodtill-cazcekdorder is in efect until ereuted or until theend of tradingin thr particular pr&r, immediately'on contract. A hll-or-kill s its ame implies,mustbe rxecutetl receiptor not at all. 'get
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2.8
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REGULATIO; '
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Futures markets in the UnitedStatesare currentlyreNlated federallyby the Compoity Futures TradingComnssion (CFTC;www.cftc.gov), which was establhed in 1974.Thisbodyis responsiblefor linsing futuresexchangesand approving contracts. ' ts and changes to existinycontractsmust be apprpved by the CFtC. To Al1new contrac be approved, the cntfact must havesomeusefuleonomic purpose. Usuallythis means that it must servethe needs of hedgersas well as speculators. TheCFTClooksafter thi public interestIt is responsibleforensuringthat pris are communicatedto the public and that futurestradersreport theiroutstandingpositions if thty are abovi rtain levels.The CFTC also linses a1lindividualswho pfer their servicesto the public in futurestmding.The backpounds of these individuqlsare investigatep,and there are minimum capital requirenients. The CFTC deals with complaintsbroght by the publk and ensuresthat disciplinaryaction is taken against indikidualswhen appropriate. It hasthe authoritj to forceexchangesto tke diciplinar action apipst membrs who al! in violation of exchapgerules, With the formationof the National Futures Association(NFA; www.nfa.futuzes. org)in 1982, someof responsibilitiesof the CFTCwere shiftedto' the futuresindustry itself. The NFA ij an organization of individualswho paryicipat: in the futures industry.Its objective is to prevent fraud and to ensure th>t the market operates in the best interists of the general public. It is authorized to monitor trading and take disciplinaryaction when appropriate. The agency has set up an ecient systemfor rbitratingdisputesbetweenindividualsand its mmbers. From time to time, other bodies,such as the Securitiesand ExchangeCommission and the (SEC; www.sec.gov), the Federal Reserve Board (www.federlreserve.gov), have clmed jurisdictional rights over US Treasury Department (www.treas.gov), someaspects of futurestradipg.Thse bodiesare conrned with the esects of futures tradingon the spot markets fr securitiessuch s stocks,Treasurybills,and Treasury bonds. SECcurrrntly has an efective veto over the approval of newstockor bonp indexfuturescontracts. However,thebasicresponsibilityfor a11futuresand options on futuresrests with the CFTC. 'fhe
Trading Irregularities ''
kost of the timefuturesmarkets operate eciently and in the public interest.However, from time to time, trading irregularitiesdo come to light. 0ne type of trading ,
Mehanics
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FutuyesMarkets
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the market'' 2 The invejtor irregularity pccurswhen an investofgroup triesto grou? takesa hugelongfuturespositionand also tyiesto exercisespmectmtrol overthe supply of the undetlying commodity. As the paturity pf the futures contracts is approached,the investorgronp does ot close out its position,so that tly nllmber of ! oqttding (utures contracts may exe d the amovnt of the commodity available for delivery.The holdersof short posktippsrealize that they will fmdit dicult to deliver and becomedesperateto lose otlt their positions. The rsult is large rise in both futuresand spot prices. Replators usua del with tilistpe of abuse of the market by margin'requirements incriasing or im/osingstricterpositionllmltsor probiting trades thatincreasea speculator'sopen position or requiringmarket partkipants to close out thiirpositions. ) Other types of tralig irregularitycan invqlvethe traders the qoor of the publicity reived when earlyin it 1989, These was announcedthat exchange. some using undercoverajentsj of tmdingon theFBI had carrie out a two-yearipvestigation, theChicagoBoardof Tradeand the CllicagoMercantileExchange, Theinvestigation wasinitiatedbecauseof complaintj flledby a largeagritultumlconrn. The alleged ofensesincluded overcharging cuqpmirs, not payingcustomers the full proceebs of sales,and tradrs using theirknowledg:of customer orders to tradeflrstfor themselves (anofence known as frontrgrzng). Stcorer
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2.9'-' ACCOUNTINGAND TAX -
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Thefulldetailsof the accounting and taxtreatmentof futures ontractd are beyondthe scopeof tilis book. A trader who wantsdetailedinfonuationon tilisshold consult experts. In this section we pyovidesome general backgroundinfonpation,
Accounting Accountingstandards mquire changes in the market value of a futurescontract to be recognizedwhen theyocur unless the contmct qualifks as hedge.If the contract does qalify as a hedge,gains or lossesare generallyfecognized for accounting purposes in the ame period in which the gains or lossesfromtheitep,beinghedgtdare recognlzed. The latter treatmentis referred to as edge Jccpgrlfrlg. Considera company with a Decemberyear end. In Septembey2*7 it buysa March 2608corn futures contract and closes out the position at the :nd of Vebrual'y 28. when the contract is entered Sppose that the futurespris are 25 cents per bushel into, 27 centj per bushel at the end of 27, and 28 cents per bushelwhen the contratis closed out. The contract is for the deliveryof 5,0 bushelk.If the contract does not qualify as a hedge,the gains for accounting purposes are @
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5,0 x (2.70 2.50)= $l, -
in 2007and
5,
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u (2.80 2.70)= $500 =
2 Pcssiblythebestkncwnexampleof thiswas theattempt bytheHunt brothersto corner thesilvermarket in 1979-89.Betweenthe nliddlecf 1979and thebeginningof 198, theiractives 1edtc a pricerisefrcm $9per cuncetc $5 per cunce.
38
CHAPTER 2
in 28. If th company is hedgingthepurchaseof 5, bushelsof crn in February of $1,500is f0r hedge ccounting, the entire 2008sg that the contract relized in 2008for aonting purposes. The treatment f hedginggains andlossesis sensible.If the company is hedgingthq purchasef 5,90 bushelsof corn in February2008,the efect of the futurescontract is to ensure that t'iie prite paid is close to 25 cents per bushel.Theaccountingtreatment refkctsthat th pri is paid in z. ln June 1998,the Financial Accounting tndards BoardissuedFASBStatement ' No. 13j (/AS 133),Aounting for DerivativeInstrgmentsapd HedgingActivities. futups, forwards,swaps,and FAS 133applis to a1ityjes of derivatives(including It requires all derivativesto be includedon the balanci sheet at fair market options). 3 It value. increasesdisclosure requirements.It also gives companies far lesslatitude than previously in using hedge neeounting. For hedge nrmounting to be used, the hedginginstrumentmust be highly.efectivein ofletting exposures and an assessment of this esectiveness is required everythreemonths. A similar standardIAS39has been is ued by the InternationplAounting StandardsBoard. 'gain
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Tax ..
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Underthe US tax rules, two keyissuesare the nature of a taxablegain or lossand the timing of the recognitiop of the gain or loss.Gains or lussesare either classifkd as capital gains or losses r alternativelyas part of ordinary income. For a corporate taxpayer,capital gains are taxed at the ame rate as ordinary income,apd the ability to deductlossesis restricted. Capit>llosses are deductible only to the eytent of capital gains. A corportion may carry back a capital loss for three yers and caryy it forwardfor up to ve years. For a noncorporate taxpayer, short-termcapital gains are taked at the same mte as ordinary income,but long-term capital gains are subject to a maximum apital gains tax rate of 15%. (Long-term capital gains are gains from thesale of a capital asset heldfor longerthan one ye&r; short-termca/tal gains are the gains fromthe sale of a capital assetheld one year or less.) Fpr a noncorporate taxpayer,capital lsses aye deductibleto the extent of nd can be carried forward capital gains pll;s ordinary income up to $3, indehnitely. Geherally,positionsin futurescontracts are treatedas if tlieyare closed ut on the kst da of the tax year. For the noncorporate taxpayer,th givesrise to capital gains and lossesthat are treatedas if theywere6% longterm and 40% short term without tt/4'' rule.A noncorporate regard to the holdingperiod.This is referred to as the y&ypiyermay elect to arry backfor threeyepr any net lossesfromthe 60/40rule to ollet n gains recognizedunder the rule i the previous threeyears. edgingtranpctions are exempt fromthisruie. Thedefnitionof a hedgetransaction ' for tax purposej is diserent from that for nemounting purposes.The tax regulations defne a hedgingtmnsactionas a tzansactionentered into in the normal conrse of businessprimarily for one of the fllowing reasons: '
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1. To reduce the risk of price changes or currency :uctuations with respect 'to propertythat is held or to be heldby the taxpayerfor the purposesof producing ordinaryincome '
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3 Previously the attraction of derivativesin some situationswas that theywere
'of-balance-sheet''
items.
Mechanics p/
39
utures Markets
2. To reduce the risk of pritr or interestrate changes or currency iuctuations with by the taxjayer respet to borrowings 'mak
The hedgig transactionmust be identied beforethe end of the day on which the taxpayerenters into the transaction.The assetbeinghedgedmust be identifedwithin 35days.Gainsor lossesfromhedgingtransactionspretreatedas ordinary income.'Fhe timingof the recognitionof gains or lossesfromhedgingtmnsactionsgenerallymatches thetimingof the recognitiopof incomeqr expenseassociatedwith thetransactionbeing hedged. .
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2.10 FORWARDvs. FUTMRESCONTRACTS Themain diferencesbetweenforwardandfuturescontracts are summarized inTable2.3. Both contracts are agreementsto buyor sellan asset for a certainprice at a rtainfuture time. A forward contract is traded i the over-the-counter larkei and there is no standardcontract sizeor stanard dlivery arrangements.A singledeliverydat is usually pecied and the contract is usually heldto the end of itslifeand thensettled.A futures contractis a standardizedcontract ttaded on an exchange.A range of deliverydatesis usuallyspecifed. It is settled dailyand usuallyclosed out prior to maturity'.
Profits from Forward and FuturesContracts Supposethatthe Sterling exchapgerate for a g-day forwardcontract is 1.9 and that also the futres oricefor a cntract that will be delivexdin xactly 90days. #this rate is Whatis the diferenc betweenthe gains and lossesundr the two contracts? Under the forwardcontract, the wholegain or lossis realized at the end of the life of the contfact. Under the futups contract, the gaih or loss.is realized day by day that investor A is lonj f 1 mtllion becauseof the dailysettlement procedures. suppose in a 90-day forWard contract and investorB is long f 1 million in 90-dayfutures contracts. (Because each futqres contract is for the purchase or sale of f 62,500, investor B mujt purchase a total f 16 contracts.) Assumethat the spot exchange rate in 90 days proves to be 2.1000dollarsper pound. Investor A makes a gain of
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Table 2.3
Comparisonof forwardad fuiqrescontracts. #'.
Forward
Fatures
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contract betweentwo parties rivate
Not standardized Usually0ne specifeddeliverydate at end of contract settled Delivery or fmalcash settlement usuallytakes place
Somecredit risk
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Traded on an exchange contract standardized
Range of deliverydates
daily settled
Contractis usually closed out prior to maturity Virtuallyno credit risk
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CHAPTER 2
$200 on the 9th day.lnvestorB inks the same gain-but dpreadout over the g-day period.0n somedaysinvestorB m>y realizea loss,whereaton other dayshe 9r she makes a gain. Howevrr, in total, when lossesare rietted against gains, thereis a gain of $2, over the g-dy pefiod. '
Foreign ExchangeQuotes
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Both fomard and futures contracts trade actively on foreip currencies. However, there is sometimes a diference in the wpy excLange mtes re quoted in the two markets.For example, futurespriceswhere one currency (s the US dollar are always qoted as the number of US'dollarsper unit of the foxip cufrency or as the number of US cents per unit of the foreigncurrency. Fomard jricesare alwys quoted in the same way as spot prices. This means that, for the British pound, the euro, the Australiandollar, and the NewZealanddollar,the fgmard quotes show the number of USdollarsper unit of the foreigncurrencyand re directlycomparablewith futures quotes. For other major currencies, fomard quotes show the number of units of the freign currency per US dollar (USD). Considerthe Canadian doltar (CAD). USD per CAD corresponds to a fomard price quote futures price quote of of 1.0526 CAD per USD (1.05261/.95). .95
=
SUMMARY very ltigh proportion of the fqtuyescontrAct.s that are traded do not lead to the deliveryof the underlying asset.Tradersusually enter into ofsetting contracts to close out their positions beforethedeliveryperiod is rached. However,it isthe possibilityof fnal deliverythat driyesthe determinationof the futures pri. For each futures contract,thereis a range of daysduringwhich deliverycan be made and a well-defned deliveryprocedure. Somecotracts, such as thosi bn stocki:dis, art settleb in cash ratherthan by deliveryof the underlying asset. Thespecifcationof tontract is an importantactivity for a futuresexchange.The two sidesto any contractmust knowwhat can be delivered,where deliverycan take place, and when deliverycan take placr. Theyalso need to knowdetailson thetradinghours, howprices will be quoted, maximum dailyprice movements,and sp on. Newcontracts must be approvedby the CommodityFuturesTradingCommissionbeforetrading starts.
Margins are an important aspect of futuresmarkets.An investorkeepsa margin accountwith his or her broker.The account is adjusteddailyto i/ect gains or losses, and fromtime to timethe brokermay require the accqunt to be toppedup if adverse pricemovements havetaken place. The brokereither musi be a clepringhousemmber or must maintain a margin acount with a clearinghousemember. Eachclearinghouse member maintains a margin aount with the exchangeclearinghouse.TVbalancein the accountis adjusteddaily to reqect gains and lossespn the businessfor which the clearinghousemember is responsible. lnformation on futures prices is collected in a systematk way at exchanges and relayedwithin a matter of seconds to investorsthroughoutthe world. Many daily Of newspaperssuch as the WallStreet Joarzal carry a summary the previous day's trading.
Mechanics 0/ Futures
.4b?'#:!$'
futures contracts in a number of ways. Fomard . Forward contracts difer from contrcts are private arragements betweentwo parties, whepas futurescontracts are traded on exchanges. Thereis generallj a single delikerydate in a forwarbcontract, whereasfuturescontracts fretnentlyinvolvea range of such dates.Becaus thy are not traded on exchnges, forwardcontracts do not need to be syandardized.A forward contractis not suallysettled untilthe end of itslife?and most contracts do in faqtlead to deliveryof the underlyingassetpr a cajh settlement at this time. In the next fewchabters we shall examinein more detl the ways in whih fomard and futurescontracts can be used fr hedging.We shall alsolook at howfomard and futurespris are determined. .
FURTHERREADING Gastineau,G. L., D.J. Smith, and R. Todd. Risk Management, Derivatives, t?a# Fnancial Analylis un#zr SFAS No. JJJ. The Research founationof AIMR and BlackwellSerksin '
Finance, 2l.
Jons, F.1.band R. J. Teweles.ln: Te utures Game,edited byB.Warwick, 3rd edn. New Yrk: MCGOW-HZ,1998. Jorion, P. stltisk Management Lessons from Long--ferm Capital Management,''Eurppean 277-300. FinancialManagement, 6, 3 (September2): Kawalkr, 1.G., and P. D. Koch. S4Meeting the HighlyEfective Expectatin Criteripnfor Hedge Accounting,'' Journal ofDeriyatibs, 7 4 (Summer2): 79-87. ,
Capital Vdmpvnl. New
Lowenstein,R. Wen Xnl?. Failed: Te Riseand Fall bflvongrhrm Y0rk2Random House) 2. .
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Qujiions4nd Problems(Aswers in SolutionsMnual)
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2.1. Distingtlishbetweenth terms open interest and trading volame. . 2.2. What is the diferene betweena localand a commissionbroker 2.3. Suppos that you enter into a short futurescontract t sell Julysilverfor $10.20 per ounce of size is Commodity Exchanze. the contract 5,000 NewYork te ounces. The on ne maintenance margin a nd mafgin change initial is $3,000.What is $4,000, the in the futufesprice will lead to a margin call?Whathappensif you do not mt the margin call? 2.4. Supposethat in September2009 a company takesa long posltin in a contract on May 2010cnlde oi1 futures.It closes out its positionin March 21. ne futuresprice (per barrel)is $68.30when it enterj into the contract, $70.50when it closes out its position, and $69.10at the end of December2009. 0ne contract is forthedeliveryof 1, barrels, Whatisthe company's totalprofh?Whenisit realized?Howis it taxedifit is (a)a hedger and (b)a speculator? Assumethat the company has a Dember 31 year-end. might it be used? What does a limit 2.5. What does a stop order to sell at $2meap?When used? mean? sell Whenmightit be order at $2 margin aounts administered by a 2.6. What is the diferencebetweenthe operation of and clearinghouse those administeredhy a broker? 2.7. What diferences exist in the wy prices are quoted in the foreign exchange futures market, the foreignexchange spt market, and the foreignexchange forwardmarket? .to
.the
CkAPTER 2
42
2.8. The party with a short position in a futurescontract sometimeshas qptions as to the pla, whep deliverywiit tak Precise sset that will be delivered,wheredeliveryFill take plac, and so on. Do thje iptionsincreaseor decreasethe futuresprice? Vxplain yur reasoning. g 2.9. What are the most imprtant aspetts of the desip f a new futprescontract? 2.1. Explainhow largins protect invesiorsagainst the possibility f default. two Julyfuturescontracts n orange juice.'Eachcontract ij for thedeliker 2.11.A traderbays pou'nds. The current fuiuresprice is 16 cents per poundkxthtinitial margini 0f 15 ,
$6,000per contract, and the maintenancemargin is $4,5* per cohtract. What prke changewould lead to a margin 0112 Under what circmstans could $2,000be withzrawn from the margin aount?
2.12. show that, if the futuresprice of a commodity is greaterthan the pot price duringthe deliveryperiod, then there is an arbitrage opportunity. Does an arbitrag opportunity existif the futuresprice is lessthan the Spot price? Explainyour answer. order and a stop order. 2.13. Explainthe diferencebetweena market-if-touched 2.'14.tplqinwhat a stoNlimit order to sell at 20.30with a limitof 20.1) means.
2.15. At the end of one day a clearinghousemember is long 10 ontracts, and the stttlement priceis $50,($0per contract. The originalmarginis $2,000per contract. Onthefllowing daythelmber becomrsresponsiblefor clearingah additional 20longcontracts, entered into at a price of $51,000per contract. The settkment pri at the end of this day is $59,200.How much does the member have to add to its margin aount Fith the exchage clearinghouse? 2.16. On July 1, 2009, a Japanesecompany enters into a forwardcontract to buy $1million on January 1, 21. On September1, 2009,it entersinto a fomard contract to sell $1million on January 1, 21. Describe the profh or lossthe company willmakt in yen as a function 1, 2409. of the forwardexchnge rates on July 1, 2009, and september franc for deliveryin 45 dys is quoted as 1.7500. The 2.17. The forward price of the swiss ftures pri for a contract that will be deliveredin 45 daysis 0.7980.Explainthesetwo francs? quotes.Whichis mol'e favorable-foran investorwanting to sell swiss 2.18. suppose you call your broker and issueinstructionsto sell one July hogs contract. Describe what happens. 2.19. ttspeculationin futuresmarkets is pure gambling. It is not in the public interestto allow speculatorsto trade on a futuresexchange.'' Discuss tbis viewpoint. 2 20 Identify the comppditie whose ftures contracts hav the highest open interestin Table 2.2. 2.21. What do you think would happenif an exchange startedtradinga contract in whichthe qualityof the underlying asset was incompletelyspecihed? 2.22. ttWhena futres contract istradedon theEoorof the exchange,it myy bethe casethat the openinterestincreasesby one, staysthe snme,or decreasesbyone.'' Explainthis statement. 2.23. suppose that, on October24, 2009, a company sellsone April 2010live cattle futures contract.It closes out its position on January21,21. The futurespri (perpound)is 91 cents when it enters into the contfact, 88.30 nts when it closes out its position, and 88.80cents at the end of Dember 2009.One contract is for the deliveryof 40,000 .2
Mechanics n.JFutuyesMaykets
43
pouli s of cattle. Whatis the total profh? How is it taxedif th company is (a)a hedger and (b)a speclator? Assumethasthe ompapyhas a Dember 31 year-rnd. 2.24.A cattle farnxr expects to have12, podq of livecattlet sell in 3 lonths. Thelive MercantieExcange is for the deliveryof 4, cattlefuturescontrqcton. the chicgo poundsof cattle. Hok can the farmeruse the contractfor hedging?From the frmer's viewpoint,Fhat are the pros and cons of hedging? 2.25. It isJuly 28. A mining ompany hasjustdiscovereda small depositof gold. It will take 6 months to construct the pine. The gold will then be extracted on a more or less cntinuous basisfor 1 year. Futurescontractson gld afe available on the NewYork Compodity Exchange. There are deliverymonths every 2 months fromAugust2008 to Deember 29. Each contract is fr the deliveryof 1 ounces.Discuss howthe mining companymight use futuresmarkets for hedging.
AssignmentQuestions 2.26. A compny eters into a short futurescontractto sell5, bushelsof wheat fof 45 cents per bushel.Theinitialmargin is $3,000and themaintenancemargin is $2,. Whatprke changewould lead to a margincall? Under what cirolmstantescould $1,500be withdrawnfromthe margin aount? 2.27. Supposethat thereare no storaj: costs for crudeoil and theinterestrate foi borrowingor tndingis 5% per annum. How could yop make money on January8, 27, by trading June 2007 and becelber 2007 contrats? Usi Table2.2. 2.28. What position is equivalent to a longforwardcontractto buy an asset at K on a certain ate and a put option to sell it for K on that date.
2 29 The author's Web page (-.rotm=.utoronto.ca/-hu/data)containsdaily closing pricesfor crude bil and future!contracts. (Bothcontracts are traded on NYMEX.) You are required to downloadthe data and ariswerthe following: (a) How high do ihe maintenance margin levelsfor oil and gold have to be set so that there is a 1% chancethat an investorwith a balante slightly abovethe maintenance marginlevelon a particlar day has a negative balance2 dayslater?How high do they have to be for a 0.1% chan? Assumedaily price changes are normally distributed with mean zero. Explain why NYMEX might be interestedin this calculaiion (b) Imagine an investor who starts with a long position in the oil contract at the beginningof the period covered by the data and keepsthe contract for the whole of the period of time cvered by the data. Marginbalancesin excess of the initial marginare withdrawn. Use the maintenan margin you calculated in part (a)for a 1% lisk leveland assume that the maintennce margin is 75% of the initialmargin. Calculatethe nllmber of margin calls and the number of timesthe investorhas a negativemargin balance.Assumethat all margin callsare met in your calculations. Repeatthe calculations for an investorwho starts with a short position in the gold 'gld
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Many of the participants in futuresmarkets are hedgers.Their aim is to use futgres to reduce a particular risk that theyfa. This risk might relate to qpctuations markets inthe price of oil, a foreignexchangerate, thelevelof the stockmarket, or some other variable.A perfect edge is one that completelyeliminatesthe risk. Perfecthedgesare rare. For the most part, therefoye,a study of hedgingusing futurescontracts is a study of the ways in whic hedgescan be constructed so that theyperform as close to perfect
aspossible.
In this chaptr we consider a number of general issues associqted with the way hedgesare set up. When is a short futures position appropriate? When is a long futuresposition apprbpriatet Whkh futufes contfact should be used? What is the optimalsize of the futures position for reducing risk? At this stage,we restrict ouf attentionto what might be termed edge-and-forgetstrategies. We assume that no attemptis made to adjust the hedgeonce it has beenput in place. The hedgersimply takesa futures position at the Leginningof the life of the hedgepndcloses out the positionat the end of the titeof the hedge.In Chapter17 we will exalnine dynamic hedgingstrategies in which te hedgeis monitred closely an frequentadjustments ar made. The chapter initiallytreats iturescontracts as forwardcontracts (thatis, it ignores that takesaccount dailysettlement).Laterit explainsan adjustment knownas of the diferencebetweenfuturesand forwards. 'stailing''
3.1
BASICPRINCIPLES When an individualor company choses to use futuresmarkts to hedgea risk, the objectiveis usually to take a position that neutralizes the risk as far 'as possible. for each 1 cent increasein the Considera company that knowsit will gn $10,000 priceof a commodity over the next 3 months andlose$1, for each 1 nt decrease in the pri duringthe sameperiod. To hedge,the company's treasurershouldtake a Short futures position that is designedto oflet this risk. The futuresposition should lead to a loss of $l, for each 1 cent increasein the price of the commodity over and the 3 months a gain of $1,00 for each 1 nt decreasein the pri duringthis period. If the price Of the commodity go s down,the gain on the futuresposition
45
CHAPTER 3 >'
ofsets the loss on the rest of the company's business.If theprke of th commodity goes up, the loss on thq futures positin is ofset by the gain n the rest bf the company'sbusiness. '
,
short Hedj:s A sort hfdgeis a hedge,suchas the one justdescribed,thatinvolvesa short position in futurescontracts. A shorthedgeis appropriate when the hedgeralready owns an asset and expects to sellit at sonie timein the future.For example, a short hedgecould be usedbj a farmerwho ownssome hogsand knowsthgt tey will be reqdy for saleat the j localmarket in two nionths. A shorthedgecan lsobe led when an aiset is not owne for example, a US right now but wi11be owned at sometimein the futu-m.Uonsider, receive who months. he she ill in knowsthat 3 The exporter will exporter euros or realizea gain if the euro increasesin value relative to the US dollarand will sustaina loss if the euro decreakesin value relative to the US dollar.A short futuresposition leadsto a losj if th euro increasesin valueand a gain ifit decreasesin value. It hasthe efect qf ofliettinj the exporter's risk. To providea mofe detailedillustrationof the operation of a short hedgein a specifc situation,we assumethatit May l5today>nd that an oilproducer hasjustnegotiated a contract to sell 1million barrelsof crude oil. It hasbeepagreedthat the pricet'hatwill oi1producer is thereforein applyin the contract isthe market price on Apst 15. for each 1 nt increasein the price of oi1over the the position where it willgain $1, f0r each 1 cent decreasein thepri duringthisperiod. next3 months and lse $1, Supposethat on May 15 te spot prke is $6 per barreland the crude oi1futuresprice on the New York Mercantile Exchange(NYMEX) for Augustdeliveryis $59per barrel. Because each futurescontracf on NYMEX is for the deliveryof l, barrels, sh.orting 1, futures contracts. If the oil the company can hedgei: xposur producercloses out its position on Aupst 15,the efect of the strategy should be to lockin a price close to $59per barrel. To illustratewat might happen,suppose that the spot price on August15 proves to be $55per barrel.The company realizes$55million for the oil under its sales contract. Because Augustis the deliverymoth for the futurescontract, the futures price on August 15 should be very ctose to the spot prke of $55on that date. The company thereforegains approximately 'rhe
.by
$59 $55= $4 -
pef barrel, or $4 million in total from the short futuresposition.The total amount realizedfromboth thefuturespositionand the sales contract isthereforeapproximately $59 per barrel, or $59million in total. For an alternativeoutcome,suppose that the priceof oil on August15 provesto be $65per brrel. The company realizes$65for the oil and losesapproximately $65 $59= $6 -
per barrel on the short futuresposition. Again,the total amount realized is approximately $59 million. It is easy to see that in all cases the company ends up with approximately$59million.
Heking Strategs UsingFutures
47
Long Hedges Hedgesthat involvetaking a long psition in a futures contract are known as Iong A longhedgeis apprbpriayeFhen a companyknowsit will haveto purchasea hedges: now. certainpssetin the futureand wants to locki a pri 1') supposethat it' is now January15.A cojper fab c4tor knowsit willrequire l, poundsof copper on May 15 to meet a certain contract.Te spot price of copper is 34 cets per pound,and thefuturesprke forMaydeliveryis 32 cents per pound. fabricator can hedgeit,spositionby takinga longlositionin fout futurescontiactson COMEXdivisionof NYMEXand closingitspositionon May15.Eachcontractis the forthe deEveryof 25, ponds of copper.Tie strater hasthe eflkctof lockingin the priceof the required copper at close to 32 cents per pound, Supposethat the spot pri of copper on May15provesto be 325 nts per pound. BecauseMayis thedeliverymonth for thefuturescontract,this jhould be very close to thefuturrsprke. ne fahricatortherefox gains approximately -l''il
l,
($3.25$3.20) $5,000
x
=
-
x $3.25 $325 for the copper, making the on the futurescontracts. It pays 1 For an alternative outcome, net cost approximately$325,000 $5,000 $320,000. supposethat the spot priceis 35 nts per pound on May 15.The fabricatofthen losesapproximately l, x ($3.2 $3.05)= $15, =
,
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=
-
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for the copper. Again,the x $3.05 $305,000 on thefuturs contract and pays lr cints pound. approximately $320,000, net cost is per or 32 Notethat it is betterfor the company to usefuturescontractsthan to buythe copper on January 15in the spot market. If it doesthelatter,it will pay 34 cents per poud of 32 cents per pound and will incurbothinterestcosts and stomgecosts. For a instead companyusing copper on a regular basis:this disadvantagewould be oflet by the convenienceof havingthe copper on hand.' However,for a companythat knws it will not require the copper until May 15 te futurescontract alternativeis likelyto be preferred. Longhedgescan be used to manage an existingshort position.Cnsid.eran investor whohas shorted a certain stock. (seeStion 5.2 for a disclpsionof shprting.) Part of the risk facedby the investoris related to the performance of the whole stock market. Theinvestorcan neutralize this risk kith a longpositioni indexfuturescontracts. This type of hedgingstrater is discussedfurtherlaterin the chapter. The exnmpleswe havelookedat assume thatthe futuresposition is closzd out in the month. The hedgehas the snme bsic flkctif deliveryis allowed to happen. delivery However,making or takingdelivry can be costlyand inconvenient.For tls reason, is not usually made even when the hedger keepsthe futurescontract until the delivery month. As will be discussedlater,hedgerswith long positions usually avoid delivery anypossibilityof having to take deliveryby closing out their positionsheforethe ' deliveryperiod. We have also assumedin the two examplesthat there is no daily settlement.In dally settlementdoeshavea small efect on the performance-of a hedje. As practice, =
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l SeeSection5.11 for a iscussionof convenienceyklds.
48
CHAPTVR 3
explainedin Chapter2,it means thatthe jayoffromthefuturescontract is realized day by dy throughoutthe lifr f thehedgerather than al1at the end.
3.2
ARGUMENTSFOk ND AGAINSTHEPGING The argumentsin favorof hed/ng are so obvidus that theyhardly need to be stated. Most companies are in the buiness of mqnvfacturing, or retailing or wholesaling, or providinga service.Theyhavenp particular skillsor experte in predicting vaiiables suchas interestrates, xchangerates, andcommodityprices.lt inaj es snse for tjyem to hedgethe risks associatedFith thesevariables as they aye. T e companies can then focuson their main activities-for wlch presumablytheydo haveparticular skillsand By hedgisg,theyavoidunpleasait surpres sch as sh&rprises ip the price of expertise. ommodity that is beingpurchased. a ln practice, many rks are left unhedged. ln the rest of th sectionwe kill explore someof the reasons. ' .
Ijedging nd Shareholders One argument sometimesput forwardis that the shareholderscan, if theywish, do the themselves.Theydo not need the company to doit forthem.Thisargumentisj hedglng lt assumes that shareholders have as much infonuation howeker, open to tuestion. aboutthe risks facid by a company as doi'the cmptly's majement. ln most instantes,this is not the case. The arpment also ignorescommissions and other costs. Theseare lessexpensiveper dollarof hed/ng for largetransactions transactions than for smalltransactions.Hedgingisthereforelikelyto belessexpensivewhen carried 0ut bythe company than when it carried 0ut by individualshareholders. lndeed,the size Of futurescontracts makes hedng byindividualshareholdersimpossiblein many sittations.
One thing that shareholderscan do far more easilythan a coporation is diversify risk.A shreholder with a well-diversifkdportfoliomay be immuneto man'y of the risksfacedby a coporation. For example,in addition to holdingsharesin a company thatuses copper, a well-diversied shareholder may holdsharesin a copper prnducer, sothat thereij yery littleoveiallexposureto the price of copper. lf companies are acting in the best interestsof well-diversifedshareholdrs, it can be argued that hedgingis unnecessaryin many situations.However,the extent to which managers are in practice insuencedby this type of arpnent is open to question. '
Hedjing and Competitors .
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lf hedgingis not the norm in a certain industry,it may not make sense for one particularcompany to choose t be diferent from a11others. Competitivepressures witln the industrymay be suchthat the pricesof the goods nd servicesprodud by the industry:uctuate to reqect raw materkl costs, interestrates, exchangerates, and so on. A oinpany that does not hedge can expect its profh margins to be roughly constant. However,a company that does hedge an expect its prolh margins to Cuctuate! To illustratetls point, consider two manufacturers of gold jewelry,Safeandsure
49
HedgingStyategiesUsingFutes
Takle 3.i
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d not htdgr.
Catlge in goldprice
f'fcf on price of goldjewelry
f#cf otlprohts of Takeachatlce Co.
lncreae
lncrease Decrease
Nne None
Decrease
,
-
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on prohts of
Cp. s'c/c/ldfs/c Increase Decrase
Companyand Takeachnce Cmpany. Weassue that most companiesin theindustry o pt hedge against movementsin the prke of gold and that Takeachane Company is no exception. However,SafeandsureCompanyhas decibedto be dferent fromits competitorsand to use futurescontmcts to hedgeits purchase of gpld over the next 18 months. If the pri of gold goes up, economicpressurej willtend to lead to a corfespn ding increaseip the wholesale pri o'f the jewelry, so that Takeachan unasected. marpin proft is By contrast, SafeandsureCompany's Company'sgrss profhmargin 111incrase aftertheesectsofthehedgehavebeentakepinto account. If the price of gold goes down,economic pressureswill tend to lead to a corresponding Again,TakeachanceCopany's prtt decreasein the wholesaleprice of the jewelry. Company'sprtt margingoes down.Ip mar/n is unafected. However,safeandsure prqf! margin cpuld becope negative as a conditions, SafeandsureCompany's extremr tthedging'' carried out! The situationis summarizedin Table3k1 result of the Tls exampleemphasizesthe importan of lookingat the bigpictmewhen hedging. A11the implicationsof price changes on a company's protability should be takeninto accountin the designof a hedgingstrategy to prptect against the pri changes. .
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Hedging Can Lead to a Worse Outcome It is importantto realke that a hedgeusing futurescdntracts can result in a decreaseor woutd be in with no an increasein a company's profhs relative to the position it oi1pfoduceyconsidered example prke of oi1 earlier,if involving the In the the hedgipg. million barfelsof oil, and the goesdown,the company losesmoney on its sale f 1 futuris position leads to an ofsetting gaip. treasurercan be congmtulated for 2 havinghad the foreslghtto put the hedgein plact. Clearly,the company is better oflthan it would be Withno hedginj.Otherexecutivesin the organization,it is hoped,will appreciatethe contribution made by the treasurer.lf the prke of oi1 goes up, the companygains fromits saleof the oil, and the futuresposition leadsto an ofsetting loss.The company is in a worseposition thanit would be with no hedgipg.Although the hedgingdecisionwas perfectly logical,the treasurermay in practice havea dicult timejustifying it. Supposethat the prke of oil at the end of thehedgeis $6%so that the comjny loses$1 per barrel on the futurescontract. We can imaginea conversation such as the followingbetweenthe treasurerand the president: rl'he
PRESIDENT: TREASURER:
Thisisterrible.We'velost$1 lilliop inthefuturesmarket inthe space of threemonths. Howcould it happen?1 want a fullexplanation. The purpoje of thefuturescontracts was to hedgeourexposureto the prke of oil, not to make a prtt. Don't forgetwe made $1 million fromthe favorableefect of the oi1prke increaseson our business.
CHAPYER3
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a:fte (jk ihetokt Ikuk.. Tte i c:ptil bakiijchaigts gid i tlt way. fi ledij thi gok loi ' as '. ..' . y .
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What'sthat got to do with it? nat'slite saying that we d not need to worry whenour salesare downin Californiabecausetheyare up in NewYork. . pri of oi1had down If the gpne I don't care what would have appenedif the price of oi1had gone down.Thefactis thatit went up, I re>llydo not knowwhat you were doing playing ihefutures markets like this. 0ur shareholders will expectus to havedoneparticularlywellts quarter. I'm going to have to explainto themthat yoly actionsreducedprofhs by $1 million.I'm afraidthis is going to mean no bonusfor you this year. That's unfair. I was only Unfair!You are luckynot to be fred. You lost $1 million. It a11dependson howyou look at it '
'fREAsuRER:
PRESIDENT:
TREASURER:
PRESIDENT: RkEASURER:
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It is easyto seewhymany treasurersare reluctant to hedge!Hedgingredus risk forthe company.Hpwever,it may increaserisk for the tfeasurerif othersdo notfullyunderstandwhat isbeingdone,Theonlyrealsolution to thisproblem involvesensuringthat a11 seniorexecutiveswithinthe organizationfullyunderstand the nature ofhedgingbeforea hedgingprogramis put in plnce.Ideally,hedgingstrategiesare set by a company's board of directorsand are clearlycommunicated to both thecompany'smanagement and the shareholders.(SeeBusinesssnapshot 3.1for a discussionof hedgingby gold mining companies)
51
Hedging S/rtdegesUsingFutuyes
3.3
BASISRISK Thehedgesin the examplescqnsideredso farhavebeenalmost top good to be tfue.The hedgir was able toidentifytheprecisedatein thefuturewhen an asset would bebought or sld. Thehedgerwas thcp,able to lse futurescontmctsto remove lpoqt a11the risk arisingfromthe price of the asset on thatdate.In praciice,hedgingis often not quite as straightfomard.Someof the reasons are as follows: 1. The asset whos priceis to be hedgedmay not be exactlythe same s the asset underlyingthe futurescontmct. 2. The hedgerpay be uncertain as to the exact date when the asset will be bought or sold. 3. The hedge may require the futurescontractto be closedout beforeits delivery month. Theseproblemsgivere to what istermedbasisrk. Tllisconceptwillnow be explained.
The Basis 2
The basisin a hedgingsituationis as follows: Basis = Spot price of asset to be hedged Futuresprice of contractused -
If the asset to be hedjed and the asset underlyig thefuturescontractare the same,the basisshouldbe zero at the expimtion of the futurescontractkPrior to expiration, the basis may be positive or negative. From Table 2.2 and Figure 2.2, we see that on January8, 2007,the basiswas negative for gold and positive for orange juice. As time passes, the spot prie and thefuturesprice do not nessarily change by the sameamout. As a result, the basischnges. An increasein the basisis referredto as a strengt ening of the basis. a decreasein the basisis referred to as a akeningof te basis.Figur 3. 1illustrateshowa basismight chanje over timein a situation where the T basisis positive prior to expiration of the futurescontract. To examine the nature of basisrisk, wewill use the followingnotation: ,
.
1.
Sp'ot price at time !1
v k'. Spot prke at time 12 >) Futuresprice at time !1 ..
F'2. Futuresprice at time tz hl : Basis at time /:1 '
bz: Basisat. time tz %z will assume that a hedgeis put in pla at time tl and close,d0ut at time !2. As an example,we will consider the cae where the spot and futurespris at the timethe hedgeis ipitiatedare $2.50and $2.20,respectively,ad that at the time the hedgeis closed0ut theyare $2.00and $1.9, respectively.Thismeans that :1 = 2.50, IL 2.20, k 2., and /-2 = 1.9. :
.
=
=
'
. .
2 This is the usual dehnition.However, the alternative deinition Basis= Futures price someum' es used, particularly when thefuturescontract is on a fmancialasset.
-
spotprice is
52
CHAPTER 3 Figure
Variation of basisoker time.
.3.1
Spt ?rice
?rice kutures
'
j
Time
1
)'
tz
From the defmitionof ihebasis,we have bL
=
,% -
and
/)
h
=
k
-
/-2
and 52= so that, i!l our example, l = Considerflrstthe situaon of a hedgerwho klpwsthatthe asset will be sld at time12 and tates a shortfuturesposition at timel1. The price realizedforthe assetis k and the prost on thefuturesposition is >) F2.Theefective pfice thatis obtained forthe asset withhedgingis therefor: 5'2+ F1 F2 FI + h .1.
.3
-
'
.
,
=
-
In our examle, thisis $2.30.The valueof >) is knownat time11 If h kere aljo known at thistime,a perfect hedgewouldresult, Thehedjingriskisthe unrtainty associated wit,hh and is knownas basisrisk. Considernext a situation wherea companyknowsit price paid for the willbuy the asset at time 12and initiatesa long edge at timel1. asset is k and thq loss on the hedgeis >) f'2. The efective price that is paid with hedgingis therefore S2 + #'1 F2 = >'1 + 52 .
'l'he
-
-
Thisis the same expression as beforeand is $2.30in te example.The value of FI is knownat time l and the term 52represents basisrisk. Note that basis risk can lead to an improvementor a worsening of a hedger's Considera short hede. If the basisStrengthens (i.e., increases)unexpectdly, position. thehedger'sposition improves',if the basisweakens (i.e.,decreases)unexpectedly,the hedger 's position worsens. For a long edge, the reverseholds.If thebasisstrengthns the hedger'spositiov worseiis; if the bsis weakehs usexpectedly, the unexpectedly, position improves. hdger's The assetthat givesriseto thehedger'sexposureis sometimesdiferentfromthe asset underlyingthe futurescontract that is used for hedging.Thisincreasesthe basisrisk. DefmeS1 a! the price of the assetunderlyingthe futurescotract at time Asbefore, k isthe price of the assetbeinghedgedat time By hedging,a company ensuresthat theprice that willbe paid (orreived) for the assetis ,
.
.
k + A'l /-2 -
Hedging StyategietUg $
53
Ituyes
Tis can be written as
knj+
v? -
hj + (.2 -
Slj
The termsS? h and k S1.represet the two components of the basis.The :2 exist beinghedged if the is would the asset the basisthat the asset term were sameas uderlying thefuturescontract. Thek S1 termisthebasisarising fromthe diserence betweenthe two assets. ,71
-
-
-
-
'
Choiceof Cqntract Onekeyfactor afecting basis rk is the choice of the futurescontract to be used for hedging.This choice has two complments: 1. The choice of the asset underlying the futurescontract 2. The choice of the deliverymonth lf the asset beinghedgedexactlymatchesan asset underlyinga futurescontract, the srst choiceis generally fairly easy. ln other circumqances, it is necessgry to carry out a f l analysis to determinewllic of the availablefuturescontracts has futuresprices careu that are most closelycorrelated with the price of th asset beinghedged. The cholce of the deliverymenthis likelyto be iniuend by severalfactors.In the exailes given tarlier in tilis chapyer, we assumed ihat,when the expiiation of the hedaeSorrwpon/s Soa deliverymonth, the contpt.with thatdeliverytonth is chosen. ln fact, a cntract with a laterdeliverymonth lsusuallychosen in thesecircumstances. The reason is that fpturespries are in someinstans quiteerrtic duringthe delivery month.Moreover, a longhedgerruns the risk f havingto takedeliveryof the physical assetif the cntract is heldduringthedeliverymonth.Takng deliverycan be expensive and inconvenient.(Longhedgersnormally preferto close out the futurescontract and buy the asset fromtheir usual suppliers.) ln general, basisrisk icreases as thetimediferen betweenthehedgeexpimtionand the ddiverymonth increases.A good rule of thumbis thereforeto choose a delivery monththat is as close as possibleto, but laterthan,the expirationof thehedge.Suppose and Dember for a futurescontract on a delivtrymonths are March, June, september, expirations Dember, in Janualyian February,the March particularasset. For hedge contract will be chosn; fer hedge expirations in March,April, and May, the June contractwill be chosen; and so on. This tule of thumbassumes that thereis sucient liquidityin all contracts to meet thehedger'srequirements.ln practice,liquiditytendsto be greatest in short-maturity futurescontracts. Therefore,in some situations,thehedge: may be inclinedto use short-maturitycontracts and roll themfward. Tls strategy is discussedlater in the chapter. fxample 3.1 lt is March 1. A US company expectsto receive50million Japaneseyen at the end of July.Yenfutufescontracts on theChicagoMercantileExchangehavedelivery months of March, June, September,and Decembef.One contract is for the deliveryof 12.5 millionyen. The company thereforeshorts four Septemberyen futurescontracts on March l Whenthe yen are rived at the end of July,the companycloses o'ut its position.Wesupposethat thefuturesplice on March1in nts per yen is 0.7800and that the spot and futuresprices when the contractis and 0.7250,Tespectively. closedput a:e .
.72
3k
CHAPTER 3 The gain on thefutumscontract is0.7800 0.7250= 0.0550centsper yen. The basisis 0.7200 0.7250= ts per yen whe the contract is closed out, The eflbctiveprice obtained in ents per yen isthefinqlspot pri plus the gain on the futures: 0,720 + 9.0550= 0.7750 -
-.5
=
This can also be written qs the initialfuturis pri
0.7800(-.5) -#.
=
plus the nl basis:
0.7754
The total amount reived bythe companyfor the 59millionyen is 50 x 0.00775 millin dollars,or $387,500.
fxample j2 It is june8 and a cpmpqny knowsthat it will net. to purchase 2, .
barrelsof oi1 crude at some timein October r Novepber.Oilfuturescontzactsare currently tradedfor deliveryvery mont,hon NYVEX an2 the contract size is l, brrels. The company thereforedecidesto use the Dember contmd for hedzingand takes a long position in 29 Dember contracts. The futuresprice on June 8 is $68.00per barrel.The company fnds that it is ready to purchase the crude oi1on November1. It thereforecloses out its futurescontract on th>tdate. spot pri and futures pri on Novemberl are $70.00per barrel an $69.14per 'fhe
barrel. The zainon the futnrescontract is 69.10 68.00= $1.1 ?er bariel.The basis whenthe contract is closed out is 7. ..L 69.10 $0.90per barrel.The eflctive pri paid (indollars per barrel)is the fmal spot pri less the gain on the f11tures or 70.09 1.1 68.99 -
=
,
=
-
:.
Tltiscan also be calculated as the initialfuturespri plus the Enalbasis, 68,99+ 9.90 68.90 =
The total price paid is 68.90x 2,
3.4
=
$1,378,000.
CROSSHEDGING In te examples consideted up to now, the asset underlyingthe futurescxtract has beenthe jame as the asset whose pri is beinghedged.Cross edgingoccurs when the twoassets are diferent. Consider,fo?example, an airline that is conrned about the futureprice of jet fuel.Becausethereis no futurescoptract on jetfuel,it might choose youse heatingoi1futurescontracts to hedgeits exposure, The edge ratio istheratioof the sizeof the positio takenin futurescontracts to the sizeof te exposure.Whenthe assetyndrlyingthe futurescontract is the sme as the asset beinghedged;it is natural to use a hedgeratio of 1.. Thisis the hedgeratio we haveused in the examplesconsideredso far. For instan, in Example3.2,the hedger's barrls of oil, and futurescontracts were entered into for the exposurewat on 2, deliveryof exactly this amount of oil. '
55
Hedging StyategiesUsingFutuyes
When cross hedgingis used, settingthe hedge ratio eqal to 1. is not always optimal.The hedgershould choose a value for the hedge ratio that minimizes the varianc of the value of the hedgedposiibn. Wenowconsider howthe hedgercan do ' this .
'
Calculting the MinimumVariance Hedge Ratio Wewilluse the followingnotation: Ak: Chage in spot price, S, duringa period of timeequal tb the life of the hedge AF: Changeinfuturesprice, F, duringa periodof timeequal to thelife f t e j)edge
c,s: Standarddeviationof LS c,: Standafddeviationof AF ' p: Coecient of correlation betweenLS and AF b' : Hedgeratio that minimizesthe varian of the hedger'sposition In the appendixat the end of tls chaptr, we showthat
.
a
cc
p
u-l
y
.
(3.1)
The optimal hedgeratiois the product of the coecient of correlation betweenLS and ; and the ratio of the standarddeviationof LS to the standard deviationof AF. Figure3.2 shows howthe variance of the value of thehedger'sposition dependson the hedgefatio choen. the hedgeratio, b', is 1 Th result is to be expected,because lf p = 1 >nd cs = price lh, the inthis case the futures nrrors the spot pri perfectly.If p = 1 and c, hedgeratio h' is Thisresult is also as expectd, becauseil this case thefuturesprice alwayschanges by twiceas much as the spot price. The optimal hedgeratio, h*, istheslopeof thebest-fhline hen LS isrepessedagainst * A& as indicatedin Fipre 3.3.Thisis intuitivelyreasonable,becausewe require to correspondto the ratio of changesin LS to changesin A17'.Thehedge:ff:clfy:rl:k?.y can be .0.
o,
=
.5.
Figure 3.2
Dependenceof variance of hedger'sposition on hedgeratio.
Variangeof
positlon
Hedzeratio,
I
..
#
y.
56
CHAPTER 3 '
.'
..
.
Regression of changein spot priceagainst chang in futuresprice.
Figure 3.3
M
@
'
@
'
'
@
@
@
@ @ @ @
@
@
@
@
@ @
*
@
@
@
a.S the proportionOf the varian that is eliminated by hedging.This is the :2 2 against AS and equals , fromthe rqression Of
defned
LS
p
or
2
h*2G..IL Gs2
The parameters p, cy, and oj in equation (3.1) are usually estimated fromhistorical asjumption (I'he On is that the futurewill in some sense be implicit datq LS and AF. likethe past.) A number o equal nohoverlappingtimeintervalsare chosen, and the valuesOf LS and AF f0r each Of the intervalsare.obderved.Idally, the lngth of each
time intervalis the same as the lengthOf the timeintervaltOr which.the hedgeis in efect. In practice, this sometimesseverelylimitsthe number of observations that are available,and a shorter timeintervalis used.
Optimal Number of Contracts Desne variables as follows:
Qg: Sizeof position beinghedged(units) Qy: Sizeof one futurescontract (units) #*: Opmal number of futurescontracts for hed/ng The futures contracts should be on
h*QA
units of the asset. ne number of futures
52
Hedging tyategiesUsingFjx?w
contractsrequired is therefgy given by #*
b*Qz
=
(3.2)
F
Example3.3 An airlineexpectslo purchase2 million gaions of jetfuelin 1 monthad decides to use heatg oi1 futuqesfor hed/ng.3 We suppose that Table 3.2 gives, for 15 succesve monthj, data on the change, LS, in the jet fuel prke per gallon and the corresponding change, LF', in the futures pri for the contract o heating 0il that would be used for hedgingpri chnges during the month. The number of observations, whkh we witl denoteby n, is 15. We will denote thefth observations o AF and LS by xi and yi, respectively.FromTable3.2,we
have
')7xi
=
)7x,?0.0138 )-1z?, 0.0097
=.13
=
l'-lyf0.003 =
-
.17
xiyi
=
Table 3.2 Data to calculateminimum variance hedge ratio when heang oi1futurescontract is used to hedge of jd fuel. purchase '
Month
i
Change f?lfatares price xj) gllor per (=
1 2 3
Changeinfal price per gallon (=yf) - ---..
-.-
'
.2
.8
..1
0.026
0.044
-0.029 -().26 -0.029
0.048 -.6 -0.036 -0.11 0.019 -0.027 0.029
11 t2 13 14 15
-.
-0.044
-0.046
'
.
0.029
0.021 0.035
4
.
-.19
-.1
-.7
0.043 (L11
-0.036 -.018 .9 -k032 0.023
3 Heating oi1futurescontracts are mom liquidthanjetfud futurtscontracts. For an acctunt of howDelta Airlinesused heatingoi1futuresto hedg' e itsfuturepumhasesof jetfuel,see A. Ness, tDelt't Winson Fne1,''
Rk, Jkne 21,
p. 8.
58
CHAPTER 3
stanardformulasfromstatists gvt'tht estimatt of c,
as
2
Ex,? (Em) -
The estimaie of o.s is
n
-
1 nn
-
1)
E g (E,f) n
The estimate of p is
-
-
1 nn
-
=
g gyj?
=
g 9243
.
2
1)
.
nExfyf2 EmE) 2 = (Ezf) E gi (Eyi) E.x?, 2(n 2 (n -
=
():928 '
-
the minimumvariance hedje ratio, h%,is therefore From equation (3.1), 0.928x
0.0263 0.78 0.313 =
Each heatingoi1contract tradedon NYMEXis on 42,000gallns of heatingoil. From equation (32) the optimal number of cntracts is .
,
0.78x 2,, 42,000
= 37.14
or, rounding to the nearest whole nllmber, 37.
Tailingte Hedge When futures are used for hedging,a small adjustment, knownas Iailing te edge, Settlement. In practice thij meansthat can be made to allowfor the impactof daily
equation(3.2)becomes4
N* =
y;#y P>-
(3.3)
whereL is the dollarvalue of tht position beinghebgedand P>-is the dollarvilue of one futures contract (thefuturesptice times Q. Supposethat in Ekample 3.3 the spot price and the futures pri are 1.94 and 1.99 dollars per gallon. Then L = 2,000,000x 1.94 3,880,0000while P>-= 42,000 x 1.99 = 83,580,s that the optimalnumber of contracts is =
0.78x 3,880, 83,55
= 36.22
If we round this to the nearest whole number, the optimalnumberof contracts is now 36 rather than 37. The efet of tailingthe hedgeis to multiplythe hedge ratio in equation(3.2)by the ratio of the spot price to the futuresprice. Ideallythe futures positionused for hedgingshould thenbe djustedas L and Vychange, but in practice this is not usuallyfeasible.
4seeProblem 5.23 for an explanationof equaticn (3.3).
Hekiht Styategis' (/.
59
Futuyes '
3.5
STOCKINDX FUTURES .
We now move on to consider stock indexfuturesand howthey are used to hedgeor manEgt exposures to equityPlis. A stock izdex trackschanges in the val.utof a hypotheticalpoftfolio of stocks.The weightof a stockin the portfolio equalsthe proportion of the portfolio investedin the stock.The percentage increasein the stck indexover a small intervalof timeis set equal to the percentageincras ip the value f the hpothetka j por tfojjooivjdends are usually not includedin the clculatin so that theindx tracksthe cajital gyin/loss fronl investingin iheportfoiio.s If thehypotheticalportfolio of stocksremainssxed, theweightsassignedto individual stocksin the portfolio do not remaip flxed.Whenthe price of one particular stockin the porfolio rises more sharplythan others, more weightis autnmaticallygivento thatstock. Someindicesare constructed froma hypotheticalportfoiio consistingpf one of each of a nnmber of stocks. The weights assigpedto the stocks are then proportional to their marketprices,with adjustmentsbeingmadewhen thereare s'tocksplits. Otheripdicesare price x constructedsn that weights are proportional to market capitalization (stock underlyig of outstanding). portfolioisthenautomaticallyadjusted number shares a nd reqect splits, stock stock dividends, to new equity issues. .
'
.
.
,
'
'
.
rfhe
stock Indices Table3.3showsfututes prices for contracts o a number of diferent stockindicesas tlwywere reported ir the FJ!l StreetJpprrlclof Jauary 9, 2007.The pris referto the closeof trading on January8, 2007. The DpwJozes Izdastrial Xvercgeis basedon a portfolio consistingof 30 blue-chip stockjin the UnitedStates.Theweightsgiven to the Stocks aie proportioal to thek prices.The ChicagoBoard of Tradetradestwo futurescontracts on the index.Oneis on$1 timesthe index.The other (theMiniDJ Industrial Average)is on $5timesthe
'index. 'fheStandarJ d: Poor's J (S&P5) Izdex is hasedon a portfolioof 50 diserent stocks:4 industrials,40 utiliiies, 20 tmnsportationcompanies, and 40 nancial The weights of the stocksin theportfolio at any gi#en tiine are proinstitutions. portionalto thelr market capitallzations. 'Thisindexaccounts for 80% of the parket of a1l the stocks listedon the NewYork StockExchange.The Chicago capitalization .
MercantileExchange(CME)trades two futurescontracts on the S&P 5. Oneis on $250times the index;the other (theMini S&P 5 contract) is on $5 timesthe
index.
The NasdaqJ is based on 1 stocksusing the NationalAssociationof Securities Service.TheCMEtradestwo contracts. Oneis on $1 DealersAutomaticQuotations timesthe ildex; the other (theMiniNasdaq 1 contract) is on szitimes the index. largestcapitalization The RaztsellJ Izdexis an indexof the prices of the l Izdex Dollar is trade-weighted stocksin the United States.The U.S. indexof the a valuesof sixforeigncurrencies (theellro, yen, pound, Canadiandollar,Swedishkrona, and Swissfranc). 6 An
exception to th a total returz izdex. Tls is calculatedby assnming that dividendson the hpothetical portfolio are reinvestedin the portfolio. .is
60
CHAPTSR 3 Tabl
3.3 Index futres quotes from WallStreet Journal, January 9, 2007: Colpmns show month, open, high, low, settle, change, and open interest, respectively. I*x F/s ? 1*e/1 Aveilpslx **-- 121r 12515
J-
uB
' :I< :1 I*-I ..--.x J-
1159l
*(*$5xex. lze
u514 ujo 17.577
4 1*
1,-
1117.39 1121.59 l1B. 1Q&1:)437.% 1426.19 IIBA
OP M(Q'$%x
MIH
.,-1-
.
41
W*O!
-%
42 jk ' sj j2 21 1r 6i
1411* 4:* 141725ll.p :ll7 1437.* .# 143.5.591- x
b
1792. Nex 1123 11113
1m
1M 1m*
6.25 15,5%
t
.3 .3
328,+ Q
1'
xa@
-$.yj> 779,75nqb
PX MG
lnlex
1W(*R'
1791.911123 1'l9. 1333
Ji*
*1,897 13,12
I*x
l>?3 112,%
ye-ul IAH
m
h
M,m
y I/
* M Iex(*$l% AM
12 42
2
%$12 (*$1
Idex
12*5 12525
*-
'''
(G$VX:
1.13 K12 Mz K*
'.G
:25,9F3 B,?16 Jux
769,15 m
3.j
79,.
-.3
21,1.1 2,921
Ie
@27 M,1
Me M.o
-.3
Source: Reprintedby permision of DowJones,Inc.,via CopyrightClearan Center,lnc., (f)2997 Dow Jones & Company,Inc. A11Rights ReservedWorldwide.
As mentioned in Chapter2,futurescontracts on stockindicesare settlezin cash,not hy deliveryof the upderlying asset. A11contmcts are marked toimayketto either the openingpri or the closing pric of the index on the last tradipg day, nd the positionsare then deemedto be closed. For example, contmets on the s&P 5 are closed out at the opening price of the S&P 5 index on the third Fliday of the deliverymonth.
Hedging an quity Pnrtfnlin Stockindexfuturescan be used to hedgea well-diversifkdequity portfolio. Dene:
/': Currentvalue ot the portfolio
.
F': Currentvalue of one futurescontract (thefuturespri
timesthe contract size)
If the portfolio mirrors the index, the ogtimal hedge ratio, h*, equals 1 and shows that the numher of futurescontracts that should be shottedis eqpation(3.3) .
N* =-
P F
(3.4)
Suppose,for example, that a pottfoli worth $5,05, nrors the siP 5. The and each futurescontract is on $250tims theindex.In this indexfuturesprice is 1 and F' = 1,1 x 25 = 252,500,so that 20 contracts should be case P 5,5, shortrd to hedgethe portfolio. When the portfolio does not exactlymirror the index,we can use the parameter beta (j) fromthe capital asset pricingmodel to determinethe appropriatehedgeratio. Betais the slopeof the best-fhlipe obtained when excessreturn on the portfolio over the risk-free rate is regressedagainst the exctssrturn of the market over the risk-iee rate.When j 1., the return on the portfolio tnds to mirror the return on the market;when j = 2., the exss return ohthe portfolio tendsto be twi as great as the it ttnds to br,half as great; and so on. return on the market',when j excess ,l
=
=
.5,
Hedging Stratqs ,
UzfA!g
,
61
Futures .
.'
.
,
')
'
.
.
A portfolio with a j of 2. is twi as sensitikttp markd tovtmtnts ps a portfolio with a beta 1.. It is thereforenetessary to use twiceas many contrcts to hedgethe is half as sensitiveto market pprtfolio. Similarly,a portfolip with a beta of with of and prtfolip should beta 1.0 movelents as a we use half many contpcts a = tp hedgeit. In geheral, ## j, so tat quation(3.3) sives .5
'as
N* = j
>
,
(3.5)
Thisformulaassume that the maturity of the futurescotract is close to the maturity of the hedge. We illustratethat this formulagivesgood results by our earlierexample. Suppose.thaia futurescontract with 4 months to maturityis used to hedgethe value of a portfolio over the next 3 nonths in the followingsituation: 'extending
index= l,
Value of S&P 5
S&P500futuresjrice= 1,1 Value of portfolio = $5,050,000 Risk-free interestrate 4% per annum =
Dividendyieldon inde: = 1t/a per annum Beta of portfolio = 1.5
0ne futures contract is for deliveyyof $250times the index.It follqwsthat >== the number of futurescontracts that 25 x 1,1 = 252,500and fromequation(3.5), shorted hedge portfoliois shouldbe to te 1.5x
5,05,00 = 30 257,500
Supposd.theindextrns out to be90 in 3 months andthefuturesprice is9 2. fromthe short futurespositionis then 30 x (ll
-
I2) x 25
e ga
$81?
=
The loss on the indexis 1%. Theindexpaysa dividendof 1% pel.annum, or 0.25% per' 3 months.Whendividendsar: takeninto aount, an investorin the indexwould thereforeearn in the 3-monthperiod. Becausethe portfolio has a j of 1.5, the pricing model gives capital asset -9.75%
'Expectedreturn n portfolio Risk-freeinterestrate = 1.5 x (Return on index Risk-lee interestrate) -
-
The risk-freeinterestrate isapproxiately 1% per 3 months. It followsthat tlieexpected return (%)on the portfolipduringte 3monthswhen the 3-monihreturnon theindex is is 1k + (1.5x (-9.75 1.)) -9.75%
-15.125
-
=
of dividends)at theend of the3 months is The expectedvalue of the portfolio (inclusive therefore $5,050,000x (1 0.15125)= $4,226,127 -
62
CHAPTER 3 '
< ,
.
Table 3.4 Performanceof stockindexhedge!
%()
Valueof indezin three months: fkturesprice o? indextoday: Futures price of index in three months: Gain on futuresposition ($):
Return on market: Expectedreturn on portfolip: Expectedportfotio value in three lonths includingdividends($): 4,286,1874,664,9375,043,6875,422,4375,801,187 Total value of position 5,096,1875,099,9375,096?187 in three months ($): 5,099,9375,103,687 -.15.125%
-7.625%
-0.125%
It followsthat the expected value of the hdger's position, includingthe gain on the hedge,is $4;286,187 + $81, = $5,996,137 Table 3.4 summarizes these calculation! togetherwith similar calculations fr other values of the index at maturity. It can be seen that te total expected value of the hedger'sposition in 3 months is almostindependentof the value of the index. The only thingwe havenot coveredintllisexampleisthe relationshipbetweenfutures pricesand spot prkes. Wewill.seein Chapter5 that the 1,1 qssumed for the futures pricetodayis roughly what wewould expectgiven theinterestrate and dividendwe are zsuming. Th sameis tl'ue of the futuresprices in 3 mnths shown in Tle 3.4.6
Reasons f0r Hedging an EquityPprtfolip k
. Table3.4 shows that the hedglngschemeresults in a value for the hedger s position at jj about 1 /: higherthan at the be/pning of te the end of the 3-monthperiod being 3-monthperiod. Theri is no surprise here.ne risk-freerate is 4%'per annum, or 1% per 3 months. Thehedgeresults in theinveqtor'spositioh grwing at the risk-free rate. It is natural to ask why thehedgershouldgo t thetroubleof usin? futurescontracts. To earn the risk-freeinterestrate, thehedgercan simplysellthe portfolio and investthe proceedsin risk-free instrumentssuch as Treasurybills. One answer to this questiol is that hedgingcan be justifkdif the hedgerfeelsthat the stocksin the portfoliohavebeen chosen well. In these circumstans, the hedger might be very uncertain about the performan of the market as a whole, but consdentthat the stocksil the portfolio will outperform the market (after appropriate adjustmentshavebeenmade for thebeta of the portfolio). A hedgeusingindexfutures ,
'
'
6 The calculationsin Table 3.4 assnmethat thedividendyieldon theinex is preictable, therisk-freeinterest rate remains constant, an the return on the inex over the 3-monthperitd is perfectlycorrelated with the returnon the portfolio. In practicq theseassumptionsdo nt hol perfectly,an thehetkeworksrather less wellthan is indicate by Tabk 3.4.
63
HedgingStyategiesUsingFutuyes
removesthe risk arkingfrommarket moves and leavesthe hedgefexposed only to the performae of the portfolio felative to the markt. nothr j reascn jcr jjugsg may be that the.hedgeris planning io hold a portfoliofor a long peyiod of time and requires shoyt-term protectien in a ucertain market situation. ne alternatike strategyof selling the portfolio and buyig it back lateymkht involveunacceptably hightransactioncojts. '
Changing the Beta bf a Portfolio In the example in Tble 3.4,the beta of thehedger'sportfoliois yedud to zero.(The hedger'sexpected return is inbependeytof the performance of the index.)Sometimes ftures contractsaf used to change the beta of a portfolioto some'value other than 'zero.ContinuingFith our earlier example: . '
.
,
S&P5 index= 1, S&P50()futuresprice Value of portfolio
..
As before,F
=
,
.
=
=
1,1
$5,()50,000
sBetaof portfolio = l.5 '
.
25 x lji
=
252,500and a complete hedgerequires
1.5x
isttx z: 30 252,500
'
' ,
.
,
to be shorted. To reduce the beta of the portfoliofrop 1.5 to 0.75,the contracts umberof contracts shorted should be 15 rather than 3; to increasethe beta of the portfolicto 2., a tonppositionin 1 coptracts sbould be taken;and so on. In general, to change the beta of te portfolio from j to j*, where j s j* short positionin ,
p (/ = /*)# contractsis required. Whenj
) (3A.3) where 'futures
.
-
LS
=
,% -
AA-= :-2 >)
an
k1
-
Because SL and N; am known at time /1 the variance of F in equation (3A.3)is minilnizedwhenthe variance of LS h AF is lninimized. ne variance of k AF is ,
-
k
2+
1) = Gs
j
y'
-
-
kpgs oy
whereo, c,, and p are as desnedin Section3.4, So that
dv -dh
=
jho'y2
-
zpooy
Settingthis equal to zefo, and noting that d 2vldk2 ispositive,we seethat the value of h thatminimizes the variance is b,= p.s/cy.
T' 'zk' )
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k 11 jQx @
e 10 Maturity(years) >
2
1
The rates we haveclculated are summarizedin Table 4.4. A chart showingthe zero mteas a functionof maturityisknownas the zero czrve.A common assumption isthat the zero curve is linear betweenthe points determinedusing the bootstrap method. (This meansthat the 1.25-yearzero l'ate is x 10.536+ x 10.681 10.6085%in ourexample.) It is also usually assumed that the zero curve is horizontalprior to the hrstjoint and horizontalbeyondthelastpoint. Figure4.1 shws the zero curve for our data using theseassumptions.By using longermaturity bonds,th zero curve wovld be more accuratelydeterminedlyond 2 years. with m>turitiesequal to exactly1.5 years, In practice,Wedo not usually havetmnds 2 years, 2.5 years, and so on. The approach often used Ly analysts is to intepolate betweenthe bond price databefortii ij usd to calclate the zero curve. For example,if theyknowthat a 2.3-yearbond with a coupon of 6% sells'for 98 and a 2.7-yearbond with a-coupon of 6.5% sellsfor 99,it might be assuled that a 2.5-yearbond with a coupon of 6.25%would sellfor 98.5. .5
.5
=
4.6
FORWARD RATES Forwardinterestrates are ti.l e rates of interestimpliedby cursnt zero rates for periods To time in future. illustrateh0w they are calculated, we suppose that a the of particularset of zer rates are as shown in the secondcolumn of Table 4.5. The rats are assumed to be continuouslycompounded. Thus,the 3% per annum rate for 1 yearX3X1means that, in yeturnfor an investmentof $100today, an inyestorreceives $13 05 in 1 year; he 4% per annum rate for 2 years means that, in ltk returnfor an investmentof $1 today,the investorreceives ltk 9.04x2$jgg ?? in 2 years; and s on. The forwardinterestrate in Table 4.5 for year 2 is 5% per annum, This isthe rate of interestthat is impliedby the zero rates for the period of timebetweenthe end of the hrstyear and the end of the secondyear, It can be calculated from the l-year zero intertst rate of 3% per annum and the z-year zero interestrate of 4% per annum. It is the rate of interestfor year 2 that, whencombinedwith 3% per annllm for year 1, gives 4% overall for the 2 years. To shok that the correct answer 12.5%per annum, suppose =
.
=
.
83
InteyestRates
that $1 is invested.A yateof 3% for the srstyear and 5% for the second year gives '
ltk
Y 3? 1
v5
e
X1
$108.33
:::F
at the end of the second year. A l'ate of 4% per anum for 2 years gives 1'1'X2
'
'
Tilis exampleillustmtesthe generalresnlt that when interestrates whichis also $108.33. continuously compounded and rates in suerassivetime periods are combined, the are overallequivalent rqte is simply th averagerate dqring the whole period. In our example,3% for tlie rst year and 5% for the second year averageto 4% over the 2 years. The result is ouly approximately true whenthe rates are not continuously compun/ed. ne fofWar rat for year 3 is the rate of interestthat k impliedby a 4% per annum z-yearzero rate and a 4.6% per annum 3-yearzero l'ate. It is 5.8% per annum. .The reasoni! that an investmentfor 2 years at 4% per annum combinedwith an investment for one year at 5.8% per annum gives ali ovemll avemgereturn for the threeyears of 4.6% per annum. ne other fomard l'ates can be calculatedsimilarlyand are shownin the third coluln of the tble. In grnerat, if RL pnd Rj are the zero l'ates for maturities T1 ad T2, respectitely, and Ry is the forwardinterestl'ate for the period,of time betweenT1and T2,then RL T1 Rb T2 T -
Rg
=
(4.5)
-
consider the calculationof the year-4 forwardl'ate fromthe To illustrateth formula, TI and the formulagives data in Table 4.5: = 3, T2= 4, R3 0.046,and R Ry .5,
=
=
.62.
'
=
Equation(4.5)can be writtenas Ry = Rj +
%
-
:l)
T1 7-2 T1
(.1.t)
-
This shows that if the zero curve is upward slopingbetwebnTj and T2,so that R > :1 then Ry > R (i.e.,the fomard rate for a period f timeending at T2is greater thanthe T2 zero rate). Similarly,if the zero curve is dwnward sloping with Rj < Rj then ,
oi lenkat te rates give iy le4.5 d thipkstht ch/ye Pslcll ye) tht) pexi yeais!Tlie.ipveFiori4 ratrF Aill l/er. . ) . u. . . The lkyeaybprrpwipjs . br:rlkd yr boyrowl-y.t fundsard investtofs-yeafq. foyfurtht lyrr piti4ds ay ih tn of tl ftrsijSicoitp? thii) nd f, lttth yrap. If ill'ylld aiproft ipterdytratr dp iay out the sam? tlkij b:vt 2.! q rys; y jj yjjy reived jypy jj ( . will 5.3X) j becaus: Ihtfist n# at yeark #. k e per (:'' j. . . .. sjeultinj tlitpts stritigy is koA: aj iild cprvplqk Te ivtstbfii,q tiqdmj jpE rie'j tii'eppt eritp iikr! theflrd i11 fptur b e d it pdy. tuiti thr lzyiqytioj k purrxaltttj tomard rfs otftvd jn thr ntkbt todj ?oq t ri y j. . ( ' y . q j g . j.. .. . . . j . j .. .g%,)... g yj . .:E ;f . . ). ; E. . . .'.. . ... . . . .r. (y . . # i.e 5t)$,.5..8Z?6.24., nd. 6..1 g E ETrepsrr liron,thi pt Qrangrcokty', ett iild crke ylpyj:ikniil) Rtfb to ive r 1997 k 1Wj. frw %'e,pfhi '''tl r. wehav, jsttktfibib ktryqjprzmjfkllyi r, . !, . .... . . . tkjrrEputy's) ciiqr Mpprtot? bdjd t 4r. citfis tip''bwnmi ttp ti re-ektyt in h t:r 11 (N pt ja i pp p A f tjoet ad Faj ' ' ' E E q E tr.4 ink' strayegywa!tpp ( Errg ), q) t ,: y k y lijtedq ik izb't yiek 19k h1s heklly lexpqne In Mr. citron' turvipky. Hi JFlrj. Thij #y rze hf iutrrei quql t,o$ hed rr p? intiwj? inut npaunk ii He also leferged his ppsltidphy borfoWingip tl l'ilj ykt' lf hrtqirl interEji fte (? a pmained the sami r c 1994 AS 1$hajpe d iterist ptes 1994 sharpl iiijg Dtcepbr rtjst i kl E/tktfolio Ecotmty lok hb ivejippt d fnje a ou cid yha:t its $15 billion d fllrd nktptcy flk dys it jrolrttio. stwrl tater an intstor slppose interest
canEorrow
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'bp
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,
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,y
,
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(i.e.,the forward rate is kss than the Tz zero rate). Takinglimits us T2 and lettingthe commonvalue ofthe twobe T, we obtain approachesTl in equation (4.6) R>-< Rz
Ry
=
R+ T
CR
whtreR is the zero rat: for a matrity of T. The yqlue of Ry obtainedin this way is rate for a maturity of T. Th is the forwardrate knownas the izstaztazeousforward applicabte to a very shortfuturetimeperiodthatbeginsat tne T. Defne P, T) thatis e-2T laturing at time T. BecauseP, T) the as the price df a zefozcoupon bond =
,
equationfor the instantaneousfofwardrate can also be written as Ry =
-
yjlnP,
T4
Assumingthatthe zero rates forborfowingandinvestingare the snme(which is closeto thetruth for a largefnancialinstitution),an invejtorcan lockin theforwardrate for a futuretime period. Suppose,for example,that the zero ratesaze as in Table4.5,lf an borrows$1 at 3%for0'C3Xl 1year and theninveststhe moneyat 4% for2 years, the investor = is cash outqowof ltk $10305 at the end of year 1 and an infow of result 0.t14X2 103.05/$5a Dtul'n eqal to the = = theendof 2.Since108.33 at 33 $108 year ltk forwardrate (5%)is earned on $103.05duringthe second yeal..Supposenext that the borrows$100for four6.016X3 years at 5% amdinyeqtsit for threeyearsat 4.68:.The investor = $1148 at the end of the third year and a cash resultis a cash iniow of ltk .
,
.
.
85
InteyestRqtes .
.
,
,
114.808662 p: $12114at the end ofthefourthyear. of ltk outqow since.122.14 rate of 6.294. moneyis beingborrowedfor the follrthyear at the torward lf an ivijtor thinksthat rates in the fture will be di#erentfromtoday'sfomard ; '5X4
,
=
.
,
'
ratesthereare many tradingStrategks that theinvestorwill :nd attrpctlve(seeBusiness Snapshot4.2).0ne pf theseinvolvesentering into a contract knownas zforwar j rate agreement.We will now discusshowthis contract worksand howis is valped.
4.7
FORFARD RATEAGREEMEjTS A orward rate agreement (FRA) is aa over-the-couter ggreement that a certain intepst rate will apply to either borrowingor lendipga certain principal during a specied future period of time. The assumption underlying the contrct is that the bonpwipgor lendipgwould normally be done at LIBOR. Considera forwardrate agreement where company X is agreeing to lend money to companyY for the period of timebetweenTj and T2. Dene: 'f.
'
,
.
The rate of interestagreed to in the FRA
RK ;
Ry : The forwardLIBORinterestmte for the period betweentimes Tj and Tz,
calculatedtoday5
.
Ru : The actual LIBORinterestrate observed in the market at time T1 for the periodbetweentimes Tj and T2
1:-. The principal uderlying ihecontract '-'- We will depart fro our usual assumption of continuous compounding and assume that the rates RK, :y-, and Ru are a11measured with a compounding frequency refkcting the length of the period to which they apply. This means that if T2 T1 they are expressedwith semiannualcompounding; if Tz Tj = 0.25, they are expressedwith quarterly compounding;and so on. tTl1isasspmption corresponds to the usual market practite for FRAs.) Normallycompany X would earn Ru fromtheLIBORloan.The FRAmeans that it willearn RK. The extra interestrate (which may be negative)that it earns s a result of RK The Ru. enteringintothe FRAis interestrate is set at timeT1and paidcattimeT2, The extra interestrate thereforeleds to a cash :ow to company X at time T2of .5,
-
=
-
'.
-
LCRK
Rul
-
T1)
-
(4.7)
similarlythereis a cash ;ow to company Y at time T2of LRu
-
'r)(T2
-
Tl)
(4.8)
and (4.8), Fromequations(4.7) we see that thereis another interpmtationof the FRA. It is an agreement where company X will reive intereston the principal betweenT1 and T2 at the fixedrate of RK and pay interestat the realizedmarket rate of Ru. CompanyY will pay intrest on the pripipal betweenTj and T2at theflxedrate of RK and reive interestat Ru. Usually FRAS are settled at time T1 rather than T2. The payof must then be 6 LIBORforwardrates are calculated as describedin section 4.6 fromthe LlBoR/swapzero curve. The latter'is determinedin the way describedin 7.6. section
86
CHAPTER 1
discountedfromtime T2to Tj. For company X, the payof at time Tl is J-(,s
,x)(r2
-
l + Rxlb ' and h for company Y, the payof
mrj)
Tj)
-
at me Tj is. . (Rx
'r)(T2
-
1+ Rxlb
..-
Tl) T)) -
fxample4.3 Supposethat a company entersinto an FRA that species it will receive a Vd rate of 4% on a principalof $1 million for a 3-mpnthperiod startingin 3 years. If 3-monthLIBORprovesto be4.5%for the3-monthperiod the cash flowto 9he ICIZCF Will be x 0.25 x (.4 1,, .45)
-$l25,
=
-
at the 3.25-yearpoiiit.'TMdis equivakntto a cjh :ow of 125, = - 1+ 0.045x 0.25
-$123,09
at the 3-yearpoint. The cash :ow to the payty on the opposite sideof the transactionwill be +$125, at the 3.25-yearpoint or +$123,609 at the 3-yearpoint. (A11interestrates in tllisexnmpleare expressedwith quarterlycompounding.)
Valuation To value an FRA we frst note that it is always worth zero when Rx Ry.6 This is 4.6,a largefnancialinstitutioncan at no cost lockin the because, as notd in section fomard rate for a future time period. For example,it can ensure that it earns the forwardrate for the timeperiod betweenyears 2 and 3 by borrowinga rtain amount of loney for 2 years and investingit for 3 years. Similarly,it can ensurethat it pays tht fomard rate for the time period betweenyears 2 and 3 by borrowingfor a certain amount of money for 3 yearsand investingit for 2 years. Comparetwo FRAS. The frst promises that the LIBORfomard mte Ry will be earnedon a principal of L betweentimesT!and Tt; thesecondpromisesthat Rx will be earned on the same principal betwn the same two dates,The two contracts are the sameexceptfor the interestpaymentsreceivedat timeT2.The excessof the value of the sond contract over the rst is, therefore,the present value of the diferencebetween theseinterestpayments, or =
LCRK
-
:y-)(T2 Tj)e -
where% is the continously compoundedriskless zero mte for a maturity T2.?Because the value of the FRA where Ry is earnedis zero, the value of the FRA where Rx is earnedis Pylt = LCRK 'y-)(T2 Tl# -a2z2 (4 :) -
-
.
6It is usually the case that Rx is set equal to Rg when the FRA is frst initiated. 1 Note that %, Ru, and RB are expressd with a compoundingfrquencycorrespondingto T2 T1 whereas R2 is expressedwith continuouscompounding. -
,
87 Similarly,for a company receivinginterestat theQoatingrate and paying interestat RK, the vlue of the FRA is PFRA
=
LRy
'k)(T1
-
-
Tl)e-ayTz
'
(4.10)
By comparing equations (4.7)and (4.9)or equations (4.8)and (4.1:,we see that an FRA can be valued if we: 1. Calculatethe payof on theassumptionthat fofwardrates are realized (thati8,on the assumptionthat Ru = R. 2. Discountthis payof at the rk-fr rate.
fxample4.4 Supposethat LIBOR zero and forwardrates are as in Table 4.5. Consideran FRAwherewe will reive a ray of 6%, measqredwith annualcompounding, on the end of ear l andthe end of year 2. In this llionbetween a principal of $1 with continuous compounding or 5.127%with case, the forward rate is 5% annual compounding. From equatlon (4.9),it followsthat th8 value of the ltl,tl,
4.8
x
(.6
.g,g4x2
0.05127):
-
=
jg $805,8
DURATION The darationof a bond, as it8 name implies,is a measure of howlong on average the holder of the bond has to wait beforereceivingcash payments. A zeroucopponbond that lasts n fearshas a durationof n years. However,a coupon-bearing bond lasting n yeart has a durationof le88than n years, becaus:theholderreives some of the cash paymentsprior to year n. Supposethat a bond providestheholderwith cash Qpwscj at timelj (1 i K n). The compounded) are related by bond price B and bond yield y (continuously n
B
'-yti
cie
=
jzzl
(4.11) ' .
The durationof the bond, D, is defned as -yj
D
n lf C,.e f=1
-
Th can be written D--
n
(4.12)
-yti
t
Cie
i=1
The term in squarebracketsis the ratio of te presentvalue of the cash :ow at time ti to the bond price. Thebond pri is the present value of a11payments. Thedurationis thereforea weighted averade of the timeswhen paymentsare made, with the weight appliydto time ti being equal to the proportionof the bond's total presentvalue providedby the cash ;ow at time ti The 8um of the weights is 1.. Note that for the purposesof the defnition of dvrational1dcounting done at the bond yield rate of interest,y. (Wedo not u8e a diferent zero rate for each cash :ow as describedin Section4.4.) .
88
CHAPTYR 4 Table 4.6
Calculationof duration.
Time (Flr#)
Cashfow
(J)
.5 1. 1.5 2. 2.5 3.
.
Total:
5 5 5 5 5 15 13
Weight
Time x weight
3.933 3.704 73.256
0.050 0.047 0.044 0.042 0.039 0.778
0..025 0.047 0.066 0.083 0.098 2.333
94.213
1.
2.653
Presezt alue 4.709 4.135 4.176
''
When a slpall change Ay in the yield is considered, it is approximately true that A:
Rom equatin
=
dB
Ay
(4.13)
citie-yti
(4.14)
-y
this becmes (4.11), hB
=
-Ay yruj
(Notethat thereis a negativerelationshipbetweenB and y. Whenbpnd yields increase, bond prices decreaje.Whenbond yields decrease,bond prices increase.)Fromequaand (4.14), tions (4.12) the keyduratin rdationship is obtained: LB This can be written
=
LB
-BD Ay
(4.15)
= -D Ay
(4.16)
is an approximaterelationslzipbetweenperc'entage changes in a bond Equation(4.16) priceand changes in its yield. lt is easy to use and is the reason why duration,which was first suggested by Macaulayin 1938,has becomesuch a popular measure. Considera 3-yearle/a coupon bondwith a facevalueof $1().Supposethatthe yield on the bond is 12% per annum with continuous compounding. Tllis means that Coupon payments of $5 are made every 6 tonths. Table 4.6 shows the y= calculationsnecessary to determinethe bond's duration.The present values of the bond's cash :ows, using the yield as the discountrate, are shown in column 3 (e.g., the -.12x.5 = 4 of cash valut Tjw numbers ;ow is 5: the Erst in prestnt sum f the s9; column3 gives the bond's price as 94.213.The weights are calculated by dividingthe numbersin column 3by94.213.The sum of the numbers in column 5 givestheduration as 2.653years. Small changes in interestrates are often measured in basispoints. As mentioned earlier, a basis point is per annun. The followingexample investigatesthe accumcyof the dumtionrelationslj in equation (4.15). .12.
.
.le;
.
89
Interest Rats
fxample 4.5 For the bond in Table4.6, the bond pri, gives 2.653,so that equation (4.15)
x 2.653Ay
-94.213
LB =
B, is 94.213and the duration,D, is
Or -249.95
hB
y
=
Whenthe yield on the bondincrepsesby 1 basispoints (=().1F:),Ay = = The durationrelationship predictsthat LB = x sp that = the bond price goes down to 94.213 0.259 93.963.How accurate is this? Valuingthe bond in terms of its yield in the usual way, we :nd that, when the bond yield increasesby 1 basispointsto 12.1%, the bd priceis -h.1.
-249.95
.1
-.25,
-
5:
-.121 x.5
-.121x
1.
+ je
l.5
-4.121x
+ je
-.121
x2.9
+ je
-.121x2.5
-.121x3.
+5e
+ 105:
=
939j? .
whichis (tothree decimalplas) the same as that predicted by the duration relationsllip.
Modified Duration .
.
' '. .
'
'
'
.
. ,
.
.
,
.
.
,
.
.
l
The preceding analysis is basd oli the ssllmptionthat y is expressedwith continuous compounding.lt y is expressedwith anual compounding, it can be showp that the becomes approximaterelationship in equation (4:15) '
BDAy
LB=
-
1+y
More generally, if y is expressedwith a colpounding frequencyof m timesper year, ' then LB =
BDAy l + y!m
-
A variable D*, defned by
.
b.
=
D 1+ ylm
is sometimes referredto as the bond's modsed daratim.It allowsthedurationrelationship to be simplifedto .
LB = -BD*Ly
(4.17)
wheny is expressedwith a compounding frequencyof m timesper year. The following exampleinvestigatesthe acuracy of the lodifed durationrelationship.
fxample4.6 The bond in Table4.6 has a pfke of 94.213and a dmqtionof 2.653. The yield, expressedwith semiannualcompoundingis 12.3673%.Themodifkd dlll-atign,D%, is given by 2.653 = 2.499 D* = 1+ 0.123673/2
90
CHAPTER 4
From equation (4.17),
.
LB =
or
.
..
.
.
,
.
x 2.4985Ay
-94.213
-235.39
XB =
ly
compounded)increasesby1 basispoints (= %), Whentheyield(semiannually = predicts that we expect LB to be wehve Ay +9.1. Thedurationielationship = pri thatthebond goes downto 94.113 0.235 -235.39x jo How accurateis th? An exact calculationsipilar tc that in the previous 93.978. showsyhat,when th bondyield (semiannually compounded)increasesby example 10bajis points to 12.4673%,thebondpricebecomes93.978,Th showsthat the ed durationcalculation givesgood auracy for smallyield changes. modif. .1
.l
-0.235,
-
=
Anotherterm that is sometimesused is dollarduration.Thisis the product of modifkd dur>tionand bond price, so that LB = -D**Ay, whep D**is dollarduratign.
Bond Portfolios The duration, D, of a bond portfolio can by dened as a weighted averageof the of the individualbondsin the portfolio, with the weightsbeingproportional durations then apply, with B beingdefned as the to (4.17) to the bond prlces. Equations (4,15) estimate of change portfolio. They the the in the value of thebond portfolio value bond for a small changeAy in the yields of a11the bonds. lt is importantto realize that, when durationis usedfol'bondportf,olios,thereis an implicitassumption that the yieldsof a11bondswill change by approximatelythe same amount.Wheythebondshavewi ely ditlkrilkmaturities, thishapptnsonly when tltt is a paralkl shift in the zero-coupon yield curve. %z should thereforeinterprei equations(4.15) to (4.17) as providing estimatesof the impacton the prke of a bond portfolioof a smallparallel shift, ?, in the zero curve. Bychoosing a portfolio so thatthedurationof assits equals thedurationof liabilhies (i.e.,the net durationis zero), a fnancialinstitutioneliminates it,sexposure to small parallel sllifts in the ykld curve. lt is still exposed tg shiftsthat ar either large or nonparallel.
!.9
CONVEXITY The duratio relationship applies only to small changes in yields. Thisis illustratedin Figure4.2, which showsthe relationship betweenthe percentage change in value and changein yield for two bond portfolios havingthe same duration.The gradients of the two curves are the same at the origin. This means yhatboth bond portfolios changein value by the same perntage for small yield changes ad ij consistent with equation(4.16). For largeyield changes, the pprtfolios behavediferently.PortfolioX has more curvature in its relationshipwith yields than portfolio Y.A factorknownas convexitymeasures this curvature and can be used to improvethe relationship in
equation (4.16). A measure of convexityis
C=-.
1d1B E'lj.j =
BJ
tze-ut cj j B
9t
Interest Rates
Figure 4.2 T*o bond portfolios iith the same duration. A#
F
$$
X F
From Taylor series expansions,we obtain a more accurateexpressionthan equagiven by tion (4.13),
: This leadsto
=
dB
'
: X =
-oy+
2,
d y+ )2 d
)zqy)
2
y
2
The convexity of a bond portfolio tends to be greatest when the portfolio provides ' payments evenly over a long perid of time. It is least when the payments are concentratedaround one particulay point in time. By choosing a portfolio of assets and liabilitieswith a net durationof zero and a net convexityof eio, a nancial institutioncan lake itselfimmuneto relativelylargepamllel silifts in the zero curve. However,it is stitl exposed to nonparallel shifts.
4.10 THEORIESOF THETERMSTRUCTURE0F INTERESTRATES It is natural to ask what determinesthe shape of the zero curve.Whyis it spmetimes downwardsloping, sometimesupward sloping, and sometimespartly upward sloping and partly downwardsloping?A number of difkrenttheorieshavebeenproposed. The simplestis expectations tbeory,which conjectures that long-terminterestrates should refkct expected fptureshort-term interestrates. Moreprecisely,it arpes that a forward iterest rate correspondingto a certain futureperiod is equal to theexpectedfuturezer
92
CHAPTER
/
interestrate for that period,Anotheridea,market segmentationteory, conjetures that there need be no relationship betweenshort-, medillm-,and long-terminterestrases. Underthe theory,a majorihvestorsuch as a largepension fundinvestsin bonds of a certainmaturity and doesnot readily Switch fromone maturity to another. The shortterm interestrate is detenninedby spply and demandin $heshort-term bond market; theinedium-terminterestrate isdeterminedby supplyand demandin the mepium-term bond market; and so on. The theory that is most pptaling is liqaiditypreference teory, whkh argpes tat forwarp rates should always be higher tan expectedfuture zero mtes. The basic assllmptionunderlying the theoryis that investorsprefer to preserve iheirliquidity andinvestfundsfor short perids of time.Borrowers,on theotherhan, usually prefer to borrowat flxedrates for'longperiods of time.Liquiditypreferen theoryleadsto a situatipnin which frward rates are greater than expected futurezero rates.It is also consisttntwith tht tmpirical result that yitld curvts ttnd to bt upard sloping mre oftenthn they are downwardsloping.
Tke Management of Net Int:rest Income To understand liquiditypreferencetheory,it is useful to consider the interestrate risk facedby bankswhentheytakedepositsand make loans.The net interestfncpr of the bankisthe exess of theinterestreceivedover theinterestpaid and needgto be carefully managed. Considera simple situation where a bank ers consumersa one-year and a fiveyear diposit rate as well as a one-year and fwe-yearmortgagerate. The rates are shownin Table4.7:We make the simplifyingassllmption that the expected one-year interest rate for future time periods to equal the one-year rates prevailingin the market today. Looselysjeaking this means tat the market considersinterestrate incre>soto be jqq as likly gs interqt.rqte dcreases.As a rjult, the rates in Table4.7 in that they refect the market's expectations(i.e.,they correspond to art expectations theory): Investingmoney for one yea'r and reinvesting fir four further reiurn as a Single Eve-yearinvestment. one-yearperiods give the same expectedoverall Sinlarly, borrowingmoneyfor one year and refnancing iach year for te next four yearslads to the same expected fmancingcosts as a single Eve-yearloan. Supposeyou havemoney to depositand agpe with the prevailingview that interest rate increasesare just as likelyas interestrate decreases.Wouldpu choose to deposit your money(or0ne year at 3% per anmlm or for fve yeafs at 3% per annum? The chances are that y0u wouldchoose 0ne year blapse this gives you more fnancial fkxibility.It ties up your fundsfor a shorter perio of time. '
dsfair''
.
Table 4.7
Exampleof rates ofered by a bank to its
customers.
Matarity year
1 5
Deposit rate
Mortgage rate
3tls 3%
6% 6%
.
':
'
'
.
'
93
Iitteytst Rates . .
.
... .
'
'
.
Now suppose that you w>nt a mortgage.Againyou agre: with the previling view that interestrate increasesare justas likelyas interestrate decreases.Wouldyou choose a onr-year mortgageat 6% or a Eve-yearmortgage at 6%?The ch>ncesare that you wouldchoosea Eve-yearmortgage becauseit flkes your bprrow ing rate fofthe next ve yearsand subjects you to lessrefpancing risk. . Whenthe bank posts the rats shown in Table4.7,it islikelyto findthat themajority of its customers opt for ope-yeaf depositsapd five-jearmortgages. Thiscreatesan asset/liabilitymismatchfor th bank and subjects it to risks. Thereis no problemif . interest rates fall. The ank will nd itself nancing th ye-year 6% loans with depositsthat cost lessthap 3% in the futureand net interestincime wilt increase. ' Howeyer,if rates rise, thedepositsthat A!'e nancing these6% loanswill costmorethan 3% in the future and net interestincone will dwline.A 3% rise in interestrates would reducethe net interestincometo zero. lt is the job of the assetliabilitymanagement group to ensure that the aturitiesof th assets on which interestis earned and the maturities of the liabilitieson which interestis paid are matched. 0ne way it can do thisis byincreasingthe ve-yearrate on both depositsand mortgages. For example,it could move to the situaon in Table4.8 wherethe five-yeardepositrate is 4% and the ve-yearmortgape rate 70:. Thiswould make ve-year depositstelativelymore attractive and one-year mortgagesrelatively moreattractive.jome customers who chose oe-year depositswhen the rates wereas in Table4.7 will switch to Eve-yeardepositsin the Table4.8 situation. Some'customers who chose ve-yearmoftgageswhn themtes wereas in Tablr4.7willchoose one-year of ssets and liabilitiesbeingmatched. lf This m>y lead to the maturities moytjages. deposltors tendingto choosea one-year maturityand thereis still an imbalancewith bdrrowrs a Eve-yearmaturity, ve-yeardepositand mortgagerates could be increased evenfurther. Eventually the imbalancewill disapper. The net reslt of all banks behavingin the way we havejustdescribedis liquidity Long-term rates tend to behigherthanthosethat would be predicted reference iheory. P by expectd futureshort-term rates. Theyieldcurve is upward sloping most of thetime. lt downwardsloping only whep the market expts a rea steepdcline in shortterm rates. Many banks now have sophticated systems for monitoring the decisionsbeing madeby customers so that, when theydetectsmall diferencesbetweenthe maturities qf the assets and liabilitiesbeingchoen by custmer theycan :ne iunethe rates the will be discussedin ofer. Sometimesderivativessuch as interestrate swaps (which Chapter7) are also used to manage their exposure. The result of a11this is that net interestincomeis very stable.Asindicatedin BusinesjSnapshot4.3,thishas not lways beenthe case. '
Table 4.8 Five-yearrates are increasedin an attempt to match maturities of assets and liabilities.
Matarity ears)
Deposit rate
Mortgagerate
5
3% 4%
6% 7%
94
CHAPTER 4 '.
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FailurespfFinanl Instittipns ip theUS
the 196s, 17s, and lq8s; Saviiij 4ndl t/n! (S&Lj) the Uited ' ' ' sllbttryefm .' tfi f tnik,tliark zpptt. stattsfiled t mapag intrfejt(ptr fi ( wgll! y j.j ando-er long-tefm xed-rate idrtgjs. Aj tfrejlj, y,tuj,r q jjysujjy jsx jj ap) jthe 11184 in $#97 82. s&l Ftre) rte inlvases 1969=7, j971,Eij70t interejt 1,98kk ih, i gtratee. by goveoment prtecpd rasop . tli failus Fat thelkpf itit rati ikt latpjmentk .oiqtcotyoorEte Vs 1;d aie ltt, of autl stim' bi i fpilv hs papr thr en .y . ' cak ' . ,. ikt Itlioil, . . . ;( . ()b' jitribpttb t k failup in theU% eiipieipl: Thelargest ristswi. btlfigqjlitjiiob'lgst? 13, it assts t? panugrltqpssrtr (i;e., fallurr
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SUMMARY .
.
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Tw0importafltinterestrates f0r derivativetrders are Treasuryrates and LIBORrates. Treasuryrates are the rates paid by a government on borrowingsin its own cu'rrency. LIBOR raies are short-termlendingrates ofel'ed by banks in the interbankmarket. Derivatives tradersassume that the LIBORrte is a risk-freerate. ne comjound ing frequencyusrd for an interestrate denes iheunits in whichit is meatured.The dference betweenan anvlly compopnded raie and a quarterly cmpounded rate is analojousto the diseren betweena distan measuredin miles and a distan measred in kilometers.Tradersfrequentlyuse continuous compounding when analyzingthe value of derivatives. Mny diferent typesofinterestrates are quoted in nancialmarketsand calculatedby analysts.The n-year zer or spot rate isthe rte applkable to an inkestmentlastingfor n years whena11of the return is realked at the endkThepar yield on a bond of a certain maturityisthe coupon rate that causesthetondto sellforits par value. Forwardrates are the rates applicable to futureperiods of timeimpliedbytoday'szero rates. The methodmost commonlyused to calculate zero rates is knownas the bootstrap method.lt involvesstartingwith short-term instrumentsand moying progressivelyto 1onger-terminstruments, making sue that the zero rates calculated at each stageare consistentwith the jrices of the instruments.It is used daily by tradinj desks to calculate a Trtasury zero-rate curve. is an over-the-counter agreement tat a certain A forward rate agreement IFRAI interestrate will apply to eitherbrowing or lendinga certin principal at LIBOR dttringa speciEedfutur! period of time. An FRA can be valued by assuming that forwardrates are realized and discouptingthe resultingpayof. An important concept in interestrate markets is dzration.Duration measures the of the value of a bond portfolio to & smallparallelshiftin the zero-coupon sensitivity
yieldcurve. specifcally,
LB
=
,-BDLy ::
,
whereB is the value of the bond portfolio, D isthe durationof the portfolio,
y isthe
'95
Interest Rates
;
sizeof a small parallel shift in the zero curve, and LB isthe resultant efect on the value of the bond portfolio. Liquiditypreference theory can be used to explain the interestrate tefm stntctures that are observed in practice. The theoryarpes that most entities liketo borrowlong andlend short. To match the maturities of borrowersand lenders,it is necessary for fmnial instittions to rae long-termrates so tht fomard interestratej alr highir thanexpected future spot interestrates.
FURTHERREADING Allen,S.L., and A.D. Kleinstein. VqluisgFixed-lscomel'rlv:q/r/la/. and DerivativeSecurities: CaskFkw Asalysis tmtf Calculatioss.NewYork:NewYorkInstituteof Finan, 1991. Fabozzi, F. J. Fixeddneotne Mathematies: Analytiealand Stattieal Teehniques,4th edn. New York:McGraw-llill,26. ta# Other:&?l#Aik Meaures. Frank J. Fabozzi Fabozzi, F. J. Duratios, Cosvexity, ssoc., 1999. nd Financial tnnovation a theRoleof Derivatives Grinblatt,M., and F. A.Longstas.d$ Secuties: AnEppirical Analysisof theTreasuryStripsprogramj''JoursalofFisasce, 55,3(2): 1415-36. Jorion, P. Big Bets (7t-: Bad: Derivativesc?l# Baskruptcy f?l Orasge Cotmty.New York: AcademicPresq, 1995. Stigllm,M., and F. L, Robinon. Mosey Marketsc?l#Bosd Calculatioss.Cllkago:Irwin,1996.
Manal) and Problems(Answersin snlutions Questions :
.
4.1.A bank quotes an interestrate of 14% per annum with quarterly compounding.What is equivalent rate witli (a)continuous compoundingand
,the
.
(b)annual comppunding?
4.2: What is meant by LIBOR and LIBID.Whichis higher? 4.3. TheGmonthand l-year zero rates are both 10% per annuin. For a bondthathas a lifeof 18 months and pays a coqpon of 8% per anmlm(withsemiannualpayments and one havingjust beenmade), the yield is 10.4% per anmlm. What is thebond's price?What is the l8-month zero rate? A11ptes are quted with semiannualcompounding. 4.4. An investor receives $1,100in one year in returp for an investmet of Tljo now, Calculatethe percentage return per annumwith: (a) Annualcompounding (b) Semiannualcompounding (c) Monthlycompounding (d) Continuouscompounding '
.
4.5. Supposethat zero interestrates with continuous compounding are as follows: .
..
.'
-.
.--.- - -
Matarity (?n(zIJ) 3 6 9 12 15 18
Rate (%per c?l?lpml
8.0 8.2 8.4 8.5 8.6 8.7
.
26
CHAPTR
4
Calculattfolard ittrtst rates for tht stcond, thirz, fourth,sfth,and sixth quarters. 4.6.kssumingthat zero ratysareas in Problem4.5,whatis thevalue of an FRA that enabks the holder to earn 9.5% for a 3-monthpeiiod startingin 1 year on a principal of The interestrate is expressedwith quarterlycompounding. $1,000,00(7 %'
.
.
4.7. The term structure of interestrates is upward-sloping. Put the followingin order of
mabitude: zero rate (a) The s-year coujon-bearing bond The yield on a s-year (b) (c) ne forwardrate conponding to theperiodltween 4.75and 5 years in the future
What.is the answert this questiovFhn the ttrm structureof interot rates is downwardsloping? 4.8. What does duration tell you about the sensitiviiy of a bnd portfolioto interestrates. What are the limitationsof the durationmeasure? 4.9. What rate of interestwith continuouscompopnding is equivalentto 15%per annumwith mtmthlycompounding? 4.10. X deposit accountpays 12% per annumwith continuous compounding, but interestis actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit? '
4.11. Supposethat f-month, lz-month, l8-mohth,24-month,and 3-month zero rates are, respectively,4%, 4.2%, 4.4%, 4.6%,and 4.8% per annum,with continuous compounding. Estimate the cash pri of a bond with a face value of 100that will mature in 30 months and paysa coupon of 4% per annumsemiannually. 4.12. X 3-yearbond provide! a coupon of 8% semiannuallyandhas a cash pri of 14. What is the bond's yield? 4.13. Supposethat the f-month, lz-month,l8-month,ard 24-month zero rates are 5%, 60/0, and 7%, respectively.Whatis the z-yeay par yield? . 6.59/01 4.14. Supposethat zero interestrates with continuous compounding are as follows: '
Haturity year 1 2 3 4 5
Rate (%per ctzlfril 2.0
3. 3.7 4.2
4.5
Calculateforwardinterestrates for the second, third,fourth, and sfthyears.
4.15. Use the rates in Problem4.14 to value an FRA where you Willpay 5% anually)for the third year on $1million.
(compounded
4.16. A lo-year8% cupon bond currently sell8fr $90.A lo-year4% coupon bond currently sellsfor $8. Whatis the lo-yearzero rate? llirltk Considertakinga longposition in two of the 4% coupon bondsand a short position in one of the 8% coupon bonds.) 4.17. Explain carefully why liquiditypreferec theoryis constent with the observation that the term structure of interest rates tends to be upward-sloping more often than it is downwrd-sloping.
97
Interest Rates
zero curve is upward-sloping,the zero rate for a particular maturhy is greater than the par yield for that maturity. Whenthe zero curve isdownward-sloping the reverse is true.'' Explainwhy this is so. 4.19.Whyare US Treasuryrate signifcantly lowerthan other rates that are close to risk-free?
4.18.'sWhente
4.29.Whydoes a loan in the repo market involvevery lhtle credit risk? rate of interestfor a flxed 4.21.Explainwhy an FRA isequivalentto the exchangeof a soating inierest. rateof compounded) pays gn 8% coupon at 4.22.A s-year bond with ;. yild of 11% (continuously end of each the year. (a) Whatis te bond's price? (b) Whatis the bond's duration? (c) Usethe durationto calculate the efect on the bond's plice pf a 0.2%drease in its yield. . annum yield and velify that (d) Recalculatethe bond's price on the basisof a 10.8%per the result is in agreement with yoqr pswer to (c). 4.23.The cash prkes of 6-monthand l-yeazTreasurybillj are 94.0and 89.0.A 1.5-yearbond bondthat thatwill pay coupons of $4every 6 months currentlysells for $94.4.A z-year monihs cuirently sells for $97.12.Calculatethe 6-month, willpay coupons of $5every6 and l .s-year, rates. zero l-year, z-year
AssignmentQuestions 4.24.A!linterestrate is quoted as 5% per annumwith semiannualcpmpounding. Whatis the
equivalentrate with (a)annual compounding, (b)monthly compounding, and (c)continuouscompounding. 4.25.The 6-month,lz-month,ls-month,and 24rmonth zero rates are 4%, 4.59/:,4.750/c,and 5%, with semiannual compounding. (a) What are the rates with continuous compounding? (b) Whatis the forwardrate for the Gmonthpeliod be/nning in 18 months? semi(c) What is the value of an FRA that promises to pay you 6% (compounded annually)on a plincipal of $1million for the 6-monthperiod startingih 18months? par yield when the zero rates areas in Problem4.252Whatis the yield 4.26.Whatis the z-year bondtht pays a coupoh equal to the par yield? ona z-year 4.27.The followingtable gives the prices of bonds: '
'
Bozdprizcipal
(y l l l l
Timeto matarity years)
coapoe zlrlrlpzcf
Bozdprice
(J)
($)
.5 .
l. 1.5 2.
.
6.2 8.
98 95 11 14
# Half the stated coupon is assumedto be'paid eve six months. (a) Calculatezero rates for maturities of6 months, 12months, 18months, and 24months.
94
CHAPTER 4
(b) What are the forward rates for the followingperiods: 6 months to 12 months, 12months io 18months,and 18,mnths to 24 months? (c) What are the Gmonth, lz-montt ls-month,and 24-month par yidds for bondsthat providesemiannual cupon payments? bondprovidinga seminnual cupon of 7% (d) Estimatethe priceand yield of a z-yiar per annm. 4.28.PortfolioA consists of a l-year zero-mupon bond with a face value of $2, and a l-year zero-coupon bond with a face.valueof $6,. Portfolio B consistsof a 5.95-year bond with &facevalue of $5,. The currentyield on al1 ondsis 1% per zero-coupon ''
.
.
.
,
anlmm.
(a) Showthat both portfolios havethe same duration. (b) Showthat the percentagechanges in the values of the two portfolios?ora per s' yields increasein the are am. annum (c) What are the prcentag changes in the values of the two portfoliosfor a 5% per annum increasein yields? .
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In this chapter we xamine hw frward pricesand ftures prices are relatid t the spt
price f the underlying asset. Forwardcntracts are easier t analyze than futures cntracts becausethere is n daily settlement-nly .a single payment at maturity. Luckilyit can be shwn that the frward prke and futuresprke f an asset are usually veryclse when the maturities f the tw0 cntracts are the same. In the flrst part f the chapter we derivesme imprtant jeneralresults n the relatinship betweenfrward (r fptures)pricesand spt prices. Wethen use the results t examin the relatinship betweenfuturespris and spt prices for cntracts n stck indis, foreign exchange, and cmmoditiej. W will consider interesi rate futures contractsin the next chapter.
5.1
ASSETS INVESTMENTASSETSvs. CONSUVPTION Whenconsidering frward and futurescontracts, it isimprtant to distinguishbetween investmentassets and cnsumption' assetj A kvestmettt sset is an asset tat is held for investmentpurposes by signifkant numbers f investors.stocks and bonds are clearlyinvestmentassets. Gold and.silverare also examples f investmentasse. Nte that investmentassets d not have to be held exclusivelyfr investment.(silver, fr uses.) number satisfy However,they do have t example,has a f industrial the requirementthat theyare heldby Signihcant numbers f investorssolely forinvestment. A consumptionasset is an asset that is heldpfimarily fr cnsumption. It.is not usually heldfr investment.Examplesof cnsumption assets are commdities such as copper, oil, and pork bellies. As we shall see later in this chapter, we can use arbitrage argunents t determine the forward and futuresprices f an investmentasset fromits spt pri and other bservablemarket variables. We cannt do this for copsumptin assets. .
5.2
SHORTSELLING some of the arbitmge stmtegies presented in this chapter involveshort selling. This invlves selling an asset that is nt trade, usually simply referred to as Ssshortinf',
99
CHAPYER 5.
l
owned.It is something that is possible for some-but not all-investment assets. We willillustratehowit works by consideringa short sale of shares of a stock. Supposean investorinstruts a brokerto short5 IBMshares.Thebrokerwill carry out the instructions.byborrowingthesharesfromanother cllent and setlingthemin the mprketin the usual way. The investorcan maintain the short position for as long as desired,prvided there are always sharesfor the broker t bprrdw.At somestage, owever,the investorFill close ou't the position by purcliasinj 500IBMshares.These arethen rellaced in the accountof the client fromwhich the shareswertborrowed. The investortakesa prot if the stockprke h% dlined and a lossif it risen.If at any time while the tontractis open the brokeris not able to borrowshaas, the investoris forcedto close out the positin, even if not ready to do so. An investor with a short positio must pay to the broker any income,such as zividendsor interest,that wpld normally be receivedon the svcuritiesthat hafe been shorted.The brokerpill transferthis incometo the account of the clknt from whom the securitieshave been borrowed.Considerthe position of an investorwho Shorts 5 shars in April when the pri per share is $120and closej out the position by buyingthembackin July when the price per Fhare is $1. Supposethat a dividendof ih April when $1per shareis paid in May.Theinvestoryceives50 x $120= $60,000 the short psition is ini. ated. The dividendleads tp a payment by tke investorof f0r sharesWhen 5 x $1 $500in Vay.The investoralsopays 5 x $10 the position is closed out in Jlzly.The net gain, therefore,is .has
J
=*$5
=
$60,000$504 $50, -
-
=
$9,500
Tabe 5.1illustmtejthis example and shows that the cash Qowsfrom the short sale are themirror imageof the cash iows frompurchasing the shares in Apriland sellingthem
in July. The investoris required to maintain a margin account w1t.hthe broker.The margin accounscopsists of cash or Marketable secmitiesdepositedby the investorwith the brokerto guarantee that the investorwill not walk away fromthe short position if the share price increases.It is similaf to the mrjin account discussedin Chapter2 for Table 5.1
Cashqowsfrom short sale
.and
purchaseof shares.
Pyrcase of sares
April: Purchase5 Shares for $120 May: Receivedividend July: Sell5 shares for $1 per Share
-sj
+$5* +$5, Nd pr:t
-$9,5
=
Sort sale of sares April: Borrow 500shares and sell themfor $120 May: Pay dividend July: Buy 50 sharts for $1 per share Replaceborrowedsharej to close Shol't position '
+$6, -$5
-$5(3,(3
Net prost
=
+$9,5
i:1
DeterminationOJFoyumydand FutuyesPrcc tere futurescontracts. An initialmargin' is requiredand k?
are adverse movements(i.e., of tht asstt that is bting shorttd, dditionalmargin may be incrtases)in thi pri required.lf the additional margin is n0t provided, the short position k closedout. Te margin aount des not represent ?,cost to the investor.This is becauseinterestis usually paid n tht balanct in margin accunts and, if the interestrate osered is unaeptable, marketable securitiessuch as Treasurybills an be usedto meet margin requirements.The proceeds of thesaleof the asset belongto theinvestorand ormally form part of the initialmargin. The SEC abolished the uptick rule in the UnitedStateson July6, 2007.Th ru e requiredthe most recent movementin the pri of a stock to be an increase for the '
l$
'
t''
'
shortingof a stockto be permitted. .
5.3 ASSUMPTIONSAND NOTATION In this capter we will assume that the followingare a11true for some mafket
participants:
1. The market participants are subject to no transactioncosts Fhentheytrade. 2. Themarketparticipants are subject to the sametax mte on all net tzadingprfhs. 3. Tht markd participants can borrowlonty at tht samerisk-frterate of interesta8 they can lend money. 4. The market participants take advantage of arbitrage oppottunities as theyopr. Notethat we do not reqte theseasumptions to be true for a1lmarket participants. A11 that we require is that they be true-or at least approximatelytrue-for a few key marketparticipants suchas largederivativesdealers.lt is the tradingjctivities of these key market particlpants and their eagernessto take advantage of arbitmge opportunities as they occur that determinethe relaonsllip betweenforwardand spot prices. The fpllowingpotation will be used tkoughout tllis chapter: T: Time until deliverydate in a forwardor futurescontrad (inyears) : Priceof the asset underlying the forwardor futurescotract today . . .
.
-'
'
,
.
kg
Fg: Forward or ftures prke today r: Zero-coupon risk-freemte of interestper annum, expressed with continuous compounding,fot an investmentmaturingat the deliverydate (i.e.,in Li-years)
The risk-free rate, r, is the rate at whichmoney is borrowedor lent when thereis no credit risk, jtj that the money is certain to be repaid. As discussedin Chapter4, institutionsand other participants in derivativesmarketsassume that LIBOR . snancial rather than Treasuryrates are the relevant risk-free rates. rates
5.4
FORWARDPRICEFORAN INVESTMENTASSET The easiest forwardcontract to value is one written on an investmentasset that provides stocksand zero-coupon bonds are the holder with no income.Non-dividend-paying of examples suchinvestmentassets.
102
CHAPTER 5
Consid. er a long forward cpntract to purchase a non-dividend-paying stock in 3 months. ssllme the current stockpri is $4 and the 3-monthrisk-freeinterest rate is 5% per annum. Supposehrst that thr forward ri is relatively high at $43.An arbitrageur can borrow$4 at the risk-freeinterestrate of 5% per annum, buy one share,and short a forward contract to sell one sare in 3 months.At the end of the 3 months, the arbitrageurdeliversthe share and reives $43.The sm of moneyrequired to pay 0f the loan is j
'
4:
=
$40.50
Byfollowig this strategy, the arbitmgeur locksin a prtt of $43.00 $40.50 $2.50 at the end of the 3-monthperiod. Supposenext that the forwardpri is relatively low at $39.An arbitrager can short one share, investthe proeds of the short sale t 5% per annum for 3 months, and take a lonc nositipnin a 3-monthforwardcontract. The proeds of the short sali t. 4: or $40.50in 3 months. At the end of the 3 months the zrow arbitrageur pays $39,takes deliveryof the share under the terms of the forward contract,and uses it to close out the short position. A net gain of =
-
-
-.5x3/l2
,
,
$40.50 $39.00= $1.50 -
is thereforemade at ihe end of the 3 months. The two trading strategieswe have consideredare summarizedin Table 5.2. Under what circumstancesdo arbitrage opportunities such as thosein Table5.2 not exist?The hrst arbitrage works when the frward prke is greater than $40.50.The
Table 5.2
Arbitrageopportunities when forwardpri is out of line with spot price for asset providig no income.(Assetpri = $40;interest rate 5%; maturityof forwardcontract 3 months.) =
=
Forward'rfc:
=
$43
Actioz npw':
Borrow$4 at 5% for 3 months Buy one unit of asset Enter into forwardcontract to sell asset in 3 monthsfor $43 Action in 3 months: Sellasset for $43 Use $40.50to repay loan with interest
Fprwer#hice
=
$39
Actioz rlpw: jhort 1 unit of asset tp realize $4 lnvest $4 at 5% for 3 months Enterinto a forwardcontract to buy asset in 3 months for $39 Actionin 3 months: Buyasset for $39 Closeshort position
Reive $40.50frominvestment Proft realized = $2.50
Proft realized = $1.50
1 Fomard contracts on individualstocks o not often arisein pradi. However,theyformusefulexamples inNovember2002. for(kveloping our ideas.Futures on inividual stocksstarte tradingin theUnite states
iike dyvtlojed a y 0 f ajnj qzrc-cpuj bbnd called a q lvesttent (jt fltkrq :derstllii cgufon-bearig Txasuff by the cash from itl str ip, Vkd a itraet stltisf.:-ise'ih jejyat Jei koikiny on s a lyingtht toqppmbeqring a ltrdinj Pebodf, hatl q rltivel? simpl: sspftpy.flewoulbbtt?strijl ad fork-iddet epatioh them in forwardparkt. th j (,1),Shwj,zmt)the ftywaidplie ifforEE qe11 prdilq tli t seckty spjqj no ipmet is alwsyj .banks
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ipterrsif4t 1s4%pti qttp ajp tli pot priqt f stfp that Se lith examjlej t 4 $7 l is $70.Te 3rgn h t rkqrd pfi t t te kj Fiy'tq 9.GM3/32 y g J tit's Kizdq Piabdfls cimnvtsyst tpdes jifk equal )Jf retkt ( '
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fai tiir fciwrdjfice ovt th jjt pri4 (s.t,itturrexainjlj.,lii t ttpfntvrcessofuoiujmqre fnktnckjkyuiatje ykttij. ui, tan tiw'ttkttof , tigs was ysk kt mtrats fwatd, Jttt kqs jjj yj pysrj,,yjjyujy uujyyq q. byrnlliqg E ' ' . . (' !,oLim. . l .. . . ' rewtt Tke as tikttkesystm polittd: plp:t #?$1 ikilin pq jitidjtrAitg ficekvd faci tiieitks E$ lss i te yggn f big bpnus) ktn (an Jett r istittltions a jit' relatively $350mit1i ?n This jhow; thai eken lfg flil simplethings Wrng! .
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We deducethat for ' secondarbitrage works when theforwardprice is lessthan $40.50. there to' be no arbitrage the fomard prke pust be xactly $40.50.
A Geeralization
T0 generalizeth exampk,weconsider a forwardcontract on an investmentasset with price k that provides no income.Usingour notation, T isthetimeto maturity,r isthe risk-freerate, and Fg is the forwardprice. The relationshipbetwtenFg and S is rT brgxz Se
,IfA'tl> Fg S d an inktstr can adopt the followingstrategy: ,
lv Borrow Sb dotlarsat an interestl'ate r for T years. 2. Buy 1 ounce of gold. 3. Short a forwardcontract on 1 ounce of gold. rT At time T, 1 ounce of gold issoldfor h. An amounthe is required to rejay theloan rT at th time and the investormakes a proft of A- he rT Supposenext that A' < he In thiscas.ean investorwho owns 1 ounce of gold can -
.
.
1. Sellthe gold for k. 2, Investthe proceeds at interestmte r for time T. 3. Take a long position in a forwardcontrqct on 1 ounce of gold. rT At time T, the cash investedhas grown to ke Thejold is repurchasedfor h and the 'T >> investormakes a proht of he t relative to the position the investorwould have beenin if the gold had beenkept. As in the non-dividend-payingstock exampk consideredearlier, we can expect the forward price to adjust so that neither of the two arbitrageopportunities we have cosidered exists. This means that the relationship in equation (5.1) must hold. .
-
.
5.5
KNOWN INCOME In thissectionwe consider a forwardcontract on an investmentasset that will provide a perfectlyprdictable cash incometo the holder.Epmples are stockspaying known divi(fends and coqpon-bearing bonds.We dopt the sameapproachas in the previous section.Wefrst look at a numerical example and then reviewthe formalarguments.
t'
105
Detetmination PJFotpakd and Futues PTce: '
.
,
.
.
.
'
'
.
Cpnsideia longforwardcontract to purch a coupon-bearingbond wose curynt will priceis $90().Wewillsupposethat the fol'wrd ctract lattlres in 9 monthj. We expectedafter of Wekssume is that 4 payment als suppose a coupnn $4 that monts. compounded) are, the 4-month and gmmonthnsk-freeinterest rates (continuously respectively,3% and 4% per annum. price is relativelyhig at $91. An Supposefrst that the forward can and short a forwardcontract. Thecoupon paymenthas a bond bdrrow$900to buy ihe -'3X4/l? = Of the $9, $39.60is thereforeborrowedat presentvalue of 4: 3%per annum for 4 months so that it can bt rejaid with the coupon paylent. The 4% per annum for 9 months. The amount pwing at $860.40is borrowedat 860.40/.*X6.75 remaining = $8866. A sum of $910is received theend of the g-monthperiodis forthe bond under the termsof thefomard contrat. The arbitrageur thireforemakes a net prost of 886.60= $23.40 91. 'arbitrageur
.$39
.
.
.
-
Supposenext that the fomard prke is relativelylowat $870.Aninvestorcan short the bond and enter into a long fpmard contract. Of the $900realized from shorting the bond, $39.60isinvestedfor 4 morghsat 3% per annum jo thatit grows into an nmount sucient to pay the coupon on the bond. Thi femining $260.* is investedfor months ard 4% Under the terms of the forWard at 9 growsto $886.60. per armum $870is paid to buythebond >nd the short position is closed out. Theinvestor contract, gains therefore -
886.60 87
$16.60
=
-
The twostrategieswe haveconsideredare summarized in Table5.3.3Thefrst strategyin whereas the Table5.3 produces a profhwhen theforwardprkt is greater than $826.60, It follows secondstrategyproduces a prtt when theforw i price.islessthan $886.60. opportunities arbitrage forkard thenthe if there be must jrice $886.60. are no that
A Generalization We can generalize from this example to argue that, when an investmentasset will incomewith a present value of I duringthelifeof a forwardcontract, we have provide F'
In our example, ss
=
9.,
I = 4e
F
=
(9.0
=
rT
(,% 1)e -
=
-
(5.2)
39.60,r =
.4,
and T
=
0.75,so that
39.60):.()4x.7j= $gg((g .
appiiesto any investment Thisisin agreementwith our earliercalculatio. Equation(5.2) a knowncash income, g assetthat provides If #' > (, IjerT , an arbitmgeur can lock in a prost by buyingthe asset and rT shortinga fo>ard contract on the asset; if F's < (,% I)e an arbitpgeur can lock in a prtt by shorting the asset and taking a long position in a forwardcontract. lf -
-
,
.
j If shorting the bond is not possible, investorswho alrea oFn the bond willsell it and buy a forward Tiis is s'imilar to the strater we contracton the bond increasingthe value of their posion by $16..60. gold Section 5.4. in describedfor '
106
CHAPTER 5
Table 5.3 Arbitrageopportunitieswhen g-monthforwardpric is out of linewith spot pric for assetproviding knowncash income.(Assetprice = $9; incomeof $4 occursat 4 mnths; 4-monthandg-monthrates are, resptively, 3% and 4% per anum.) '
'brped/rfce
Iwapd price = $910
$870
=
'
Actioz,1t?w:
,
Actionnpw: Borrow$9: $39.60for 4 months for 9 mnths and $860.40 Buy 1 unit of asset Enter into f4rwardcontract to sell asset in 9 months f0r T91
Short 1 unit of assetto realize $900 Invest$39.60f0r 4 months and $860.49 for 9 mnths Enter into a forwardcontract to buy asset in 9 monthsfot $870 ctiozfrl4 monthsl Reive $4 from4-monthinvestment Payincope of $4 on asset
Action in 4 mpnfzs: Receive$4 of incomeo asset Use $4 to repay rst loan
withinterest
Action in 9 monts Sellasset for $910 Use $826.60 to repay second loan
Actios iz 9 mprlf':
Prot reahzed
Prot realized $1.0
Reive $226.60 fromg-monthinvestment Buyasset or $ Closeont short position
withinterest =
$23.40
=
'
,
,
shortsalesare ot possible, investprswho ownthe asset will fmdit profhable t sell the asset and enter into longforwardcontracts.4 'fxample 5.2 Consider a lo-month forwardcontract on a stockwhen th stock price is $50.We compounded) is 2% per assumethat the rijk-free rate of interest(continuously annum for a1lmaturities. We also assume that dividendsof $0.75per share are expectedafter 3 months, 6 months, and 9 months. The present value of the dividends,1, is .
I = .g yj:-.8x3/ .
12
j + g yjy-.sx /l2 + g yj:-.08x9/l2 2.162 .
=
.
The variable T is l months, so that thefoiwardprice, F'(),fromequation (5.2), is givenby 2.162): .8x1/t2 $51.14 F' '
.
=
(5
-
=
If the forwardpri were lessthan this, an arbitrageur would short te stock and buyforwardcontracts. If theforwardprice were greater than this, an arbitrageur wouldshortforwardcontracts and buythe stockin the spot market. 4 For another way of.seeingthat equation (5.2)is correct,considerthi followingstrater: buyoneunit of the assetand enttr intp a short fomar contract to sellit for f' at timeT. This cost,s an is certain to lead to a cashivow of J' at timeTJ'e-rT. an an incomewith a present value of 1. The initialoutqow is k. ne present of Hence,k l + J'e-rT', or equivalentlyA' (% flerT. value thr iniowsis l + S
=
=
-
17
Deteymination :.JFoyzayd and FutuyesPyices
5.6 KNOW! YIELD W nowconsiderthe situaiionFkere the ajsd underlyipga fomard cpntzactprovides a knownyield rather than a knowncashincome.Tilismeansthat the incomeis known whenexpressed as a percentageof the asset's price at the tile the incomeis paid. Supposethat an assetk expectedto providea yieldof 5% perannum.Thiscouldmean thAt incomeis paid oncea year andis equal to 5% of the assetpri at thetimeit is paid, inwhkh case the ykld wouldbe 5% with annualcompuunding. Alternatively,it could meanthat inome is paid twi a year and is ekual t 2 5% of the asst priceat thetim it is paid,in which case the yieldwould be 5% per a,nnu!n ith semiannual compounding.In Section4.2 we explained that we will normallymeasureinttrest ratts with compounding. Similarly,we willnormglly measure yklds with continuous continuous Formulas for translatinga yield measured with one compunding compounding. to a yield measured
=, is given by .1,
.5.
=
f
5.7
=
(().10-4.0396)x0.5
25:
=
$25.77
VALUINGFORWARDCONTRACTS 'Thevalue of a fomard contract at the tim it is srstentered into is zero. At a later stage,it may prove tn havea positive or negativevalue. It is importantfor banksand pther snancial institutionsto value the contract each day. (This is referred to as markingto market the contract.) Usingthe notation intzoducedearlkr, we suppose K is the deliveryprice for a contmct that was negotiated sometime ago, the delivery dateis T years fromtoday,and r isthe T-year risk-freeinterestrate. The variable F is theforwardpricethat would be applicableif we negotiated the contract today.Wealso dCfme
J: Valueof forwardconsract today It is important to be clear about the meaning of the variables >), K, and J. At the beginningof the life of the fomard contract, the deliverypri, K, is set equal t the forwardprice, 1$, and the value of the contract, J, is 0..As thne passs, K stay the
108
CHAPTER 5 it is part of the defnitin of the contract), but the forwardprice changes same(because value of and the the contract becomeseither positive or negative. A general result, apjlicableto a11lng fprwardcontracts (boththose on investment assetsand those on consumption assets),is /
1F
=
V)-VT
-
(5.4)
T0 seewhy equation (j.4)is correct, weuse tn arpment >nalogousto the one we used for fomard rate agreementsin Section4.7.We compare a' longforwardcontract that has a deliverypri of f' with an otherise identkallongfor:ard contract that has a deliverypriceof K. The diferenctbetweenthe two is only in the amotmtihatwill be paid for the underlying asst at time'T. Underthe frst contract, tls amount is /$; under the second contract, it is #. A cash outior diference of f' K at time T K4e-'T today.The contract with a deliveryprice F is translatesto a diexp of (F therefore less valuable than the coniract with. deliverypri K by n amount (1$ K4e-rz The value of the contrat that has a de1iverypr j of g yy ojujtion pri o'f K is (1$ K4e-rT Zefo. lt followsthat th8 value of the contmct with a delivery Similarly,the value of a short forwrd contract with delivery Thisprovesequation (5.4). pri K is -
-
-
.
-
.
CK /$# -
-rT
fxample5.4 stock was entered ipto some longforwardcontract on non-dividend-paying ntaturity. risk-free currently The time go. lt has6monthsto rate of interest(with continuouscompounding) is 1% per annum, the stock pric is $25,and the T= deliverypri is $24.ln this case,.s = 25, r = and K N. Ftom equation(5.1), the f-monthforwardpri, &, is given by .1,
.5,
=
f'
25: .1x.5
=
=
$2j 2g .
the value of the forwardcontpct is From equation (5.4), -.1x.5 J (26.2824): =
Eiuation
=
-
$2jy .
(5.4)shows that wecan value a longforwaidcontract on an assetby making
the assumptionthat the price of the assetat thematurityof theforwar contract equals the forwardprice /$. To see this, note that when we make that assumption,a long forwardcontract provides a payof at time T of f' K. This has a present value of -rT (F K4e Whichis the value of f in equation (5.4). Similarly,wecan value a short fotwardcontract on the assetbyassumingthat the current forwardpri of the assetis realized.These results are analogonsto the result in section 4.7 that we can value a forwardrate agreementon the assumptionthat forwardratts are realized. givesthefollowingexpression Usingequation (5.4) in conjunctionwith equation (5.1) for the value of a forwardcontract o an ivestment assetthat provides no income -
-
,
f
=
Jk
-
Ke-'T
Similarly,using equation (5.4)in conjunction with equaon
(5.5)
(5.2)gives the following
Dettmination
p./
109
Foywaydand Fufure; Pyes
'''''' .'' ''. ''.','
k' .'
Busiss Sliapsbot5.2
g
j
rrpr? skstims
zjtm
A foriignerchaje tyadi.J'wrkig fr ba?k inteyj jyju g jjsyu jtytraj o buy 1 million jud kerling t n chagt rte f 1k90 ili 3 mnths. .Atthe pm Q tkt pex!de/ ttkqs a lopy ptsittpp in 16 copcts ?pr time,npther fwtes p jlirlip The ftprrs (prici % 1.9 4z achicohtig ij on 3-month a3/'ttfes trdrrk hefiibfe 62,50 puu4t. Theposiiipq tte'byqte'ftkrkzd y r liekkg p' ta/k: tlit frwf:rndfte tturg' t sgme.kithi'pittts Qf ti tyitkp syj4is'w,itat irhi iccteasr to 1.904:. hs th futurit tratkr soth m ptt k $t,94 frtd l: ke it r'si if a ( lmmiiartrty clk tt qStk j qyqtiyk s drja jts : jt tg Tr foqwardtidef k y. q t?iwfEEtfaet,l' ppjtli/' E E 90F 1he qy (q v a ykli ; ( q ) tEtttuis:t , typdjrjirq thfyhifuttusj ' The gnkirq is pp! Thi ppilyjttknittt iiimdilitjfp tziiekipij'tp ftkllcitj jk,tfifutre, an tinujt tfer E1fdtm fofiaidlilr qy'thiE j'pFitipnlywytfilk itts siitptract ttsi te jfiek rltlliiljtktul j't 1,9406 4h ffwfd iidir ktktd lf4tdrd tpby l.9 ' i ! . olyl' batl at sell 1 niilliq pbs l.'Ikt 3 popths t? 4 moilkkEtty. in pitt-cltlt iEfoikafd j ij jfpi i iijer's ie'jresent t . st . E Va. y luedf $40 This is cqsistrt Witlirqtion($,4). Tlle frward trqderca yaln qmr tonolpti frol tr fd tht ud lo ses It jric drjpeb Cplktle l injyad of aretreted fmpetriatly.thrforward/ftltr fomardE ?? tke loss ld 0 tht theptheftyis l $4 lradi!.wp t a rising trqikrwu td'take loss of (ml !!,%, ( q q .tfkder
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expressionfor the vplue of a longforwardcontracton an investmentassetthatprovides a knownincomewith present value J:
J
uc
S
-
I
-rT
kle
-
(5.6)
Finally, using equation (5.4)in conjunction with ekuation (5.3)gives the following expressionfor the value of a longforwardcontract on an investmentassetthat provides a knownyield at rate ql -qT -rT Ke J S@ (5.7) '
=
-
Whena futuresprice changes,the gain or losson a futurescontractis calculatedas the changein the futuresprice multipliedby the sizeof the position. This gain is realized almostimmediatelybecauseof the wayfuturescontractsare seqleddaily.Equation(5.4) showsthat, when a forwardprice chankes, the gain or loss the present value of the changein the forward price multipliedby the size of the positipn. The diference the gain/loss on forwardandfuturescontracts cancauseconfvsionon a foreign between exchangetradingdesk (seeBusinessSnapshot5.2).
5.8 AREFORWARD PRICESAND SUTURESPRICESEQUAL? The appendix at the end of this chapterprovides an arbitragearplment to show that whenthe risk-free interestrate is constant and the same for al1maturities,the fomard
110
CHAPTER 5
Price)ora contract with a rtain deliverydateisin theorythe sameas the futurespri for a contrat with thatdeliverydate.The argumet in the appendix can be extended to coversituationswhere the intemstrate is a kpownfunction f time. Whn interestraies vary unpredictably (ajjhey do in the real world), fomard and futuresprici are in theoryno longerthe snme.Wecan grt a senseof te nature of the the situationwhere the pri of the underlying aset, S, relationslpby considering w1t11interestrates. When S increases,a investorwhQ p ositively correlated strongly holds longfuturesposition makesan immediategain becauseof the dailysettlempt procedure.The positivecorrelation indicatestht it is likelythat inteyestrates havealso increasid.The gqin will thereforetend to be investedat a higherthan averagt rate of interest.Similarly,when S decreases,theinvestorwill inur an immediateloss. loss Wi11tend to he finnted at a lowerthan average rate of interest.An investorholding a forwardcontract rather than a futurescntract is not afected in tbis way byinterestrate movements.It follpwsthat a longfuturescontract will be shghtlymori attractive than a similarlong fomard contract. Hen, when S is stronglypositivelycorrelated with interestrates, futurespiices will tend to be slightly higherthan forwardpris. WhenS is strogly pegatively correlatedwith interest ratesj a similar >rgument shows that fomard prices will tend to be slightlyhjgheythan futuresprices. The theoreticaldiferens betweenforwardand futuresprices for contracts that last nly a few months are in most circumstans sllciently small to be ignored. In practi, there are a numbei of factorsnot refkcted in theorrtkal models that may causeforwardand futuresprices to bediferent.Thesrincludetaxej, transactionscosts, and the treatmentof mar/ns. The risk that the counterparty will defaultis generally lessin the caseof a futurescontract becauseof thi role of the exchangeclearinghouse. Also, in sme instances,futxs contracts more Equidand easier to trade than Despite a11 thesepoints,for most purposes it is reasonable to ajsllme fomarrdcontracts. that forwardand futuresprices are the same. This is the assllmptionwe will usually makein tllisbook.Wewill use thesymbolF to representboththefutnresprice and the forwardpri of an asset today. 0ne exception to the rule that futuresand forwardcontractscan e assumed to be $e same concernsEurodollar futures.This will be'discussedin Section6.3. 'fhis
'are
'
5.9
.
FUTURESPRICESOF STOCKINDICE! '
.
Weintroducedfutureson stock indicesin Section3.5 and showed how a stock index futurescontractis a useful tool in managing equityportfolios. Table 3.3showsfutures prkes for a number of diferentindis. Weare now in a position to considerhowindex futuresprices are determlned. A stok indexcan usuallybe regarded as the pri of an investmentasset that pays dividends.sThe investment' asset isthe portfolio of stocksunerlying theindrx,and the dividendspaid by the investmentassetare the dividendsthat would be receivedby the holderof th portfoli. It is usuallyassumed that thedividendsprovide a knownyield ratherthan a knowncashincome.If q isthedividendyield rate, equation (5.3) givesthe futuresprice, 1$, as >- = ke(r-lz (j g) '
o
.
5 Oasionally thisis not the case: see Business 5.3. snapshot
11i
DeteyminationPJFoyzlmydand Futures P?tw
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tte >fjtltts ip thifcapter p howlzdexfutrejprkes are detrrminedyttlire that tb tldkxtlwvale'f l in&tdtitui swikTis mtus tht it must bt $hivaluegfa '
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tmikflfing the ChicagoMercantik f tht can be ir/ed. Pprtftlil g cikyattym 22$Idir,gg Ayikp'k fpwlts thr ktkkrl po$tallfy,And t% ps.n (??s ake Nlkkiilislkx! Thijij the fally of hyis titkwbtlii'spjptttlk tyd, ptljp tli CME ipaywky jtktti Jjyptkr Thevyykble str tjie t2# p oserbpldst 215 np,ie yfiy nifqyt tlp futltyj Nikk k dbllk of $Sil f j Ecifat in apd krilt iaii is wjttd ytl! Srrat i s ingl.it is dllais. ikis a, ilwk' frtto alwayssebt dolkrs. Te best we 1li 'Wiruti it pse valuiwill cattrkiEtp iiyk, )' E/qys' inqqe'tt ls kqtt 55'ye lrqin c thqt is llq Wrth dplktij ktty' Q je' dtttf yalpr Ml ye. The fibte L dllts is nt, L t . ilseifte th,piki f 44 iktkmri s/ ? pikatio ($.8jE2cis ppi ajjly, CMj,j xltkii E E2i fktufip tptiyqt ls k e/Tplb of a qaatltb.A quhto ij a eviife wire th: ttlirl/lkajsiy is inwsuridinE(me urrency atd the aynt is in (liiusst. Fill furtif in hapter29. zfficy. Eotgnts a4tlkir ,
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This shows that tht futurtspriceincrtasesat ratt r q with the matmity of the futures contract.In Table 3.3the June futurs pri of the S&P5 is about 0.9% more than tlwMarchfutmw jrict. This indicattsthat on Januarys,207, tht short-term risk-free rate, r, was greater than the pividendyield, q, by aboui 3 6 n per year. -
.
fxample 5.5
Considera 3-monthfuturescontract on the S&P 5. Supposethat the stocks underlyingthe indexprovide a dividendyield of 1% per annum, that thecurrent value of the index is 1,3, and that the continuously compounded risk-free interestrate is 5% per annum. ln this case, r = s' 1,3, F 0.25,and given by .1: Hence, the futurespri, fL,is .5,
=
F'
=
1,3(k
(.5-.qI)x.25
=
=
$j ? j? gy ,
.
ln practice, the dividendyield on the portfolio underlying an indexvariesweek byweek throughoutthe year. For example, a largeproportion of the dividendson the NYSE stocksare jaid in the firstweekof February,May,August,and Novembereachyear. The chostn value of q shouldrepresentthe averageannualized dividendyieldduringthe life of the contract. The dividendsused for estimatingq should be thosefor wllich the ex-dividenddate is duringthe life of the futurescontract.
Index Arbitrage ronts can be made by buyingthe stocks underlying the indexat the If >- > ke,P ker-oT spot jrice(i.e.,forimmdiate delivery)and shortingfutures nntracts.lf f' < prohts can be made by doing the reverse-that is, shorting oi silling the stocks underlyingthe indexand taking a long position in futurescontracts. nesestrategks ke-T indexarbitrage is oftendoneby a are knownas index arbitrage.When F k ker-oT it might be pension fundthat ownsan indexedportfolioof stocks. Whenf'a > ,
,
,
112
CHAPTER 5 .
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futtrtsEonyfpt idi: indet fbittagr,, a trir musfbi able ip tykiskh, lllliii thli j'rie yt in : thepitfolionoipof alsttkj hdC iqrkei cpndjiik 1) ppiilpyiy'p'gjr.kdiji iikil In ? ihtxrkt. ijt@iz theyilt knjliij ip equatiqh'(5.tj tkk'keil tl niket'-slqk:Aj ) y r.,j,',pkd,.l,i,.k. 1'9t.'t,.,@,:p.,:'kf,.ji.'ikitr :a y b ni-su, are tjts y ll8t,'wi.iE tEtktbykkjr Cli'kk 2tE
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jtpt'rtkl'jl'e''jllf; k 11 b;tiu' oidst ) tnkthtNw Ytkt 'qtiilqkbil'iu',''ilkjlyiqseay # y skie tfdi yyj, syjti kik y y jyjyjjzyzj ( ne d/cv 4lk, (ikik'u y t, q igvld bt dql/e'by' kf kki' ; ),. tt ihat lok'c? Octttf,i', 1?8y ?ut tq jii gjj j ((tyjj kgjtt j (kjjt jyjjy j43( r j ; ( ytj t( t t,ltj,,',ji't,'stl,ipk pf ipuexkEEyf' k t itti ) 4 tytklik.ktl; xlpjti,' r uptkrlgg 578 gdylj iij' listtltbtt iljky J/ilrtiltfi, kl 225,ttdpwh ktkt ..
j .. rttfttsms oil the ombir20,21987, jfoyr'kmti ttt'-f, tbixkfltr very thewaykndwhic qttlie tikebfekdow ditil ,lik>jr yt t#e tj stqckudius dqstpk ) dilt . idx fture tontiptz. i oik tiit ttkiifs'yfit,ifor(jttkibremtri intict ia' ekitytfi'itw s t pvki, t iepintd to l: lrss thp ie s&,p5t0 lvdtxkljk, ' jttblEiat 'rlikiurk kattpE)(5,'t'j(jverne t:e q tl a1 d te ciiyitks f r ?.htt retipnit'.betAee yttp pt lttcs.' f ) F (. ,
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done by a corporation holdingshort-termmoney market investments,For indis involvingmqny stocks, indexarbitrage sometimesaomplhed by trading a ltlativelysmall representativesample of stocks whosemovements closely milxr thoseof the index.Oftenindexarbitrage is implementedthroughpogra tradizg. Thisinvolves using a computer systemto generate the tl-ades. Most of the timetke acvitis of arbitpgeursensure that equation(5.8)holds,but oasionally arbitrageis impossibleand the futurespri doesget out of line with the 5.4). spot price (seeBusinrss snapshot
5.10 FORFARD ND FUTURESCONTRACTSON CURRENCIES We now move on to consider forwaldand futumsforeigncurrencycontracts fromthe perjpectiveof a US investor.The underlying asset is one unit of the foreip urrency. We willthereforedefnethe variable h as the ctlrrent spot pri in dollars f ne unit of the foreigncurrency and A-llas theforwardor futurespri in dollarsof one unit of the foreigncurrency. Tltisis constent with the waywe havedened k and F9 for ther 2.1, assetsunderlying forwardandfuturescontracts. However,as mentionedin section exchange it does not ncessazilycorrespondto the w@yspot and forward ratey are quoted.For majorexchangerates otllerthantheBritishpound,euro, Australiandollar, and New Zealanddollar, a spot or forwardexchange rate is normally quoted as the numberof units of the currency that are equivalent to 0ne US dollat.
113
Determination p/ Forward and Fttlurd! Prcc!
Figure 5.1 Two ways of convertipg lj units of a foreigncurrency to dollarsat timeT. Here, is spotexchangerate, f isforwar exchangerate, and r and r/ art tke dollar and foreignrisk-freerates. k
l0 unitsof fofeip carrency at zzrc 'tlmt
10:'JT
li% dcllars attimezefc
anitscf
fofeigncurrency attme T
1OCCCSJ/F : dollafs at timeT
lotl'nzf dollars at;me T
A foreigncurrency hasthe propely that theholderof the currencycan earn interest at the risk freeinterestmte prevailing in the foreigncountry. For txample,the holdtr bond.Wedefne ry as the value of the can investthe currency in a foreigmdenominqted risk-free when foreign interestrate monej is investedfor time T. Th variable r is the US dollar risk-free rate when moneyis investedfor this period of time. Th relationship betweenF' and is -
k
F'g=
5'gE'(r-ry)T
(5.9)
.
This is the well-knowninterestrate parity relatignship frominternaonal snance. The that an individualstarts with reason it is true is illustratedin igure 5.1. suppose units of the foreigncurrency.Thereare twowaysit can be converted to dollarsat l, time T. Oneis by investingit f0r T years at r/ and entering into a forwardcontract to sellthe proceeds for dollarsat time T. Tilisgenerates 1, erl'F () dollars.The other is by exchanging the forign currency for dolkrs in the spot market and investingthe rT proceedsfor T years at rate r. This generates 1,000,%e dollars.ln the absence of arbitrageopportunities, the tw0 stmtgies lust give the same result. Htnce,
so that
r/r-() 1,()(k F
=
=
S
1
kgdr:r
,
/V-T.f)T
Exapple 5.6
Supposethat the z-yaiinterestratesip Australiaand the UnitedStatesare 5% andt%,respectively,and the spot exchange rate betweenthe Australiandollar
114
CHAPTER 5
(AUD)and the Us dollar(UsD)is 0.6200UsD per AUD. From equatip the ) cyepr forwardexhange rate shouldbe
(5,9),
0.62:((o:-(oj)x2= g jjj; .
forwardexchange rate is lesi thn this, say Suppose rst that the z-year An arbitrageurcan:
.63.
AUD at 5% per annumfor 2 years, convertto 62 UsD and 1. Borrowl, investthe UsD at 7% (boihl'ates are continuouslycompounded). 2. Enter into a forwardcontract to buy 1,105.17AUD for 1,105,17x 0.63= 696.26USD. = The 62 UsD that arq investedat 7% grow to 620:b 713.17UsD in 2 years. Of this, 696.26USD are used to purhase 1,105.17 AUD under the terms of the forward contract. This is exactly enough to repay principal and AUD that are borrowed(1 tk ksxz j jgj jy). Tjw interest on the l, r,of . 713.17 696,26 16.91 strategy therefore gives rise to a riskless proft exciting,consider followinga similr strategy UsD. (If this does not soundvery whereyou borrow1 million AUDI) fomard rate is 0.6600(greater suppose next that the z-year than the 0,6453 value given by equation (5.9)). An arbitrageurcan: '
97x2
=
,
.
=
-
' = for 2 years, convert to 1,/.6'2 1. Borrow l, UsD at 7% per annum 1,612.90AUD, and invst the AUD at 59:. 2. Enter into a forwardcntract to sell 1,782.53AUD for 1,782.53x 0.66= '
.
.
g jjxa =
. 1,612.90AUD that are investedat 5% The 1,782.53 to 1,612.9% zrow AUD in 2 years. The forward contract has the ellct of convertingthis to 1 176.47 USb. The nmount neeed to payoF the USb borrwings is = l,tk 0.07x2 1 15 27 Uso. The strategy thereforegivesrise to a risklessproft of 1,176.47 1,150.27= 26.20UsD. '
,
.
-
Table5.4 shows currency futuresquotes on January8,2007.The qutes are Us dollars (r cents) per unit of the foreigncurrency. Thisis the usual quotation convention fr futures cntracts. Equation(5.9)applieswith r equal to the Us risk-free rate and rj equal to the forekn risk-free rate. 0n Janury 8, 2007,interestrates on the Japaneseyen, Canadiandotlar, British pound, swiss franc,and euro were lowerthan the interestrate pn the Us dollar.This correspondsto the r > rj situation and explainswhy futurespris for thesecurrencies increasewith maturity in Table 5.4. Foy the Australiandollar and Mexicanpeso, interestrates were higherthan in the United states. This correspondsto the rj > r situationand explainswhy thefuturesprices o thesecurrenciesdecreasewith maturity. Example5.7 In Table 5.4the June settlementpri for the Canadiandollaris 0.28%Mgher than the Marchsettlementprice. Thisindkatesthat the short-term futuresprices thisis are increasingat about 1.12%per year with maturity. Fmm equation (5,9) which estimate of exceededshortby short-term US interest rates amount the an term Canadianinterestrates on January8, 2007.
115
Deteymination:./' Foywaydand FutuyesPrices Table 5.4 Foreip exchange futuresquotes from the WallStreetJt?vrr on January 9, 2007. (Columnsshow month, open, hlgh, l0w, settle, change, lifetimehigh,lifetimelow,and open interist,respectively.) --.
A Foreign Currency as an Asset Providinga Known Yield equation (5.) with q replaced by ry. Th is not a coinciden. A foreigncurrencycan br regarded as ap investmentaqsd paying a known yield;The yield is the risk-free rate of interestin the foreigncurrtncy. T0 undeystand t, we note thai the value of interestpaid in a foreip currency that the interestrate on British dependson tht value of the foreip currency. suppose poundsis 5% per annum, To a Us investortheBritishpoundprovidesan incomeequal to 5% of the value pf the Britishpound per annum. ln other words it is an asset that provides a yield of 5% per annum. Equation
(5.9)is identicalto
5.11 FUTURESON COMMODITIES %z now move on to consider futurescontracts on commodities. First we consider the futuresprices of commodiiies that are investmentassets such as gold and silver.6 We then move on to cosider tite fuilzrs prices of constlmption assets. *
.
lncomeand Storaje Costs As explainedin Businesssnapshot 3.1,the hedgingstrategks of gold producers leadsto a requirement on the part of investmentbanksto borrowgold. Gold owners such as centralbanks charge interestcinthe formof what is knownas the gold leaserate when theylend gold. The same is true of silver.Goldand silvercan thereforeprovide income to the holder.Like other commodities theyalso haveStorage costs. Equation(5.1)shows that, in the absence of storage costs and income,the forward 6 Recall that,for an asset to be n ilwtstmentasset, it nee,dnot be hdd solelyforinxtstmtntpurposts. What is requirqd is that some individualsholdit forinvestmrntpurposes and thattheseindividualsbe prepared to selltheirholdingsand go long forwardcontracts, if thelatter look more attraitive. This explainswhy silvef, althoughit has signifcant industrialuses, is an inYestment asset.
116
CHAPTER 5 price of a commodity that is an investmentasset is gvenby F
icz
'T
ke
(5.10)
Storagecosts can be treated as negative income.If U is the presentvalue of a11the storage costs, net' of income,durilg tLelife of a forward contract, it followsfrom equation(5.2)that rT F- = (5.11) t5'+ U)e fxample t.8
Considera l-yearfuturescontract on an investmentassetthatprovids no income. It costs $2per uhit to storethe asset,with the payment beingmade at the end of the year. Assumethat the spot pri is $450per unit and the risk-free rate is 7% Sb= 450, T = 1,and per annllmfor all maturities.This corresponds to r .7;
=
U = le
1
-.7x
j
=
.
gjj
From equation (5.1 1),the theoreticatfuturesprke, F, is given by J'
=
(450+ 1.865#mxl
=
$484j? .
If the actualfuturesprke is greatr than 484.63,an arbitrageurcan buythe asset and shrt l-yearfuturescontracts to lockin a prolh. If the actualfuturesprice is. lessthan 484.63,an investorwho alreadyowns the assetcan improvethe return by sellingthe asset and buyingftures contracts. If the storage costs net of incomeincurredat any timeare proportional to the price of the commodity, theycan be treatedas negltive yield. In tilis case, frop equation(5.3), F'
=
(r+l4r
he
(s jy .
whereu denotesthe storage costs per annum as a proportion of the spt price net of any yield earned on the asset.
ConsumptionCommodities Commoditiesthat are consumption assets rather than investmentassets usually provideno income,but can be subject to sipilkant storage costs. We nqw review the arbitrage strategies used to detemine futuresprkes from spot prices carefully.? 1), we have Supposethat, insteadof equation (5.1 A- >
rT
t5'+ Uje
(5.13)
To take advantageof this opportunity, an arbitrageur can implementthe following '
strategy;
..
1- Borrowan amountS + U at the rk-free rate and use it to.purchaseone unit of the commodity and to pay storage costs. 2- Short a fomard contracton one unit of the commodity. 7 Fcr some commodities the spot price depends on the deliverylocation. We assume that the delivery for spot and futuresare the same. location
117
Deteyminatim p/ Foywaydand FutuyesPyices
If we regard the futurescontract as a forwardcontract, this strayegyleadsto > prt D):'T at time T. Thereis no problem in implementingthe jiptegy for 0f F (S + vh . any commodity. However,as arbitrageurs do so, there will be a tendency.for to is no longertrue. We conclude that increaseand f' to decreaseuntil equatio (5.13) sipifkant of iime. length hold equation(5.13) for cannot any . . next that juppose rJ):rT A- < (s + (514) -
.
,
,
,
.
Whenthe commodity ij an investmentasset, wecan argue that manyinvestorsholdthe commoditysolely for invtstment.Whentheyobserve the inequplityin equation (5.14), they will Endit profhable to do the following: 1. Sellthe commodity, save the storagecosts, and investthe procreds at the risk-free interejt rate. C. Take a long position in a forwardcontract. LljerT
J'g relative to the position The result is a riskless proht at maturity of (,% + If followsthat commodity. would held the in if had investors been they have ' the cannot hold for long. Because neither equation (5.13) equation(5;14) nor (5.14) can rT + Uje holdfor long, we must have f' = (5'(y Tiis argument cannot be used for a commodity that is a consmption asset rather thanan investmentasset. Individuals and companies Who own a consumption commodityusually plan to use it in someway. Theyare reluctant to sell the commodity in and buy fprwr or futurescontracts, becauseforwardand futures the spot contractscannot be consumed(forexample, ()il futures cannot be used to fd a fromholding,and a11we Thereis thereforenothing to stop equation (5.14) rehneryl). fQr canassert a consumption commodity is -
.
emarket
'
jz
%tS
rT
+ Uk
(5.15)
If stomge costs are expressedas a proportion a of the spot pri, the equivaknt result is
f' %S (r+g)z
(5 jji .
Cpnvenience Yields and (5.16) becauseusers of a We do not nessarily haveequality in etuatios (5'.'15) physicalcommodity ownershipof commodity provides the feelthat consumption may benefhsthat are not obtained by holdersof futures 1.ntracts. For example, an oi1 is unlikely to regard a fqturescontract on crude oil in tht sameway as crude oi1 refmer heldin inventory.The crude oil in inventry can be an input to the rening pross, whereasa futurescontract cannot be used forthis purpose. In general, oFnership of the physical asset enables a manufacturer to keep a production process nmning and ptrhaps profh from temporarylocal shortages. A futurescontract does not do the same. The benefhsfrom holdingthe physical asset ale sometimes referred to as tht yield provided by the commdity. If the dollaramount of storage costs is convenience knownand has a present value U, thenthe convenien yield y is desnedsuch that A-g: zz=
(sg + vjer
118
CHAPTER 5
If the storagecos'tsper uhit are a constantproportion,R, of the spot pri, then y is defned so. that w,yggr+:lr # gg . or (r+If=y)r .
.
F'll=
.
.
, .
.
.
,
,
(j
ke
.
1g)
The convenien yield simplymeasuresthe extent to whichtheleft-handsideislessthan ( ' For investmentassets the convemence 5) of (5.16). the right-hand side in equation (5.1 yieldmust be zero; otherwise,thereare arbitrage opportunities.Figure2.2 of Chapter2 showsthaythe futuresprke of orange jui decresd as the ttmr to matmityof the ccntractincreasedon January8,2007.Th patternsuggtststhattheconvenienceyield, y, is greater than r + a for orangejui on tllisdate. The convenien yield reiects the market'sexpecttions conrning the futureavailabiltyof the commodity.The greaterthepossibilitythatshortageswill our, thehigher yield. If users ofthe commodityhavelligl inventorks,thereis very little theconvenience shortagej in the near future and the convnience yield tends to be low.If chance of inventoriesare low,shortages are morelikelyandthe convenien yieldis usually ltkher.
5.12THEcojy oF cRRy The relationsip betweenfuturespricesand spoi pris an be sllmrnarizedin termsof the cost ofarry. Tl1ismeasuresthe storagecost p1u8the interestthat is paid to snance the asset lessthe incomeearned on the asset. For a non-dividend-payingstock, the cost of carry r, becausethere are no storagecosts and no incomeis earned; for a stockindex,it is r q, becauseincomeis earned at rate q on the asset.For a,currency, it r ry; for a commoditythat providesincomeat rate q and retuires storagecosts at rate a, it is r q + If; and s0 0n. Dehne the cost of carry as c. For an investmentasset, the futurespri is -
-
-
,
Fg= For a consumption asset,it is
F
=
,g:
cT
(5.18)
J'/c-l'
(5.19)
wherey is the convenienceyield.
5.13 DELIVERYOPTIONS Whereasa forward contract normally specihes that delivery to take place on a particular day, a futures contmctoften allowsthe party with the short position to the party has to givea chooseto deliverat any timeduringa certain period.t'Typically choice noti of T he deliver.) introdus its intentionto fewdays' a complication into the detenminationof futurespricestShouldthe maturity of the futures contract be to be the beginning,middle, or end of the deliveryperiod?Even thoughmost assumed futurescontractsare closed out priorto maturity, it isimportantto knowwhen delivery wouldhavet>kenpla in order to calculatethe theoreticalfuturesprice. If the futups priceis an increasingfunctionof the time to maturity,it can be seen the asset (including that c > y, so that the benestsfrom fromequation (5.19) 'holding
119
Deteyminaton 0.JFoywaydand FutuyesPyices
yield and net of stomgecosts) are lessthan the Ik-free rate. It is usually convenience optimalin such a case for the party with the shol't position to deliveras early as posible,becausethe interestearned on the cash reived outweighs the benefts of the asset.As a rnle, futuyespris in thesecircumstancesshouldbe calculated holding on the basis that deliverywill take pla at te beginningof the deliveryperiod. If futufspris are decreasingas timeto maturity increases(c< y), the reverseis true. It is then usually optimal for the party with the short position to deliveras late as apd futuresprices should,as a yule,be calculated on tlt assmption. possible,
FUYURESPOT PRICES 5.14 FUTURESPRICESAND EXPECTED Werefer to the market's averageopinion about hat the spot pri of an asset will be at thatit a certainfuturetimeas the expectedspot priceof the asset at that time. suppose is nowJune and the Septemberfuturespri of corn is 35 nts. It is interestingto ask whatthe expectedspoi price of corn in Septemberis. Is it lessthan 35 ents, greater than 35 cepts, or exactly qual to 35 nts? As illustmtedin Figure2.1, the futures prke converges to the spot pri at maturity. If the expected spot price is lessthan futurespri to decline,so that 35 cents, the market must be expectingthe september with log positios lose.If the expected traders with short positions gain and tmders price be true. must market must be is than 35 greater nts, ihe spot reverse expecting the Septemberfuturesprice to increase,so that traderswith long positions . gin while thosewith short positions lose. -'rhe
Keynesand Hicks ' EconomistsJohn MaynlrdKeynesand JohnHicksargued that,if hedgerstend to hold short positions and speculatorstend tp holdlongpositions,thefuturespri of an asset willbe belowthe expectedspot price.8Thisisbecausespeculatorsrequire compensation for the risks theyare bearing.Theywili tradeonly if theycan expect to make moneyon Hedgerswill lose money on avemge, but they are likly to be prepared to average. accepttltisbecausethefpturescontract redus theirrisks. If hedgerstend to holdlong while speculatorshold short positions, Keynesand H.icksargued that the positions p rice futures w1l1be above the expetd Jpot prke for a jimilar reason.
Riskand Return
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The model'n approach to explaining the relationship betwen futures pr ces and spot prices is based on the relationship betweenrisk and xpected return in expected theeconoml. Ip general,the higherthe lisk of an investment,the higherthe expected returndemandedby an investor.Readersfamiliarwith the capital asset pricingmodel willknowthat thereare twotypesof risk in the economy:systematicandnonsystematic. Nonsystematicrisk should not be important to an investor.It can be almost completely eliminated by holding a well-diversifed portfolio. An investor shold not therdore requirea higher expected return for bearingnonsystematicrisk. Systematic risk, in contrast, cannot be diversifkdaway. It arises from a correlation between N.
B
.
see:J. M. Keynes,zt Treatise oz Mozq. London: Macmillan,1930,.and J. R. Hkks, Vale c?WCapital. Oxford: Clarendon Press, 1939.
120
CHAPTER 5
returns from the invstment and returns from the whole stock market, An investor generallyrequires a higherexpectedreturn than the risk-free interestrte for bearing positiveamounts of systematicrisk. Also, an investoris preparedto accept a lower expectedreturn than the risk-free interestrate when the systematicrisk in an investmet is nejtive.
The Risk in a Futured Pesitin Let us consider a speculatr who takesa longposition in a futurescontract that lastsfor T years in thehopethat the sppt priceof the asset will he above the futurespriceat the end of the life of the futurescontract.Weipore dailysettlementand assllme that the futurescontract can be treatebas a folard contract.We supposethat the speculator puts the present value of the futuresprice into a risk-free investmentwhile simultaneouslytaking a longfuturesposition.The proceeds of the risk-free inyestmeptare. used to buy the asset on the deliyerydate. The asset is then immediatelysold for 1ts marketprice.The cash jows to the speculatora!t s follows: Today: s/): -rT End of futurescontract: +%
wheref' isthefuturesprice today,Sz isthe pri of the asset at timeT at the ed of the futurescontract, and r is the risk-fr return on fundsinyestedfor time T. Thediscountrate we shoulduse for the expected Hw do we valpe thisinvestment? cash:ow at time T equals an investor'srequired return on theinvestment.Supposethat k is an investor's required return for this investment.The present value of this investmentis -rz - J't: + gs r y-kz where E denotes expectedvalue. We can assume that a11investmentsin securities marketsare priced so that theyhavezero net present value. This means that ->):
-rC
+f
0r
>()
:,:
se
Eser-tT
-lT
=
'
(5 2(j) .
As we ave justdiscussed' the returns investorsrequire on an investmentdependon its risk. ne investmentwe havebeenconsideringis in essencean investmentin systematic underlying the asset the futurescontract. lf the returns fromthis asset are uncorrelated with the stock market,the correct discountrate to use is the risk-frer rate r, so we should set k = r. Equation (5.2)ten give ,
J'
=
Es:.,
This showsthat the futurespriceis.an unbiased estimate of the expected future spot price when the retrn fro!n the underlying assej is uncorrelated with the stock market. If the pturn fromthe asset is pitively correl.atedwith the stockmarket,k > r and equation (5.2)leads to F' < Es. This shows that, wen the asset underlyinp the futures contract has positive sytematic risk, we shoutd expectthe futures pri to the expected future spot pri. An example of an asset that has positive understate
Detevminationp.f Fovud
121
and FutuvesPvices
systematicrisk is a stck index. Theexpectedrdurn ofinvestrs on thestocksunderlying an indexis generallymore thanthe risk-freerate, r, Thedividendsprovide a return of q. 'l'he expected increas in the indexmust therfore be more than r q. Equation(5.8) is thereforeconsistent with the prediction that the futres pri understates the expected ' futurestockprice for stockindex. If the retrrn fromthe asset is negativelycorrelatedwith the ktockmarket, k < r and Tilisshowsthat, when the asset underlyingthefutures equation(5.2)givesh > contracthas negative systematicrijk, we shouldexpectthe fqturedpri to overstatethe expected futurespot price. -
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'(&).
Normal Backwardationand Contanjo Whenthe futuresprice is belokthe expectedfuturespot price, thesituationisknownas zormalbackwardatioz;and when the futuresprice is above the expected future spot price,the situatiohis knownas coztazgo. However,it shouldbe noted that somtimes theseterms are used to refer to whethr the futurespri is belowor above the current spot pri, rather than the expected futurespot pri.
SUMMARY For most purposes,the futuresprice of a contract with a certain deliverydate can be to b the sameas the forwardpricefor a contract with the samedelivery considered date.It can be shownthat in theorytlie two shonldbe exctly the samewhen interest rates are perfectly predictable. For the purposes of understanding futures(orforward)prices, it is convenient to dividefuturescontracts into two catejories: thosein which the underlying asset is held for investmentby a signifkantnumber of investorsand thosein which the underlying asset held primafily for consumption purposes. In the case of investmentassetsnwe haveconsideredthreediferentsituations: 1. The asset provibes no income. 2. The asset provides a knowndollarincome. 3. The asset provides a knownyield. e .
.
The results are summarizedin Table5.5. Theyenable futuresprices to be obtained fol. contractson stock indices cufrencies,gold, and silver. Storagecosts can be treatedas negativeincome. In the case of consumptionassets, it is not possible to obtainthe futurespri as a functionof the spot price an2 otherobservablevariables. Herethe pafameter knownas the asset's onvenien yield becoms important.It measurej the extent to which users of the commodity feelthat ownershipof thephysicalasset providesbenefhsthat are not obtainedby the holdersof the futurescontract. Thesebenets may includethe ability to prost from temporarylocal shortages or the ability to keep a production process running.We can obtain an upper boundfor the futurespri of consumptionassets usingarditrage arguments, but we cannot naildownan equality relationsp between futuresand spot prices. The conpt'of cost of carry is sometimesuseful. The cost of carry is te storagecost ,
122
CHAPTER 5 of results for a contract'with timeto maturity T n an investment Tablv 5.5 sllmmary ' assetwith price k when the risk-freeinterestrate for a Fyearperiod is r. Asset r Providesno income: Providesknownincome
) and :2 are two futmescontracts on the samecommoditywith timesto' 5.16. suppose maturity,fl and f2, where 12 > fl Prove that 'and
.
'suppose
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F2 6 F 1/f2-fl) '
,
124 ' x
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CHAPTBR 5 .
,
constant) and there are no storagecosts. For the where r is the intrest rate (assumed of problel, assllmethat a futurescontract is the same a8 a fomard purposts this contract.
5.17.When a knownfuturecash outiow in a foreigncurrency is hedgedby a company usipp a forward contract, there is no foreignexchange rk. When it is hedgedusing ftures contractsjthe marking-to-market procesj doesleavethe compjny expose to kome risk. Eplain the natuye of tllis risk. Ir particular, consider whether the'companyis better oflung a futurescontpct or a fomard conact when: (a) The value of the foreigncurrency fallsrapidly duringthe lifeof the contract. (b)The value o? the foreigncurrency rkes rqpidly duringthe lifeof the contract. (c) The value of the foreip currency flrstrises and then fallsback to its initialvalue. (d)The value of the foreigncurrency flrstfallsand then res back t its initialvalue, Assume that the forwardprke equals the futuresprice. forwar/ excbange rate is an unbiased predictorof future 5.18.It is qometimesargued that acirolmsiances is this s0? rates; Under what exchange 5.19. Showthat the growth rate in an indexfuturespri equals the excessreturn of the index over the risk-free rate. Assumethat the rk-fr interestrate and the dividendyield are constant. is true by consideringan investmentin the asset combined Witha 5.20. Showthat equation (5.3) short position in a futurescontract. Assumethat a11incomefromthe asset is reinfested in the asset. Use an argumentsimilar to that in footnotes2 and 4 and explin in detailwhat did not hold. an arbitrageurwuld do if equation (5.3) 5.21. Explaincarefully what is meant by the expectedprice of a commodity on a particular futuredate. Supposethat the futurespri for crude oil declineswith the maturity of the contract at the rate of 2% per year. Assumethat speclators tend to be short crude oil futures and hetkers tend to be long.Whatdoesthe Keynes and Hicksargumept imply about the expected future price of oil? 5.22. The Value Line Indexis designedto reiect ch>ngesin the.vale of a portfolioof over 1,600equally weighted stocks. rior to March9, 1988,te change in the indexfrom one day to the next was calculated as the geometricaverage of thechnges in the prices of the crrectly relate stocks underlying the index.In thesecircumstances,does equation (5.8) casll prke? If not, does the equation overstateor the futures price of the index to its understatethe futuresprice? '
...
.
5.23.A US company is interestedin using thefuturescntracts tradedon theCME to hedgeits maturities) on the US dollar Australiandollar exposure. Defme r aj the interestrate (al1 and ry as the interestrate (allmaturities)on the Australiandollar.Assumethat r and ry j are constant and t at the company uses a contract expiring at time T to hedge an exposureat time t l- > f). (a) Showthat the optimalhedgeratio is er-T-t' (b) Showthat, when t is 1 day,the optimal hedgeratio isalmostexactly klh, where k is the current spot price of the currency and Fn is the current futures price of the 'currencyfor the contract maturingat time T. (c) Showthat the company can take aount of the dailysettlementof futurescontracts for a hedgethat lastslonger than 1daybyadjustingthehedgeratio so that it always equalsthe spot price of the currency dividedby the futuresprice of the currency. '
.
125
Determinationof Ftvrma?'tfand Futuyesp?'itw
AssignmeptQuestions of $1per sharein 2 months and in 5 months. The 5.24.A stock is expected to pay a stock prke is $50,and the risk-free rate of interestis 8% per annum with continuous compoundingfor a11maturities. An investo?hasjusttaken a short position in a Gmonth fomard contract on the stock. . (a) What the forwardprice and the initia'lvale of the forwardcontract? (b) Three months laser,the price of the stock $48and the riskzfreerate of interestis still8% per annum. What are thefomard pri and the value of theshort position in the forwardcontracs? ' borrowingcash at 11 per annum and 5.25.A bank ofers a corporate client a choi between borrowinggold at 2% per annum. (If gold is borrowed,interestmust be repaid in gold. Thus, 1 ounces borrowedfodaywould require 12 ounces to be repaid in 1 year.) The risk-freeinterestrate is 9.25%per annllm, and storagecosts ye0.5%per annum. Discuss whetherthe rate of intereston the gold loan too highor too 1owin relation to the rate of intereston the cash loan.The interestrates on the two loansare expressedwith annual with continuous compounding.The risk-free interestrate and storage costs are rxpressed compounding. 5.26.A company that uncertain bout the exact date when it will pay or receive a foceign currencymay try to negotiate with its bank a forwardcontract that species a period duringwhich deliverycan be made. The company wants to reservethe right to choose the exactdeliverydate to fit in with its own cash :ows. Put ourself in the position of the bank. How would yotl price the product that the.companywants? portfolio. The trader can buy gold 5.27.A trader owns gold as part of a long-termilwestment 'fhe trader an borrowfundsat 6% per fpr $550per oun and sellit for $549per onc. year and investfunds at 5.5% per year ( oth interestrates are expressedwith annual compounding).For what range of l-year forwardprices of gold doesthe trakr haveno arbitrageopportunities? Assumetherei8 no bid-ofer spred for forwardprkes. 5.28.A company enters into a forwardcontract with a bank to sell a foreigncurrency for rl at timi Tj The exchangerate at timeT1proves to be & (>rl). Tliecompny asks thebank if it can roll the contract forwarduntil time T2(> T1)rather than sittle at time T1 The bank agrees to a new deliveryprice, r2. Explainhow r2 should be calculated. 'dividend
'are
.
.
126
CHAPTER 5
APPENDIX PROUF THATFORWARDAND FUTURESPRICESbRE EQUAL WHEN INTERESTRAJqS RECONSTANT This appendix demonstratesthat forWard and futurts prkes are equal when interest rates are constant. Supposethat a futures ontrat lastsfor rl daysand that F'i is the futures prke at the end of day i ( < i < #. Defme gs the risk-f'reerate per day (assmed constapt). Cosider the followingstrategy:9 '
j
1-Take a longfuturesposition of e at the
nd of day0 (i.e., at thebeginnig .of the
contrct). 2. Increase longpositionto e23at tjje end (jf day 1. 3. lncrease longpositionto eyj at tjje entj oj ;ay y, Ad so 0n.
Tls strategy is summarizedin Table SA.I.Bythebeginningof day i, theinvestorhas a lnngposition of e ne prot (pobly ntgative) frpmthe position on day i is .
ti (6 6.-1): -
Assumethat the prot is compottned at the risk-free rate until the end of day rl. Its value at the end of day rl .is
f (6. ' 6.-1)ee(n-i)3
=
-
n ' #'f=lle
(6. -
The value at the end of day rl of the entire investmentstrategyis therefore (F
nb
6-1)e
-
i=1
Tizis is
((6, -
>'n-1)Y (F,-1
-
F,-2) Y
'
'
Y (F1 Fk
'
-
nt (y, =
g
-
,
yn
'is
BecauseF'n the same as the terminalasset spot price, &, the terminalvalue of the strategy can be writttn investment tSr
'k -
nb
/
.
An investpent of f' in a risk-free bond combinedwith the strategy involvingfutures justgivenyields
>-en 1 tr hlen' -
vy'
=
described.It at time T. No investmentis require,dfor al1 the long futurespositions nj followsthat an amount f' can be investedto give an amount Sye at time T. Supposenext thattheforwardpri at the end of day0 is G. InvestingG in a riskless n bond and taking a longforwardposition pf e forwardcontracts also guarantees an aniount Sye'' at time T. Thus, there are two investment strategks-one rtquiring an ; Ja J-
Reprinted by permissionof Dow Jones, Inc.,via CopyrightClearanceCenter,lnc. 2997Dow Jones & Company,lnc. Al1Rights ReservedWorldFide.
SourceL
7
133
InteyestRate Futuyes
Cpnversipn Factprs As mentioned. the Treasurvbond futurescontract allows the party with the short positionto choose to deliverany bondthatha a maturityof lore than 15years andis not callablewithin 15.years. Whena particularbondisdelivered,a parameterknownas its cozgersionfactor defnesthe prke receivedfor the bqndby Jh party wiih the short The applicablequoted prke is the productof the cnversion factorand the position. mostrecent settlementptce for the futures contract. Taking accrued interestinto for each $100facevalue of bond 6.1,the cash received acount,as describedin section delivereis
.
'
(Most recent settlementprice x Conversionfactor)+ Arued interest face value of bonds. suppose Each contract is for the deliveryof $10, that the facior conversion for the bond deliveredis most recent settlementprice is 9-, the 1.38, and the accruedintereston thisbond at thetimeof deliveryiq$3per $1 fa value.The cash received bf the party with the short position (andpaid by the party with the long position) is then (1.38 x 9.)
+ 3.
$127.20
=
per $1 facevalue. A party with the short position in one contrad wuld deliverbonds and receive$127,20 wit a facetalueof $1, The cnversion factorfor a bondis set equal to the quoted pricethebondwould have per dollarof principal on thefrst dayof thedeliverymonth on the assumption that the semiannualcompoupding). interestl'ate for a11maturities equals 6% per annum (with Thebond maturitf and the iimesto the coupon payment datesare toundeddownto th: nearest3 months for the purposes of the catculation.The practie enables theCBOTto producecomprehenjive tables.If, after rounding, thebondlastsfor an exact mlmber of 6-monthperiods, thefrst coupon is assumedto be paid in 6 months. If, after rounding, the bonddoesnot last for an exact number f 6-mopthperiods(i.e.,thereare an extra 3 months), theflrstcoupon is assumed to be paid aer 3 months nd accrued interestis subtracted. As a flrst example of theje rules, consider a 1% coupon bond with 20 years and 2 months to maturity. For the purposesof calclating th conversionfactor,thebondis assumedto haveexactly 20 years to maturity.Thefrst coupon payment is assumed to be made after 6 months. Couponpayments are then assumed to be made at f-month intervalsuntil the end of the20 years when theprincipalpaypent ij made. Assllmethat the face value is $1. When the discotmtfate is 6% per annum with semiannual compounding(or3% per 6 moths), the value of the bondis .
'
,
4 i=1
j
jgg
= $146.23 + 1.0311.034
Dividingby the.fae value gives a conversion factorof 1.4623. As a secondexample of the rules, consider an 8% coupon bond with 18 years and 4 months to maturity. For the purposes of calculatingthe conversionfactor,thebondis assumed to have exactly 18 years and 3 months to maturity. Discountinga11the
134
CHAPTER 6 paymentsback to a point in time3monthsfromtodayat 6% per annum (compounded semiannually)gives a vale of
4+
36 4
Yl1.03
i+
=l
tgg 36
=
1.03
(
$121.83
The interestrate fr a 3-lhonthperiod is 1.03 1, or t Hence,discounting backto the jrejent/vesthebond'svalueas 125.83/1.014889= $113.99. Subtractingthe of accruedinterest 2., th becomes$121.99. Thecopyersipnfactoristherefore1.2199. .4889%.
-
Cheapest-torDeliverBond At any given time during the delivev month, there are many bonds that can be in the'CBOTTreasurybond future contract.These vary widelyas far as delivered couponand maturity are concerned. party with the short position can choose to deliver.Becausethe party wit the short whichof the'availablebondsis 'lie
'scheapest''
positionreceives
(Most recent settlementpri
x Conversionfactor)+ Arued
interest
and the cost of pyrchasing a bond k bond price + Accruedintefest Qqnted the chepest-to-deliver bond is the 0ne foi which bondpri Quoted
-
(Most recentsettlementprice x Conversionfactor)
isleast. Oncethe party with the short position has decidedto deliver,it can determine thecheapest-to-deliverbond by examining eachof the deliverablebondsin turn. Example6.J
'
The party with th short position has decied to deliverand is tryingto choose the threebondsin yhrtablebelow.Assume the most recent settlement between is 93-08 , or 93.25. ' price Bozd
Qaotedprl# price ($)
Cozyersioz
factor 1.0382
99.50 143.50 119,75
1 2 3
1.5188 1.2615 '
.
The cost of deliveringeach of the bondsis as follows: Bond 1: 99.50 (93.25 x 1.9382) $2.69 Bond2: 143.50c (93.25 x 1.5188)= $1.87 Bond3: 119.75 (93.25 x 1.2615) $2.12 =
-
-
=
The cheapest-tcdeliver bond is Bond2.
135
Inteyet Rate Futuyes
''.''' ' '..'.' :'''
('E' .' ''''
.''' ;'
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BusipesiSpapshet6.2 Th: Wild ard Ply
,
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,
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:
qq
7
CBU''/Trisry
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;
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, bond futryescoptfad at 2:p.ni.inrlytil ClilajpE mljiq t time.Hwtver, Tratlty lndd thelipike otilp traij in tht dlb't ::. lqil 8:t>jm.: toq jit kshrt ttm ptigii lg,s 4:()j.ni, tuittrmpf, ttdii ) L ( j nott ?f j , j ju tt dtlikr. Jf tjye poti iyiswt ijsp t the ckkiiglwljse k iti L*k rlykq E thk )pV0 i jri: id calulid on tlle ajisy y tlie ttttlenttlkiyyujjy ); ) y,y kjjjjs y , jyjjjsyy t;: ( ) jfy t wlh kdinj :s cvdgcted jttjtbefflr jjj : C'filisEprliiceipvs; klk i/ ilt ris to(a jtiil play bodlijpctq tli .Ejay/itli dky.fttritltveiftupt,' t' tt deileaftrr litps. p ii.flrst t s&y,'3:k$t,':,('od: difiyerkt to f pjitioat isskt;, 'jE 7:*3p'(iJf/iid jfl! th, llitkl dt1 i jjlii'teejjEie icp jqfofwEt'Lptigiryq ri 'ppill Se'itlEiy': t party pjep jif'itji shrt ftjlti' E i : E 5 3 5 ; kile the .salii qftejy i jjq gjjd) j )q ( Ty i )t (;(, ( ) y : q E q ( y r q ojenEt Ethr tht ktty kithfte sft ppiyi jr with oty ojtij isEreqtciez' in,ir ftyltCjti ii Atlki lkei is tl k ' i'idti'rieItt play ' ( y )( q beAithputih o/ion.
Trading in te
,
.
,
.
.
'
.
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;
'
.
'ilttrtigp
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,
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:i?
'pr
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lli
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A number of factorsdeterminethe cheapest-to-deliverbond.Whenbond yields are in excessof 6%, te conversion factor systemtendsto favorthe deliveryof low-coupon long-maturitybonds. When yields are lessthan 6%, the system tends to favor the deliveryof high-coupn short-maturitybgnds.Als?, when the yield curve is upwardsloping,there is a tenpeny for bonds wit a long time to matmity to be favored, whereaswhen it is downward-slojlng, thereis a tendencyforbondswith a short timeto maturityto be delivered. In addition to $hecheapest-to-deliverbond option, the party with a short position has an optionknownas the wild card play.This is describedin Businesssnapshot 6.2.
Determininj the Futures Price
An exact theoreticalfutures pri fof the Treasury bond contractis dicult io becausethe short party's options conmrned with the timing of delivery determine andchoi of thebondthat is deliveredcannot easilybe valued. However,if weassume thatboth thecheaptjt-to-deliver bond and te delivetydate are known,the Treasury bondfuturescontract is a futurescontract on a tradedsecurity(thebond)that provides thehoiderwithknownincome.lEquation(5.2)then showsthat thefuturespri, F, is relatedto the spot price,S, by '' (6.1) t5' Ikr' .
=
-
whereI is the pfesentvalue of the couponsduringthe lifeof the futurescontract, T is the time until the futurescontrad matures, and r isthe rk-free interestrate applicable to a time period of length T. fxample 6.2
Supposethat, in a Treasury bondfuturescontract, it is knownthat the cheapestwith a conversion factorof 1.4. to-deliverbond will be a 12%coupon tbond
1 In practice, for the purposes of determiningihecheapest-to-deliverin this calculation,analysts usually that zero rates at the maturity of thefuturescontract willequal today'sforwardrates. assume
136
CHAPTER 6 '
.
.
Timechart for Example 6.2.
Fijure 6.1
Matutity cf
Coqpon paymem
C0uP0n pament'
Current time
60 days
Coupon ontract payment
futuDs
'
148
l22 days
'
35 days
days
Supposealso that it is knwn that deltverywilltakeplacein 27 days.Coupons arepasbk stmiannually on thebnd. Asillustfatedin Figure 6.1,thelastcoupon dat kas 60 daysago, the next coupon dateis i 122days,and tlle coupon date thereafteris in 305days.The term structure is sat,anb the rate of interest(with ' . continuouscompounding) is 1% per annum.Assumethat the currtnt quoted bond prict is $120.The cash price of the bond is obtained by adding to tllis quoted price the proportion of the next coupon paymentthat ares to the holdtr. The cash priceis therefore x'
,
60 x 6 :cJc121.978 60+ 122
12 + A coupon of $6will be reived of this is
l22days(=0.3342ytars). Theprestntvalue
'after
'
64-.1
x2.3342
=
.,
5.803
The futurescontract lastsfor 27 days(=0.7397years). The cash futuresprice,if the contract were writtenon the 12%bond, would thereforebe (121.978
5.803): .1
x2.7397
-
=
j)j
.
g94
At delivery,thereare 148daysof arued interest.The quoted futuresprice,if the contractwere writtenon the 12%bond,is calculated by subtracting the arued interest 148 = 120.242 125.094 6 x 148+ 35 ' -
From the defmitionof the conversion factor,1.4000standard bonds are considered equivalentto each 12%bond.The quoted futurespricesould thereforebe
120.242 85.887 1.400 =
6.3
EURQDOLLARFUTURES The most popular interestrate futurescontract in the UnitedStatesis the 3-month Euroollar futures contract tladtd n the Chicgo MercantileExchang (CME).A Eurodollar is a dollardepositedin a US or foreignbankoutsidetheUnitedStates.The Eurodollar interestrate is the ratt of interestearned on Eurodollarsdepositedby 0ne bank with another bak. It is essentiallythe same as the LondonInterbankOfered Rate (LIBOR)introducedin Chapter4. Three-monthEurodollarfutures contracts are futures contmcts on thq 3-month
137
InteyestRate Futuyes
(g-day) Eurdllar interestrate. Theyallw an investr t lck in an interek rate n !1 millin fr a futurr 3-mnth perid. The3-mnth perid t which the interestrate applies starts on the third Wednesdayof the deliverymonth. The contracts have and Decemberfr up tp l years intp deliverymonths of March,June, september, the futur. Thismeans that in 2008an investorcan use Eurdeljr futurest lockin an interestrate fr 3-monthperids that are as far intothe futureas 2018.Shrt-maturity and December. contracts trade for months thet than March june, september, However,ihesehaverelativelylW open interejt. To understand how Eurdollar futures contmcts wrk, cnsider the June 2007 cntract in Table6.1.The quotedsettlementprke on January8, 2007,is 94.79.The cntract endson thethirdWednesdayf thedeliverymonth. Inthe case of thiscontract, the third Wednesdayof the deliverymnth June2, 2007.Yhe contract is marked to marketi the usupl wayuntil thatdate.However,on June2, 2007,thesettlementpri R, where R the acual 3-monthEuroollar ipterestrate on thatday, set equal to 1 expressedwith quarterlycmpounding and an actual/36 day counycnvention. (Thus, if the 3-monthEurodollar interestrate on June2, 2007,turnedout to be 4%, thefmal settlementprice would be 96.)nere a snalsettlementrefkctingthis sdtlementpriee and a11contracts are declaredclosed. The contract is designedso that 1basispoint (= mve in the futuresqute correspgndsto a gain or loss of T25per contract. When a Eurodollar futuresquote increasesby 1basispoint, a traderwho k longone contract gains $25ad a traderwho is whenthe quote decreasesby l basispoint a trader short one contract loses$25.similarly, whois long one contract lose $25and a trader wh' ii hrt one contract gains $25. suppose,for examkle, that the settlelent pri changes frm 94.79to 94:90beyween January8, 2007,and January9, 2007.Traderswith lng positionsgain 1l x 21 $275 per contract; tfaderswith short positions lose$275per contract. The $25per basispoint ruleis consistentwith the point made ealkr that the cntract locksin an interestrate on $1million dollarsfor 3 months. When an interestrate pe?year changes by 1basispoint, the interestearned on 1 million dollarsfor 3 months changes by '
,
-
.1)
=
.l
l,tq
x
x 0.25 25 =
the futuresquote is 1 minus the futuresinterestrate, an investorwho is or $25.since lng gains when interestrates fall and ne who is short gainswhen interestrates rise. Exampk6. 0n January 8, 27, an investorwants to lock in the interestrate that will be earnedon $5millign for 3 months starting on June2, 2007.The investorbuys :ve June7 Eurodllat futurescontracts at 94.79.On June2, 2007,the 3-month LIBORinterestrate is 4%, s that the fnal settlementprice proves to be 96.00. The investorgains 5 x 25 x (9,6 9,479) $15,125 n the long futuresposition.The interestearned n the $5million fr 3 months at 4% -
5,,
x 0.25x
=
.4
=
5,
o?$5,. The gain on thefututescontract bringsth up to $65,125.Th the interestthat wuld have been earred if the interest rate had been 5.21% (5,, x 0.25x 0.0521 65,125).Th illustrationshws thatthefuturestrade of lockingin an interestrte equal to (1 ellkct 94.79)%, r 5.218/:. hasthe =
-
138
CHAPTE
Theexchange dehnesthe contrat prke as l,o Fhere
x
(l
-
0.25x (100 Q)) -
(t6.:1)
Qis the quote.Thus,the settlementprke of 94.79fr theJune 2097 contract in
Table6.1 correspond! to a contract pri of l,
x
tl
9.25x (l
-
94.79)) $986,975 =
-
ln Exmple 6.3,the fmalcontrpct pri is 1t),
x
(1
-
9.25x (1
-
96)) $990,600
and the diferencebetweenthe initialand fmal contract pri is $3,925,8o that an investorwith a longpositioninfivecontracts gains 5 x 7,925dollars,or $15,125. Tllisi8 consistentwith the t1$25 per 1bas point move'' rule used in Example6.3. We can see frol Table6.1that the interestrate termstructurein the UnitedStates wa8 downward-sloping on January 8, 2007. The futres rate for a 3-monthperiod in January17,27, wa8 5.3575%.fqr a 3-mgnshperiod benning June 2, bepinning 2 . 2907,it was 5.210:; for a 3-lonth period beglnningSeptemberi9,2007,it was 5.0450/0., and for a 3-monthperiod beginningDember 19,2007, it was 4.91%. Othercontracts similar to thi CMEEurodollarfuturescpntract tradeon interejtrates in other countries. 'I'he CME trades Euroyencontracts.The London lnternational Finaial Futures and OptionsExchange(partof Euronext)trades3-monthEuribor conttacts (i.e.,contracts on the 3-monthLIBOR rate for the euro) and 3-month Eurosws futures.
' Forward vs. Futures Interest Rates The Eurodollarfutures contract is similar to a forward rate agreementIFRAC. see Section4.7)in that it locksin an interestrate for a futureperiod.For short matulies (up to a year or s0), the two contrads can be assumed t be the same and the Eurodollarfuturesinterestrate can be apumed to be the same as the corresponding forward interest rate. For longer-datedcontracts, diferens betweenthe contracts becope important.Comparea Euroddlarfuturescontmct on an interestrate for the period betweentimes T) and T2 with an FRA for the same period. 'TheEurodollar futurescontract is settleddaily.Thefinalsettlementis at timeTj and reects the realized interestrate for the periodbetweentimesT1and T2.By contrast the FRA not settled and thefinalsettlement refkctingthe realizedintertstrate betweentimesT1and T2 daily * is made at time T2.2 There are thereforetwo dferens betweena Eurodollarfuturescontract and an FRA. These re: .
1. Thediferencebetweena Eurodollarfuturescontract and a similarcontract where thereis no dailysettlement. Thelatteris a forwardcontract where a payof equal to the diferencebetweenthe forwardinterestrate and the realized interestrate i8 paid at time T1. .
2 Asmentionedin 4.7,sdtlement may occurat timeTl lmtit islhenequl to the present valut of the section normalfomard contract payof at timeT2. ,
139
InteyestRate Futuyes
2. Thediferencebetwen a forwardcontract wheretheieis settlement at timeTl and a forwardcbntract where thereis settlement ai time T2. Thesetwo comjonents to thediferencebetweenthe contracts cause some confusion in practi. Both decreasethe forwardrate relative to the futuresrate; but for long-dated contractsthe reduction caused by the secpnd dference is much smaller than that causedby the Erst. Ti reason wy the rst diflbrence(dailysettlement) decreases the forward rate followsfrom the arpments in Section5.8. Supposeyou have a contractwhere the payof is Ru Rt- at time Tj, where Rg is a predetermined rate forthe period betweenTl nd T2,apd ,Ru is the realized rate for this period, and mu havethe oktionto switch to dailysettlement.In this case dailysettlement tendsto lead ' to cash infows whn rtes a!e llighand cash outsowswhen rates are low.Youwould therefore:nd switching to daitysettlement be attractive becuse you tend to have moremoney in your margin aount when rates are high.As a result the market would thereforeset RF higherfor the daily settlementalternative (reducing your cumulative round, switclng fromdailysettlement to expectedpamfl). To put thisthe other way settlementat tne T1reduces Rg. To understand the reason why the secpnd diference redus the fomard r?te, of Ru .L RF is t time T2insteadof T1 (asit is for a regular supposethat the gyoflFRA). If Rx is hlgh,the pyof is positive.Becauserates are high,the cost to yu of havingthe payof that you receiveat timeT2rather than timeT1is relativelyhigh.If Rx is low,the payof is negative.Because rates are low,the beneft to you of havingthe payof you make at time T2 rather than time Tj is rlatively low.Overallyou would rather have the payof at time T1 If it is at time T2 fgther than you must be 3 ' compensatedby a reduction in Rt.. Analystsmake what is known as a cosvexityadjustment to aount for the total diferencebetweenthe two rates. One popular adjustmqnt is4 -
'to
.
,
'
f'orwar d rate Futi res rate =
-
ic2T17-2
'
'
(6.3)
2
where,as bove,T1is the timeto maturity of thefuturescontract and T2is the time to thematurity of the rate underlying the futurescontract. The variable c is te standard of the change in the short-term interestrate in 1 year. Both rates are expressed deviation withcontinuous compounding.s A typical value for c is 1.2%or .12.
Example 6.4 and we wish to calculate the forwardrate Considerthe situation where c when the s-year Erodollar futures pri quote is 94. In this case Tl = 8, T2 8.25,and the convexityadjustmenf is .12
=
=
ix 2
.12
2
x 8 x 8 25 = .
.
00475
basispoints). Thefuture!rate is 6% per annum on an actua1/36 or 0.475%(47.5 3 Quantifying the efect Chpter 29.
of thls tpe of timingdference on the vaiue of a erivativeis discussedfurtherin )
uJ
4 SeeTechnicalNote l on te author's websitefor a proof of this. 5.Tls formuiais ase on theHo-.l.,eeinterestrate model,whichwill be iscussedinChapter3. seeT. S.Y. pricing nd contingent claimj,'' Ho and S.-B. Lee, interest Jourzal of rate structure movements Finance,41 (December1986),11l-29. . rf'erm
-
140
CHAPTER 6
basiswith quartirly compounding.This corresponds to 1.5% per 90 days or an ln 1.15 6:038%1t,11continuous compounding and an annualrate of (365/90) actual//dsday count. The estimateof the fgrwardmte given by equation (6.3), =
therefore,is 6.038 0.475p: 5.563%per annum w1t11continuous compounding. The tahlebelowthows howthe size of the adjustment increaseswith the time to aturity. -
'
.
'
.
.. ...
.
. ............ ........... . . .
Matarity offatares
.
. ..
Convexityaiustmests bastspointsj
yearsj
3.2 12.2
l 4 6
27.0
47.5 73.8
8 lg
We can see fromthis table that the sizeof th adjustmeni is rougl proportional to thi square of the tim to paturity of the futurescontract. Thus the contract is approximately 16timesthat for convexity adjustment for the s-year a 2-year contract.
Using Eurodollar Futures to Extend the I-IOR Zero Curve .
,
'
The LIBORzero curveout to 1 year is determied y the l-montk 3-month,6-month, and lz-moyth LIBOR rates. On the convexityadjustment just describedhas been made,Eurodollarfuturesare often used to extendthe zero curve. Supposethat the th Erodollar futurescontract maturesat tim T. i = 1 2, ). It is usually assumed that the forwardinterestrate calculatedfromthe fth futurescontract applies to the period 6 enables a bootstrp produre to be used io 6.+1(In practice thisis close to true.)Yhis to determinezero rates. Supposethat Fi is the forwardratt calcutated from the ith Eurodollar futurescontract and Ri isthe zero rate for a maturity %.. Fromequation (4.5), ,
.
.
.
,
A'f
=
&+lTf+I
RiTi Tj+1 Ti -
-
so that
Fi(Tf+! Ti)+ hTi &+1 T2+l -
=
(6.4)
Other Euro rates such as Euroswiss,Euroyen,and Euribor are used in a similar way, Example 6.5 LIBOR zero rate has bttn calculate as 4.80% with continuous compoundingand, from Eurodollarfuturesquotes,it has been calculated that (a) the forward rate for a g-day period beginningin 400 daysis 5.30% with contiuous compounding,(b)tht forwar rate for a g-day periodbeginningin 491daysis 5.50%with cpntinuus compounding, and (c)the forwardrate for a g-day period eginning in 589daysis 5.60%with continuouscompouhding. We
The 4-day
141
InterestRate Futures can us eqution
(6.4)to obtai
the491-dayrate as
0.053x 91+ 0.048x 4 491
= g.gjs93
4.89304.Similarlywe can use the secgndforwardrate to otain the 589-day rate as 0.055x 98+ 0.04893x 491 = 0.04994 589 Or
or 4.994% The next forwardrate of 5.60%would be used to determinethe zero curveout to the maturity of the xt Eurodollarfuturescontract. (Notetht, even thoughthe rate underlying the Eurodollarfuturescgntract is a g-day r'ate, it assumedto apply to the 91 oi 98 days elapsing betweenEurodollarcontract maturities.) .
'is
64 *
DUCATION-BASED USING FUTURES HEDGINGSTRATEGIES We disussed durationin Section4.8. Considerthe situation where a posion in an asset that is interest rate dependent,such as a bond portfolio or a money market security,is beinghedgd using an interestrate futurescontract. Defme: Fc : Contractpricefor the interestrate futurescontract Dy: Duration of the asset underlyipgthe futres contract at the maturityof the fpturescontract
f': Forwardvalue of the portfolio beinghedgedat the maturity of the hedge(in practice,thij is usually assumed to be the same as the value of the portfolio todayj Dp: Dufation of the portfolio at the maturity of the hedge If we assume that the change ip the yield, y, is the snmefor a1)maturities, which meansthat only parallel shiftsin the yieldcurve can occur, it is approximatelytrue that
; It is also approximately true that
;
y
=
,-PDP
o-yyss
y yy
'l'he number Of contracts required to hedgeagainst an uncertain
#*
PDP =
Fclv
y,
therefore,is (6.5)
This is the duratioz-basededge ratio. It is sometimes also called the prke sezsitity dge ratio.4 Usingit has the efect of making the durationof the entire position zero. Whenthe hedginginstrumet is a Treaqurybond futurescontract, the hedgermust baseDy on an assumption that one particular bond will be delivered.This means that thehedgermust estimatewhich of theavailablebondsis Ekelyto be cheapest to deliver 6 For a more detailed discussionof equaon (6.5),see R. J. Rendleman, sDuration-BasedHidging witil Treasury Bond Futuresj'' Joarzal ofFixed Iscome9, 1 (June1999):84-91. '
142
CHAPTER 6
at the time the hedgeis put in place. lf, subsequently,the interestrate environment changesso that it looksas thougha diserentbond will be cheapst to deliver,thenthe hedgehas to be adjusted and its performace may be worse than anticipated. When hedgesare constructed using interestrate futpres,it is important to bear in mind that interst rates and futurespris pove in opposite directions.Wheninterest rates go up, an interestrate futurespricegoes down.Whenintrest rates go down,the reversehappens, and the interestrate futuresprice goes up. Thus, a company in a position to lose poney if interestrates drop shouldhedge taking a long futures position.Similarly,a company ip a position to losemoney if interestrates rise should hidgeby taking a shol't futuresposition. The hedgertriesto choose thefutults contract so that thedcrationof the underlying assetis as tlose as poible to theduratio?of the asset beinghedged.Eurodollarfutures tend to be used for exposuresto short-terminterestrates, whereasTreasury bond and Treasurynote futurescontracts are used for exposuresto longir-termrates. .by
fxample6.6 lt isAugust2 anb a fundmanager with $1 millioninvestedin governmentbondsis concernedthat interest rates are expected to be highly volatile over the next 3 months. Thefundmanager decidesto use theDecemberT-bnd futurescontract curr nt futurespriceis 93-02 or 93.0625. to hedgethe value of the porttolio Becauseeach contract isforthedeliveryof $1, fa valueof bonds,thefutures price is $93,062.50. contract Supposethat the durtion of the bondportfolio in 3 months will be 6.80years. The cheapest-to-deliverbondin the T-bondcontrlct is expected to be a z-year 12% per annum coupon bond. The yield on this bond is crrently 8.80% per annum, and the durationwill be 9.20years at maturity of the futurescontract. The fund manager requires a short position in T-bondfuturesto hedgethe bond portfolio. lf interestl'ates go up, a gain will be made on the short futures position,but a losswill be made on thebond portfolio. If interestrates decrease,a loss will be made on the short position, but there will be a gain on the bond portfolio.The number of bond futnrescontracts that should be shortedcan be calculatedfrom equation (6.5) as rfhe
.
,
6.80 1,, 79.42 x 93,062.50 9.20 =
To the neafest whole number, the portfolio manager shouldshort 79 contracts.
6.5
HEDGING PORTFOLIOSOF ASSETSANPLIABILITIES Financialinstitutionssomtimes att.emptto hedgethemselvesagainst interestrate risk by ensuring tlkatthe averagedurationof theirassets equals the averagedurationof their liabilities.(Theliabilitiescan be regarded as short positionsin bonds.)This strater is known as daratiosmatchisg or porblio immasatios. Whenimplemented,it ensures that a small parallel shift in interestrates will havelittle efect on the value of the portfolio of assets and liabiliiies.The gain (lo$s) on the assets shouldolhet the loss (gain)on the liabilities. N
.
143
Interest Rate Futures .
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.
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..
. .
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BusinessSnapdot 6.j Aslqt-labiliiy Mahajlppt by Bqnks .
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.
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(
and l Asdesqribedln BusinessSnqj; o443 in t 196s 1970s4: , llpny savings State mde thi ijtaki f fidic locz som banksi tlieiited ' an Loans and ', ''rhis iiltts y tese termlons wlth,shoftrtee' (dpositskj , le,,. t sii jpicutr ) ... . j y .. . F , fnancialiiptitutlpj. ; yekr( ittieq anks oF innii The asei2utitit y iaginpp, (LM) qs/ts pnl exj,ojufe 40 ipyyesyftis veif arefvlly.Mtthinp tl dytidp: ityyaiiil g>i't llttj libilitii; ij a pt k8f, 15 ut this does;ynot prutwt a Sk 'rhiEivolgej ; Ai anqglitt inihevildcurvi, nppularappioach is knit a tlw ziro :coupon 1rld tukviiio gkpysj kp o bycts thr flrst dividipg r Elply, mqtk, j'rt'g,'/. e LM rqtld seod 1 3 tnight tol he to tt the ntkt lliklitlij : one i ired pn tse vlqo tn ipyrstigayes cgmmiitet t ttj r, ,ppi, EAllleflt Ft tqilhpipg, y ll i ptes cgfftslmljdipg'to bket l.'7E ,
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Duration matching does not immunizea portfolio against nonparallel shifts in the zerocurve.Th a weakness f the approach.In practi, short-term rato are usgally it moreplatile than, and are not perfectlycorrelatedwith, lont-termrates. sometimes shorteach in and lonterm rates move opjosite directionsto even happens that other. D uratkn matching ij thereforeonly a frst step and nancial institutins have developedother tools to help them manage theirinterestrate exposure. seeBusiness
snapshot6.3.
SUMMARY Two very popular interestrate contracts are theTreasurybond and Eurodollarfture,s In the Treasury bond futurescontracts, the that trade in the United states. contracts number with short position of interestingdeliveryoptions: the has a party
1. Delivery can be made on any dayduringthe deliverymonth. 2 Thereare a number Qf alternative bondsthat can be delivered. *
3. On any day duringthe deliverymonth, the notke of intentionto deliverat the 2: j.m. settlement pri can be made any time up to 2:00p.m. Theseoptins all tend to redu the futurespri. The Eurodollar futures contmct is a contract on the 3-monthrate on the third Wednesday of the deliverymonth.Eurodollarfuturesare frequentlyused to esmate LIBOR forwardrates for the purpose of construcng a LIBOR zero curve,When longdated contractsare used in this way,it importantto make what is termeda convexity adjustmentto allowfor the marking to market in tbe future contract. The conceptof durtion importantin hedginginterestrate risk.It enables a hedgey
144
CHAPTER 6
to ssess the sensitivityof a bondportfolioto small parallelshifts in the yield curve. It alsoenablesthe hedgerto assessthe snsitivity of an ihterestrate futuresprke to small, changesin te yield.curve. Thenumber offuturescontracts nessac to protectthebond portfolioagainst spall parallel shifts in the yield cyrvecan therefore be calculated. Thekeyassumption underlyingthedumtion-basedliedgingschemeisthat al1interest rates change by the same amount. Tis means that only parallelshifts lnthe term structure are allowedfor. In practii short-term interest rates are generally more volatilethan are long-termipterestrates, and hedgeperformanceis liabletp be poor if the duration of the bond underlying the futurescontmct difers markedly fromthe duption of the sset beinghedged.
.
,
FURTHERREADING Burghardt, G., and W. Hoskins. 63-79.
trf'he
ConyexityBias in EurodollarFuturesj''Risk, 8, 3 (1995):
Dlle, D. SGDebtManagement and Interest Rte Risk? in W. Beaverand G. Parker (eds.), Risk Mazagemezt: ChllezgesJrl: Solutiolu.NewYork:McGraw-ltll, 1994.. ''
Ri-k Xl-wreky. Frank Fabozzi ASSoc., 1999. azd Other Cprlvcxfly, Grinblatt, M., and N. Jegadeesh. $$TheRelative Price of EurodollarFutures and Fomard Contractsj''Jovnal of Fizance, 51, 4 (September1996):1499-1522.
.:prl#
Fabozzi, F. J. Datioz,
and Question
roblems(Anslers in SalutiansManual)
6.1. A US Treasurybond pays a 7% coupon on January7 and July 7. How much interest tie arues per $100of principal to bondholderbetweenJuly7, 2009,and August9, 2009:? How would your answerbe diferent if it were a corporte bond? 6.2. It is January9, 2009. The price of a Treasurybond with a 12% coupon t at matures on October12, 2020, is quoted as 12-7. Whatis the cash price? 6.3. How isthe converon fqctorof a bondcalculatedbytheChicagoBoard of Trade?How is it used? 6.4. A Eurodollarfuturesprice changes from96.76to 96.82.Whatis the gain or loss to an investorwho is two contracts? 6.5. Whatis the purpose of the convexityadjustmentmade to Eurodollar futuresrates? Whyis the convexityadjustment necessary? 6 6 The 35-day LIBOR rate is 3% with continuous compoundinj and the fomard (ate calculatedfrom a Eurodollarfutures contract that matures in 35 daysis 3.2% with continuouscompounding. Estimatethe Mo-dayzero rate. 'long
.
'
6.7. It is January 3. You are managing a bond portfolio worth $6million. The duration f the portfolio in 6 monthswill be 8.2yars. TheSeptemberTreasurybondfuturesprice is currently18-15, and the cheapest-to-deliverbond will have a duration of 7.6 years in September.How should you hedge against changes in interest rates over the next .
mont s
6.8. The price of a g-day Treasu billis quoted as 1.. What continuously cotpounded return (onan atua1/365 basis)doesan investorearn on the Treasurybillfor the g-day period?
145
Inteyett Rate Future
6.9. It is May 5, 2008.The quoted pric of a government bond with a 12% coupon that matureson, July 27,21 1, is 11-17. Whatis the cashprice? 6,1. Supposethat the Treasurybond futurespri is 11-12. Whkh of the followingfour bondsis cheapest to deliver? Ctzverue?l factor
Bozd
Price
1 2 3 4
125-05 142-15 115-31 144-02
.
1.2131 1.3792 1.1149 1.4026
-
6.11. lt isJuly3, 29. Thechtapest-to-deliverbondin a September2009Treasurybondfutures contractis a 13%coupon bond,and deliveryis expecyedto bemadeon September3, 2009. Couponpayments on thebond are made on February4 and August4 tach year. The term structureisCat,and the rate of intrest withsemiannualcqmpounding is 12% per annum. Theconversionfactorforthebondis 1.5.Thecurrent quotedbond priceis $11. Calculate the quotedfuturesprice for the contract. 6.12. An investoris lookingfor arbitrage opportunities in the Treasurybopdfuturesmarket. Whatcomplicationsare cyeatedbythefactthatthe party with a short position can cloose to deliverany bond with a maturity of ver 15 years? 6.13. Supposethat theg-monthLIBORinterestrate is 8% per annumand theGmonthLIBOR interestrate is 7.5% per annum (bothwiyhactual/365ad. continuous compounding). Estimate the 3-monthEurodollar futuresprie quote for a contract maturing in 6 months. 6.14 Supposethat the 3-day LIBORzero rate is 46 and Eurodollar quotts fof contract,s . maturingin 3, 398, and 489 days are 95.83,95.62,and 95.48.Calculate398-day and betweenforwardand futuresrates for 489-dayLIBORzero ratts. Assume.no of calculations. the purposes your .
'difkren
6.15. Supposethat a bond portfolio with a durationof 12 years is hedgedusing a futures contractin wlch the underlying asset has a durationof 4 years. Whatis likelyto be the impacton the hedgeof the fact that the 12-yearrate is lessvolatile than the 4-year rate? realizej that on July 17the company will 6 16. Supposethat it is February20and a treasurer have to issue $5 million of commercial paper with a maturity of 18 days.if the paper (In other words, the company were issuedtoday,the company would realize $4,820,000. wouldreceive $4,820,000 for its paper ad have to redeem it at $5,0()0,000 in 18 days' quoted Eurodollar futures price is time.) The September as 92.. How should the treasurerhedgethe company's exposure? 6 17 O upst 1, a portfolio manager has a bond portfolio worth $1 million. Theduration of the portfolio in Octoberwill be7.1years. TheDecemberTreasurybnd futurtsprice is currently91-12and the cheapest-to-deliverbond will have a duratin of 8.8 years at maturity.How shouldthe portfoliomanager immunizethe portfolio against changesin inttrest rates over the next 2 months? '
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.
6,18. How can the portfolio manager change the'duration of the portfolio to 3. years in Problem6.172 6.19. Between October3, 2009,and November1, 2009,you havea chice betweenowning a US govtrnment bond paying a 12% coupon and a US corporate bond paying a 12%
146
CHAPTER 6 .
2
coupon.Cnsider carefullythedaycount conventionsdiscussedin this chapter anddecide whicllof the two bonds you would preferto own. Ignorethe risk of d.efault. 6.20. Supposethgt a Eurodollarfuturesquote is 88for a contract mturing in 60days.Whatis the LIBORforward rate for th 6- to ls-day period?Ignorethe diferepcebetween futuresand fomards for the purposesof this question. 6.21. The 3-monthEurodollarfuturej prke for a contract matring in 6 years is qvoted aj 95.20.The stadard deviationof the hangein the short-term interestrate in 1 year is 1.1%. Estimatethefomard LIBORinterestrate fortheperiodbetween6. and6.25years j in the fu ure. 6.22. Explainwhy the forwardinterestrate is lessthan the corresponding futqresinterestrate calculatedfrom a Eurodollarfuturescontract.
AssignmentQpestions 6.23.Assumethat a bank can borrowor lend money at the same interestrate in the LIBOR markej;The g-day rate is 1% per anum, and the l8-day rate is 1.j% jer anum, both expressed with continuous compounding and actal/actual day count. The Eurodollarfuturespricefor a contract maturingin 91daysis quoted as 89.5.What arbitrage opportunitiesare open to the bank? ' 6.24. A Canadiancompany wishes to create g, CanapiapLIBORfutuiescontractfrom a Us Eurodollar futures contract and fomard contracts on fgreip exchange. Using an example,explain how the company should proceed.For th purpose of this problem, assumethat a futurescontract is the same as a forwardcontract. 6.25.The futuresprice for the June2009CBOTbond futurescontract is 118-23. (a) Calculatethe conversioil factorfor a bnd maturing on January 1, 2025,paying a coupon of l%. (b) Calculatethe conversion factor fo'r a bond maturing on October1, 2030,paying a coupon of 7% (c) suppose that the quted pricesof the bondsin (a)and (b)are 169.00apd 136.0, respectively.Whichbondis cheaper to deliver? (d) Asslrmingthat the cheapest-to-deliverbond is actuallydeliveredon June 25, 2009, whatis the cash price receivedfor the bond? 6.26.A portfolio manager plans to use a Treasurybond futurescontract to hedge a bond portfolio over the'next 3 months. The jortfoliois worth $1 lnillion and will have a durationof 4.0 years in 3 months.Thefuturespriceis 122,and each futurescontract is on of bonds. The bond that is expected to be cheapest to deliverwill have a $1, duration of 9. years at the maturityof the futurescontract. What posion in futures contractsis required? (a) What adjustments to the hedge are necessary if after 1 month the bond that is expectedto be cheapest to deliverchanges to one with a durationof 7 years? (b) suppose that al1rates increaseover the next 3 months,but long-termrates increase less than short-tenh and medium-term rates. What is the efect of this on the performanceof th hedge? '
The first'swapcontractswere negotiateg in the early 198s. Sincethen the market has seen phenomenal growth. Swapsnow oupy a positionof ntral nportan in the over-the-counterderivativesmarket. A swap is an agreementbetweentwo companiesto exchangecash iows in thefuture. The agreementdefmesthe dates when the cash :ows ar to be paid and the way in Whichthey are to be calculated.Usuallythe calculation of the cash iows involvesthe futurevalue of an interest rate; an exchangesrate,or other market variable, A forward contract can be viewed as a snple example of a swap. Supposeit is March1, 29, and a companyenters into a forwardcontract to buy 1($ ounces of gold for$900per ounce in 1 year. The company c4n sell the gold in 1 yearas soon as it is tl The forwardcontract is thereforeequivaknt to a swap wherethe company receive agfeesthat on Mfh 1, 21, it will py $9, and reive ls', where S is the marketprice of 1 ounce of gold on that dt. Whereasa forwardcontract is equivalent to the exchangeof cash iows on justone futuredate, swaps typicallylead to cash tlowexchangeson severalfuturedates.In this chapterwe examine how swaps are desiped, how they are used, and how they are valued.Most of this chapter focuseson two popular swaps: plain vanilla interestrate swapsand fixed-for-faedcurrencyswaps. Othertpes of swaps are briey reviewedat theend of the chapter and discussedin more detailin Chapter32. .
'
7.1
MECHANICSOF INTERESTRATE5WAP5 vanilla'' interest rate swap. In this swapa The most common type of Swap is a agrees to pay cash iows equal to interestat a predeteyminedflxedrate on a company principat for a number of years. In return, it receivesinterestat a qoatingrate notional principal for the snmeperiodof time. notional onthesame tdplain
LIBOR Theioating rate in most interestrate swap agreementsistheLondonInterbankOfered Rate(LIBOR).Weintroducedthisin Chapter4. It isthe rate of interestat which a bank is preparedto depositmoney with other banksin the Eurocurrencymarket. Typically, l-month,3-month, f-month,and lz-monthLIBORare quotedin a11majof currencies. 147
CHAPTkR 7
148
Just as prime is often the referen rate o? interestfor qoating-rateloans in the dopesticfnancial market, LIBOR is a referenl rate of interestfor loans in intermarkets. To understand howit is used, consider a s-earhond with a nationalinancial j of interestspecied rate per annum. Thelifeof the bod as 6-monthLIBORplus is dividedinto l periods, each6 months in length.For each period, the mte of inttrest is setat 0.5% per anum abovetheGmonthLIBOR rate at thebeginningof the period. Interestis paid at the end of the period. .
.5%
,
.
'
Illsttatio Consideta hypothetkal3-yearswap initiatrdoh March5, 2007,betweenMicrosoftand Intel.We suppose Microsoftagreesto pay Intel an interestrate of 5% per annumon a principal of $1 million,and in retrn Intel agreesto pay Microsoftthe 6-moth LIBOR rate on the sameprincipal. Microsoftis Ljed-ratepayer; Intelistjkatizgare to be exhanged every ratepayer. %z assumetheagreement specifes that gayments lth enannual quoted months and that the 1%interestrate is compounding.This 6 Fipre is dkgrammatkally in 7.1. represented ' ' swap The frst exchangt of jaymentswould take place on Septembtr5, 2067, 6 months afterthe initiationof the agreement.Microsoftwould pay Intel$2.5million.Thisis the intereston the $1 million principal for 6 months at 5%. Intel would pay Micrsoft intereston the $1 million principal at the f-montliLIBOR rate prqvailing6 months is, on March 5, 27. Supposttht the f-month priorto September5, 27-that LIBOR rate on March 5, 2007, is 4.20:. Intel pays Miciosoft x 0.t42 x $1 $2.1million.i Note that thereis no uncertainty aboutthis flrst exchangeof paymnts it is determinedby the LIBO: rat at the tlme the contract is entered intp. because seond exchage of payments would takeplace on March5, 28, a year afterthe The of the agreement. Micrsoft would pay $2.5million to Intel.Intelwould pay initiation interest n the $1 million principal to Microsofjat theGmonthLIBOR rate prevailing 6 months prior to March5, zs-that is, on September5, 2007. Supposethat the 6-monthLIBOR rate on September5, 2007, is 4.8B/n. Intel pays x 0.948x $1 = $2 4 million to Microsoft. In total- there are six exchangesof payment on the swap.The flxedpayments are always$2.5nllion. The Coating-ratepayments on a payment date are calculated using the 6-month LIBURra: pvailing 6 months beforethe payment date. An j interestrate swapis generallystrnctured so that one side rents t e diserencebetween the two payments to the other side. In our example, Microsoftwould pay Intel $0.4million (=$2.5million $2.1million)on September5, 207, and $0.1million (= $2.6million $2.4rnillion) on March5, 2008. .5
=
.5
-
-
Figu/e 7.1
Interestrate swapbetweenMicrosoftand Intel. 5.9%
Intd
.
Microsoft LIBOR
!
i The calculations ere are simpiied in that theyignoredaycount conventions.This point is discussedin moredetaillaterin the chapter.
:149
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of dollars)to Microsoftin a. $1 nllion 3-year /.1 Cash Qows(millipns
iterestrate swa.pwhen a.flxedrate of 5% is paid and LIBORis reived. Date
Mar. 5, 2007 sept. 5, 2007 Mar. 5, 2002 Sept.5, 2008 Mar. 5, 2009 sept. 5, 2009 Mar.5, 2010
Table 7.1 provides a completeexampleof the paments npde under the swapfor one particulprset of f-monthLIBORrates. The tableshowsthe swapcash :ows fromthe perspectiveof Microsoft.'Note that the $1 million princijal is used only for the calculationof interestpayments. 'I'heprincipal itselfis not exchaged. Fof this reason it is iermed the notionalJrfncll/, or justthe notional. lf the principal were exchanyed at the end of the life of thr swap, the nature pf the deal would nt be changed in any way.The principal isthe samefor hoth$e fixedand Poatingpayments. Exchanging$1 millioh for $1 millp at the end of thelifeof the swapis a transactionthat wuld haveno Enancialvlue to either Microsoftor Intl. Table 7.2 Shows the cash Eowsin Table7.1 witha nal exchangeof principal added in. 'hisprvies an interestingway of viwing the swap. The casf flowsin the tlrd of thistable afe the cash iows froma longposition in a ioating-ratebond.The column cash:ows in thefourth column of thetableare the cash Qowsfroma short position in a fxed-ratebond. The table showsthat the swapcan be regarded as the exchange.f a bond for a ioating-ratebond. Microsoft,whose positio is describedby flxed-ratt Table 7.2, is longa ioating-ratebond and short a fixed-ratebond.Intelis longa flxedrate bond and short a :oating-ratebpnd.
Table 7.2 Cash iows (millions of dllars) from Table 7.1 when there is a final of principal. exchange Date
Mar. 5, 2007 Sept.5, 2007 Mar. 5, 2008 sept. 5, 2008 Mar. 5, 2009 Sept.5, 2009 Mar. 5, 21
Thischaracterizationof the cashiows in the swap helpsto explainwhy the qoating ratein the swapis set 6 months befortit is paid. On a qating-rate bond, interestis set at thebeginningof ie period to which it will apply and is paid at the end gen:rally perod. vanilla'' interest Thecalculatknof theCoytinrate p¥ts in a 0f the jnch as theone in Table7.2 reiects this. swaj pte ddplain
Usinjthe Swap to Transforma Liaility For Microsoft,the swapcouldbe used to trarnsfolpa ipatingrrat loaninto a Exed-rate that Microjofthas alranged to brrow $1 million at LIBOR plus loan.suppose points. (One basis point is one-hundredthqf 1%, so the rate is LIBOR 10basis AfterMicrosofthas enteredinto theswap, it has the ibllowing three sets plus cash of Pows: .1%.)
1. It pays LIBOR plus 0.1% ypi1spulde lenders. 2. It pceives LIBOR under th termsof the swap. 3. It pays 5% undei the termsof the swap. Thesethree sets of cash Powsnet 0ut to an interestrate payment of 5.1%. Thus, for Microsoft, the swapcould haveth efect of transformingborrowingsat a floatingrate of LIBOR plus l basispints irltoborrowingst a xed mte of 5.10:. For Intel, the swap could havethe efect of transforminga ed-rate loan into a ioating-rateloan. Supposethat Intel has a 3-year$1 million loan outstanding on whkh it pays5.29:. After it haj enterd intothe swap,it haS thefollgwingtkee sets f cashCows: 1. It pays 5.2% to its'outside lenders. 2. It pays LIBORunder the termsof the swap. 3. It receives5% under the termsof the swap. Thesethree sets of cash flws net 0ut to an interestrate paymentof LIBORplus 0.2% (or LIBOR plus 20 basis points). Thus,for Intel, the swap could Lavethe efect f transformingborrwings at a flxedrate of 5.2%into borrowingsat a foating rate of LIBORplus 20 basispoints. nesepotentialuses of the swap by Intel and Microsoft are illustratedin Figure7.2.
Usinj the Swap to Trapifvrm an Asset ,
.
'
swapscan also be used tl) transfoe the nature of an asset. ConsiderMicrosoftin our example.The swap could havetheefect of tmnsformingan asset earning a flxedrate f interystinto an asset earning a ioating rate of interest.suppse that Microscft owns $1 millin in bondsthat will provideinterestat 4.7%per annum over the next 3 years. Figure 7.2 Microsgft and Intel use the Swap to transfrm a liability. 5%
5.2%
Microsoft
Intel LIBOR
'
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151
Snmps Fijure 7.3
Microsoft and Intel use the swap to transforman asset. .:
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AfterMicrosoft has entered into the swap,it hasthe followingthreesrts of ashIlows: 1. It receives4.7% on the bonds. 2. It reives LIBORunder the termsof the swap. 3. It pays 5% under the terms of the swap. Thes threesets of cash llowsnet out to an interestrate lnfowof LIBORminus 30basis points. Thus, one pssible use of the swap for Microsoft is to transforman asset earnig 4.7% into an assetearning LIBORminus 30bas points. Next, consider Intel.Theswap could havethe efect of transfgrmingan asset earning that Intel a qoatingrate of interestinto an asset earning a xed rate of intepst. suppose yields minus 20 o f million basispoints.After that LIBOR investment it has $1 haja enteredinto the swap, it has the followingthree sets of cash Cols: 1. It receivesLIBORminus 20 basispints on its investment. 2. lt pays LIBORunderthe terms of the swap. 3. It t:ceives5% under the termsof the swap. Thesethree sets of cash fows net out to an interestrate infow of 4.8%. Thus, one possibleuse of the swapforIntelis to transforman asset earning LIBORminus 20 basis points into an asset earning 4.8%. These potential uses of the swap by Intel an Microsoft are illustratedin Figure 7.3.
Roleof Financial Intermediary Usuallytwo novnancial companies such as Intel and Microsoft do not get in touch directlyto arrange a swap in the way inicate in Figures7.2 and 7.3.Theyeach deal with a fnancial interpediary Such as a bank or other fnancial inqtitution. vanilla'' flxed-for-foatingswaps on US interestrates ale tpually structured so that the fnancial institutionearns about 3 or 4 basispoints (0.03.% or on a pair of ossettingtrnsactios. Figure7.4showswhat the role of theEnancialinstitutionmight bein the situationin Figure7.2.The Enancialinstitutionenters into tw oflietting swaptransactionswith 'tplain
Intl and Microsoft.Assllmingthat both companies honoy their obligations, the slancial institutionis certain to make a prot of 0.03% (3 basis points) pqr yeqr gmounts multiplid by the notional principl of $1 million. to $3, per year of 5.1%, as in for the 3-yearperiod. Microsoftends up borrowingat 5.115%(instead of at Figure7.2),and Intel ends up borrowingat LIBORplus 21.5bas points (instead plus points, in Figure7.2). LIBOR 20 basis as Figure7.5illustratejthe role of thefmqnial institutionin thesituationin Figure7.3. The srap is the sameas beforeand the nanial institutionis rtain to mke a proft of 3 basispoints if neither company defaults.Microsoftends up earning LIBORminus of LIBORminus30basispoints, as in Figure7.3),and Intel 315 bas points (instead of 4.8%, as in Figure7.3). ends up arning 4.785%(instead Note that in each case the fmancialinstitutionhas two separate contracts: one with Intel and the other with Microsoft.In mostinstances,Intelwill not even knowthat N'a fi-ncial institutionhas entered into an ofletting swapwith Microsoft,and vice versa. If one of the compales defanlts,the nancial institution still has to honor its agreementwith the other company. The 3-basis-pointspreadearned by the hnncial institutionis partly to compenste it for the risk that one of the two companies will dfault on the swappaymepts.
tls
*
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Market Makers In practice, it is untikelythat two companiesWillcontact a fnancialinstitutionat the same time and want to take opposite positions in exactly the same swap. For this reason,many largefnancial institutionsact as market makers for swaps,This means that they are prejared to enter into a swap without havingan ofsetting swap with anothercounterparty.2 Market makers'must carefullyquantify and hedgethe risks they are t>king.Bonds, forwardrate agreements,and interestrate futuresare examplesof the instrumentsthat can be used for hedgingby swap market makers.Table 7.3 shows quotesfor plain vanilla US dollarswapsthat might be posted by a marketmaker.3 As mentionedearlier, thebid-ofer spreadis 3 to 4 basispoints. The averageof thebidand ofer flxed rates is known as the swap rate. Th is shownin the al column of Table7.3. Considera new swap where the flxed rate equals the current swap rate. We can reasbnablyassume that the value of this swapis zero. (Whyelsewould a market maker choosebid-tder quotes centered on the swaprate'?)In Table7.2we sawthat a swapcan .
2 This is sometimesreferred to as warebousingswaps. 3 The standard swap in the United States is one where flxedpaymentsmade every 6 monts are exchanged f0r ioating LIBOR payments made every 3 months. In Table 7.l we assumed that 5zed and ioating rate should be almost exactlyte same in bot cases. paymentsare exchangedevery 6 monts. Te sxed
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153
Swaps .'....' .'
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Bid and oser flxedrates ili the.swapmarketand swap rates (percent per annum).
Table 7.3
Mturity
earsj
2 3 4
5 7 l
le characterized as ) efne:
Bid
Ofet
Sap rate
6.03 6.21 6.35 6.47 6.65 6.83
6.06 6.24 6:39 6.51 6.68 6.87
6.045 6.225 6.770 6.490 6.665 6.850
the dference betweena hed-rate bond and a ioating-ratebond. '
'
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We will use tls rewlt laterin the chapterwhen discussinghowthe LlB0R/Swap
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7.2
zero
DAYCOUNT ISSUES We discussedday count conventions in Section6.1 The day count conventions afecj paymentson a swap, and some of the numberscalculatedin te exampleswe havegiven do not exactlyrefect theseday count coventions. Consider,for example,the b-month LIBORpaymentsin Table7.1.Becauseit is a US money marketmte,6-monthLIBOR ij quoted on an actual/36 basis.Theflrstioating payment in Table 7.1,basedon the LIBOR rate of 4.294,is shown as $2.10million.Because there are 184daysbetween March.5, 2007,and September5, 2007,ij sjjoujtj s .
l
x
0.042x
1s4= millio $2.1467 360
In general, a LlBoR-based foating-mtecash ;ok on a swap payment dqteiscalculated whereL istheprincipal,R isthe relevant LIBORl'ate, and n isthe number asf'n/3, of days since the last payment date. The flxedrate that is paidin a swap transactionis similarlyquotd with a particular day count basisbeingspecifed. As a result, thehxedpayments maynot be exactlyeqgal on each paymentdate.TheExedrate ij usuallyquoted as actua1/365or 30/360.lt is not thereforedirectlycomparablewith LIBOR becauseit appliesto a fullyear. To make the rates approximatelycomparable, either the 6-mopthLIBORrate mnst be multiplied by 365/360or the flxedrate must be multipliedby 360/365. For clarity of exposition, we will ignoredaycount issuesin the calculationsin the rest of this chapter.
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CONFIRMATIONS
A cohhrmationis thelegalagreementunderlyinga swap nd issiped by representatives f the two parties. Thedraing of conhrmations hasbeenfacilitatedby the work of the InternationalSwapsand DerivativesAssociation(ISDA;www.isda.org) in NewYork. Thk organization has produced a number of Mastr Agreementsthat cnsist of clauses dehningin some detailthe terminologyused in swap agreements,what happensi the eventof defaultby either side, and so on. ln BusinessSnapshot7.1,weshow a possible extractfromthe conhlpatipn for the Swap Shown in Figure7.4betweenMicroft and here to be GoldmanSachs).Almost rtainly, the full institution(pssumed a snancial con6rmationwould state that the provisions f an ISDAMasterAgreementapply to the contmct. Theconhrmationspecihesthatthefollowingbusinessdayconventin is to beused and that the US calendardetermineswhichdays are businessdays and which days are holidays.This meansthat, if a payment date fallson a weekendor a US holiday,the The paymentis mpde on the next businessday.4 September5 2009 is a saiurday. -
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4 Anotherbusines (lay convention that is sometimes jpecifedis the modsed followisg business(lpy d ay convention following wllich when bnsiness that, convention, is the same as te expt the next business day fallsin a diserentmonth from the specifed day,the payment is made on the immediatelypreding businessday. Precedisg and modsed precedingbusiness(lay conventionsare dened analogously.
155
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exchangeof paymentsinthe swapbetweenMicrosoftandGoldmanSachsis penultimate therefore on Tuesday,September8, 2009(Monday,Srptembei'1, i Lbor day, a holiday).
7.4
THE COMPARATIVE-XbVANTAGE ARGUMENJ An explanation put fomard to expln the popularityof swapsconcerps comparativeadvantages.Considerthe use gf an interestrate Swap to transform a liability.Somecompanies, it is argued,havea comparativeadvantagewhen borrowing in flxed-flte markess, whereasother coinpanies have a comparative advatage in ' ie Maikets To btain a new loan it makestense for ioating-ra to go to the market where it has a comparativeadvantage.Asa result, the companymayborrow fued when it wants qoating,or borrow:oating when it wants fxed. The swapis used to tmnsforma fxed-rateloan into a :oating-rateloan,and vice versa. Suppose that two companies, AAACOP and BBBCorp,both wis to bortow $10lnillion for 5 years and havebeen osered the mtesshown in Table 7.4.AAACOP has a AAAcrebit rating; BBBCO/ has a BBBcredit rating.s Weassumethat BBBCOP wantsto borrowat a xed rate of interest,wereas AAACOP wantsto borrowat a qoatingrate of interestlinkedto 6-monthLIBOR. Becauseit has a worse credit rking in both flxed than AAACorp,BBjCO/ pays a higherrate of intetestthan AAACO/ and :oating markets. A keyfeatureof the rates olered to AAACOPand BBBCOP is that the diferen the two flxedrates is greater thanthediferencebtwten the two ioating rates. betwetn BBBCO/ pays 1.2% more than AAACOP in flxed-ratemafketsand only 0.7% more AAACOP in qoating-ratemarkets. BBBCOP appears to have a comparative than in qoating-ratemarkets, FhereasAAACP ppefj 'tJ h'fe a clparative advantage in fued-ratemarkets. It is this apparentanotaly that can lead to a swap advantage beingnegotiated. AAcorp borrowsflxed-ratefunds at 4% per annum. BBBCOP ioating-ratefunds at LIBOR plus 0.6% per annum.Theythen enter into a borrows agriement to ensure that AAcorp ends up with qoatinj-rate funds and swap BBBCOP ends up with sxed-rate funds. 'commonly
'
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,
.
Table 7-4 Borrowingrates that provide a basisfor the
argl3ment. comparative-advantage Fixed
AAACOP BBBCOP
4.0% 5.2%
Fkatiq
6-monthLIBOR 0.1% 6-monthLIBOR 4. 0.6% -
5 The credit ratings assignedto companiesb s&P (inorder of decreang
creditworthiness)are AAA, AA, A, BBB,BB,B, CCC,CC, and C. Te correspondingratings assiped by Moo's are Aaa, Aa, , Baa, Ba, B, Caa,Ca, and C, respectively. . 6 Note that BBBCOT'S comparativeadvantagein jloatinprate markets does not implythat BBBCO!Ppays lessthan Acorp in this market. lt means that te extra amount that BBBCOPpays over theamount paid byAAACOP isless in tMsmarket. One of my studentssummarizedthesituationas follows:sAco!.p pays morelessin hed-rate markets; BBBCOPpay less more in hoatinpratemarkets.'' 2
156
CAPTER
7
Swapagreementbetweep Acorp and BBBCOrPFhen'rates in Table 7.4
Figre 7.6 ZPPIY.
.
corp
4%
4,35% BBBCO/
LIBOR+ 9.6%
LIBOR
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To understand ilow this swap might work, we flrst asjume that AAAcorp ahd BBBCOrPget in touch with each other directly.The sort pf swaptheymightnegotiate is shownin Fipre 7.6.Tls is similarto or examplein Fipre 7.2.AAACOP ayreesto pay BBBCOPinterestat f-monthLIBORon $1 million.In return, BBBCOPagrees to rate of 4.35%per annum on $1 million. pay AAACOrPinterejt at a flxed j AAACOrPhas three sets of nterest rate cash gows:
1. It pays 4% per annum to outsidelenders. 2. It receives'4.35%per annum from BBBCOr/. 3. It pays LIBORto BBBCorp. 'fhe net efect of the three cash flowsis thayAAACOP ppyj LIBOR minus 0.35%per annllm.Th is 0.25%per annum lesstan it would pay if it went directlyto :oatingrate markets. BBBCOPalso has threesets of interestrate cash :ows: per annum to outside lenders. 1.lt pays LIBOR + 2. It receivesLIBORfrop AAcorp. 3. It pays 4.35%per annum to AAACOP. .6
The net 'eflkctof the yhreecash flowsis that BBBCOP pays 4.95%per annum. This is 0.25% per annum lesstha it would pay if it went irectlyto hxed-ratemarkets. In th exapple, the swap hasbeenstructuredso that the net gain to both sidesisthe This need tlot be the case. However, the total apparent gain fromthis same, of type interestrate swap arrangement is alwaysa b, wherea isthe diferen between the interestrates facingthe two companies in flxed-ratemarkets, and b is the diference betweenthe interestrates facingthe two companies in Qoating-ratemarkets. In this so that the total gainis jy n. case,a 1.2% and b = If AAACOPand BBBCOP did not dealdirectlywith each other and used a snancial institution,an arrangement such as that shown in Figure 7.7 might result. (This is similart the example in Figute7.4.) In this case, AAACOP ends up borrowingat .25%.
-
.7%,
=
.
'
.
,
.
Fi! ure 7.7 SwapagreementbetweenAAACOrPand BBBCOPwhen rates in Table 7.4 applyand a snancial intermediaryis involved. 4.33%
4% AAACOrP
LIBOR
4.37% Financial jnstiutjon
BBBCOrP LIOR
LIBOR+ 0.6%
157
Swaps .
'
.
LIBOR minus 0.330:, BBBCOI'Pends up borrowingat 4.970:, and the snancial institutionearns a spriad of 4 basispoints per year. The gain to AAkcorp is institutionis 0.04% The the gain to BBBCOP is 0.230:,.and the gain to the snancial total gain to all three parties is 0.50%as before. .23%;
.
Criticijm of the Arjument The pmparative-advantagearNment we havejustoutlined for exjlaining the attracof interestrate swaps is opin to question. Whyin Table7,4shouldthb spreads tiveness the rates ofered tp AAACPrPand BBBCOrPbe diferentin flxedand fbating between market hs beenin ettence for some tipe, w: might Now that the markets? expect thesetypesof diferencesto hayebeenaybitraged away. reasonably The reason that spreaddiferentialj appear to exist due to the nature of the available to companies in flxedand lloatingmarkets. 4.0% and 5.2% contracts rates (e.g., ratesavailable to AAcorp and BBBCOPin flxed-ratemarkets are s-year companies which flxed-rate b onds). TheLIBOR the can issues-year therates at AAACOPand BBBC:P available LIBOR markds in Qoating-ratt 0.60/q r ates + to and m arket, thelenderusually hasthe opportunity to areGmonthrates. In theCoating-rate If months. the creditworthiness of AAACOP or 6 the ioating rates evry review BBBCOP has declined,the lenderhasthe option of increasingthe spreadover LIBOR thlt is charged. In extremecircumstances,the lendercan refuse to roll over the loan at all.The providers of flxed-ratesnancing do not havethe option to change the tenps of 7 the loan in this way. The spreadsbdweenthe raies ofered to AAACOP and BBBCOP are a re:ection of the extent to which BBBCOP is more likelythan AAAC'OITto default.During the next 6 months, thereis very littlechance thateithr AAACOIPor BBBCOPwill default.As we look further ahead, the probability of a defaultby a company with a retaiively1ow credit rating (suchas BBBCOP) is liableto increasefasterthan the probability of a defaultby a company with a relativelyhigh credit rating (such This is as AAACOPI. whythe spreadbetweenthe s-year rates is greater thanthespreadbetweenthe6-month 'skp
'he
.1%
-
rates.
After negotiating a Coating-rateloan at LIBOR + 0,6% and entering into the swap loan at 4.970:. The shownin Figure 7.7, BBBCOTPappears to obtain a sxed-rate argumentsjustpresenttd show that thisis not rtally tht cast. In practice,the rate paid is 4.97% only if BBBCOP can continue to borrowlloating-ratefunds at a spreadof 0.6% over LIBOR. If, for example,the cyeditrating of BBBCOPdeclinesso that the Coating-rteloanis rolled over at LIBOR+ 1.60:,the rate paid by BBBCOP increases to 5.97%.The mafket eipects that BBBcorp'sspread over GmonthLIBORwill on BBBCOP'S expectedaverageborrowingfatewhen it averageriseduringthe swap'slife. 4.97%. enters into the swap is therefolt greater than The swapin Figure7.7 locksin LIBOR 0.33%for AAACOP for the whole of the next 5 years, not just for the next 6 months.TMs appearsto be a good deal for AAACorp.The dwnside is thpt it is bearingthe risk of defaultby the fmancial institution,If it borrowedioating-rattfvndsin tht usual way, it would not be bearing this risk. '
,
-
.
7 If theqoatinprateloansare structure so that te sprea over LIBOR is guarantee in avance ofchanges in credit rating, tbe spread iferentials isappear.
regardless
158
CHAPTER 7
7.s
THE NATURi oF SwAp RATES
' At this stage it is appropriateto examinethenatureof swaj rates and the relationship betweenswapandLIBORmarkets.We. explainedinSection4.1thatLIBORisthe rate of interestat which AA-ratedbanks'borrw for periods betwqn 1 and 12 months from otherbanks.Also,as indicatedinTable7.3,a swaprate i the averageof (a)theflxedrate that a swapmarket maker is prepared to pay in'exchangefor receivingLIBOR(itsbid rate)and (b)the flxedfate thatit is prepared to rceive in return for payinz LIBOR(its ofer rate). LikeLIBOR rates, swap mtes are not risk-fpe lendingrates. However,theyare close to risk-free. A nancial institutioncan earn the s-year swaprate on a certain principal by doingthe following: 1. Lendthe principal for theflrst6 months to a AAborrower nd thenielendit for successive6 month periods to otherAA borroWir; d 2 Enter into a swap to echange the LIBOR inome for the s-year swaprat. '
This showsthat the s-year swaprate is Anintepst rate with a credit risk cbrrespopding where situation l consecutive Gmonth LIBOR loans to XA companies are to the made.Sinlilarlythe 7-yearswaprate is an interestrate with a tredit lisk corresponding to the situation where 14 consecutive 6-monthLIBOR loahs to AA companiesare made.Swaprates of other maturitiescan be interpretedanalogously. AAborrowingmtes,It is muc more Notethat s-year swap rates are lessthan s-year attractiveto lend money for successive6-monthperiods to borrowerswho are always AA at the beginningof the periods tn t lendit to 0ne borrowerfor the whle 5 years whena11we can be sure of is that the borroweris AA at the beginningof the 5 years.
7.6
DETERMININGLIBOR/SWAPZERO RATCS '
.
Weexplained in Section4.1that derivativetraderstendto use LIBORrates as a proxies for risk-free rates when valuing derivatives.0ne problem with LIBOR mtes is that directobservations are possible only for maturities out to 12months. As describedin Seqtion6.3, one way of extending the LIBORzero curve beyond12monthsis to use EurodollaLftures. Typically Eurodollar futuresare uwd to produce a LIBOR zero' curveout to 2 years-and sometimesout to as far as 5 jears. Tradersten use swap ratesto extend the LIBORzero curve further.The resulting zero cqrve is sometimes referredto as the LIBOR zero curve and sometimesas the swapzero curve. To avoid any confusion, we will refer to it as the LlBoR/swapzero curve. Wewill now describe h0w swaprates are used in the determinationof the LlBoR/swap zero curve. The rst point to note is that the valle of a newly issuedqoating-mtebond that to its principal value (orpar value) when the pays Gmonth LIBORis alwaysequal ' 8 LlBoR/swapzero curve is used f0f discounting.The reasn is that the bond provides a rate of interestof LIBOR,and LIBORis the discountrate. ne intereston thebond exactlymatches te discountrate, and as a result the bond is faifl priced at par. In equation (7.1), we showedthat for a newlyissuedswapwhere theflxedrate equals = the swap rate, :x Wehavejust argued that equals the notional principal. It '
.
' The same is of course tnx of a newly issuedimndthat pays l-month, 3-monyh, or lz-month LIBOR.
159 rates thereforedefmea that 'flx also etuals the swap'snotional principal. swap follows example, of yield For fromthe in Table7.3, bonds. rates we can dedu swap set par LIBOR/swap par yield k 6.945%,the 3-yearLIBOR/swap par yield is tat the z-year and so on.9 6.225%, section4.5 sikowed howthebpotstrapmethod c'anbeusedto determinetheTreasury zerocurye fromTreajurybond prices.It can be used withswaprates in a similarway to extendthe LIBOR/swap zero curve.
'
fxample7.1 supposethat the f-month,lz-month,and l8-monthLIBOR/swapzero rates have beendeterminedas 49:, 4.50:,and4.8%with cotinuous compounding andthat the z-year swaprate (fora swapwhere paments are pade semiannually)is 5%. Thk 5% swaprat means that a bond with a pricipal of $1 and semiannual zero rate, cpuponof 5% per annumsellsfor par. It followsthat,if R isthe z-year then 1.5 1. -.04x0.5 2.5: + 2 j:-.48x + 102.5: = l j + 2 j:-.45x -2;
.
.
.
Solvingthk, we obtain R = 4.953%.(Notethat this calculation is simplifkdin that it doesnot takethe swap'sdaycount conventions nd holidaycalendars into 7.2.) account.seesection
VALUATION OF INTERESTRATE SWAPS Wenow move on to discussthe valuation of interestrate swaps.Aninterestrate swapis worth zero, or close to zero, when it is flrstinitiated.Afterit has beenin existen for some time, its value may' Vcomepositive or negative. There are two valuation approaches.Thefirstregards the swapas thediferencebetweentwo bonds;the second regardsit as a portfolio of FlGs.
Valuation in Terms of Bond Prices Principalpayments are n'pt exchanged i an interestrate swap.Howevr, as illustrated in Table 7.2, we can assumethat principal paymentsare both receivedand paid at the end of th swapwithout changing its value. By doingtls, we fnd that, fromthe point of viewof the ioating-ratt payer, a swapcan be regarded as a longposition in a ftxedrate bond and a short positioqin a ioating-mtebond, so that PCWaP :flx
-2
=
wherehwap is the value of the swap, is the value of the Eoang-rate bond (corresponding to paymentsthat. are made), and 'flx is the value of the ftxed-ratebond fromthe point of view of (correspondingto payments that are received). similarly, the fixed-rte payer, a swap is a long position in a :oting-rate bond and a short .
9 nalystsfrequentlyinterpclate betweenswap rates beforecalculatingthezerocurve,so tat theyhave swap intervals.For example,forthedatain Table7.3 the2.5-yearswap pte would rates'for mamrities at and so cn. the 7.5-year swap rate would be assllmedto be be assumedto be -month
.135%;
.9%;
160
CHAPTER 7
positionin a flxed-ratebond, so that the value of the swap is '
>
.
The value of the flxedrate bond, 'fyx,can bedeterminedas describedin Setion 4.4.To notionai principal va1uethe qoating-ratebond, we note that the bond is wol'th th afttr an interestpayment. Tiis is becase at thistimethe bod is a immediatily deal''were the borrowerpays LIBORfor each subsequentarual period. Supposethat the notional principalis L, the next exchange of paymentsis at time !*, was determinedat the last andte qoatingpayment tht will bemadeat timet%(whkh ::: L after the date)is Immtdiately payment V. as justexplaied.It follws payment = tht immediatelybeforethe payment Bg L + V. ThePoating-ratebond can therefore beregarde as an instrumentprovidinza sigle cash ipw of L + k' at time!#. Discountingthis,the mlueof the ioatipg-ratebondtodayis Sfair
'te
CL+
-r*!#
wherer' is the LlBoR/swapzero rate for a matcrity of t'. Exampi 7.2
Supposethat a hnancialinstitutionhas agreed to pay f-month LIBO: and semiannualcompounding) on a notional principal receive8% per annum twit.h of $10 million. The swaj has a remaining life of 1.25 years. The LI0R rates with continuous compounding for 3-month,g-month,and ls-month maturities are 10:, 1.50:, and 110:, rspectivelyk The f-month LIBOR rlte at the last semiannualcompounding). Paymentdate was 10.i% (with The calculations for valuinj the swap in terms bonds are summarized in Table 7.5. The flxed-ratebond has cush iows of 4, 4, and 14 on the three paymentdates.ne disount factorsfor these cash fbws are, respectively, x0.25 '5x().7 e e e-.1 -of
-.1
-.11x1.25
,
,
and are shownin the fourth column of Table7.5.Te tableshowsthat the value of the flxed-ratebond (inmillins of dollars)is 98.238. In th example,L $100million, k*= $5.1million, and x 0.102x 1 = valued b ond be Coating-rate as thoughit producesa t 0.25, so that the can cash ;ow of $105.1million in 3 months.The table shows that the value of the ioating bond (inmillions of dollars)is 105.100x 0.9753 102.505. .5
=
=
#
'
=
Table 7.5
Valuinpa swapin termsof bonds(Tmillions).Here,Sf:xis flxed-rate bond undellyingthe swap,and Bg is Poating-ratebond underlying the Swap.
Time
&x cas
0.25 0.7j 1.25 Total:
'
/pw
4.0 4.0 104.0
Sg casjkw
105.100
Presezt valae
factor
Presezt valae Qxcasbsow
0.9753 0.9243 0.8715
3.901 3.$97 90.640
102.505
98.238
102.505
Discoazt
cas
/pw
11
Swaps The value of the swapis the diferencibetweenthe tw0 bond prkes: Pswap=
18.238102.505 -
-4.267
=
-4.267
milliondollap. If the hnancialinstitutionhd ben in the o/posite psition of paying flxed and receiving qoating,the valui of the swap would-be +$4.267 millidn.Note that thess calculations do not take account of day count conventions and holiday calendars.
or
'
y,
in Terms of FRs valuation A swap.can be characterized as portfolioof forwardrate agreements. onsider the swapbetwee Microsft and Intel,inFigure7.1.The swapis a 3-yeardeal entered into on March5, 27, with seniiannual layments.Thefirstexchangeof paymentsisknown at the timethe swapis negotiated. The oiher 5veexchangescan be regarded as Flts. The exchange on Varch 5, 28, is an FRA where interest ai 5% is ektlinge'd ff interestat the f-month rate obseryed in the market on September5, 27; the exchange on September5, 208, is an FRAwhereinterestat 5% is exchangedfoi interestat the f-lonth rate observed in the marketon Mamh5, 2008,.and so on. As shon at the end of Section4.7,an FRA can be valued by assllmingthat forwafd interestrates are realized. Because it notlng more than a portfolio of forwardrate agreements,a plain vanilla interest rate swap can also be valued by making the assupptionthat forwardinterestmtes are realized.The procedureis aj followj: '
1. Usethe LIB0R/sFap zero curve to calculte fomard rates for each of theLIBOR wilt determine swapcash zows. rates that 2. Calculateswap ash qowson the assumption that theLIBORmteswillequal the forwardrates. the LlBok/swap zero turvel 3. Discpunttheseswap cash Qows(using to btaintht vale. swap fxample 7.3 Consideragain the situation in Example7.2. Under the terms of the swap, a nancialinstitutionhas agreed to pay 6-mnth LIBORand receive8% per annum (withsemiannualcompounding)on a noonal prinipal of $1 million.Theswap has a remaining lifeof 1.25years..rfheLIBOR rates with continuous compounding for 3-month, g-month,and ls-monthmaturities are 1t%, 1.5%, and 11%, respectivly.The (-fnonthLIBOR rate at thelast payment date was 10.2%(with copkou semiannual
gj frst fowof thetableshowsthe Thecalculationsare summarizedinTable7.. Yhe cashqowsthatwillbe exchnged in 3months. Thesehavealreadybeendetermined. = Theflxedrate of8% willleadto acashiniowof 1 x $4million.Tlle x will ago) of 3 10.2% cash months lead to a was set qoatinkrate ouow of (wltkh miltion. Thesecondrow ofthetableshowsthe cash fows 100x 0.102x = $5.1 months assumingthat forwardrates are realized.Tlle that will excgnged in 9 casiiin:ow is $4.0million as before.To calculte the cash outQow,we must jrst the forwardrate corresponding t the period between3 and 9 months. calculate jy
.
.8
.5
.be
.5
162
CHAPTER 7
Tabl, 7..6 Valuinggwap in Mrls f FRAS ($mt.llions).Floting cash Egwsare calculatdby assumingtha,tforwardrates willbe realized. Time
Fltiq
Fixed cassow
0.25
casbjkw
4. 4. 4
'
0.75 12
.
Net
Discouzt
casbjkw
factr
-5.1
-1.1
-5,572
.
valae of zet casnow
0.975j 0.9243
-1.572
Z-6.051
pcKcrlf
-2.051
=1.073 .. -1.407
.8715
-1.787
Total:
-4.267
' .
'
'
'
.
.
thisis From equation (4.5), .15 x 0.75 .5 -
.1
x 0.25
= 0.1075 .
Fith continuops compoundig. Fromequatin (4.4), the fomard rate or 10.75.% ' () 11.044/k with semiannualcompounding. The cash outqow istherefore becomes 1 x 0.11044x = $5.522million. The third rw similarly shows the cash Cowjthat will be exchanged in 15 months assumingthpt fprwardmtes are realized.The discountfactorsfor the three paylent dates are, resjectively,
(
.5
.
e
-0.l05x.y5
-.1 x0.25
C
,
1x 1.25
-.1
,
C
million.Thevaluesof Thepresentvalueof th exchangein threemonthsis FRAS corresponding the exchangesin9 months and 15months to the are respectivelf million, Thetotal value of the swap is million. and Thisis in agreementwith the value we calculatedin Exnmple7.2bydecomposing theswap into bonds. -$1.73
-$1.47
-$1.787
-$4.267
.
Thefed rate in an interestratr swap is chosen so that the swap is worth zero initially. This means that at the'outsetof a swap the sum of the values of the FRAS underlying the swap is zero. It does not mean that the value of each individualFRA is zero, In general,some FRAS will havepositive values whereasothers havenegativevalues. Considerthe FRAS u'nderlying the swap betweenMicrosoftand Intelin Figure7.1: Valueof FRA to Microsoft > when fomard interestrate > 5.0% when forwardinterestrate = 5.0% Valui of FRA to Microsoft Valueof FRA to Microsoft < 0 when forwardinterestrate < 5.0% =
Supposetat the term structure of inrest fates igupward-slopingat the tne the swap is negotiated. This means that the forwardinterestrates increaseas the maturity of the FRA increases.Sin the sum of the alues of the FRAS is zero, the forwardintrest rate must be lessthan 5.0%for the early payment datesand greater than 5.0%for the laterpayment dates. The value to Microsoft of the FRAS corresponding to early datesis thereforenegative, whereas the value of the FRAS corresponding to payment laterpayment dates is positive. lf the term structure of interestrates is downwardat te timethe swap is negotiatd, the revefseis true.Theimpactof the shape of sloping the term structure of iterest rates on the values of the forwardcontracts underlving a swapis summarized in Figure7.8. ' '
163
Swapt
Fijure 7.8 Valuingof forwarzrate agreementsunderlying a jwap as a functionof In (a)the term structure of intrrestrates is pwafd-slopingand we receive maturity. and we reive qoating;in (b)the term structure of flxed,or it is downward-sloping iFupward-slopingand rates and we receiveqoating,or it isdownward-goping interest We receivefued. Valueof fotward tontract
Matrity
Valueof fotward contract
Maturity
7.8 CURRkNCYSW4PS Anotherpopular type of swap is knownas a currcncyswap. In its simplestfon, th involvtsexchangingprincipal and interestpayments in one currency for principal and interestpayments in another. A currency swapagreement reqires the principal to be specifkdin eachof the two cufrendes. The principal amounts in each currncy are usually exchanged at the beginningand at the end of the life of the swap. Usuallythe principal amountsare chosento be approximatelyequivalentusing the exchangerate at the swap'sinitiation. Whenthey are exchanged at the end of the lifeof the swap,their values may be quite diferent.
164
CHAPTF 7 .
.
.
.
.
.
'
Fijuri 7.9 A currencysrap. Dollars6%
.
Britiqh putrcjuum
IBM
sterling5%
,
Illustratipn ' .
Cnsider a hypoihtical s-yarcurrencyswap agmementbetweenIBM and Mritish PetroleuM entered into on February1,2007.We supposethat IBM pays a flxedrate of intrest of 5%in sterlingand receivis a flxedrateof interestof 6%indollarsfromBritish Petroleum.Interestrate payments are mde once year and the principal alun are $18million Eand f 1 million. This is termedzhxed-for-jed currency swapbecausethe interestrafe in both currencies is flxed.The ywapis shownin Fkur 7.9. In'ially, the principalamounts :ow in iheopposite directiont te arrowsin Figure7.9.The interest paymevtsduripgthe life of the swap nd the fnal principal payment :or in the snme directionas the arrows. Thus, at the outset of the swap,'IBM pays $,18 million and receivesf 1 millin. Each year during the' life of the swap contract, IBM receives 0?$18 million)and pays f.5 million (= 5% of f 1 million).Atthe $1.08millio (= 6% end of the lifeof the swap,it pays a principal of f 1 million and receivesa principal of $18million. Tese cash Cowsare shown in Table 7.7. '
'
.
.
'
'
.
Use af a Currenc Swap to Transfprm Liabilitiesand Assets A swap such as the one justconsidered can be used tu transfgrmborrowingsin one
...
currencyto borrowingsin another. Supposethat IBM can issue $18million of USbonds at 6% interest.The swap hasthe efect of transformingthis dollar-denominated transactioninto one where IBM has borrowedf l million at 5% interest.The initial exchangeof principal converts the proce ds of the bond issue iomUS dllars to sterling.The subsequent exchangesin the swap havethe efect of swapping the interest and principal payments fromdollarsto sterling. The swap can aljo be used to transformthe nature of assets. Supposethat IBM can investf 1 million in the UK to yield5% per annum for the next 5 years, but feelsthat Table 7.7
the Us dqllaf will strengthr against sterling and prefers a Usldollarrdenominated The swap has the efect of transformiagthe UK ivestment into a investment. $12million investmentin the Us yielding6%.
. '''''
ComparatiyeAdvantage
'
Currencyswaps can be motivated by comparative advantaggkTp illustratethis, we cpnjider anothr hypotheticalexample. suppose the s-year hed-rateborrowipgcosts . i n dellar.s(UsD) GeneralElectricand Airways Us to an Australiandollars Qantas (AUD)are as shownin Tble 7.8.Thedatainthetablesuaast thatAujtrlap rates are higherthan UsD interestrates, and also thatGeneralElectrlci more'creditworth.ythan Airways,becauseit is ofeted a more favorablerate of interestin both currencis, Qantas From the vkwpoint bf a swap trader,the interestingaspect of Table7.8 is that the spreadsbetwventhe rates pd by Gelgal Eletrk and QAntas Airwaykin the twp marketsare not the same. Qantas Airwayspays 2% more than Qeperal Electrkin the Us dnllarlarket and only 0.4%mor,ethan GeneralElectricin the AUD parket. This situation is analogousto that in Table7.4.GeneralElectrkhas a comparative AirFays has a comparative advantage adkantagein the UsD markt? whereai Qantas 7.i, ete a, plaln vanilla interest rate swap waj in the AUD market. ln considered, we argued Shatcomparativeadvantagesare largelyillusory.Here we are the comparing rates ofkred in two diferent currencies,and it is mcre likelythpt the advantages are genuine.Onepossible source of comparativeadvantageiq comparative tax.GeneralElectric's positionmight be suchthat UsD borrowingskad to lowertaxes Airways'position might bethe on itkworldwide incop than AUD borrowings.Qantas shwp interest in Table 7.8 have been rates that th reverse.(Note that we assume adjustedto re:ect thesetypes of tax advantages.) We suppose that GeneralElectrk wants to borrow20 mlion AUD and Qantas Aimayswnts to borrow15n'lillionUsD and thatthe current exchangerate '(UsD per AUD)is This creates a perfect situatin for a curreny swap. GeneralElectric and Qants Aimays each borrow in the market where they have a comparative Airwaysborrows advantage;that is, GeneralElectric orrows UsD whereas Qantas AUD. They then use a currency Swap to transformGeneralElectric'sloan into an AUD loan and Qantas Airways'loan into a UsD loan. already the diferencebetweenthe UsD interestrates is 20/c,whereas mentioned, s the d'erence betweenthe AUDinterestrates is 0.4% Byanalor with theinterestrate 1.6%per annum. swapcase, we expect the total gain to all parties to be 2. Thereare many ways in which the swapcan be arranged. Figure7.l showsone way swapsmight be entered into with a fnancialinstitution.GeneralElectrk borrowsUsD and Qantas AirwaysborrowsAUD. The eflct of the swap is to transformthe UsD '''
'''''''
...g
.
.
1
'lje
'.-.
'...1
.
.
.
tF*
.''
'
.
'
.7-
'
.75.
.
.4
=
-
Table 7.8
Borrowingrates providing basisfor currency swap.
GeneralElectric Airways Qantas
USD.
XUD'
5.0% 7.0%
7.6% 8.0% ...
'
-
# Qotedrates havebeenadjusted to reiect thediferentiaiimpactof taxes.
interestmte of 5% perannumto a AUDinterestrate of 6.9%per annumfor General Electric. As a result, deneral Electricis0.7%per annumbetterofl-thanit would beif it exchangesan AUD loan at 8% per markets.. Similarly,Qantas directly AUD to went ' apnum for a UsD loan at 6.3% per anplm and ends up 0.7% per annum better oflthan it would be if it went directlyto UsD markets. The fnancial institution'zain' s 1.3%per annum on its UsD cajh IIOF!and loses1.1%per annum on itS AUD jows. If we ignorethe diferencebetWeenthe tw currencies,the fnancialinstitutionmakes a net gain of 0.2% per amplm. As predkted, the total gain to all parties is 1.6% per '
almllm
t
.
.
,
. Each year the fnancial instltutionmkes a gain of UsD 195,000 (= 1.3% of 15 million) and icurs a loss of AUD 220,000 (= 1.1% of 20 million). The fnancial institutioncan avoid any foreip exchangerisk by buyingAUD 220,000per atmumin the forwardmarket for each year of the life of the swp, thus lockingin a net gain
inUsD. it is possible to redesign the swp
so that the fmancialinstitutipnmakes a 0.2% spreadin UsD. Figures7.11 nd 7.12 present two altenmtives.Thesealternatives are unlikeljto be usedin practice because10theydo not leadto GeneralElectricand Qantas being free of foreign exchnge risk. ln Fipre 7.11, Qantas bears.some foreign risk becaudeit in exchange pays 1.1% per almum AUD and pays 5.2% per annum in UsD. ln Fipre 7.12, GeneralElectrkbearj some foreip exchangerisk btcauseit receives1.1% per annum in UsD and pays 8% per annuin in AUD.
7.9
VXLUAJION OF CURRENCYSW4PS Likeinterestrate swaps,fixed-for-flxed cprrencyswapscan be decomposedinto either the diferencebetweentwo bonds or a portfolio of forwardcontracts. Figure 7.1 1 Alternativearrangement for currency swap:Qantas Airwaysbears some foreignexchangerisk. USD5.2%
USD5.0%
General USD5.9%
Electric
Aj.o (.9%
Financial institpticn
Qantas
Ato
.9o
Airways
Aun g .
o
?
10usually it makes sense fcr the fmancialinstitutionto beartheforeignexchangerisk, becauseit is in the bestpcsiticn tc hedge the risk. '
Ssc>:
167
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Figufe 7.1 2 Alternativearrangemnt for currencyswap: General Electricbears soe exclnge foreign
'
risk.
USD6 1% . .
Genergl Electric
' DSD 5.9%
.
USD6.3%
Finarcial injtittion '
Aun g.ntj)
Qantas
. Aa
ffp Maykets Fijure 8.2
Pri
Ptoht frombuyipga Europeanput option on one shareof a stock.Option $7;strikeprice $70. '
=
=
Profit ($)
30
20
10 Terminal
0
stcckprice ($) 40
50
60
70
,
90
80
10
-7
in 3 months, and the price of an optiop to sell one shax is $7,Theinitl investmentis $700.Becaus the option is European,it will be exercised (mly if the stock priceis that the stpck priceis $55on thisdate.The below$70on the expiration date. supposr the tenm of the put option, invstor can buy 100shares for $55per share and, sell the same shares for $70to realize a gain of $15per share, or $1,500.(Agai, transactionscosts are ipored.) When the $7(%initialcost of the option is takeninto account,the investr's net prot is $800.There is no guarantee that the investorwill makea gain. If the fnal stock pri is above$70,the put option expiresworthless,and the investorloses$700.Figure8.2 shows the way in whkh the investor'sprot or loss on an option to sell one share varies with the terminalstock pricein thls example. 'under
EarlyExercise As alfeady mentioned, exthange-traded stock options are generally Americanrather thanEuropean.Thismeans thattheinvestorintheforegoingexampleswouldnot haveto waituntil the expirationdatebeforeexercisingthe option. Wewillseethatthereare some when it is optimal to exercise merican options beforetheexpirationdate, circumstances
8.2
OPTION POSITIONS Thereare two sidesto every option contract. 0n one side is theinvestorwho hastaken thelongposition(i.e.,hasboughtthe option). 0n the other side istheinvestorwho has taken a short position (i.e.,has sold or writtez the option). The writer of an option receivescash up front,but haspotentialliabilitieslater.Thewriter's prot or lossisthe reverseof that for the purchaser of the option. Figures8.3and 8.4showthe vaziationof the profh or loss with the final stock prke for writers of the options considered in Figures8.1and 8.2.
182
CHAPTER 8 Figure 8.3 Proft fromwriting a Europeancall option on one share of a stock. Option pri = $!; strikt prict = $100. ' Prcfit ($)
' 5
110
0
:0
80
90
l0
120
130
'
.
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prjx
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There are four typesof option positions: 1. A long position in a call option 2. A long position in a pt option 3. A short position in a call option 4. A short position in a put option It is often useful to chracterize a Europeanoption in terms of its payof to the purchaserof the option. The initial cojf of the optiop is then not includedin the calculation,lf K is the strike prke and ST is the Enalprice of the underlying asset, the Fijure 8.4 Prost from.writing a Europeanput option on one share of a stock. Option = pri $7;strikepri $70. =
Pifit
($)
7 40
50
Terminal stcck price ($)
6 70
-10
-20
-30
80
90
100
Mechanics p/
O>lpAt
183
Maybets
Fijure 8.5 Pyofl from positions in Europeanoptions:(a)1ongca11 ; (%short ca'11; = #; pric of assetat mgturity = &. (c)long put; (d)short put. Strikepri Payoff
ypnff
l
.r
KJT
K
K
( Payoff
Payoff
H
iz K
K
(c)
payof from a long position in a Europeancall optin is
maxtkr -
K,
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Tlt iefkctsthe factthat the option Fillbe exercisedif S > K and will not be exerced if %K. The payof to the hlder of a short position in the European call optionis kr
-
maxti
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K,
)
=
mintf
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-
in a Eurpean put optionis
The payofl-to the holderpf a long psitin
maxtr &, ) -
and the payof from a short position in a European put ption is -
Stock b ptlons themain exchanges Mosttradingin stock options is n exchangej.In the Unitedstates th Philadelphiastock arethe Chkago Bpard OptionsVxhange(www.cboe.com), Exchange American the Exchange(mvw.phlx.com), the (www.xmex.com), stock Exchange (www.iseoptipns.com), and the Boston Options International securities Optionstrade on more than 1, Exchange(www.bostonoptions.com). diferent right gives 100 buy sell the holderthe shares One contract the specied to at or stocks. strikepri. This contract size is convenientbecausethe Shares themslvesare nofmally traded in lots of l.
Foreign Cqrren
Options
Most currency options tradingis now in the over-the-countermarket,but thereis some exchangetrading. The major exchange for trading foreip currency opons in the United states is the Philadelphia Stock Exchange. It ofers bo'th Eurpean and merican contractk on a variety of diferent currencies. The ie of one cntract dependson the currency. For example, in the case of the Britishpound, one contract givesthe holderthe right to buy or sell231,250;in the case of the Japanese yen, one contmct gives the holder the right to buy or sell 6.25million yen. Foreigncurrency optionscontfacts are discussedfurthtr in Chapter 15.
Index Options -
.
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Manydiferentindexoptions currently tradethroughoutthe world in boththe over-thecounter market and the exchange-trded parket. The mpst popularexchange-tmded contractsin the United states are those on the s&P 500 Index(sPX),the s&P 1 and the Dow Jones IndustrialIndex Index (0EX), the Nasdaq 10 Index (NDXI, (DJX). A11of these trade on the ChicagoBoard Optins Exchange. Most of the contractsare European. An exption is the OEX contmct on the s&P 10, which is American.One contract i8 usually to bu) or sell 100 mes the indexat the specihed strike price. settlement is always in cash, l'ather than by deliveringthe portfolio underlyingthe indx. Consider,for example, one call contract on the s&P 1 with a strike price of 980.If it is exercisedwhen the value of theindexis992,the writer of the $1,2. Tls cash payment isbasedon the contractpays theholder(992 980)x l index value at the end of the day 9n which exercise instructionsare issued.(Not surpringly, investorsusually wait until the end of a daybeforeissuingtheseinstructionsa)Index options are discussedfurtherin Chapter15. -
=
Futures Options When an xchange trades a particular futurescontmct it often alsotrades optionson thqt contract. A futuresoption normally maturesjustbeforethe deliveryperiodin the futurescontract. When a call option is exrcised the holderacquiresfromthe writer a longpositionin the underlying futurescontract plus a cash amopnt equal to the exss of the futurespri over the strikeprice.Wh en a put option is exercised,the holder underlying short position futurescontract plus a cash amount equal to acquiresa in the the excess of the strike prke over the futuresprke. Futpres options contracts are discussedfurtherin Chapter16. ,
185
MechanicsPJOptions Markets
8.4
SPECIFICATIONOF STOCKOPTIONS In the rest of th chapter, we willfocuson stockoptions. As alreadymentioned, an stock option in theUnitedStatesis an American-styleppiioncontract exchange-traded to buy or sell 1 shares of the stock. Detailsof the contract-the xpiration date,the strikeprice, whpt happenskhen dividepds are declared,howlargea positioninvestrs and specihed by the exchangq. canhold, so on-are
ExpirationDates One of the itemsused' to describea stok option is the month in wllich the expiration d>te ocurs. Thus, a January call tradingon lBM is a call optioh on IBM with an expirationdate in January. The precise expiration date is the Saturdayimmediately followingthe thirdFlidayof the expirationmonth. The lastdayon wllichoptionj trade is the third Friday of the expiration month. An investorwith a long position in an optionnormally has until 4:30p.m.CentralTime on that Flidayto instructa brokerto exercisethe option. The brokerthin has until 1:59 p.m. the next day to complete the PaPerworknotifying.the exchange that exqrciseis to take place. Stockoptions are on a Januars Februars pr Marchcycl. The Januarycycleconsists of the months of January,April,July,and October.The Februarycycleconsistsof the months of February,May,Aupst, and November.The March cycle consts of the monthsof March,June,September,andDember. If the expirationdateforthe cprret month has npt yet beenreached,optionstradewithexpirationdatesinthecurrent month the followingmonth, and the next twomonths in the cycle.If the expiratlondate of the currentmonth has passed, options tradewith expiration datesin the next month, the next-but-onemonth, andthe neyt twomonths of the expirationcycle.For example,IBM is on a Januarycyle. At the beginninyof January,options are tradedwith expiration datesin January,February,April,and July;at the nd of January,theyare tradedwith expirationdatesin February,March,April,and July;at thebeginningof May,theyare traded with expiration datesin May,June, July, and October;and so on. When one optionreachesexpiration, tradingin anotheris started. Longer-termoptions, knownas equity anticipation seculities),also tmde on about 5 stocks in the LEAPS(long-term United Stayes.These have expiration dates up to 39 months into the future. The expirationdatesfor LEAPSon stocks are alwaysin January. ,
StrikqPrices
'
.
The exchangenormally chooses the stlike pricesat wllichoptions can be written so that theyare spaced $2.50,$5,or $1 apart. Tpically the spacing is $2.50when the stock price is between$5 and $25,$5 when the stock prke is between$25and $2, and $1 for stock prices above $200.As will bt explained shortly, stock splits and stock dividendscan lead to nnstandard strike prices. Whena new expiration dateis introduced,the two or threestrikepricesclosejt to the ctkrrentstock price are usuallyselectedbythe exchange.If the stockprice moves outside the range dened by thehighestand loweststrikeprice, tradingis usually introducedin an option with a new strike price. b illustrate'theserules, suppose that the stock price is $84 when trading beginsin the October options. Call and put options would probablysrstbe ofered withstrike prices of $80,$85,and $9. If the stock price rose
186
CHAPTER 8
above$90,'it is likelythat a strikeprice of $95would be ofered;if it fellbelow$80,it is likelythat a strike priceof $75would e ofkred;and so on. .
Terminology
'
. . . .. ..
,
F0r any given asset at any given time,many diferentoptioncpntracts may be trading. Considera jtock that has four expiration datej akd ve strike prkes. If call and put optionstrade with every expiration date and every strike price, there are a total of 40diflkrehtcontracts. A11optionsof the sametype (calls or ppts) are referpd to as an 0ne calk class,whereas example, IBMputs ar ahother class. IBM are optiosclts. For An optioz series consists of a11tht optiovsof a givenclss with tht sameexpiratin date nd strikeprice.In other words, an option seriesrefers to a particularcontract that is traded.For example, IBM 70Octobercalls would cnstitute an option series. Optionsare referred to as itl te pm/ley, at te moty, or out of te moty. If v is the stockpri and K is the strike pri, a call option ij i the moneywhen S > K, at the moneywhen v = K, and out of the money when S < A put optionis in the money whenv s K, at the money when v = K, and out of the money when v > K. Clearly,an optionwill be exercisedonly when it is in the money. ln the absence of transactions costs,an in-the-moneyoption will alwaysbe exercisedon theexpiration date if it has notbeen exerised previously. The iztritic vl/lfc of an option is defned as the maximum of zero and the value the ojtion would haveifit were exercisedimmediately. For a call option,theintrinsicvalue is thereforemaxt, #, ). For a put option,it is maxtf S, ). An in-the-money Ameiican option must be worth at least as much as its intrinsicvalue becausethe holdercan realize the intrinsicvalue by exerising immediately. Oftenit is optimalfor the holderof an in-the-mope Amrican option to wait rather than exerdse immediately.The ption is thl jid ti hgyetin vcl>. The total value of an option can be thought of as the sum of its intrinsicvalue and its time value. 1
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FLEXOptlpns The Chicgo Board OptionsExchangeofkrs FLEX (shortfor iexible) options on equitiesand equity indices.These are optionswhere the traders on the :oor of the exchangeagree to nonstandard terms.Thesenonstandard terms can involvea strike price or an expiration date that is diferent from what is usually ofered by the exchange.lt can alsoinvolvethe option beingEuropeanrpther thanAmerican.FLEX options are an attempt by option exchanges to regain businessfrom the over-thel contracts) for FLEX countermarkets. The exchangespecifkqa lninimum size (e.g., tion trades. op
Dividendsand Stock Splits The early over-the-cognteroptionswere dividendprotected.lf a company declareda cash'dividend,the strike prke f0r options on the company's Stock was feduced on the ex-dividendday by the amount of the dividend.Exchange-tradedoptilms are not usuallyadjustedfor cash dividends.ln other words, when a cash dividendours, thereare nb adjustmentsto the termsof the option contrat. An exceptionisspmetimes madefor large cash dividends(seeBusinessSnapshot8.1). Exchange-tradedoptionsare adjustedfor stock splits. A stock split occurs when the
Mechans
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For a written ngkid put option,it is the greater f 1. A total of lo: of the procerds of the sale plus 20% of the underlying shar prkelessthe mbupt, if pny;by which the optionis dut of the money 2. A toyalof 1% of the optionproceedsplus lth of the exerise prke
.
T.he 2% in the preceding calculations is repld by l 504 for optigns on a broadly badedstock indexbecausea stck indexis usually less volatile than the price of an individualstock.
fxample8.3 '
.
Aninvestorwrites four naked call option contracts on a stock.Tht option price is $5 the strikeprice is $4 and the stpckprke is $38.Becausethe optionis $2out of the money,the rst calculation jives $
.
,
400x (5+
x 38 2) = $4,240
.2
-
The secondcalculationgives 400x (5+
.1
x 38) $3,520 =
Thi initialmargin requirement is therefore$4,20. Note that, if tht ojtion had been a put, it would be $2in the moneyan7 the margin requifementwould be
40x (5+
.2
y 38) $5,040 =
ln bot,hcases the proeds of the sale, $2,00, can be used to form part of the margin account. A calculationsimilr to theinitialmarginltaltulation (butwith.the current parket price replacingthe proeds of sale)is.repeate every day.Fundscan be withdrawn fromthe marginaccout when the calculationindicatesthat the marn required is lessthan the current balancein the margin aount. Whenthe calculationindicatesthat a signicantlygreater mrgin is required, a margin call will be made.
Other Rules ln Chapter1, we willexamineoption tradingstratgiessuchas coveredcalls, proteciive puts, spreads,combination, straddles,and strangles.ne CBOEhas special nlles foi determiningthemargin requiremets when thcsetradingstrategiesare usz. Theseare describedin the CBOE Margin Manual,which is available on the CBOE website (www.cboe.com). As an example of the rules, consideran investorwho writes a coverebcall.This is written call option when the sharesthat might have to be deliveredaw already a owned.Coveredcalls art far less risky than naked calls, becausethe worst that can happen is that the invejtoris required to sell shams already owned at belowtheir marketvalue. No marginis required on te written optn. However,the investorcan Kj, rather than the usual borfow an amount equal to on the stock position. .smint5',
.55',
192
CHAPTER 8
8.8 THE OPTIONS CLEARINGCORPORATION The Optitns ClearingCorporation(0CC) perfofins much the jame function for option! markets as the clea#nghpase does for futures markets (seeChapter 2). It guaranteesthat options writers will fulflltheir bligptionsunder the terms f options contractsan keepsa record of a11longan'd short ppsitions. The0CC has a number of members,and a11option tradesmust be cleared througha member. If a brokeris not itself a member of an exchange's OCC, it must anange to clear its tmdes with a member.Membersare required t have4 certain minimum amount.of capital and to contributeto a specialfund that can be used if any membei defaultson an option obligation. When purchasing an option, the buyermust pay fof it in fullby the morningof the ext businessday.The funds are depositedwith the OCC. ne writer of the option ' maintains a margin aount with a broker,as describedearlier.l brokermaintains a marginaccountwith the 0CC member that clears itstrades.The 0CC member in turn maintainsa margin accountwith the OCC. 'l'he
Exercisingan Option whenan investornotifks a bpker to exercisean option, the brokerin turn notifks the Occ member that clears its traLes. member thrn places an exemiseorder with 'this
the OCC.Th 0CC randomly selectsa member with an outstanding short position in the same opyiqn, The member, using a produre established in advance,selects a particularinvstor who has written the option. It the option is a call this investoris required io sellstock at the strike pri. If it is a put, th investoris requiredto buy stockat the strikeprice. Theinvestoyissaidto be assigzed.Whenan option is exemised, the open interestz goes downby one. At the expiration of the option, a11in-the-moheyoptions shouldbe exercisedunless the transactionqcosts are so high as to wipe out the payof from the option. Some brokerswill automaticallyexerciseoptions for a client at expirationwhen it is in theif client'sinterestto do so. Manyexchanges lsohaverules for exercisingoptions that are in the money at expiration. ,
8.9
REGULATIQN Optionsmarketsare regulated in a number of diferent ways.Both the exchangeand its OptionsClearingCorporationhaverules governingthebehaviorof traders.In addition, thereare both federl and state regulatory authorities.In general,options marketshave demonstrateda willingnessto regulate themselves.nerehavebeenno major scandals or defaultsby 0CC members. Investorscan havea highlevelof conflden'in the way the larket is run. The Securitiesand ExchangeCommissin is responsible for regulating options markets in stocks, stock indis, crrncies, and bonds at the federallevel. The 1The margin requirements
escribe in the previoussection are the minimumreqtliremcntsspecifk by the A OCC. brokeragehousemay reqtlire higher mar/ns from its clients. However, it cannot require lower brokeragehousesdo not allow theil'retail clients to write uncoveredoptions at ali. margins.some
Mechanicsp/ OptionsMaykets
.
j93
CommopityFutures TradingCommissioni responsible f0r regulating markets f0r optionson futures. The major options markets are in the states ot Illinois and New York. These states actively enfor their own laws on unacptable trading
practies.
8.10 TxATls Determinivgthe tax implicationsof'option strategies can be trkky, and an investor who is in doubt about this shouidconsuit a tax specialis. ln the Unhed States,the generalrule is that (unless the taxpayeris a piofessional trader)gains and lossesfrom the tradingof stock optionsare taxed as cajital gains or losses.The way that capital gainsand lossesare txed in the Uniteb Stateswas discussedin Section2.9. For both the holder and the writer of a stockoption, a gain or lossis recopize when(a)the option expires unexercised or (b)the option positionis closed out; If the option is exercised,the gain or lossfromthe optionis rdlled into theposition takenin thestock and recognizedwhen the stockpsitin is closed out. Fr txample, when a call option is exercised, the party with a long position is deemedto have the stockat price. used Tls is then the strike price plus the call ak a basis for calculating this gain when eventually sold. Similarly,the party with the loss the stockis party's or shortcall positionis deemedto have sold the stock at the styikepri plus the call pri. When a put option is exercised, the seller of the option is deemedto llave bopghtthe stockfor the strikeplice lessthe original pt price and the pmchaser of the option is deemedt havesoldthe stockfor the strikeprke lessthe ori/nal put 'purchased
price.
Wash Sale Rulq is the wash salenzle. To One tax consideration in optiontrding in the United states understand this rule, ima/ne an investorwho buys a stockwhen the price is $6 and . plansto k.eep it for thelongterm.If the stockprice dropsto $40,theinvestormight be it, so that the $2 lossis temptedio sellthe stockand then immediatelyrepurchase ' realized for tax purposes. T prevent tltis sori of thing,the tax authoritieshaveruled thatwhen the repurchase is within 30daysof the sale(i.e.,etween 30daysbeforethe saleand 30daysafter the sale),any losson the saleis not deductible.Thedisallowan alsoapplies where, within the 61-daf piriod, the taxpayerenters into an option or similarcontract to acquirethe stock. Thus,sellinga stockat a loss and buyinga call optionwithin a 3-day periodwill leadto thelossbeingdisallowed.Thewash salerule doesnot apply if the taxpayeris a dealerin stocksor securitiesandthe lossis sustained in the ordinarycourse of businejs.
Constructive Sales Prior to 1997, if a United States taxpayershorte a securitywhileholdinga long positionin a substantiallyidenticalsecurity, no gain or loss was recognizeduntil the short positionwas closed out. Tltismeans that short positions could be psedto defer recognitionof a gain fortax purposes.Thesituationwas changed bytheTaxReliefAct
194
CHAPTER 8 q'
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1. Enters into a short sale of the same or substantipllyidentkal property 2, J ptrrs into a futures or forwardcontractto deliverthe same or substantially ldenticalproperty 3 Enters into 0ne or morepositionsthat eliminatesubstantially a11of the loss and opportunityfor gain *
lt should be noted that transactionsreducing oplythe risk oflossor only the opportunity for gain should not result in constructivesales. Thereforean investorholdinga long positionin a stock can buyin-the-money put options on tlie stock without triggeringa contructive sale. Tax practitionerssometimes use options to minimize tax costs or maximize tax 8.2), Tax autilties in many jurisdictions beneEts (seeBusinesssnppshot have proposedlegislationdesignedto combat the use of derivativesfor tax purposes,Before enteringinto any tax-motivatedtransaction,a corporate treasureror privateindividual shouldexplore in detailh0wthe stnlcture could be unwound in the event of legislative changeand hoF costly tllis processcould be,
STOCKOPTIQNS?ANDCONVERTIBLES 8.11 WAFRANTS?EMPLOYEE Warrants are options issuedby a fnancialinstitutionor novnancial coporation. For example,a fnancialinstitutionmight issueput warrants on one million ounces of gold and then proceedto create a market for the warrants. To exercisethe warrant, the institution.A common use of warrants by a holder would contact the snancial novnancial corporationis at the time of a bond issue.The corporation issues call warrantson its own stock and then attachesthem to the bondissueto make it more
attractiveto investors.
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motivatethem to act in the best interestsof the company's shareholders.They are usuallyat the money at the timeof issue.Theyare now a cost on theincomestatement of the companyin most colmtries,making thema lessattractiveform of compensation than they used to be. (Sei BusinessSnapshot8.3.)
196
CHAPTER 8 often simplyl'efel'l'ed to as conyertibles,412bonds issuedby a that can be conveld into equityat certain times using a predetermined company ratio. They are thereforebonds with an imbeded call option on the exchange Ompany's stock. isthat a predetetOnefeatumof warrants, employeestockoptigns,and ionyeftibles Qptions minednumber ofoptionsart issued.Bycontrast, thenumber ofexchange-traded (As morepeopletradea particularoptionseiks, the outstandingis not predetermined. Warrantsissuedby a company on its own numberpf optionsoutstandingincyeases.). convertibles are diferent from ekchgnge-traded stock,eecutive stock options,and Whentheseinstnlmets i n anotherimportant are exrrcised,the company way. options issutsmoreshares ofits on stock andsellsthemto theoptionholderf0rthe strikeprice. The xerciseof theihstrumentstherefre leadsto an increasein the number of shares of thecompany'sstok that are outstanding.By contrast, when an exchahge-traikdcall optiopis exercisedit party with the short positionbnysin the market shari! that have alreadybeenissuedand sellsthemto thepartywith thelongpositionforthe strike prke. The company whose stck underlks theoptionis not involvedin any way. Conyertible
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4.12 OVR-TMq-CQUNTER OPJIONS MARKETS -
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Most of this chapter has focusedon exchange-tradedoptionsmarkets.t'heiver-theimportarntsincetheearly1980jand countermarketfor optionshasbecomeincreasingly market. As explainedin Chapter1, in the overis n0w largerthan the ex hang the-countermarket, fmanial institutkns,corporatetreasurers,andfundmanggerstrade overthephope.Thereis a wide rahge of assetsunderlyingtheoptions.Over-the-counter options on foreign exchange and interestrates are particularlypopular. The chief Potentialdisadvantageof the over-te-countermarketisthat optionwriter may default. Thismeansthat thepurchaseris subjectto some credit risk. In an attempt to overcome requiringcounterparties to post thisdisadvyntage,market partkipantsare increasingly 2.4. collateral.Thiswas discussidin section The instrumentstraded in the over-the-countermarket are ofteh stnlctured by fmancialinstitutionjto meet the predseneeds of their clients. Smetimes tis involves choosingexercisedates,strikepris, and contract sizesthat are diferent fromthose traded by the exchange. In other cases the structure of the option is diferent from standardcallsapd puts. The pption is ten referred to as an exotic option. Chapter24 of exotic optins. describesa number of diferent iypes .
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-traded
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Thereare twotypesof options:callsand puts.A call optiongivestheholderthe right to buy the underlying asset for a certain prke by a certain datt. A put optiongivesthe holderthe right to sell the underlying assetby a certn datefor a rtain prke. There are fourpossiblepositionsin optidnsmarkets:a longpositionin a call, a short position in a call, a longpositionin a put, and a short positionin a put. Takinga short position in an option is known as writing it. Optiops are currently traded on stocks,stock indices,foreip currencies, futnrescpntracts,and other assets. An exchangemust pecify the termsof te optiopcontracts it trades.In particular,it must specify the size f the contract, thepreciseexpirationtime, and the strikeprice.In
Mechans
197
OJO>foxs Maykets
the UnitedStatesone stockoptioncontractgivs the holderthe right to buy or sell 100 shares.The expiration of a stockoption contractis 10:59p.m. CentralTime on the SaturdayimmediatelyfollowingthethirdFridayof the expkation menth. ptiops with severaldiferent epifation months trade at any given time.Strikeipricesare at $212, $5, strike pri intervals,depending the stockprice.'I'he is generallyfairlyclose or $10 on to the stock price when tradingin an optionbegins. The terms Qf a stock optionare not normally adjsted for cashdividends.However, they are adjustedfor stock dividends,stock splits, and rights issues.'I'he aim of th adjustme is to keep the positions of both the writer ad the buyerof a contract
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Most option exhanges use marketmakers.A mrket makeris an individualwho is to quote both a bid pri (atwhich he or she is prepard to buy)and an oflbr prepared which he or sheis prepared to sell).Marketmakers improvetheliquidityof the pri (at market and ensure that there is never any delayin executin/ market orders. ney make a prgfh fromthediferen betweentheirbid and oflkrpris (known themselves as theirbid-ofer spread). The exchangehasrulesspecifyipguppey limitsfor the bidrofer sPread. Writers of options have pdential liabilitiesand are required to maintain marns With their brokers.If it is n0t a memberof the OptionsClearingCorporation,tke brokerwill paintaina margin aeztount wit,ha fll'mthat is a member.ThisEl'mwill in turnmaintaina marginaount with the OptionsClearingCrporation. ne Options ClearingCorporationis responsiblefor keepinga record of all outstandingcontracts, exerciseorders, and so on. handling N0t all options are tradtd on exchanges.Manyoptionsare tradedby phone in tht over-the-counter market.An advantage of over-the-counteroptios is that theycan be tailoredby a fnancial institutionto met the particular needs of a corporatetreasurer Or fund man ger.
and Problems (Answers in SolutionsManual) Questions 8.1. An investorbufs a Europeanput on a share ff $3.Thestockprke is $42and the strike ' price is $40.Under what circumstancesdoes the investormake a prtt? Under what circumstanceswill the option be exercised? Drawa diagramshowingthe variation of the ai investor'sprofh with #hestock pli the maturityof the option. '
8.2. An investorsells a Europeancall on a share f0r $4.The stockpli is $47and ihestrike price is $50.Under what cirolmtances does the investormake a proht? Under what circumstanceswill the option be exercised?Drawa diagramshowing the variation of the investor'sproft with the stock priceat the maturityof the option.
198
CHAPTER 8
8.3.An investorsellsa Europiancall option with strikepriceof K and maturity T and buysa putwith the same strikepri and maturity. Describe the investor'sposition. 8.4. Explainwhy brokersrequire margins whenclients writeoptions but not when they buy options. 8.5. X stockoption is on a February,May,Xugust,and Novembercycle.Whatoptionstrade on (a)April 1 and (b)May 32 8.6. X comjany declaresa 2-for-lstocksplit.Explainh0wthe termschange for a call option with a strikepriceof $60. '
Stock optios issuedby cmpany are diferentfromregular exchange-traded 8.7. options company's stockbecaujethey can afect 4hecapital structure of the tall on the Simployee
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company.''Explin this statement.
8.8.A corporatetreasureris desiging a hedgingprograni involvingforeigncurrecy What are the pros and cops of using (a)the Philaelpia market for trading? over-the-counter
89
options. stockExchangeand (b)the
that a Europeancall option to buya share for $1. suppose
costs $5.00and ishelduntil of option what circumstanceswill make a proft? Under holder the the maturity.Under what ircumstanceswilltht option be exercised?Draw a diagramillustratinghowthe prot from a long position in the option dependson the stock pri at maturity of the option.
8.1. Sppose that a European put option to sell a sharefor $6 costs $8 and is held until maturity.Under Fhat circumstanceswill the sellerof the option' (thepprty with the short position)make a prot? Under what circumstans will the optionbe exercised?Draw a diagramillustratinghow the prot frm a short position in the option dependson the stockprice at maturity of the option. 8.l1. Describethe terminalvalue of the followingportfolio: a newly entereb-into longforward contract n an asset and a longposition in a Europeanput option on the asset with the samematurity as the fomard contract and a strike pri thatis equal to the fma-rd pri that theEuropeanput optionhasthe of the asset at the timethe portfolio is set up. show Fith the same Strike pri and maturity. samevalue as a Europeancall option 8.12. X traderbuys a call option with a strike price of $.45and a put option with a strike price call costs $3and the put costs $4.Draw of $4. Both options havethe same maturity. of a diagramshowing the variation the trader'sprot with the asset price. rfhe
8 13 Explainwhy an Americanoption is alwaysworth at leastas much as a Europeanoption
on the same asset with the same strike price and exercisedate. 8.14, Explainwhy an Amerkan option is alwaysworth at leastas much as its intrinsicvalue.
8.15. Explaincarefullythe diferencebetweenwriting a put optionand buyinga call option. 8.16. Thetreasurerof a corporgtion istryinzto choosebetweenoptionsand forwaidcontracts to hedgethe corporation's foreignexchangerisk. Discussthe advantages and disadvantages of each. -
8.17.Consideran exchange-tradedcall optioncontract to buy5 shares with a strikeprice of $4 and maiurity in 4 months. Explain howthe termsof the option contract changewhen thereis: (a)a lFa stock dividend;(b)a l% cashdividend;and (c)a 4-for-1 stock split. 8.18.$$Ifmost of the call optionson a stock are in the money,it islikelythat thestockprice has risen rapidly in the last fewmonths.'' Discuss this statement.
199
Mechanics:./' Options Maykets
8.19.Whatis the efect of an unexpectedcashdividendon (a)a calloption price and (b)a put opyionprice? 8.20.Optionson GeneralMotorsstockare on a March,June,september, andDecembercycle. What options trade on (a)March1, (b)June3, and (c)August52 8.21,Explainwhy the market maker's bid-fer spreadrepresentsa real cost to options invtstors. 8.22.A UnitedStatesinvestorwrites ve naked call option cotracts. The option plice is $3.50, the strike price is $6.0, ad th stock pri is $57.00,What is the initial marcin rtquirement? '
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AssignmentQuestions 8.23. Tht priceof a stnckis S4. The price of a l-yearEuropeanput optionron thestockwith a strikeprice of $30is quoted as $7and the priceof a l-yearEuropeancall option on the stockwith a strikepriceof S5 is quotedas $5.Supposethat an investorbuys 100shares, shorts l call options, and buys l put options.Draw a diagramillustratinghowthe investor'sprost of lossvarits with the stock priceover the next year. How doesyour answer changeif the investorbuys l shares, shorts200call options, and buys200put options? 8.24. $$Ifa ompay does not do better than its cpppetitors but the stock market goes up, makes no senst.'' Discussthis executivesdo very well from their stock options. viewpoint.Can you think of alternatives to the usual employeestock option plan that take the viewpoint into account. 8.25.Use DerivaGemto calculate the value of an Americanput option on a non-dividendpayingstockwhen the stock price is $30,tllestrikeprice is 532,the risk-free rate is 50/:, thevolatility is 3%, and thetimeto maturity is 1.5 years. (Choost lBinomialAmerican'' type'' and 50 timesteps.) forthe (a) Whatis tht ption's intrinsicvalue? (b) Whatis the option's time value? (c) What would a time value of zero ipdicate?Whatis the value of an option with zero time value? (d) Usinga trial and error approach, calculatehow1owthe stock price would haveto be for the time v'alue of the optionito be zero. 8.26.On July2, 24, Microsoftsuprised the marketby announcing a $3dividend.ne exdividenddate was November17, 2004,and the payment date :as Dember 2, 24, Its stock pr ice at the time was about $28.It also changed the terms of its employeestock optionsso that each exercisepri was adjusteddownwardto 53.00 . x Closingpri Predividendexerciseprice clcsingprice rfhis
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Numberof shares predividend x
Closingprice Closingprke S3. -
closing price cf a share of Microsoft tdclosingPrice'' means the odal NASDAQ tlie comrnonstockon the last'tradingday before ex-dividenddati. Evaluatethis adjustment.Compareit with the System ustd by exchangesto adjustfor extraordinary dividends ts BusinessSnapshot8.1).
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Amount of Future Dividends Dividends havethe eflkct of reducing the stock prke on the ex-dividendbate. is good value of options and put options. The news for the bad ews for the value of call aniicijated negativelyrelated of of size call therefore the optionis future to value a an and the value of a put option is positivelyrelated to the size of an anticipated dividend, 'rhis
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In thiq hapter wewill make.assumptionssitnilarto thosemadefor derivingforward andfuturespricesin Chapter5. We assume that there are jome marketpartkipants, suchas largei vestmrnt banks,for which the followingstatementsare true: 1. Thereare no transactionscosts. 2. A11tyadingprohts (netof tradinglosses)are subjt to the same tax rate. 3. Borroing and lendingare possibleat the risk-free interestrte. We assumethat thesemarket.partkipantsare prepard io take advantage f arbitrage opportunitiesas they are. s discussedin Chapters1 an 5, this means that any availablearbitrage opportunities disappearvery quickly.For the purposes of our analysis,it is thrrefcrereasonabk to assume that there art no arbitragt opportunities. We will use the followingnotation: h : Currentstok price #: Strike price o? option T: Time to expiration of option s'r : Stockprice on the expiration date r: Continuouslycompounded risk-free rate of interestfor an investmentmaturinj in tipe T Valueof Amerkan call option to buy one share Valueof Americanput option to sellone share Valueof Euiopeap call option to buy one shap selt p: Valueof European put option to one share
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It jhould be noted that r is the nominal rate of intrest, not the real rate of interest.We can assume that r > Otherwise,a risk-freeinvestmentwould provide no advantages overcash. (Indeed,if r < 0, cajh would be preferableto a risk-free investment.) .
9.3
UPPERAND LOWERBOUNDS FOR OPTION PRICES In this sectionwr deriveupper and lowerboundsfor option pris. Theseboundsdo not dependon any particular assumpons about the factorsmentioned in Section9.1 price is abovethe upper bound or belowthe lowerbound, (exceptr > ). If an then there are prohtable opportunities f0r arbitrajeurs. 'option
Upper Bounds or Epropean all option givesthe holderthe right to buy one shareof a n merkan certaln pri. No matter what hajpens, he option can never be worth more stockfor a siock. than the Hene, the stockprke an upper bound to the optiop price:
c %k and C %h If theserelationships were not true, an arbitrageurcould easilymake a rklesq proft by buyingthe stockand selling the iall option..
206
CHAPTER 9
An Amerkan or Europeanpu4option givestheholdertheright to sellone shareof a stockfor #. No matter howlowthestockpri bcomes, theoptioncan neverbe wortli more than K. Hence, . p %K and P %K Fdr Europeanoptins, we knowthat at maturity theoptio: cannot e worth more than K. It followsthat it cannot be worth more than the preqent value of K todaj: '
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20 18e
Ke
-
=
3.71
or $3.71.Considerthe situation where the Europeap call prke is $3j, whkh is less thanthe theoreticglminimllm of $3.71.An arbitrageur an short the stock andbuythe callto provide a cash iniow of $20.00 $3.00 $17.. If investedfor 1 year at lb perannum, the $17.00grows to 17 '1 $1879. At the end of the year, the option If the stock price is greater than $18., the arbitrageur exercises$heoption for exkires. $18., closes 0ut the short position, and akesa prtt of =
-
=
.
$18.79 $18k -
=
$0.79
If the stock price is lessthan $18., the stock is boughtin the market and the short positionis closed out. The arbitrageur then makes an eve greater prtdt. For example, if the stock price is $17., the arbitrageur's profh is
$lt.00 $18.79 -
=
$1,79
For a more formal argumenkwe consider the fllowing two portfolios: one Europeancall optionplus an amountof cash equal to Jbrt/lfp :: one share Porblio
.
:
Ke-rT
In portfolio A, the cash, if it is investedat the risk-freeinterestrate, will grow to K in timeT. If Sz > #, the call optionisexercisedat maturityandportfolio A is worth Sz. If s'r < #, the call optionexpiresworthless and the portfolio is worth #. Hence,at time T, portfolio A is worth
maxts'r,
'l
A is alwaysworth as much as, and PortfolioB is worth Sz at time T. Hence,jortfolio
j07
Pyopeytie p./ StockO>lip?l:
canbe worth more than, portfolio B at the option's maturity. It followsthat in the absenc:of arbitrage oppoftuitij this ust lsobi tfe tod. Hence, '
-rT c + Ke 7 h
.
0r
ck S
Ke-rT
-
Becausethe worst that can happento a cll option isthat it expireswrthless, its valu: cannotbe aegative.This means that k and therefore c k maxtk
-rT
Ke
-
,
g)
(j j) .
Example 9.1
Considera European call option on a non-dividend-payingstockwhen the stock pri is $51,the strikeprict is $50,thetimeto maturity is 6 months, and the riskand freerate of interestis 12%per annum. In thiscase, h = 51, K = 5, T = -/r Fromequation (9.1), a lowerboundforthe option price is S Ke or r -.12x.5 = 51 5: $39j .5
.12.
=
-
,
-
.
Lower Bound for European Pqts on Non-Dividend-payingStocks For a European jut option on a non-dividend-payingstock, a lowerbound for the priceis -rr
y:
-
.
sg
Again,we srstconsider a numericalexampleand thenlook at a niore formalargument. Supposethat k $37,K $40,r = 5% per annum, and T = years. In this case, .5
=
=
K
-rT
-
s
=
4g:-.5x.5
-
)
=
$2,.g)
Considerthe situation where the Europen put price is $1., which is lessthan the minimumof $2.1. An arbitrageur can borrow$38.00foi 6 months to buy theoretical and the stock.At the end of the6 months, the arbitrageur will be required boththe put '5X'5 $3s?96 If thestockpriceisbelow$4. the arbitrageurexercises torepay 38: theoption to sellthe stock for $4., lepays the loan, and makes a prost of =
,
.
$40.00$3.96 $1.04 -
=
If the stock price is greater than $40., the arbitrageur discardsthe option, sellsthe stock,and repays the loan for an even gfeater proft. For example, if the stock priceis the arbitrageur'sprost is $42.00,
$42.0j-$38.96 $3.04 =
For a more fprmalargument, we consider the followingtw0 portfolios: PorlfblioC: one European put option plus one share ' Porolio D1 an amount of cash equal to ke -r
If ST < K, then the option in portfolio C is exercisedat option maturity, and the portfoliobecomesworth K. If ST > #, then the put option expiresworthless,and the
208
CHAPTER portfoliois worth
ST
at thistime.Hen, portfolio C is worth
maxts'z, & in time T; Assumingthe cash is investedat the rk-free interestraye, poitfolio D is worthK in time T. Hen, portfolioC is alwayjworth as much as, and can sometimes beworth more than, portfolio D in time T. It followsthat in the absenceof arbitrage opportunitiesporyfolio C must be worth at leastas much as portfolioD today.Hen, '
r
.
.
7 Ke
S , p+
or
'-rT p k Ke
-rT
h
-
Becausethe wol'stthat can happe to a put option is thatit expiresworthless, its value cannot be neyative, Tilismeans that > p maxtf:
-rT
y j)
-
(pz;
,
.
fxample9.2 Considera Europeanput ption on a non-dividend-payingstock when the stock prkeis $38,thi' strike pfice is $40,the time to maturity is 3 months, and the risk-freerate of interestis 1 ea per ann m In tilis case s' = 38, K = 4, From equation (9.2), ' T = 0.25,and r = a kwerboundfor the option price .
.1.
is Ke-rT
9.4
-
S, or
.
4: -.
jxg.zj
-
?g
.sj-gj
PUT-CALL FARITY We now deriveal importantrelationshipbetwernp and c. Considerthe followingtwo portfoliosthat wereused in the previous section: Porlfolio :1 : one Europeancall option plus an amountof cash equal to Porblio C: one Europeanput option plus one share
Both are worth
Ke-rT
maxts'z,#)
at expirationof th op.ons. Becausethe options are European,theycannot be exeyised prior to the expiration date.The portfolios must thereforehpveidenticalvalues today. Tbs means that . -rT (9.3) c + Ke = p + s' '
Til relationship is knownaspat-call parity. It showsthatthe value of a Europeancall with a certain strike price and exercisedate can be deducedfrom the value of a Europeanput with the same strikeprice and exercisedate,and vi versa. If equation (9.3)does not hold,there are arbitrageopportunities. Sppposethat the stockprice is $31, the strike pri is $30,the rk-free interestrate is 1% per annum, the price of a 3-month European call option is $3, and the price of a three-month Europeanput option is $2.25.In this case, ?o-.lx>l2 c + Ke-rz ? + =
=
$?224 .
209
Propeyties0.JStock O>ffp,
and
* '
r
p+
=
k
2.25+ 31 = $33.15
PortfolioC is overpricedrelativeto portfolio A.An rbitrageur can buythe securitiesin A and skort the securitks in porttolio C. strategyinvolvesbuyingthe call Portfolio siock, geerating a positive cash;ow of andshorting bth the put and the 'fhe
.
'
'
,
,
-3 4: 2. .25
..h
31 = $30.25*
wheninveitedat the risk-ffeeinterestrate, tilis amount grows to upfront. x9.25 30.25:.1
$3j g2
=
.
in 3 monthsklf thestock pri at expirationof the optiopis greater than $30,the call willbe exercised;and if it is lessthqn $30,the ptlt willbe cxercised.In either case,the ar bitrageurends up buyingone share for $3 Tls sharecan be used to closeout the short ppsition. The net proft is therefore .
$31.92 $3.
=
=
$1.02
For ap Alternativetpatimr.pppose that the call pri
ln this case,
c + Ke
and
p+
=
h
3 + 3: =
is $3and the put pri is $1.
$32.26
=
1+ 31 = $32.00
PortfolioAis overpricedrelativeto portfolio n aritrajeurcan short the securitiesin A and buythe ecuritiesin portfolio C to lockin a proft. The strategyinvolves portfolio .
Table 9.2 Arbitrageopportlmiykswhep plzy-call paritydoesnot hold.Stock = = price $31;interestrate 1%; call prke $3.Bothput and all havea strikeprice of $3 and 3 months to maturity.
Three-month put price
=
$2.25
Three-month put price = $1
cffpnn0w: Buycallf0r $3 short put to realize $2.25 short the stock to realize $31 Invest $30.25for 3 months
Xcfbn nw:
Actioz frl mozt. !/ Sz > 3:
Actioz frl morts Tkz> 3: Call exercised:sell stock for $3
.?
Receive$31 frominvestment Exercisecall to buy stock for $30 Net profh = $1.02 Actioz frl mozts (JS < 3: Receive$31 frominvestment Put exercised:buy stock f0r $3 Net proft $1.02 .2
.?
.2
=
Borrow$29for 3 months Shortcall to realize $3 Buyput for $1 Bu the stockfpr $31 .?
Use $29.73to repayloan Net proft $0.27 =
Actioz rl mos (fS < 3: Exerciseput ttj sell stock for $3 .?
Use $29.73to repayloan Net proft = $0.27
211
Pyopeytiesp./q Stock O>/ftm.
American Options PutQa11parity holdsonlyfor European options.However,it is possible t derivesome resultsfor American cptin prices. lt can be shown(seeProblem9.18)that, when there are no dividends, . h K %C P %h Ke (9.4) .,y
-
-
-
'
'
'
.
.
.
Exalnple 9.3
'
6
'
' call option on a non-dividend-payingstockwith strikeprit $20.00 An Amrican and maturity in 5 months is worth $1.50.Supposethat the current stockpri is $19.00and the risk-free interestrate is 1% per annllm.From equation (9.4), we
haVC
'
.g
19 20 %C P %19 2: -
-
or
1k P
-
-
,
j x j/ j 2,
C k 0.18
showingthat P C lies between$1. and $0.18.With C at $1.50,P must lie between$1.68and $2.50.In other words, upper andlowerboundsfor thepri of an Americanput with the samestrikeprice and expiration datr as theAmerican call are $2.50and $1.68. -
9.5
STQCK EARLYEXERCISE: CALLSON A NON-DIVIDEND-PAYINC Thissectio demonstratesthatit is neveroptimal to exercisean Americancall optionon stock beforethe expiration date. a. non-dividend-paying To illustratethe general nature of the arpment, consider an Americancall option on non-dividend-pqyingstock with 1 month to expimtionwhen the stock pri is $5 and a the strikeprice is $40.The option is deepin the monry, and theinvestorwho owns the optionmight well be temptedto txtrce it immediately.However,if the investorplans to hold the stockobtaihed by exercisingthe option for more than 1 month, thisis not the best strategy.A bettercourse of action is to.keepthe option and txtrcise it at the endgf the month. The $40strikeprice isthenpaid out 1month latrr thanit would beif thr option were exercised immediately, jo that interesiis earned on the $4 for 1month. Because the stockpays no dividends,no incumefromthe stock is Sacriflced. A fufther advantageof waiting rather than exercisingimmediatelyis that thereis some c'han (howeverremote) that th stockjri will fall below$4 in 1 month. In this case, the inyestor will not exercisein 1 month and willbe glad that the decisionto exerciseeqrly was nt taken! This argument showsthat there are no advantagesto exercisingearly if the investor plansto krepthe stockfor the remaininglift of the option(1month,in thiscase). What if the investorthinksthe stock is currently overprid and is wondering whether to exercisethe option and sellthe stock?ln this case, the investoris betterofl-selling the option than exercising it.1 The option will be bought by anotherinvestorwho does want to hold the stock. Suchinvestorsmust exist: othemise the current stock prke would not be $50.The prke obtained fo: the option will be grtater than its intrinsic value of $1, fcr the reasons mentioned earlier. '
j
.
..
,
'
As an alternative stratqy, te investcr can keepthe option and shcl't the stcck to lock in a etter proft than $l.
1t1
CHAPTER 9
Variationof priceof an American or Europeancall option on a nondividend-payingstock withthe stockpri, k.
Figure 9.3 Call option
price
z # 2 .
2
# #
z
# z
z #
#
#
#
z e
K
price,So stock
'
For a more formalarpment, we can use equation(9.1): ck h
-rT
Ke
-
Becausethe owner f an Americancallhas all the exemiseopportunitiesopen to the ownerof the corresponding Europeancall, we must have Htnce, Givenr
Ck
k
-
-rT
Ke
#. lf it were optimal to ezerce early, C would it followsthat C > eqal s's K. Wededucethat it can never be pmal to exerciseearly. Figure9.3 shows the genel'alwayin which'the call pri varits with h. It indicates thatthe call priceis alwaysaboveitsintrinsicvalue of maxtk K, ).As r or T or the increases,the line relating the call pri to the stock pri moves in the volatility indicatedhy the arrows (i.e.,farther awayfromthe intrinsicvalue). direction To summarize, there fe two reasons an Americancall on a non-diyidend-paying stockshouldnot be exercisedearly. Onerilates to the injurancethat it provides.A call of the stockitself,in eflct insuresthe holder againstthe option,when held lnstead price strike price.Oncethe optionhas beenexercisedand the fallingbelowthe stock strikepricehas heenexchangedfor the stock price, thisinsuran vanishes. The other reasonconcerns the time value of money.Frop the perspectiveof te option holder, thelater the strike pri is paid out, the better. >
,
kg
-
-
-
9.6
EARLYEXERCISE: PUTSON A NON-DIVIDEND-PAYINC STOCK It can be optimalto exercisean Americanput option on a non-dividend-payingstock early.Indeed,at any given timeduringits life,a put optionShould alwap beexercised earlyif it is suciently deepin the money.
?13
oj StockO>lfpx, Pvopeyties
that the strike prici is $1 Tp illustratetls, consider an extreme situation. suppose andthe stock price is virtually pero.By exercisingimmeditely,an investormakes an gain of $10.If the investorwaits, the gain from xercisemightbe lessthan immediate $10,but it cannot be m''ore than $10becausenegative stock prkes are impossible. Furthermore,recelving $1 now is preferable to receiving$10in the future.lt follows thatthe option should be exercisedimmediately. Likea call optionj a p'ut option can be viewedas providing insuran. A put option, whenheld in conjgnction withthe stock,insure!the holderagainstthe stock pri fallingbelowa certain level.However,a put optionisdiferentfroma call option in tat it may be optimal for >n investorto forgotllisinsuranceand exerciseearly in order to reali/e the strike price immediately.ln general, th early exemise of a put option becomesmor attradive as S decreases,as r increases,and as the volatility decreases. that It will bt recalkd from equation (9.2) -rT
p 7 Ke
=
h
#or an Americanput with'price P, the strongercondition PkK
-
S
must plwayshold becauseimmediateexemiseis alwayspossile. Figure9.4'showsthe general way in whkh the prke of an Americanput varks with h. Providedthat r > it is alwaysoptiml to exercisean Americanput immediately whenthe stock price is suciently low.Whenearly exerciseis optimal, the value of the optionis K k. The curve represinting the.value of the put threfore mergesinto the 9.4 tls value put' s intrinsicvalue, k -L k, for a sllciently smallv'alue of k. ln f'igure pri of k is shown as point :. Thelinerelating the put to te stockprice movesin the directionindicatedby the arrows when r decreases,when the volatility increases,and when T increases. Becausethere are some cirolmstans when it is desirableto exerce an American put option early, it followstat an Americanput option alwaysworthmore than the ,
-
,
Figure 9.4
of an Amerkanput option with stockpri,
Variation of pri
American put price
I I I
l l l
I I I
l
I
x
xN
N
xN
N N N N N
x
N N
x
K
stcckprice,
::
%.
2i4
CHAPTER Fijure 9.5
Varition f prke of afEuropeapput option wit the stock price, k. '
'
'
'
.
.
'
.
h
'
Enropean putprice N
E
N
*
N
-
N
N
N
N
,
xN
N
N
N
x
N
N
N l N' l j
I I
.
N
N
N
'
N
xN
l '
l
.
N
.
xN
'
K
B
'
stockprice, s
correspondingEuropeanput option.Furthermore,becausean Americanput is some:.timesworth its intrinsicvalue (seeFipre 9.4),it followsthat a Europeanput option must sometimesb worth lessthan itsintrinsicvalue. figure9.5 showsthe variation of the Europeanput price Fith the stok pricr. Note that point B in Figure9.5, at which the price of the optipn is equal to itsintrinsicvalue, must representa highervalue of the stock price than point z4 in Figure9.1.Point E in Figure9.5 is where s' = ()and the -rT Europeanput pri is Ke .
9.7
EFFECTOF DIVIDENDS The results prpduced so far in th chapter have assumed that we are dealingwith ojtions on a non-dividend-paying stock.In th sectionwe examine th: impact of dividends.In the UnitedStatesmost exchange-tradedstockoptionshavea life of less than 1 year and dividendspayable duringthelifeof the option can usually be predicted with reasonable accuracy. We will use D to denotethe present value of the dividends duringthe life of the optipn.ln the calculationiofD, a dividendis assumed to occurat the time of its ex-dividenddate.
' Lower Bound for Calls and Pts We can redefne portfolios A and B as follows: Porlfolioz4 : one Europeancall option plus an amountof cash equal to D + lbrt//fp ;: one share
Ke-rT
showsthat A similar argument to the one used to deriveequation (9.1) C
V
i
-'
P
-
Ke-fT
(9.5)
215
Properties t)./*StockOptions We can also redesne portfolios C and D as follows: '
.
Porfolio C: one European put option plus oe shaze Porfolio D:.. an amount of cash equal to D + ke ,
-r1%
.
' ,
('
.
showsthat A similr argument to the ne used t deriveeqpation (9.2) =rT
p 7 D + Ke
'
S
-
(!,.t)
.
EaflyExercise When dividendsare expected,wecn no longerassertthat an merkan call option will it is optimal to exercisean Americqncall immediately not be exercisedearly. sometimes prior to an ex-dividend date.It is never optimal to exrrdse a call at other times.Th
point is discussedfurtherin section 13.12.
Put-call Parity Comparingthe value at option maturity of the reene portfolios A and C showsthat, with dividends,the jut-callparity result in equation (9.3) becomes -rT c + D + Ke
Dividendscause equation (9.4) to be mdifkd
(9.7)
h
p+
=
(se Problem9.19)to
e
k
-
D
.
-
K %C P %h -
y
.y
Ke
-
(!1.11)
SUMMARY '
Thereare sixfactorsafecting the value of a stockoption: the current stockpri the strikeprice, the expiration dpte the stockpricevolatilhy,the risk-freeinterestrate, and the dividendsexpected during the lii-e of the option, The value of a call generally increases as the current stockprice, the timeto expiration, the volatility,and the rkfreeinterestrate increase.The valve of a call decreasesas the strikeprice and expected increase.The value of a ptt genemllyincreasesas the strikeprice, thetimeto dividends the volatility, and the expected dividendsincrease.The value of a put expiration, decreases as the current stock price and the risk-free interestrate increase. It is possible to reach some conclusions about the value of stock options without makingany assumptionsabout the volatilityof stok prkes. For example,the pri of a call option o a stock must alwaysbe worth lessthan the price of the stockitself. similarly,the price of a put option on a stock must alwaysbe worth lessthan the option'sstrikeprice. A Emopeancall option m a no-dividend-paying stck be worth more than ,
.
'must
maxts
-
-rT
Ke
,
)
wherek is the stockpri, K is the strikeprke, r is the rkk-free interestrate, and T is thetimeto expiration. A European put option on a non-dividend-payingstockmustbe
216
CHAPTER 9
worth more than
maxtfe -rr
s
-
,
g)
present value D will be paid, thelowerboundfor a European call Whendividendsw1t.11 optitmbecolies PaX( Ko-ry g) s -
.p
-
,
and the lowerbound for a European put optionbecomes maxtf:
urT
+D
-
k, )
Put-call parity is a relationship betweenthe price, c, of a European call option on a stockand the pri, p, of a European put optionon a stcck. For a non-dividend-paying stock it i! -rT p+ y c + Ke =
stock, the put-call parity relationship is For a dividend-paying '-Y
c + D + Ke
=
p+ S
Put-call parity does not hold for Americanoptions.However,it is possible to use arguments yo obtainupper and lowerboundsfor the diferencebetweenthe arbitrage of price an Americancall and the prke of an merican put. Ih Chapter 13,we will carry the analysesin this chapter furtherby making specifk assumptionsabout the probabiltic behaviorof stock prices.Theanalysiswillenable us to deriveexct pricing formulasf0r European stock options.In Chapters11 and 19, we willsee how numerical prcedures can be used to pri Americanoptions.
FURTHER REAOING Black F., and M. Scholes. srfhe Pricing of Options and CorporateLiabilitiesr'' Journal of : Polltical Ecohomy,81 (May/lune1973):637-59. . Option Valuation:Nw Bods, Ajjrximatins, and Broadie, M., alid J, Detetple. dsAmerican of FinancialStudks,9, 4 (199$:1211-5. of ExistingMethods,'' .Rdvw Comparison a Merton, R. C.. t$On the Pricing of CorporateDebt: The Risk Structureof Interest Ratesy'' 449-70. Journql of Finance,29, 2 (1974): of Rational Option Prkinp'' BellJournal o/f'tztlrrlc.y and Management Merton, R. C. Sciezce,4 (Spring1973):141-83. Merton, R. C. s-f'he Relationship betweenPut and CallPris: Comment''Journal of Fizance, 28(March 1973):183-84. Relatinship betweenPut and Call Option Prices,''Joarnal of Finance, 24 Stoll,H. R. 1 69): (December 801-24. trfheory
-f'he
and Problems(Answersin SolutionsManul) Questions pris. stock 9.1. List the six factorsthat aYect 9.2. What is a lowerboundfor the pri of a 4-month all option on a non-dividend-jaying stockwhen the stockprice is'$28,the strikepri is $25,and the risk-free interestrate is 8% per annum? .option
217
Poperties t)./'Stocb O>ff:l.s
9.3. What is a lowerbound for the prke of a l-month European put optjon oh a nondividend-paying stockwhen thestockprice is $12,thestfikepri is $15, hd the risk-free interestrate is 6% per annum? 9.4. Give two reasons why the early exerciseof an merican call option on a npn-dividendpayingstockis not optimal.Thehrst reason should involvethetimevalue of mony. The secondshould apply even if intrestxrtes re zero. 9.5. $$Theearly exercise of an merkan put is a tradezofbetweenthe time value of money and the insufancevalue of a put.'' Explainth sttement. stock is alwayswrth at least 9.6. Explainwhyan Americancall optionon a dividend-paying as much as its intrinsicvalue. Is the same true of a Europeancall optin? Explainyour aIISWCI'.
9.7. The pri of a non-divided-paying stockis $19and theprke of a 3-monthEuropean call option op the stock with a strike prke of $2 is $1.The riskzfreerate is 4% per antmm. What is the price of a 3-monthEuropeanput option with a strike pri of $202 9.2. Explainwhy the arpments leadingto put-call parity for European pptions cannot be used to give a similar result for Americanoptitms. 9.9. What is a lowerboyndf0r the pri of a Gmonthcall option op a non-dikidend-paying stockwhen the stock pri is $8, the strike pri is $75,and the risk-free interestraye is 1% per almum? of Europeanput optiop on a non., 9.1. What is a lowqrbound for the price a z-month stock when the stock prke is $52,the strlke prie is $65,andthe risk-free dividend-paying interestrate is 5% per aknum? stock 1S currently se call option dividend-payipg 1. European A 4-month for $5. on a 9.1 The stock prke is $64,the strike pri is $60,and a dividendof $0.20is expected in 1month. The risk-free interestrate is 17%per annup for all maturities. What opportunitiesare therefor an arbitrageur? stock is currently selling 9.12.A l-month European put option on a non-dividend-payipg prir and strikt stock price is the th T he is riskrfreeinterestrate is $50, $47, for$2.50. 4%per annum. What opportunities are therefor an arbitrageur? '
tling
9.13. Givean intuitiveexplanation of why the earlyexerciseof an Americanpt becomesmore attractiveas the risk-free rate increses and volatility decreases. The price of a Europeancall that expires ip 6 months and has a strike prke of $3 is $2. The underlying stock prke is $29,and a dividendof $0.50is expcted in 2 months and againin 5 mpnths. The term structure is Qat,with risk-free interst rates being1%. Whatis the pri of a Europeanput pption that expiresin 6 monshsand as a strike price 'all
Of
$302
9.15. Explain carefully the arbitrageopportunitiesin Problem9.14if the uropean put pri is $3. 9.16. The price of an Amerkan call on a non-dividend-payingstock is $4.The stock pri is $31,the strike price is $3, and the expiration dateis in 3 months.The risk-free interest rate is 29/0.Deriveupper and lowerboundsforthe price of an Amerkan put on the same stock with the same strike ptice and expiration date. 9.17. Explaincarefully the arbitrageopportunities in Problem9.16if theAmericanput price is greaterthan the calculated upper bgund.
218
CHAPTER 9
9.18. Prove the result in equation(9:4).Hittt: F0r the rst part of the relationslp, consider (a) a portfolio consistingof. a Europeancall plus an amount of cash equal to #, and (b)a portfolio consistingof an Americanput option plus one share.) 9.19. Provethe result in eqtlation(9.8):Hittt: F0r the flrst part of te relationslp, consider (a) a portfdlio consiitingof a. Europeancall plus an aount of cash equal to D + #; and (b)a portfolio consistingof an Amerkan put option plus one share.) employzestockoption n a non-diviend-payingstock.Theoptioncan 9.2. onsidera svyegr aftefthe end 9f the frst year. Unlikea regular exchange-traded beexercisedat any call option,the employeestockoptioncannot' be sold.Whatis the likelyimpactof this decision? restrktion on the early-exercise 9.21. Use the softwareDerivaGemto verify that Fiprej 9.1 and 9.2 are correct. .
.the
AssignmentQuestions 9.22.A European calloptionand put optionon stockbothhavea strike prke of $2
and an risk-free interestrate 1% per annum, expirationdatein 3months.Bothsell fr $3.The the current stock prici is $19,and a $1 dividendis expectedin 1 month. Identifythe
arbitrageopportunityopen to a tradef 9.23. Supposethat c1, c2, and c3 are the prices of European call optionswit strikeprices #1, #2, and #3, respecvely, where #3 > #2 > 11 and #3 #2 = #2 #1 A1loptionshave the samematury. Showthat + c3) c2% liittt: Considera portfolio that is long one optionwith strikepri #l, long one option withstrikeprice #3, and short two optionswith strike prke #2.) 9.24.ih at is tht result crresponding to that in Problem 9.23for Europeanput optionst 9.25.Supposethat you are the manager and sole ownerof a lghly leveragedcompany.A11the debtwill mature in 1 year. If at that timethe value of tliecompanyis greater thapthe face plp.pf the debt, yu will pay ofl-the debt.If the value of the company is lessthan the facevalue o? te debt, you will declarebaknlptcy and the debt holderswill own the company. (a) Express your position as an option on the value of the company. (b) Express the position of the debt holdersin terms of opticns on the value of the .
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(c) What can you d to increasethe value of your position? an option on a stockwhen te stockprice is $41,the strike prie is $40,the riskconsider 9.26. is 350:, and the time to maturity is 1 year. Assumethat a freerate isof6%, thtis volality $0.50 expectedafter 5 months. dividend (a) Use DerivaGemto value the option assuming it is a European call. @) Use DerivaGemto value the option assuming it is a European put. (c) Verifythat put-call parity holds. (d) Explore tsing DerivaGemwhat happensto the price of the options o the time to pattlriiy becomesvery large. For this purpte, assllmethere are no dividends. Explain the results you get.
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The proft pattern from#n investmentin a Single option waS in Chapter3. In th chapter we cover more fully the range of proft patterns btainable using options. We assume that the upzerlying asset is a stock. Similar results can be obtainedfor other underlying assets, such as foreigncurrencies, stck indices,and futurescontrads. The options used in the strategieswe discussare European.American options maj lead to stkhtly diferent outcomes becauseof the pqssibilityof early exercise. In the flrst section we consider what happenswhen a position in a stockopsion is combinedwith a position in the stockitself.Wethep move on to examine the profh patternsobtained when n investmentis madein two or more diferent options on the samestock. One of the attractions of options is that theycan be used to create wide rangeof diferent payof functions.(A payof functionisthe payof as a functionof the stockprice.) If European options were availablewith everysinglepossibl strikeprke, anypayof functioncould in theorybe created. For ease of exposition the fgures and tables showingthe profh from a trading will ignorethe time value of money. Te prot will be shown the fnal strategy payof minus the initialcost. (In theory,it should be calculated as tht present value of the fnal payof minus tht initialcost.) 'discussed
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INVOLVINGA SIjGLE OPTION AND A STOCK 10.1 STRATEGIES Thereare a number of diferenttradingsttategies involvinga singleoption on a stock and the stockitself.The profhs fromtheseare illustratedin Figure1p.1.In this sgure and in other sgures throughoutthis chapter, the dshed line shows the relationsp between proft and the stock pri for the individualsecuritiesconstituting the portfolio,whereas the slid lin shows the relationship betwee prost and the stock Pricefor the wholeportfolio. In Figure10.1(a),tlie portfolio consists of a long positionin a stock plus a short longstock position positionin a call option. Thisisknownas writizga coyeredcall. payof short call that becomes investorfrom the the the on or protects necessaryif thereis a sharp rise in the stock price.In Fipre 10.1(b),a shortposition in a stockis combined w1t11 a longpositionin a call option. Thisisthe reverseof writing 'l'he
Sovers''
219
220
CHAPTER l
Fijufe 10.1 Prtt patterns (a)longposition in a stock combinedwith short positipn in a call; (b)short position in a stock combied with longposition in a all; (c)long positionin a put combinedwith longposition in a stock; (d)ihoft position in a put combinedwith short position in a stk. Profit
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a coveredcall.In Figure1.1(c), theinvestmentstrategyinvolvesbuyinga put option on a stock and the stock itself.The approachis sometimesreferred to as a protectiveput strategy.In Figure1.1(d), a short position in a put option is combinedwith a short positipnin the stock. Thisis the reverse of a protective put. The proft patterns in Figures1.1 havethe samegeneralshape as the proft patterns discussedin Chapter8 for short put, longput, long call, and shrt call, respectiyely. Put-call parity provides a way of understanding why this is so. From Chapter 9, the
221
TyadingStyategiesfxtlprixg Options ' .
,
-put-calljarity relationship is p+ k
rrT
c + Ke
=
(10.1)
+D
wherep isthe price of a Europeanput, c isthe stockprice, c isthe price of a European call, K isthe strike price of both clll and put, r isthe risk-fr interestmte, T isthe me to maturity of both call and put, and D isthe present value tliedivideds.antkipated duringthe life pf the options. Equation(1.1)jhowsthat a longposition in a put combinedwith a longpositin in is equivalent to a long call position plus a certain amount(= ge + pj of the cash.This explais why the prost pattern in Fipri 1.1(c) is similarto the profh patternfroma longcallposition. The position in Figure1.1(d) isthe reverseof that in Figurel.1(c) and thereforeleadsto a profh pattern similar to that from a short call pos t on. Equation(1.1)can be rearranged to becotne 'of
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This shows that a long position in a stock combine with a short position n a callis equivalentto a short put position plus a certain amount (= Ke-rT + D) of cash.This equalijy explains why the prost pattern in Figure 1.1(a) is similar to the prpt pattern from a short put position. The position in Figqre1.1(b) ij the reverse of that in Figure 1.1(a) and thereforeleads to a prot pattern similar to that from a long put position.
10.2 SPREAMS A spread tradingstrategy involvestakinga pojition in two or more options of the same type (i.e.,two or more callsor two or more ppts). '
BullSpreads One of the most popular typesof spreads is a ballspread.Thiscan becreatedbybuying a call option on a stock with a certainstrikeprice and sellinga caltoption on the same '
Figure 10.2
Profh frombull spread createdusing call options,
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222
CHAPTER 10
Payof from a' bull spread creatid usig calls.
Table 10.1
Payoffrom sort call plfip?i
J'cyp/s/'/t?ln
Stockprice rcrre
lozgcall plfp?i
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stock with a hkher strike prkt. Both optionshave the same expirationdate. The strategyis illustratedin Figure1:2. The prtts fromthe tw0 option positions taken separatelyare shown by the dashedlines.The prtt fromthe whole strater is the sum of the prots given by the dashedlinesandis indicatedby the solid line.Becausea call jrice alwaysdecreastsas the strikepri increases,the value of theoptionsold isalways less than the value of the option hought.A bull spread, when created from calls, thereforereqyires an inisialipvestment. Supposethat fj is the strikepri of the call optionbought,#2 is the strikeprke of th call opljon sld, and s'ris the Stock pri on the expiration date of the options. Table 1.1 shows the total payof that will be realized from a bull spread in diferent ciicumstances.If the stock prke doeswell nd is greater than the higherstrikeprice, the payof is the diferencebetweenthe two strike prkes, or #2 #j. If the stock price on tht tkpiration date liesbetweenthe two strike prkes, the payof is s'r #l. If the stock price on the expiration date is belowthe lowerstrike price,the payof is zero. The prtt in Figure1.2 is calculatid by pbtracting the initialinvestmnt from the -
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Payog
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Abull spread strategylimitstheinvestor'supside as wellas downsiderisk. Thestmtigy can be describedby saing that theinvestorhas a calloptionwith a strikepri equal to Kj and has chosen to give up some npsidepotenal by selling a call option with strike prke #2 Kz > #!). In return for giving up the upsidepotential, the investorgets the Figure 10.3
Profhfrombull spread creAted using put options.
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223
TyadingStyategiesInvolving O>!i0-
priceof theoption Fith strikeprice #2. Threetpes 9fbullspreads can bedistinpished:
1 Both calls are initially ut of the money.
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2. 0ne callis inially in the mony; the othercallij initially ut of the money. 3. Both calls are initiallyin tllemoney. '
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The'mst are thoseof tpe 1.Theycostviry littleto set up and have a sinall probability of giving a relativelyhigh pamf (= K1 = #1). As we movr fromtype 1 to type2 and ffomtpi 2 to tpe 3,thespreadsbecomemoreconservative. aggressivebull jpreads
Example/:./ An investorbuysf0r $3a callwith a strikepriceof $3 and sellsf0r $1a callwith a strike price of $35,The payof fromth bull spreadstrategyis $5 if the stock Pri is above $35,and zero if it isbelw $30.If thestockpri isbrtween$3 and $35,thepayof isthe amount by whkh ihestockpri exds $30.Thecostof the strater is $3 $1= $2.The prot is thereforeas follows: -
Prost
Stqck price Fcng:
-2
Sz %30
30< s'r< 35 s'r k 35
Sr
32
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3
Bu11spreadscan also becreatedbybuyipga put with a l0wstrikepri and sellinga put witha highstrikepri, as illustmtedin Fipre 1.3. Unlikethebullspreadcreated from calls,bullspreadscreated fromputs involvea positiveup-front cas qowto theinvestor (ignoringmarginrequirements)and a payof that is either hegative or zero.
Bar Spreads '
..
An investprwh0 enters into a bu11spread is hopingtltat thestockpricewillincrease.By contmst,an investorwh enters into a bear spread is hopingthat the stockpri will decline.BTar spreadscan be created bybuyinga put with 0ne strikepri and sellinga put with anotherstrikepri. The stfikepri of the optionpurchased is greater than the strike pri of the option sold. (Tilisis in contrast to a bullspread,where the strike ,
Figure 10.4 Profit
N
Prtt frombear spread created using put options. N N
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224
CHAPTER 10
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Table 10.2 Payof frm a bear spfeadcreated Fith put options. .
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#1 n are liableto lose money. '
ButterflySpreads A battersy spread involvespositions in options with threediferent strike pris. It can be created by buyinga call optipn with a relatively1owstrikepri, #j , buyinga call option with a relativelyhigh strike pri, Kj, and selling two call options with a strike price,#z, halfw'aybetween#1 and Kj. GenerqllyKz is close to the current stok pri. The pattern of profitsfrom the strategy is shown in Fipre 1.6. A butter:y spread leadsto a prost if the stock priceslayscle to Kj, but givesrise to a small losj if there is a signifcant stock pri move in either direction.It is thertfore n appropriate strategyfor an investorwho feelsthatlargestockpri myes are unlikely.The strater requiresa small investmentinitially.The payof from a buttey spreadis shown in Table 1.4. Table 10.3
PayoFfrom a box spreap.
Stockprice range
fkypff/rp?n ball call spread
S %#1 Kz S k Kz
#1 ff0Al-r Fijure 10.9 Profit
Prost (rom a calendar spread created using two puts.
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To understand the proht pattern froma calendar spread, flrstconsiderwhat happens the if stock price is very lowwhen the shri-maturity option expires.The short-maturitf optionis worthless and the value of the long-maturityoption is close to zero. The investorthereforeincursa lossthatis close to the cost of setting up the spread initially. Considernext what happensif the stock pli, &, is very lgh when the short-maturity K, and the longoption expifes. The shorbmatulity option costs the inv:stor sr where strike prke of the options. sr option K worth close is the = maturity is to Again,the inkestormakes a net lossthat is close to the cost of sestipg up the spread short-maturityoption costs the investoreither a small If initially. r is close to #, the nmount or nothing at all. However,the long-maturity option is still quite valuable. In '' thls ca e signiscant net prt ij made. In a zeutral calezdar spbead,a strikepri close to the current stock price is chosen. A bullishcqlezdar spread involyesa higher strike pri, whereas a bearishcalezdar spreadinvolvesa lowerstrike prici. Calendarspreads can be createdwith put options as wellas call options. The investor buysa long-maturityput option and sells a short-maturity put option. As shown in Figure 1.9, the proft pattern is similar to that obtained from using calls. A reversecalezdarspread isthe opposite to thatin Figures 1.8 and 1.9. The investor buys a short-maturity option and sells a long-maturityoption. A small proft arisesif the stock plice at the expirationof the short-maturityoption is wellaboveor wellbelow tht strikt prict of the short-maturity option. HoWever, a signiEcantloss results if it is closeto the strike price. -
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DiagonalSpreads ' .
Bull bear, and calendar spreads can a11be created froma longposition in one call and a short position in another call. In the case of bull and bear spreads, the calls have diferentstrike prkes and the same expirationdate.In the case of cakndar spreads, the callshavethe same strike price and dferent expirationdates. In a diagozalspread both the expirationdate and the strike pri of the calls are of proft patterns that pl'epossible. diferent.This increasesthe .rage
230
HAPTER 10 .'
.
Figu/e 10.10 Proft from a straddle, J
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10.3 COMBINATIONS A combizatiozis an option tradingstrater thgtinvolvestakinga positionin both calls andputs on the same stock. Wewill considerstraddles, strips, stpps, and strangles. '
StrAddle One popular conibintion' is a straddle,which involes buyinga call and put with the samestrike prke and expirationdate.The proft pattern is shown in Figure10.1. The strikeprke isdenote by K. If the stockpri is close to tls strikepri at expirationof th8options,the straddle leadsto a loss.However,if ther is a suciently largemove in eithefdiretion, a signifcant prot will result. The payofl-froma straddle is calculaied in Table 10.5. A straddle is appropriate when an investoris expectig a largemove in a stock prke but des not knowin which directionthe move will be. Cnsider an investorwhofeels that the pri of a rtain stock, currently valued at $69by the market, will move signifkantlyin the next 3 months. Theinvestorculd create a stmddle by buyingboth a put and a call with a strike pri of $7 and an expimtion datein 3 months. Suppose that the call costs $4and the put costs $3.If the stock pri stays at $69,it is easy to see that the strategy coststhe investor$6.(Anup-front investmentof $7is required, the call expiresworthless, and the put expiresworth $1.)If the stock pri movesto $70,a loss of $7 is experienced. tTls is the worst that a happen.)However,if the stock pri jumpsup to $9, a prot of $13is mad; if the stock movesdownto $55,a proft of $8 ismade; and so on. Asdiscussedin BusinessSnapshot10.2an investr should carefully Table 10.5 Razge of stockprice S %K r > K
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tl/pliiklilitt 8f 'Jk' jiiiqifytlj tji q ylyjjg jyrkuj jttjkyy. ilitiitiklr jtyjyjykyj,j ittyjtiirjtis yjjjfj k ) y (uyg q yijjj yjjj, jyj-,yy,,,,y,,,y-j,j.ryy,,kjy,,;yyy Eljikk4jyt i-tkklrfkEjl yjk, ? , kitj',i) jy ijjc ttt iijt,tys i yjgttu, y jjyjj r tqtt j j r q,: #f qk.ql,(l i? bt q i tlrkn yktpjy ytj gyjjyjjjt ut)p ;r qjy ), ) x, s j j, jy ilirt jte sf jty jem: oksjjj? jr, ern. T1 Draw a diagramshowing the profh when (a)#2 > #1 and (b)#2 < #l. 1.21. Draw a diagramshowingthe variation of an investor'sprofh and losswith the terminal stockprice for a portfolio consistingof : .
(a) 'One shareand a shortpositionin pne call option (b) Two shares and a short positionin one call option @) One share and a shortpositionin two call options (d) 0ne share and a short positionin fourcall options In each case, assumethat the call option has an exerciseprice equal to the current stockprice. stockis $32,its volatility is 30B/:, and 10.22.Supposethat the price of non-dividend-paying the risk-free rate for a11maturities is 5% per annum.UseDerivaGemto calculate the cost of settingup the followingpositions: (a) A bull spreadusing Europeancall options with strikepris of $25and $3 and a maturity of 6 months (b) A bear spreadusing Europeanput options with strikepris of $25and $3 ad a maturity of 6 months (c) A butterqy spiead using European call options with strike pris of $25,$30,and $j5 and a maturityof 1 year
2j
CHPTER
10
(d) A butter spreadusingEuropean options with strikepricesof $25,$30,and $35and a iaturity of 1 year (e) A straddleusinpoptions with a strikepri of $30and a Gmonthmaturity (f) A strangleusingoptions with stlikepris of $25and $35and a Gmonthmatuiity .put
In each caseprovidea tableshowingthe relptionshipbetweenprt and snalstockprke. Ignorethe impactof discounting. 10.23.What tiading positionis createdfrol a longstrangleand a short stryddlewhen both havethe sametime to matulity? Asjumethat the strikepri in the stiadle is halfrc betweenthe two strikepricesof the stzangle.
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j.& $j:z, js,r.NIj 'dl' 4,.1 tz : 1, or downfromh to a newlevel, kd, where d < 1. The percentage increasein the stock price when there is an up movementis z 1',the percentage drease when thereis a downmovement is 1 d. If the stockprice movesup to hz, we suppose that the payof fromthe option is J.; if the stock price moves down to hd, we supposethe payof fromthe optionis h. The situationis illustratedin Figure11.2. As before,we imaginea portfolio consisting of a long positiop in A sharesand a short position in one option. We clculate the value of A that makesthe portfolio riskless.If thereis an up movement in thestockprie, the value of the portfolio at the end of the lifeof the optionis jj: h j. -
-
-
the valuebecomes
If thereis a dwn movement in the stock pri, k%#A
h
-
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Stock and (1ption pris in a genefal one-steptr. ka Ju
,% J kd J#
(11.1)
CRAPTER 11
240
In this case, the portfoliois riskless and,for thereto be no arbitrageopportunities,it ear the risk-free interestrate, Equation(11.1) shpws that is the ratio of the . changein the option price to the change in the stock price as we move betweenthe nodesat time T. lf we denote the risk-free interestrate by r, the presentvalue of the portfoliois .pwst
vrT
sut fuje -
The coss of setting up the portfolio is lt followsthat
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equation (11.1) for and simplifying,wecan redu thls equation to Substitutingi-rom
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d
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u-d
enable an optionto beprid when stock pri movements Eqaons (11.2) and (11.3) are given by a one-stepbinomialtree. Theonlyassumption needed for the equation is thatthere are no arbitrage opportunities. In the numerical example consideredpreviously(seeFigure11.1), u = 1.1, d = = = = we have r 0.12, T 0.25, J. 1, ad fd From equation (11.3), .9,
=
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=
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and, ffom equation (11.2), we have
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e
x1 (0.6523
0.3477x
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=
0.633
The result agrees with.the answer obtainedearlier in tls section.
Irrelevance of te Stock's ExpectedReturn The optionpricingformulain equation (11.2) boesnot involvethe probabilitiesof the stockprice moving up or down.For exnmple,we get the same option pri when the Thisis surprising and probabilityof an upward movementis 0.5as we do when it is counterintuitive. natural of an upward that, probability is the It to assume as seems movementin the stock priceincreases,thevalue of a call option on the stock increases and the value of a put option on the stk decreases.Thisis not the case. The key reason is that we are not valuingthq option in absoluteterms.We are calculatingits valuein terms of the pri of the underlyingstock. ne probabilitiesof futureup or downmovementsare alreadyincorporatedinto the stock price:we do not needto taketheminto aount againwhenvaluingtheoptionin termsofthe stock prke. .9.
ZMF::J l/inov%ial
VALUATION 11.2 RISK-NEUTRAL We do not need to make any ssumptionsabout the probabilitiesof up and dorn movementsin order to deriveequation (11.2).Al1wi require isthe absen of arbitrage opportunities However,it is natural to interpretthe variable p in equation (11.2) as the probabilityof an up movementin the stock prie. The variable 1 p is then the probabilityof a down movement, and th# expression .
-
:
.
pfv+ (1
pjfd
-
is the expected payof fromthe option.Withthis interpretationof p, equation(11.2) then statesthat the value of the option todayis its expectedfutup payof discountedat the risk-free rate. We now investigatethe expectedptur fron thestockwhen the probability of an up movment is p. The expected stockpriceat time T, Es, is givenby
'(&)
pjkd
(1
phu +
=
-
Or
&Sr)
=
phu
-
$ + Sbd
substitutingfromequation(11.3) for p, we obtain S )= S f (r
'
eVT
(11.4)
showingthat the stock prke grows n avefaje at the rk-free rate. setting the probability of the up movement equal t p is thereforeequivaknt to asuming that thereturn on the stock equals the rk-fr rate. In a risk-neatral worlda11individualsal'eindiferentto risk. In such a world, investors requireno compensation for risk, and the expected return on a11securities is the rkkshowsthat we are assllminga rk-neutral world when freeinterestrate. Equation(11.4) weset the probabilityof an up movementto p. Equation(11.2)showsthat the value of theoptionis its expectedpayo- in a risk-neutralworld discountedat the risk-free rate. This resuli is an exampleof an importantgeneralprinciple in iption pricing known as risk-neatral valaation. The principle states that we can with complete impunity assumethe world is risk neutral when prking options. I'he resultng jrices are correct notjustin a rk-neutral world, but in other worlds as well.
Thq One-step BinomialExampleRevisited We now return to the examplein Figurel 1.1and illustratethat risk-neutral valuation vesthe same answer as no-arbitrage arguments.In Figur 11.1,the stock pri is $2 and will move eitherup to $22or to $18at the end of 3 months. currently call with considered ojtion is a European a strike price of $21and an ke option date in 3 months. The rkk-free interestrate is 12%per annum. expiration Wedesne p as the probability of an upward movementin the stockprice in a risk' neutral world. Wecan calculatt p fromequation (11.3).Alternatively, wt can arpt that worid must be the rk-free rate the expectedreturn on the stock a rk-neutral of 120:.This meansthat p must satfy .down
'in
. 22p + 1841 pj = 2: -
g. j2 x jj j2
Binomia'lTyees
243
the expectedpaypf in the real world. A position in a call option is riskier than a positionin the stock. As a result thediscountrate to beappliedto the payof froma all optin is gteaterthan 16%.Withoutknowingthe optin's value',Wi do not knowhow much greater than 16% it should be.l Using risk-neqtral valuatn is convenient becuse we' knowthat i a risk-neutral world the xpected ieturnon a11assets(and thereforethe discountrate to use for a1lexpectedpayofl) is the risk-free rate,
11.3 TWO-STE BINOMIALTREES Wecan exyendtheanalysisto a two-stepbinomialtreesuch as that shownin Figure11.3. Herethe stock prke starts at $2 and in eachof two time steps ma go up by 1% or downby ln. Wesuppose that each timestep is3 monyhslongandthe risk-freeinterest rate is 12% per annum.As before,we consider a!l option with strike yrieof $21. The ohjectiveof the apalysijis t calculate the optionprke at theimtialnode of the tree. This can be done by repeatedly applyingtht principlesestablishedearlier in the chaptzr.Figure 11.4shows the same tree as Figure11.3,but with both the stock prk and the option price at each node. (Thestockprice isthe upper numberand the option pri is the lowernumber.) The option prkes at the snalnodes of the tree are esily calculated.Theyare the payofs fromthe option. At node D the Stock pri is 24.2ant the option pri is 24.2 21 3.2;at nodes E andF the option is out of the money and its value is zero. -
Figure 11
.4
=
Stockand ption prices in a twstep tree. The upper number at each and the lowernumber is the option price.
nodeis the stock pri
D
22
24.2 3.2
B
2.0257 E
20 1.2223
18
19.8 -
C
. F
I
16.2 0.0
Becausethe correct value of the option is 0.633,we can euce that the correct iscountrate is 42.5894. This is because0.633= 0.7041:-0.4258x3/12
244
CHAPTER 11
Fisure 11
Evaluationpf optionpri at node B.
.5
D
,
24.2 :j 2
22
B 2.9257
E
.
19.8
.
At node C the optionpriceis zero, becausenode C leadsto either node E or node F and at both of those nodes the option priceis zero. Wecalculatethe option prke at nodeBbyfocusing0ur attentionon the part ofthe treeshownin Figure11.5.Usingthe notationintroducedearlkr in the chapter,a = 1.1,d = and T = (.25,so r= givesthe value of the option at node B as that p = 0.6523,and equation (11.2) .9,
-.12x3/12
e
(g jjz?
.
.
x 3.2+ g.3477x
)
.12,
=
2.0257
It remainsfor us to calculate$heoption pri at th initialnode A.Wedo so b focusing on the srststep of the tree.Weknowthatthe valu oftheoptionat node B is2.257 and thereforegvesthe value at node A as that at node C it is zero. Equation(11.2) -.12x3/l2
e
x 2,g2j7+ g,3477x 0) = 1.2823
(g jjg? .
The value of the option is $1.2823. Note that this exsmplewas constructedso that a and # (theproportionalup and downmovements)were the same at eachnode of the treeand so that thetimesteps were of the samelength.As a result, therisk-neutralprobability,p, as calculatedbyequation (113) is the same at each node. * '
*
A Generlization Wecan generalizethe case of two timesteps by consideringthesituationin Figure11 The stockprice is initiallyk. Duringeachtime step,it either movesup to a timesits initialvalue or movesdownto d timesitsinitialvalue. Thenotationfor the value of the optionis shownon the tree. (For example, after tw0 up movementsthe value of the optionis fuu We suppose that the risk-freeinterestrate is r and the length of the time Step is ht years. Because the lengthof a timestep is n0w ht rather than T, equation (11.2) and (11.3) become f = e-rht(#i + (j p)ydj (jj.s) .6.
'
.
e
p=
rzk/ tj '-'
R-#
(11.6)
245
l?inovlialzqree: Stockan2 option prices in general two-steptree.
Fijure 11 .6
s02 u
fu J0
soad
f
yu
.
SJ fd .,2 fdd
gives RePeated applicationof equation (11.5)
'
-rAl
fuu+ (j -rAl fd e Er/: + (j J e-oj Lpfu (j
L
=
d
.
=
-
p)yl j
(11.:)
p)ysdj
(11.8)
.gjgvj
(11.9)
.y
=
into (11.9), Substitutingfromequations (11,7) and (11.8) Fe get 2 Ep1u'+ 2p (1 /' e-2r,, =
-
p)yu d +
(1 -
vzjj
(11.10)
This is consistent with the principle of risk-neutralvaluation menoned earlier. The ,2 p)2 are the probabilitiesthat the upper, middle,and ' Var ia.bles 2p(1v p), pnd (1 lowerfmalnodes will be reached. The option pri is eiual to itj expectedpayof in a risk-neutralworld discountedat the risk-free interestrat. As we add more steps to the binomialtree, 4herisk-neutral valuation principle continuesto hold. The option price is alwaysequal to its expected payof in a riskneutrl world discountedat the risk-iee interestrate. -
j
'
,
11.4
PUT EXAMPLE The procedures describedin this chapter can be used to prke puts as well as calls. European put with a strikeprice of $52on a stock whose current Considera z-year priceis $50.We suppose that thereare two time steps of 1 year, and in each time step the stock price either movesup by 20% or movesdownby 2%. Wealso suppose that the risk-free interestrate is 5%.
.
246
CHAPTER 11 ..
'
.
.
.
Usinga two-steptreeto value a Epropeanput option. At eachnode, the upper nutber is the stock pri and the lowernumber is the option price.
Fijufe 11
.7
72 0
2E
.
(j
1.4147
50 4.1923
48 4
'
40 9.4636
32 20
At = 1, and r The tree is shown in Figure 11.7.In this caseu = 1.2,d = value risk-neutral equation 1.6) of From probability,p, is given by the (1 the .8
,
p
e =
,5x 1 .
1.2
-
gg= .
.8
-
0.6282
The possible nal stock prices are: $72,$48,and $32.In tllis case, fuu an d > f 20 From equation (11.1) =
=
,
fad
4,
=
,
.
( 62822x ./' e-2x'5xl =
.5.
=
.
+ 2 x 9.6282x 0.3718x 4 + 0.37182x 2)
=
4.1923
The value of the put is $4.1923. Thisresult can alsobe obtained using equation (11 and working back throughthe tree one step at a time. Figure 11.7 shows the intermediateoption prices that are calculated. .5)
11.5 AMERICANOPTIONS Up to now a11the options we hay consideredhavebeenEuropean.WenoF moveon to considerhowAmerkan options can be valued using a binomialtree such as that in Figure 11.4 or 11.7. ne procednreis to work backthroughthe treefromtheend to the beginning,testingat each node yoseewhether earlyexerciseis optimal. Thevalue of the option at the fnal nodes is ihesameas for the Europeanoption. At earliernode,sthe value of the option is the greater of '
1. The value given by equation (11.5) 2. The payof from early exercise
247
BinomialTrees
Figure 11.8 Usinga two-steptreeto value an Americanpgt option. At eachnode, the upper number is the stock price and the lowernumber is the option price. 72
60
1.4147
50 5.0294
4s 4
40
C
12.0
32 20
Figure 11.8 shows how Figur 11.7is >fl-cted if the option under consideration is American rather than European. The stock prkes and their probabilities are unchanged.The values for the option at the nal nodej are also unhanged. At node B, equation(11.5) gives the value of the option as 1.4147,whereas the payof from early Clearlyearlyexerciseis not optimal at node B and the value exercise is negative(= of the option at this node is 1.4147.Ai node C, equation (11.)givesihe value of the optionas 9.4636,whereas the payof fromearlyexerciseis 12. In thiscase, earlyexercise is optimal and the value of the option at the node is 12.At iheinitialnode A, the value givenby equation (11.5) is -8).
,
x 1.4147 (0.6282 e-.05xl .1..
?y18x 12 g) 5.9894 .
=
and the payof from early exerdse is 2. In this case early exerciseis not optimal. The valueof the option is therefore$5.0894.
11.6 DELTA At tltis stage it is appropriate to introducedelta,an importantpammeter in the pricing andhedgingof options. The delta of a qock option is the ratio of the change in the pri of the stock option to the change in the price of the undrrlying stock. It isthenumberof pnits of the stock weshould hold for each option shortd in order to create a risklessportfolio. It is the sameas the A introducedearlier in this chapter. The construction of a risklesshedgeis sometimesreferred to as deltahedging.Thedeltaof a call option ispositive,whereasthe delta of a put option is negative.
248
CHAPTER 11 From Figure11.1, we can calculate the value of the delta o? the ca11pption being considered as 1 = 25 22 18 .
-
.
-
' ,
Thisis becausewhen the stockprke changesfrom$18to $22,the option prke changes
from $0 to $1. In Figure 11.4the delta corresponding to stock prke mvements over the hrst time step is 2.0257 0.5064 22 18 = .
-
-
The deltafor kock pri movementsover the secondtime step is 3.2 0.7273 24.2 19.8= -
-
if thereis an upward movement over the flrsttime step, and 19.8 1-6.2= -
if thereis a downwardmovementover the Ersttime step. From Figure11 deltais .7,
1.4147 9.4636 = 60 40 -
-0.4024
-
at the end of the frst time step,and either - 4 72 48 = -
4 20 = or 48 32 -
-0.1667
-1.
-
at the end of the second time step. The two-stepexamples showthat delta changes over time. gn Figure 11.4,delta changesfrom0.5064to either 0.7273 r0; and,inFigure11.7,it changesfrom to either Thus,in order to maintaina risklesshedgeusing an option or andthe underlying stock,weneed to azjustour holdingsin the stockperiodically.Thisis featureof options that wewill return to in Chapter 17. -0.4024
-0.1667
-1..)
WITH p AND d 11.7 MATCHINGVOLATILITY In practke, when constructing a binomialtr to representthe movements in a stock pri, we choose the parametersu andd to match the volatilityof thestockprke. To see howthisis done,we suppse thai theexpectedreturn on a stock(inthe real world) is p and its volatility is c. Figure11.9showsstock pri movementsoverthe frst step of a binomialtree.The stepis of length l. The stock pri startsat h and moves either up to hu or downto kd. Theseare the only two lossibleoutcomesi boththe real world and the risk-neutral world. The prbability of a up movementin the real world is denotedby pe and, consistent wii our earlier notation, in the rk-neutral worldtllis probabilityis p.
249
Binomial Tyees
Fi! ure 11.9 Changein stock pri in time t in (a)the real world and (b)the riskneutrl wrld. soa ,stjj;
'
.
P*
J) '
,
-
s'o
s'tj l
-p%
1-p
hd
(a)
hd
(b)
The expectedstock prie at theend of theErsttime step in the real world is heht On the tree the expectedstock pri at thistimeis p'ls)d
p'shu+ (1 -
In order to match the exjtep thertforehave
or
return on the stock with the tree'sparameters, we must
p'S
-
e
p'
heiht
p'lSbd =
(1
u+
=
pLt
d
'-
(11.11)
u d -
As we willexplain in Chapter13,the volatilityc of a stockprice isdefned so that c Al is the standard deviationof the Dturn on the stock price in a short period of time of lengthit. Equivalently,the varian of the Dturn is c2A!. Onthe treein Figure11.9(a), the variance of the stock pri Dturn is2
pu
2
+(1
p
-
2
*
#
)# Epu ..F(1 -
-
'
p
#
2
)(f!
In order to match the stock price volatilitywith thetree'sparameters, we musttherefore
have
p
2 u + (j
.
p
y2 yu
(j
.j.
-
pigjz
-
=
g2j
tjj,jy
gives substitutingfromequation(11.11) into equation (11.12)
epst u 4.
-
ud
ppst
.gzaj
-
Whentermsin l2 and higherpowersof l areignored,one solution to thisequation is3 ca u= e d = -cW7
(jj (jj
e
2
'I'llis uses the result that the variance of a variable X equals EX 2)
value. 3 We are hereusing the seriesexpansion
E''
=
1+ x +
x.2
+
x
+
'
.
.
-
('(:j
2 ,
.
.
jy j4)
where E denotesexpected
250
CHAPTER 11
These are the values of u and d propgsed y Cox,Ross,and Rubinstein(197$fr matcllihgvolatility. teplnea the tree in Figm 11 The analysisi Section11 shows that we by the tree in Fipre 11.9(b),where the probabilityof an up movementis p, and then behaveas thoughthe world is risk neutral.Thevariable p is given by equation (11.6) as '
.2
.9(a)
'can
a.d
p=
(11.15)
p-#
where J
=
6
rAl
(11.16)
In Figure11.9(b),iheexpectedstockpriceat theendo? thetimestepis Sernt as shown in equation(11.4). The valiance of the stockpri return in the risk-neutral worldis '
2
pu + (1 -
pjd
.
2
'
-
.
'
(pl+ (1 -
p)ff12 =
,
rl
+ #) ud -
-
6
2rAl
)
this equals ht Substitutingf0r u and d fromequations(11 and (11.14), we End when:termj in A and higherpowersof A! are ignored. Thisanalysisshowsthat when we movefromthr real world to the risk-neutral world the expectedreturn on the stockchanges, but its volatilityremainj the same(atleastin the limit as A! tends to zerol. Tllisis an illustmtionof an important generl result known as Girsanoy'stbeorem.When we move from a world with one set of risk preferencesto a world with another set of risk preferences,the expectedgrowth rates in variables change, but their volatilitiesremain the same.Wewill examinethe impact of risk preferens on the behaviorof marketvariables in more detailin Chapter 17. Movingfromone set of risk preferences to anotheris,sometimesreferred to as cbanging .13)
American put option when the stock Fijufe 11 Two-steptree to value a z-year risk-free strike pri is 5, pri is 52, rate is 5%, and volatility is 3%. .10
91.11
67.49 0.93
50 7.43
50
;
37.04 14.96
27.44 24.56
251 '
.
.
fc measte. The real-world mepsureis sometimesrefeqed to as the P-measzre, while 4 therisk-neutral worldmeasure is referred to as the Q-measure.
Consideyagain the Americanput optionin Figure11.8,where the stck prke is $5, the strikeprice is $52,the risk-freel'ate is 5%,thelifeof theoptionid2 yearskand there that the volatility c is 3%. Then, are two time steps.In this cas, Al l suppose from equations (11.13)te (11..16), have =
.
.w
li
=
= 13499 e '3x1 .
and
1
d=
,
=
1.3499
gsx!
0.7408,
a = e ..
1.j13
=
1.053 t.7408 p 1.34990.7408 0.5097 -
=
=
-
The tree is shownin Figure11.1. The value of the put option is 7.43,(Tl1isis diferent fromthe value obtainedin Figure11.8by assumingu = i.2and d = .8.)
11.8 INCREASINGTHENUMBEROF STEPS simple. Clearly,an analystcan The binomialmodel pfesented aboveis nnrealistically approximation obtain only assuming that rough to an optiop pri expectto a:very stockpri movementsduringthelifeof the option consistof oneor t*o binomialsteps. Whenbinomialtrees are nsedin practi, the lifeof the optionis typicallydivided stock prke movement. into30 or more time steps.In each timestepthereis a binmial 3g and abcut pris 2 With30tipe steps yherearr 31termlnalstock 1billion,possible cr considered. paths pri implicitly stock are The equations dehningthe tree are equations (11.13) .to (1l.lf'regardless of the example, of that for there time steps. suppose, are fivestepsinsteadof two in nllmber theexample we consiered in Figure11.1. The parqmeters would be Al = 2/5 = nesevaluesgivea = :.3xW7 = 1.2089d 1/1.2089 0.8272, r :.5x0.4qnd c 1.22, and p = (1.22 0.8272)/41.20890.8272) 0.5056. a .by
'
.
,
.4,
.3.
.5
=
.
=
=
,
=
,
-
-
=
=
=
Usinj DerivaGem The software aonpanying this book, DerivaGem,is a useful tool for becoming comfortablewith binomialtrees. Afterloadingthe softwarein the way describedat the end of thisbok, go to the kuity-Fx-lndex-Futures-options worksheet. Choose Equity as the UnderlyingType and select BinomialAmerkan as the bptionType. Enter the stockprice, volatility,risk-freerateitimeto expimtion,exerciseprice, and tree steps,as 5, 3%, 5%, 2, 52, and 2, respectively.Clickon the Pat buttonand then on Calczlate.Thejrice of the option is shown as 7.428in theboxlabeledPrice.Nowclick on Display Tree and you will see the equivalent of Figure11.1. (The red nllmbers in the software indiate the nodes where the opon is exercised) Returnto theEquity-Fx-lndex Futuretoptions worksheetand change the number of time steps to 5. I'Iit nter and chck on Calczlate.You will fmdthat the value of the option changes to 7.671.By clickint on Display Tree the fve-steptree is displayed, tpgetherFith the values of u, #, a, and p calculated above. '
4 Withthe notation we havebeenusing, p istheprobabilityunder the Q-measurej while f isthe probability underthe P-measure.
2j2
CHAPTER 11
DerivaGemcan displaytreesthathaveup to l steps,but the calculgtions n bedone for up to 500 steps.In 0ur example,500stepsgives the optionprke (totwo decimal plnees)as 7.47.Tllisis an accurateanswer.By changing the OptionTypeto Binomial Europeanwecan use the treeto value Europeanoption.Using5 timestepsthe value of a Europeapoptiop with the snmepammeters>s the Amerkanoption is 6.76.(By changing pption typeto AnalyticEuropeanwe can displaythr valu of the option usingthe Black-scholesformulathat will bepresentedin Chapter13. Thisis also6.76.) B? changing the UnderlyingType, we :an consider options nn assets other than stocks.Thesewill n0w be discussid. '
'
.
.the
11.9 OPTIONS ON OTHERAS#ETS Weintrodd optionj on indis, currencies,and futurescontracts in Chapter8 and willcover themin moredetailin Chapters15 anp 16. It turnsout that wecan construct and use binomialtrees for theseoptionsin exactly the same way as for optins on stocks except that the equations fpr p change. As in the case of options on stocks, equation (11.2) applies so that the value at a node (before the possibility of early exerciseis considered) is p timesthe value if thereis p.nup movement plus 1 p times the value if thereis a downmovement,discountedat the risk-fr rate. -
Options on Stocks Paing a Contineous DividendYield Conjider a Stock paying a knowndividepdyield at rate q. The totalreturn from and capital gains in a risk-neutralworld is r. dividendsprovide a return dividends reiurn gains of r q. If the stockstarts at 5', its must thereforeprovidea of q. Capital of after value Al length time be ker-nt This meansthat step must one d expecte 'fhe
-
.
r'?z + (1 p l5'd= her-ct -
so that
(r-)A!
p=
-
d
u-d
As in the case of options on non-dividend-payingstocb, we match volatility bysetting and d 1/u. Thismeansthat we can use equations(11.13) 4=d I'VZ'E except to (11.16), = of ert. er-' instead thatwe set a a =
:::q
' .
Options on Stock lndics When calculating a futurespri for a stockindexin Chapter5 we assumedthat the stocks underlying the indexprovideda dividendyield at rate q. We make a similar assumptionhere.The valuation of an optionon a stockindexis thereforevery similar to the valuation pf an option on a stdckpayinga knowndividendyield.
fxample11.1 A stockindexis currntly 81 and has a volatility of 20% and a dividendgeld of 2% The risk-free rate is 5%. Figure11.11 showsthe output fromDerivaGem f0r valuing a EuropeanGmonth call option with a strikepri of 3 using a two-steptree. .
253
BmomialTrees
Two-steptr to valu a European(-month call option on an when the indexlevrl is 81, strike ?riceis 8, risk-free rate is 5%, index i 2%, and dividendyield is 2% (DerivaGemoutput). volatility
Strikeprice 800 Discountfador per step 0.9876 Timestep, tt 0.2500years,91,25 tays Grnwthfactcrper slep, a 1.0075 Prnbabilityof upmnve,p 0.5126 Upstep size, u 1.1052 Dnwn step size,d 0.9048 =
=
=
=
=
=
=
09.34 895.19 1cc.66 810.00 53,39
810.*
732.92 5.06 663.17 0.00 Nnde Time:
0.0000
0.2500
ln this case,
.zx4l7J j jgjz, 0.25, a= e 1/If 0.9048, a = e(.5-m)x.25 j ggyj (1.75 0.9048)/(1.10520.9048)= 0.5126 Lt
d= p
0.510
=
=
.
=
=
-
.
-
The value of the option is 53.39.
Options on Currencies 5.1, a foreip currencycan be regarded as an asset providing As pointed 0ut in section a yil.d at the fpreignrisk-freerate f interest?rj. Byanalor with the stockindexcase a tree for opons on a currency by using equations (11.13)to (11.16) we can construct (r-r/)! and settingc-= e .
fxample 11.2 ne Australiandollaris currentlyworth 0.61 U.s. dollarsand tltisdchange rate has a volatilityof 12%.TheAustraliaprisk-freeratt is 7% and the U.s. risk-free rateis 50/0.Figure11.12showsthe otput fromDerivaGemfor valuing a 3-month using a three-steptree. Americancall option with a strike prke of .600
.
254
CHAPTER 11
Fijure 11 2 Thr-step tree to value an American3-monthcall option on a ' when the value of-thecurren y is 6100strikepri risk-ffee is , currency risk-frierateis (berivaGem volatility is5%, is129:,andforeign 7% output). rate .1
.6
.
,
:
.
&
At each node: Asset Price Uppervalue= Underlying = Lowervalu: OptbnPrice is exercised Shadingindicateswhereqoptbn '
.
Strikeprice= 0.6 Discountfactor per step 0.9958 Time step, dt = 0.0833 years,30.42days Growthfactor per step, a = 0.9983 Probabilityof upmove,p 0.4673 Upstep size, u = 1,0352 Downstep size, d 0,9660 =
=
=
; r. . ;
. 0.610 0.019
0.632
0.654 j''k ' '
(.) j .kt' '.
:
t.
.,
0.632
$i',) ! 1( ) ,rj1
0.033
'
0.610 0.015
t.589 0.007
0.589 0.000
0,569 0.000
0.550 0.000
NodeTime:
0.0000
'
g ;
,/
.
,
0t77( 'L'
0.1667
0.0833
0.2500
ln this case, 9.98333 = 1.0352 A! ' 0.08333, u = e (.5-,7)x.8333 = 0.9983 d = 1/If 0.9660, a e .12x
=
=
=
p=
(0.99830.9660)/(1.0352 0.9660) -
-
=
0.4673
The valgeof the optionis 0.19.
(lptionson Futures It costs nothing to take a longor a short positionil a futurescontract. It followsthat in an expectedgrowth rate of zero. (We a risk-neutral world a futurespri should discussthis point in mori detail in Section16.7.)As abve, we dehne p as the probabilityof an up movementin the futuresprke, u as the percentage up movement, 'have
BinomialTrees .
..
'
25
andd as the pelcentage downmovement.lf '- is the initialfuturesprie, the expected futuresprke at the end of one tim step of lengthAl shouldalsobe /$. Thismeansthat
phu+ (1c pjhi
sothat
F= f'
.
,
,
,
1-J
p=
R-d
with a = 1. and we can use equations (11.13) to (11.16)
fxample11..3 A futuresprke is currehtly 31and llas a volatilityof 3%. The risk-free rate is 5%. Figure 11.13 shows the outpct frbm Derivadem fo/ valuing a g-month mericanput option with a strikepri 30 sing a three-steptree. 'f
Fijure 11 3 Three-steptrre to value an Amerkang-monthput option on a futurescontract when thefuturespri is 31,strikeprke is 3, rk-free rate is 50:, and volatility is 30%(DerivaGemoutput). .1
At qachnode: Uppervalue UnderlyingAsset Price Lowervalue = OptionPrice Shadingindicateswhere optbn is exercised =
Strik price = 30 Discountfactor per step = 0.9876 Timestep, dt s 0.2500 years, 91.25days Qrowthfaclor per step, a = 1.000 Probability of upmove,p = o4*6' Up step size, u 1.1618 Downste size, d 0.8607 ,
1.1618 d = 1/: = 1/1.1618= 0.8607, a = 1, p = (1 0.8607)/41.16180.8607)= Al = 0.25, a
=
e
'
zm
'0.4626
-
-
The value of the optionis 2.84.
SUMMARY Thischapterhas provided a rst look at the valuation of optionson stocksand other assets.In the simplesituationwheremovementsip the pri of a stockduringthe lifepf an option are governed by a one-stepbinolal tree, it is possible to set p a riskless Portfolioconsistingof a stockoptionand a rtain nm'ount of the stock.In a world withno arbitrageopportunities,risklessportfolios mustearn the risk-freeinterest.This enablesthe stockoptiont be priced in termsof the stock.It is interestingto note that no assumptionsare required abouttheprobabilitiesof up and downmovments in the stockpri at each node of the tree. When stockpri movementsare governedby a mulstep binomialtr, wecan treat eachbinomialstep separatelyand work backfromthe end of the lifeof the option to the begnning to obtain the current value of the option. Again only no-arbitrage arpments am used, and no assllmptionsam required about te probabilities of up and downmovementsin the stock price at each node. A very importantprinciple statesthat we can assumethe world is risk-neutrl when valuing an option. This chapter has shown,throughboth numerkal examples and algebra,tat no-arbitrage arpments and risk-neutralvaluation are equikalent and led to the same option prices. The delta of a stock ption, A, considersthe efect of a small change in the underlyingstockprice on the change in the option pri. It is the ratio of the change in the option price to the change in the stockpri. For a risklessposition, an investor shopldbuy A shams for each qptionsold.An inspectionof a tpical binomialtDe showsthat delta changes duringthe life of an option.This means that to hedge a particular option. position, we must change our holding in the underlying stock Periodically.
Constructkngbinomialtrees for valuing options on stock indis, currencies, and futures contracts is very similar to doing so for valuing options on stocks. In Chapter 19, we will return to binomialtrees and provide more detailson how they can be used in practi.
Pricing:A simplied Cox,J. C., s. A. Ross, and M. Rubinstein. Approach.''Joarzal of Fizazcial Ecozomics7 (October1979):229-64. Rendleman, R., and B. Bartter. $
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Any variablewhose value changes over timein an ncertin way is said to followa stocastic process. Stochtic prosses can be classifkb as discretetime or coztizuous time.A' discrete-timestochastic prcess is one wherethe value of tht vriabk can stochastic change only at certain ftxed poinss in time, whereas kontinuous-time place where changes time.Stochastic at any processs can alo can take processis one beclassised as contihuousvariableor discretevariable.ln a continuous-variableprocess, theunderlyingvariable can take any valuewithin a certain range, whereasin a discretevariabk ross, only rtain discrdt valqes art possibk. This chapterdevelopsa continuous-variable,continuous-timestochasticprocess for stockprices. Learningabout th pross is the srststep to understanding the priing of options and other more complicated deritvatives. lt should be noted that, in pmctice,we do not observe stock prices followingcontinuous-variable,continuousmultiples of a cent) timt proctssts. Stockpricts art rtstrictrd to diqcrdt valurs @.g., only khen the exchangeis open. Nvertheless, the and changes can be continuous-time pross proves to be a: ujeful model for many continuous-variable, xobserked
Ptlrposes. Many peoplefeelthat cotinuous-time stochasti processesare so complicated that scientistf'. Thisis not so. 'Fhebigges'thurdleto they shpuld be left entirely to understandingtheseprocessesisthe notation. Herewe present a step-by-stepapproach aimd at getting the reader over thishmdle.We>lsoexplainan importantresult known as It's lemmathat is central to the pricig of derivatims. dsrocket
)
12.1 THEMARKOVPROPERTY A Markoyprocess is a particular typeof stochastk processwhereonly the present value of a variable is relevantfor predictingthe future.The pastlstory of the variabk acd the way that thi presnt has emerged fromthe past are irrelevant. Stock prics ari usually assumed to followa Markov process.Supposethat the pri of lBM stock is $1 now. lf the stock prke followsa Markov process,our predictionsfor the future should be unafected by the prke one week ago, one month 2!9
'
,
260
CHAPTER 12 'p''
ago,or 0ne year ago. The only relevant piee of informationis that the priceis n0w $1.1 Predictionsf0r Jhe future are unrtain and mult be expressedin terms of
ppbability diqributions.The Mrkov propertyHjlies that the probabilitydistribution of the pri at any particularfuturetimeis n0t dependenton the partkular pyth followedby the jricein the past. ' The Markov propertyof stockpricesis consistent with the Feak Brm of market states thas the presentpri of a stockimpoundsall the ilbrmation eciency. in iecordof pastprices.If theweak formofmarketeciency werelpt true, contained analysts could makeabove-ayemge returns by interpretingcharts of the past technical vidente that theyare in factablt to do this. i very little historyof sttk'ltis. competition that tends in marketplace is the It to ensure that weak-formmarket eciency holds.Therearemanyinvestorswatchingthestockmarket losely.Tryingto make>prtt'from it leadsto a situationwherea stockprice,at anygiventime,reqectsthe information past pris. suppose that it was discoveredthat a particularpattern in stockpris alwajsgave a 65% chan of subsequentsteeppri rises. Investorswould attemptto buy a stock s soonas the patternwas observed,and demandf0r the stock would lead to an immediate re in its prke and the wouldimmediately would eliminated, be observedefect opportunities. as would any prttable irading rl-hi
'there
'
.
.
.in
'rise.
'th
12.2 CONTINUOUS-TIMESTOCHASTICPRQCESSES that its current Considera vprlablethat followsa Markovstpchasticprocess.suppose valueis 1 and that the chang: in its value during 1 year is 4(, 1), where km, p) denotesa probabilitydistributionthat is normally distributedwith mean m and varian p!z What is the probabilitydistribution0 y the change in the value of the 2 years? variable The change in 2 years is the sum of tw0 normal distributions,each of which has a meanof zero and variance of 1.. Becausethe variable is Mafkov,the tw0 probabilhy dtributionsare independent.WhenFe add twoindependentnorml distributions,the resultis a normal distributionwhere the meanisthe sumof the means andthe varian is the sum of the variances. The meanof the change during2 years in the variale we zero and the variance of this chanpe is 2.. Hence,the areconsidering is, fherefore, hangein the variable over2 years has the distribution4(, 2). ne ptandard deviation of the distribqtionis Xi. Considernext the change in the variable during 6 months.The variance of the changein the value of the variable during 1 year equals thevariance f the change duringthe Erst6 months plusthe variance of the change duringthe second6 months. We assumethese are the snme.It followsthat the varian of the chang during a Equivalen, thestandarddeviationof the change is 0.5. f-monthperiodmust be Theprobabilhydistributionf0r the chanpe in the value of the varkble during6 months is 4(, 'during
,
.
.5.
.5).
1 Sutistkal properties of the stnck price ilistnryof 1BMma7 be useful in determining the characteristicscf its volatility).ne point linz made hereis that the the stochastic pross followedby thestockpri @.g., particularpath followidbf the stock in the past is irrekvant. 2 Varian is thesquare of standard deviation.TheVarian of a l-yearchanze ipthevalue of thvariable we are consideringis therefnre1..
261
Wner Processesand .IJ% Lemma
argumentshows that the probability distributionfor the change in the value f the.variableduring3 months is (b 0.25).Mox genemlls the changi during anytimeperiod of lengh T is 940,f).In paricular, the change duringa vel'yshort time periodof lengthht is #(, A(). Note that, when Markovprocessesare considered, the variances of the hangesin time periods are attditive.The standard deviationsof the changes in successive time periods are not additive.ne variance of the change in the variable successive in our example is per year, so that the varian of the hangein 2 years ij 2. and thevarian of the change in 3 yep is 3.. The standarddeviatiopsof the changes in respectively,Strictlyspeakinj, we should not refer to the 2 and 3j years are ,Cfand stndar deviationof the variable as 1. per year. Thersults exjlain why Jmcertaintyis sometimesreferred to as beingproportioal to the squareroot of time. A
Similar
'1.
,/j,
Wiener Prpcesses The pross followedby the variable we hav been considering is knownas a. Wietr process.It is a particulartypeof Markovstochasticprocesswith a meanchange of zero and a variance rate of 1. per year.It hasbeeppsedin physic!to describeth motion f of smallmolecular shocksand is sometimes a particlethat is subjectto a larg rumber referredto as Browniattmotion. Expressedformally,a variable z followsa Wiener process if it hasthe followingtwo Properties: PROPERTY 1.
Te cbange Az darizga smallperiod of time Al
L'
Az =
6X
(12-1)
wheret: has a standardizedtrmal distribatioz. #(, 1). .
PROPERTY 2.
Te
of lz
yalues
frl#dld?l#dhlf.
for
Jhly fw0
dfrent sort itltervalsof
ffrrld,
Al, are
It followsfromthe srstpropertythat Az itse has a normaldistribuon with mean of Az = standarddeviationof Az = X-ht vriance of Az = A! The secondpropertyimpliesthat z followsa Markov process. Considerthe change in thevalueof z during relativelylongperiod of time,T. Tls can be denotedby z(T) z(). It can be regardedas the sllm of the changes in z in N smalltimeintefvalsof lengthAl, where -
N=-
Thus,
T A! N
z(z) .=,
z()
=
)-l ./''-At i
(12.2)
f=1
N) are distributeb #(, 1). We know from the second where the i i 1,2, Wiener propertj of prosses that the i are independentof each other. It follows .
.
.
,
22
CHAPTER lj
Fiju/e 12.1 Hw a Wienerpross is obtained when !
->
in equation(12.1).
z
Relavely largevalueof
f
z
t
Smallervalneof
l
t
.
ne trueprocessobtained as &
-y
0
Fcxr
263
tmtf It's Lemma Pyocesses
from equation
that z(T) (12.2) -
z() is normally distributed,with
meanQf z(T) z()) varianm fEz(T) z()) -
-
standarddviation' of (z(T) z()) -
= =
N !
=
Uf
=
T
Tltisis consistint withthe discussionearlier in this section. '
' ,
Example12. Stlpposethat the value,.z, of a variablethatfollowsa Wienerpross isinitially25 and that timeis measured in jearj. At the epd of 1'year,the value of the vible' is normallydistributedwith a mean f 25 and a standarddeviationof 1.. At the endof 5 years, it is normally distributedwith a mean of 25 and a standard of U!,or 2.236. Our unrtainty about the value of the variable at a deviation certaintimein the future,as measured by its standarddeviation,increasesas the squareroot of howfar we are lookingahed.
'
In ordinarycalculus, it is usual to proed frop smallchnges to the limitas the small changesbecomecloser to zero. Thus, tlz = adt is the notation used to indicatethat x a l in the limit a ! 0. Weuse similar notational conventions in stochastic So, when werefer to dz asa Wienerprocess,wemean thatit hasthe properties calculus. for z given above in the limit as l t.t Pigur'e12.1illustrateswhat happensto thepath followd by z as the limit l is quite of This is becausethe thgt path is Note the size apppached. a in z in time ! is proportionalto W and, when ! is small, W is much movement biggerthan !. Two intriguingpropertiesof Wienerptocesses,related to th W are as follows: property, -+
=
.
-+
Sjagged''.
1. The expected lengthof the path followtdby z in any timeintervalis infnite. 2. Thr expected numberof timesz equals any particularvalue in any timeintervalis infnite.
Generalked Wiener Pracess Th.emean change per .unit timefor a stochasticpross is known s the drft rate and the varian per uit timeis knwn as the variasce rate. ThebasicWienerprocess, dz, that has bn developedso far has a drift mte of zero and a varian rate of 1.. The drift rate of zero means that the expectedvalut of z at any futuretimeis equal to its currtnt value. The variance rate of 1. means that the prian of the change in z ip a for a variable x can be timeintervalof lengthT equals T. A gezeraled Ffcncrtrocess defnedin terms of dz as J.z = ait + bdz (1z.3) wherea and b are constants. it useful to consider the two components on the To understand equation (12.3), sideseparately.Theadt termimpliesthat x has an expecteddriftrate of a per right-hand unit of time. Without the bdz term, the quaon is Jz = adt, whichimplks that dxjdt= a. Integratingwith resptct to time,we get .x
=
xg + at
264
CAPTER
12
wherex is the value of at time0. ln a period of time of length T, the variable x increasesby an amout aT. Thebdz termon the right-hand side of equation (12.3) can be regarded as adding noise or variabilityto thepathfollowedby x. Theamount of this noise or variabilityis b times a Wienerprocess.A Wienerprcess has a standar deviationof 1.. It followsthatb timesa Wienerprocesshas a standarddeviationof In a smalltimeintervall!, the change lx in the value of is givenbyequations (12.1) and (12.3) as lx = c1/ + bkv-.x
.
.x
where, as before, 6 has a standard normal distribution.Thus A.x has a normal distribptionwith meanof Ax = 1/ stapdarddeviationof lx jxs 2 varianceof. A.x= b A! q
=
Similarargumentsto thosegivenfor a Wienerprocess sow thatthechangein the kalue of x in any timeintervalT is normally distributedwith meanof change in = aT standarddeviationof changein x b varian of change in = b2z .x
=
.x
has an expeted dri rate Thus,the generalizedWienerjrocessgivenin equation (12.3) (i.e., avtmge drift per unit of time)of a and a variance rate (i.e.,variane per unit of time) of b2 It is illustratedin Figure12.2. .
Figure 12.2 Valueof
variable,x
GeneralizedWiener pross with a
.3
=
and b
=
1.5.
oeneralized
Wknerpress tfz= c dt + b#:
tfz =
qdt
Wienerpress, dz
'I'tpe
' 1.
265
Wney PyocessesJ?lJ It's Lemma Exmple 12.2
Considerthe situaon where the csh'yositio: f a conpany, measurtdin thousandsof dollars,followsa generalizedWienerprocesswit,ha drift of 20 per year anda variance rate of 9 per yearkInitklly,the cash positionis 5. At theend of 1 yqar the cash positionwill havea hormal distributionwith a mean of 70 and a standarddeviationof 9, or 3. At the enb of 6 monthsit willhavea normal 21.21. 0ur distribuon with a mean of 60 and a stadard deviationof 30 unrtainty aboutthe cash positionat sometimein thefuture,as measured byits standarddeviation,increasesas the suap root of howfar ahead we are lookinj. Note that the cash position can becomenegative. (We can interprettllis as a situationwhere tht cbmpany is borrowingfunds.) .5
=
It Process A further type of stochastic process,knownas an It process,can be desned.illisis a generalizedWiner processin wlch thepanmetersa and b are functionsof the yalue of the underlying varible x an' tie t n It processcan be wfitten algebraicallyas .
d.x c(x, 1)dt + :(x, 1)dz
(12.4)
=
Both the expecteddrift rate and variance rate of an It pross are liableto change over time.In a small time intervalbetweent and t + Al, the variable changes from x to x + Ax, where J(I, l)! + g/, .l = '
.
sys
Thisrelationship invlves a small approximation.It assumesthat thedriftand varknce rate of x remain constant, qual to c(x, 1) and bx, 1)2 respec ve1y, duringthe time intervalbetweent and l + Al, ,
12.3 THEPROCESSFORA STOCKPkICE In this section we discussthe stochasticpross uiually assumedfor thepri of a non-
dividend-paying stock. It i!temptingto suggestthat a stockpricefollowsa gener>lizedWienerprocess;thatis, that it has a constant expecteddrift rate and a constant variance rate. However, this' modelfailsto capture a keyaspect of stock ptices.Tllisis that the expectedpercentage returnrequired byinvestorsfroma stock isindependentof the stock's pri. If investors requirea 14% per anhum expectedreturn when the stock prke is $1, then, ceteris paribas,theywill also require a 14%peracnumexpectedreturn when it is $50. Clearly,th assumption of constant expectd drift rate is inappropriateand neds to be replne-edby the assumptionthat the expeted return (i.e.,expecteddrift dividedby the stockprice) is constant. If S is the stockpiice at time 1, thenthe expecteddrift rate in S should be assumedt be PS for some constantpammeter p.. Thismeans that in a shortintervalof time, Al, the expectedincreasein S is P.SA!. The parameter Jz is the expectedrate f return on the stock,expressedin decimalform. If the volatilityof the stockprice is alwayszero, then tllis model impliesthat LS = pst
264
CHAPTER 12 ln the limit, as A!
->
,
dz
psdt
=
0f
dS
-
pdt
=
' .
.
lntepating betkeentime0 and time T, we get :T
sT t =
(125) .
'
.
shows that, whereh and St are the stockjri at time and time T. Equation(12.5) variance price continuosly rate is zero, thestock whenthe compoundedrate growsat a Of unit of time. p per In practice,of course,a stockprice doesexhibitvolatility.A reasonable assumptionis that the variability of the percentagereturn in a short period of time, A!, is the same regardlessof the stock price. In other words, an investoris justas unertain of the pementagereturn when the stockprice is $5 as when it is $1. This suggeststhat the standarddeviationof the change in a shor!period oftime A! shouldbe proportional to the stock price and leadst the lodel '
.
dz
=
pfzdt + o'zdz
or
T=p
dt + cJz
(12.6)
Equation(12.6) isthe most widelyusedmdel of stock pri behavior.The vqriable c is th volatility of the stock prke. The variable p is its expectedrateof return. The model in equation (12.6) can be regarded as thelimitingcase of the random walk represented by the binomialtrees in Chapter11 as the time step becomessmaller.
Discrete-TimeModel The model of stockpri behaviorwe havedevelopedis knownas geometricBrowziaz motioh.The discrete-ti version of the model is
hs phtsv'-i T = .
0r
hs
=
gs'l-hcsk:-'
(12.7) (12.8)
The variable LS is the chanze in the stock price, S, in a smalltimeintervalA!, and 6 has a standardnqrmal distribution(i.e., a normal distributionwith a mean of zero and standarddevition of 1.). The parameterp is the expected rate of return per unit of tiine from the stock and the parameterc is the volatility of the stock prke. In this chapterwe will assumetheseparnmeters are constant. The left-hapdsideof equation (12.7) is the return provided by the stockin a short periodof tim, A!. The term p A! is the expectedvalue of tllis return, and the term cfa/'-f is the stochasticcomponentof the return. The variance of the stochastic component(and,therefore,of ihewhole return) is t. This is consistentwith the definitionof the volatility c given in Section11.7; that is, c is such that c,/-f is the standarddeviationof the return in a short timeperiod A!.
WieneyPypc-:
and
Equation
267
f/J': Lemma
XS/S is normally distribttd with mtan p, ! and words, other In
(12.7)shows that
standarddeviationcu-.
ss itl, S 'v'
c20
(12.9)
-
fxample12.3 Considera stotk that pays no dividends,has a volatilityof 30% per annum, and providesan expetd return of 15% per annum with continuouscomppunding. Ip apd c Thi pross for the stockpfi is t1118 case,Jz = k15
.3.
=
dS
T=
0.15:1+ ().3#z
lf S is the stock pri at a particular timeand XS istheincreasein te in the next smallintervalof time, XS
.15l
=
+
Stock
price
0.306X
where6 has a standard normal distriution. Considera timeintervalof 1 week, or 0.0192year, so that l 0.0192.Then '
'
=
XS
T
0r
k
=
= 0.00288+ 0.04166 0.90288:+0.0416,%
Monte Carlo Simulation MonteCarlo simulationof a stochastkprocess is a produre for samplingrandom outcomesfor the process. Wewill use it as a way of developingsome understandingof the nature of the stock price process in equation (12.6). Considerthe situation ih Example 12.3 where the expected return from a stock is 15% ptr apnumand the volatiliyyiq 30% per annum. The jtock price change over 1 week was shown to be XS = 0.00288,+ 0.0416,% (12.19) A path for the stockprice over 1 weekscan be simulated by samplingrepeatedlyfor e The xpression=RANDI from44, 1) and Substituting into equatin (12.1). ) in Excel producesa random snmplebetween and 1.ne inversecumulativenormal distribution is NORMSINV. The instHction to produce a random sample froma standard normal distributionin Excel istherefore=NORVSINVIRANDI )).Table 12.1shows one path initl stock plice is assumedto be for a stockpri that was sampledin this way. equation period, sampled $1, Forthefirst as 9.52.From e is (12.1),the change during thefrst timeperiod is XS = 0.00288x 1 + 0.0416x 1(# x 0.52 2.45 '
'rhe
.
,
=
Therefore,at the beginningof the second timeperiod, the stock price is $102.45. The valueof E: sampledforthe next period is 1.44. rom equation (12.1),the change during the second time period is XS = 0.00288x 102.45+ 0.0416x 102.45x 1.44 = 6.43
'
28
'
..
..
' .
.
. .
CHAPTER 12 of stockpri when p Table 12.1 simulation during l-weekperiods. c=
.15
=
and
.3
Stockprice at start of teriod l. 102.45 108.88 15.3 112. 19.11 106.06 17.3 102.69 16.11 111.54
Rasdom sample Caqe fl stock price dariq period for6
2.45 6.43
0.52 1.44 -.86
-3.58
1.46 -0.69 -g,yj 0.21
6.% -2.89 -3.04 1.23 -4.6
-1.1
0.73 1.16 2.56
3.41 5.43 12.20
and so on. Notethat, So, at the beginningof the next period, thestockpri is $108.88, becausethe processwe ari simulatingis Markov,thesamplesfor 6 shouldbe independentof each ?ther.3 Table12.1assumes yhatstockprices are measured to the nearest nt. It is important So realize that the table showsonly one possible pattrn qf stock price niovements. Dferent random sampleswould lead to difereni price movements, Any smalltime intervalht can be used in the simulation.In the limitas ht a perfect description ofthe stochasticprocessis obtained. ne nal jtock price of 111.54in Table12.1can be regardedas a random snmplefrom the distributionof s'tockpris at the end of l weeks. By repeatedly simulatingmovementsin the stock pri, a complde prlbabilitydistributionof the stockprice at the end of tls timeis obtained. MonteCarlo is discussedin mo4 detailin Chapter19. simulation -+
,
12.4 THE PARAMETEFS The proqs for a stockpri developedin tls chapter involvestwo pammeters, p and earned by an investorin a short 0..The parameter p isthe expted return (annualized) periodof time. Vostinvestorsrequir lgher expected returnsto inducethem to take higherrisks. It followsthat the value of p shoulddependon te risk of the retuyn from the stock,4 It should also dependon the levelof interestrates in the economy. ne lgher the levelof interestrates, the gher the expecte return required on any given
stock.
Fortunately,we do not haveto concern ourselveswith thedeterminantsof p in any detailbecausethe value of a derivativedependenton a stockis,in general, independent 19.6. in section 3 In practice, it is more ecknt to sampk lnS rater thn S, aj will bedipcussed 4 More precisely,p depeds on that part of tile risk tilat cannot be diversed away by theinvestor.
269
Fexer Processesand Its's Lemma
ofp,. The parameter the stockprice volatilit'y,is,by contrast, critically nportant to the determin>tionof tlte value of many derivatives,We will dijcuss procdures for Of c in Chapter13.Typicalvalues c for a stockare in the range 15to 0.60 estimating 0/ ( i .C., 15b/() tii 10 ()) ,
.
.
.
The standard deviationof the proportional chnge in the stock price in a small intetkalof time Lt c/i. As a rough approximation,the standarddeviationof the proportionalchange in the stotk price overa relativelylong period f time T is cu/. Th means that, as an approximation, volatility can be interpretedas the standard deviationof the change in thestockpri in 1 year. In Chppter13,we will show that the volatilityof a stock pri is exactlyequal to the standard deviatio of the continuously compoundedreturn provided by the stockin 1 year.
12.s IT'5
LEMMA
The price of a stock option a functionof the underlyingstock's price and time.More genetally,we can say that the price of any derivtive a functionof the stochastic variablesunderlying the derivajve and time.A Serious student of derivativesmust, therefore,acquire some und.erstanding of th behaviorof functions of stochgstic variables.An important result in this area was discoveredby the mathematkian K. It in 1951,5 arld is knownas It's lemma. Supposethat the value of a variabk x fllows the 1t process J.x = c(x, !) dt + :(x, tj dz
(12.11)
variable x has a wheredz is a Wiener proess and a and b are functionsof x and !. shows 7 of variance of and b G of x and t It's lemma that function rate a driftrate a a fllowsthe process 'l'he
.
dG
=
'G
V
a+
3G
V
32G 2 G bdz + 12 xl b dt + .V
(1:.12) .
.
.
.. ;
'.
s
.
'
.
.
1).Thus, G also followsan wherethe dz isthe sameWiener processas in equation (12.1 1t process, with a drift rate of 'G
17
a+
and a variace rate of
J 2G 2 + 12 x2 b 3F 96
2
(.)' bc
A completely rigorous prof of It's lemmais beyondthe scopeof th book. In the appendixto this chapter, we showthatthelemmacan be viewedas an extensionof wellknownresults in diferentialcalculus. Earlier, we argued that ds p'sdt + o'sdz (12,13) I
,
=
5 Ste K. lt, 1-51. 4 (1951):
on StochasticDitlbrtntial Equitions,''
Memos
t).f
tl Xmcrctm McfemnffcnlSociety,
270
CHAPTER 12
with p and c constant, is a masonablemodelof 'stock prite movements. From Ii's lemma,it followsthat the process followedby a funcon (7 of S and l is 'G
dG =
S
. '
.
.
,
347 ) 320 2 2 'G o'zdz + 2 2 c S dt + 3.$ bS t
pS +
.
(12.14)
'
,
Note that both S and G are pfectedby the pmeunderlying sour of uncertaints dz. importantin the derivationof the Black-cholesresplts. This proves 4obe 'very
APPIicationto Forward Contracts To illustrattIt's kmma, condtr a fmard contmct on a non-dividend-payingjtock. . Assumethat the rijfree rate of interestis constant and equal to r for a1lmatqrities. Fromequation(5.1), rT F'g= 5'g: '
x
.
.
whereF is the fomard price at timezero, k is the spot pric at timezero, and T is the time to maturityof the forwprdcontract. ' We are interestedin what happensto thefomard price as timepasses. WedehneF as theforwardprice at a general time1, and S as the stock pri at timet, with t < T, The reltionship between# and S is given by F
serT-tj
=
(jz
.
jg)
Assumingthat the process f0r S is givenby equation (12.13), wecan use It's lemmato determinethe process for F. From equation (12.15), 'F --
=
bs
2,
rtz-t)
e
sz
j
og
j
JA= bt
.rserT-t)
=
the pross fol' F is given by From equation(12.14), r(T-l)
#>- h =
Su bstitutingF for
SerT-t'
p
s
?.pr(T-l)j
-
a+
e-tqgsdg
gives dF
=
p
Fdt + cF#z
-
(12.16)
Like S, the forwardpri F followsgeoletric Brownianmotion. lt has an expected growthrate of p r rather than p. The growth rate in l is the exss rturn of S over the risk-free rate. -
12.6 THE LOGNORMAL PROPERTY Wenowuse It's lemmato derivttht pross followtdbyln S whtn Sfollowythe process in equation (12.13). Wedefme c = jn s Since ' 2 J G 1 'G = 'G = 1 = 31 'S S ps' S1 '-
,
a
=
-
-
y
FdAld?'
271
Ptncessesand ftl': Lemma
it followsfromequation(12.14) that the process followedby G is 2
dG =
c dt + dz c p. 2 -
(12.17)
-
*
.
.
.
.
h.
.
.
.
;. . .
.
.
.
7
.
'
'
Sinceg and o' are constant, thisequationindicatesthat = ln S fllows a generalized Wienerprocess. It has constant drift rate p. c 2/ 2 and constant varian rate c2. The changrin lnS betweentime and somefuturetimeT isthereforenormally distributed, with mea (# c2/2)T and varian c2F. This meanstikat -
'
-
lns,
-
or
ln Sz ew
ln-t
-
pgts
4g1n k+
-
(Jz -
))z,Jzj
(12.18)
f)z,pzj
(12.19)
whereSz is the stock prke at a future timeT, h is the stockprke at time0, and as beforekm, 1l) denotesa nrmal distributionwith mean m and variance 1l. Equation(12.19) showsthatlnSzis normqllydistributed.A variable has a lognormal distributionif the natural logarithm f the variable is normally distributed.The model f stpckprice behaviorwe havedevelopedin th chapter thereforeimpliesthat a stock's Priceat time T, given its price today,islopormallydistributed.Thestandarddeviaiion f the logqrithmof the ttock pri is o'X.It is proprtional to the square root of how far ahead we are looking.
SUMMARY processes describethe probabilistic evoluon of the value of a variable stochastic throughtime.A Markovproces is one where only the presentvalue of tt' variable The past histeryof the variable and the way in is relevant for predicting the whichthe present has emerged fromthe past is irmlevant. 'future.
A Wienerprocess dz is'a process describingthe evolution of a normally distributed Thedrift of the pross is zero and the variance rate is 1.0per unit time.This variable. nrmatly meansthat, if the value o.f the variable is z at time then at time T it is d with mean x and standarddeviationX. distribute gneralizedWiener process describesthe evolution of a normallydistributed with a drift of a per unit timeand a varian rate of b2 per unit time where variable of the variable is zt at a and b are constants. This means that if, as before,the and w1t.* normally aT dtributed a standard eviation of a mean of z + time0, it is b at time T. An It prcess is a pross whire thedrift and varian rate of x can be a functionof bothx itself and time. The change lnz in a very short periodof timeis, to a good normally distributed,but itS change overlongeryriods of timeisliable approximation, o benonnormal. Oneway f gainig an intuitiveunderstandingof a stochasticprocess for a variable is to simulate the behaviorof the variable. This involvesdividinga time intervalinto many smalltime steps and rqndomly samplingpossible paths for the variable. The future probability, distributionfor the variable can then be calculated. Monte Carlo simulationis discussedfurtherin Chapter19. ,
,
'value
:.
)y
CHAPTER 12
It's lemmais a way of calculatingthe stochasticprocessfollowedby a funkn of >. the stpchasticprocess fotlowedby the variable itself. s we sall see in variable'from Chapter13,It's lemmaplays a very importantpart in the pricing of derivatives.A key pointis that the Wienerpross dz underlyingthe stochasticpross for the variable is ihesame as theWienerprocessunderlyingthe stochasticprocessfor thefunctio exqctly variable. of the Both are subjectto the same underlying sourceof uncertainty. The stochasticprocess usually assumed for a stock prke is geometric Brownian motion.Under th process the returnto the'holderof the stck in a Small period of time is normally dtributed and the returns in tw ponoverlappingperiods are independent.The value of thestockpri at a futuretimehas a lognormaldtribution. TheBlack-scholesmdel, whichwecoverin the nert hapter,isbsed on the geometrk Brownianmotion assumption. '
FURTHERREADING On ffficiel)t Makets and theMarkopPoperty o StockPrice: Brealey, R.A. z4< Itttroductitmto Ak ald AeflrrlfromCt?plr?*:Stock, 2nd edn. Cambridge, MA:MIT Press, 1986. Te 'tmffpm Caracter ofstock MarketPrics. Cambridge,MA:MITPress, Cootner,P. H. (ed.) 1964,
On StochasticProcee: Cox,D. R., and H. D. Miller,Te Teory ofStocastic Processes.London: Chapman&Ha11,1977. Feller,%. lrl/rtWlcnbrlto Probability Teory cplffIts Applicatioas.New York: Wily, 1968, Karlin, S.: and H.M. Taylor. zl First Coarsefrl Stocastic Processes, 2nd edn. New Ynrk: AcademlcPress, 1975. Neftci,s. Introdactioz to Matematics 0./' Finatial Derivatives,2nd edn. NewYork:Academic Press, z.
and Problems(Anjwersin SolutionsManual) Questions 12.1.What would it mean to assert that the temperatureat a rtain place fllows a Markov pross? Do you think that temperaturesd0, in fact,followa Markovpross? 12.2.Can a tradingrule based on the past ltory of a stock's pri ever produ returns that j are constent y above average?Discuss. 12.3. company's cash position, measured in millions of dollars, followsa generalized Wienerprocess with a drift rate f per quarterand a varianw rate of 4. per quarter. H0whighdoestht company's inhialcashposition hayeto bef0r the company to havea lessthan 5% chance of a negativt cash position by tv tnd of 1 year? Wienerprocesses,Fith drift rates Jz) an# /12 and VariablesLX nd Xj followyeneralized o'1 and 2 What pross does Zl + X2 followif: variances changes in :1 and j in any shortintervalof time are uncorrelated? (a) The (b) Thereis a correlatin p betweenthe changesin Xl and z in any shrt timeinterval? 12.5.Considera vafible S that followsthe process .5
.
'
dS = p, dt + tr
273
WienerProcessesand It's Lemma
For the hrst tkee years, p' = 2 and c = 3;for the next threefears, y' = 3 and ?.= 4. If probability value of variable distributionof the value of the is 5, whatis the theinitial thevarkble at the end of year 6? that oj and cc are the that G is a funcoc of a stock prlct S andtime.suppose 12.6. suppose volatilitiesof S and G. show that, when the expted return of S increasesby koj, the groFth rate of G increasesby cc, where is a constant. 12.7.jtockA and stock B both fdllowgeometric Brownianmotin. Changesin any short intervalof timeare uncorrelatedwith eah other. Ddesthe value pf a portfolioconsisting of one of stock A and one f stockB followgemetrk Brownianmotion? Ex/lain yof is 12.8.The process fof the stock pricein equation (12.8)
hs
+c.f7
gsnt
=
,
wherep' and c are constant.Explaipcarefullythediserence etweenthis mode1and each of the following: =
hs h. h.
g
at + cxu-j
=
gvht-
=
ght-
o'fu-f
cs'eu''
Whyis the model in equation (12.8) a more appropriate modelof stock pricebehakior than any of thesethree alternatives? 129 It has bee u'ggested that the short-term interestrate r followsthe stochastic process
dr
=
ab
r) dt + rc #
-
wherea, b, c are positive constantsand #2is a Wienerprocess. Describethe nature of this process. 12.1. suppose that a stock prke S followsgeometrk Brownianmotion with expectedreturn y' and volatility c: dv 2: P'Sdt + o'vdz What is the process followedby the variable Sn Brownianmoon.
that Sn also followsgeome4ic show
that is the yield to matmity with continuouscompounding on a zero-coupon 12.11.suppose ' bond that pays ofl-$1 at tlme T. Assumethat followsthe process .x
.x
dx
=
Jtx
-
# dt + ix
z
and s are psitive constants and Jz is a Wienerprocess. Whatis the process wherea, followedby the bond prite? ' .p,
AssignmentQuestions 12.12.suppose that a stockprice has an expectedretuzn of 16%per annum and a volglity of 30% per annum. Whenthe stock prke at the end of a certainday is $5, calculatethe following: (a) The expected stock friceat the end of the next day. (b) The standard deviationpf the stock price at the end of the next day. (c) The 95% covdence limitsfor the stpck prke at the end of the pext day.
274
CHAPTER 12
12.13.A company'scash position, measured in millionsof dollars, followsa generalized Wienerprocesswith a drift rate of per month anLa varian rate of 0.16per month. The initialcash positionis 2.. (a) Whatare the probabilitydistributionsof thecashpositionafter 1 month, 6 monyhs, and 1 year? (b) What are the probabilitiesof a nigaiive cashpositionat the end of 6 months and 1 year? (c) At what time in the futttreis the probability of a negative caqhpositiongreatest? 12l4. suppos'tatis the yield on a:perpetualgovernmentbondthat pays interestat the rate of $1/er annllm.Assllmethat is expressedwith continuouscompounding, that interest is paid continuouslyon the bond,and that followsthe pross Jz = c(.x # dt + sx dz .1
.
.x
.x
.x
-
wherec, and s are positive constants,and dz is a Wienerpross. What the process follwd bythe bond price?Whatisthe expectedinstantaneos retup (ipqluding injerest . and capitalgains)to the hqlderof the bond? what is the 12.15.If S followstht geometric Brownianmotion process in equajion (12.6), folloWed by process (a) y = IS (b) y S2 @) y = e efT-tsjs (d) y In each case express the coecients of dt and dz in terim of y rather yhanS. 12.16.A stock price is currently 5. lt$ vxpected return ad volality are 12% afld 3%, respectively. Whatisthe probability thatthestockprke 111be greater than8 in 2 years? > () when ln ST > ln8.) llistl .x,
=
=
,z
WienerProcessesand It's
27!
JMF?IF/IJ
APPENDIX DERIVATIONoq IT'5 LEMMA
'
.
.
,
In th appendix, we showhowIt's.lemma can be regarded as a natuml extension of Considera continpous and diferentiablefunction G of a 0 ther, jimpler variablex. If Ax is a smallchange in and G k the resulting smallchange in G, a well-knownresult from ordinary calctllusis 'results.
.x
dG AG r -Ax
(12A.1)
In other words, AGis approximatelyequal to the rate of change of G with respect to x multipliedby Ax. The error involvestermsof orderAx2 If moreprecision is required, a Taylorseriesexpansion of AG can be used: ' .
.
'
AG
=
dG tfx:
Ax + ;1 u
.
.
.
d jc
2
Ax + j6
a2 .
.. .
-.
;3g
a3
n
3
Ax +
.
.
.
:
r. . For a continuous and diferentiablefgnctionG ot two variables x and y, the result analogousto equation (12A.1)is .
.
.
;
.
,
.
'G JG AG rks Ax + Ay
17
(12A.2)
-y
and the Taylorseriesexpansion of AG is LG
=
G
G
Ax +
y
.z
2G
hy + j2
2
.z
2
A.z +
#2G y
.z
Axhy + j2
2G 2
y
2
hy
.y
:
.
.
(jax.y
In the limit, as Ax and Ay tend t9 zero, equation (12A.3)becomes dG
=
'G
tfx +
JG
VV
dy
(12A.4)
(12A.4)to coverftmctionsof variablesfollowingIt processes. variable Supposethat a x followsthe lt process
WC n0W Cxtend Cqtlatioll
dx c(.z,t) dt + =
(x,t)dz
(12A.5)
and that G is some functionof x and of timet. By analogy with equatioq (12A.3),we
Can Write
LG =
'G
V
Ax +
G
V
2G
ht + )2
.t
2
2
Ax +
J2G i.z t
hxht + )2
32G 2 Ap +
t
.
.
.
(jax
.
j)
Equation(12A.5)can be discretizedto Ax
or, if arguments are dropped,
=
c(x, t) ht +
Ax = aht
tx,t)6,/'-E
+1:G
(12A.7)
276
CHAPTER 12 This equation revealsan importantdference betweenthe situation in equation (12A.6) and the situationin equation (12A.3).Whenlimisihgarguments wefe ujed to move fromequation (l2A.3) to equation (12A.4),terms in A.x2 wereignored becausethey Fere second-orderternp. From.equation(12A.7),we have 2ht Ax2= :2: 4 termsof higherorderin Al
(12A.8)
This showsthat the terminvolvingA.x2 in equation (12Ak6) has a comjonentthat is of order ht pnd cannot be ipored. The variance of a standardized normal distributionis 1.0.Tltis means that
'(:2) gf(4)2 1 =
-
2
E denotesexpectedvalue. Sin F(4 0, it followsthat f(6 ) 1. The expected where 621t V terefore, is At. It can beshownthat th varian of 6214 is of order aluecf 2 =
=
,
' that, As a result,we can treat6 and
,
Al as nonstochasticand ual to its expectedvalue, A(, as ht tends to zero. It followsfrom equation (12A.8)that A.x2 becomesnonand equalto b3dtas ht tendsto zero. Taking limitsas A.xand Al tendto zero stochastk equation (12A.6),and usingthis last result, we obtain in dG =
G
z
dx +
JG
t
dt + j2
2G 2
3z
2
b dt
(12A.9)
This is It's lemma.If we substitute for #.xfrom equation (12A.5),equation (12A,9)
beconles
'
dG
=
G
a+
G
2
+
j
2
G 2
2 b 31+
G
bdz
pla ksetqs-
Merton Mdel
In the early 197s, Fischer Black, MyrpnScholes,and Robert Vertonachieved a ajor breakthroughin the prking of stockoptions.1Thisinkolvedthe developmentof 1. Wht has becomtkngwnas the Black-scholes (orBlack-scholes-Merton) odel. The modelhas had a hugeiniuence on the waythat tiadersprice and hedgeoptions. It has lso been pivotal to the growth and suess of fmancialengineeringin the last 30 years. In 1997,the importanceof the model was recognized whenxobert Merton and Myronscholes FischerBlack were awatded theNohelprize foreconomks.sadly, diedin 1995,otherwisehe too Fould undoubtedly havebeen one of the recipients of 'this prize. This chapter shows howthe Black-scholesmodel for valuing Europeancall and put options on a non-dividend-payingstock is derived.It explains how volatility can be eitherestimatedfromMstoricaldata or impliedfrom option prics using the model. It showshowthe risk-neutral valuation arpment introducedin Chapter11 can be used. It.alsoshows howthe Black-scholesmodel can beextendedto dealwith Europeancall stocks and presents some results on the pricing of and put options on dividend-paying stocks, Americancall options on dividend-paying .
13.1 COGNORMALPROPERTYOF STUCKPRICES The niodel of stock prke behaviorused Y Black, Scholes,and Mertonisthe model we in Chapter 12.It assumes that perntage changes in the stock price in a developed shortperiod of time a?enormally distrilmted.Dene Expectedreturn o stock per year c: Volatilityof the stock price per year
Jz:
The mean of the return in time i
l
is JzA! and ihestandard deviationof the return is
Pricing of Optionsand Corporate Liabilitiesy''Jourzal of Political ts-rheory of Rational OptionPridngs'' BellJoarnal of Economy,81 (May/lune 1973):637-5%R.C. Merton, 1973):14143. Science,4 (spring Ecozomicsand Mcplcge?l'lepl/
seeF. 3lack and M. scholes,
rf'he
277
278
CHAPTER 13 so thai
c:'-f
(S
. .
'
. .
ewjjy
ht,
g
2
h j;
(13.1)
.
whereLS is tlle changein the kockprke S in time ht, and (m, p) denotesa normal distributionwith mean m and varian p. 12.6, th model impliesthat As shown in section J
.
2
' -))z,
tnsr
gtp.
.-lns
-
From ti s, it followsthat
s lnt
pj
'-gjislz,
and
2 J-j-lz,
pglil.g
lns,.
+
-
pzj
(/z
(13.2) czj
(13.3)
-
is the stock price at a futuretime T and h is the stock price at time shows that ln Sy is normally distributed,so that has a lpgnormal (13.3) /2)T and the standard dviation distribution.The mean of ln is ln h + p is cu/.
where
ir
.
Equation
iz
'
-
iz
fxample1.11 Considera stock with an initialprice of $40,an expected return of 16% per the probability nnum,and a volatilityof 20% ptr annum. Fromequation(13.3), of the stock pri S in 6 months' timeis givenby distribution ln Sz .
.
(ln4+ (.16 /3.759,
'x,
-
.
0.22/2)x
.5,
0.22x 0.5) '
.7)
ln
'w
Kz
Thereis a 95% probability that a normally distributedvariable has a value within 1.96 standard deviationsof its mean. ln this case, the standard deviationis 0.02= Hence,with 95%consdence, .141.
3.759 1.96 x -
This can be written e
lnS
.141
11n(&)), s0 that c2/2.) leadsto Exj < Jt. (As pointed out abovi Exj =
,
'llnts'r/,%ll
+ rF > Xsc tend t zero, h apd k tend,to . Thisimpliesthat ln t,%/& . d #/2) tend to 1. and equation (13.2)become! +x, so that #() .
c
=
Sg
-C
'r
Whenh < Ke-rr it follws that lntS/m + rF < As c tends to zero, dj an d2 tend '' to so that Ndjj and Nd tend t zero and equation (13.2)givesa call price of Ke-rT h T ) as g tendsto zer. Similarly, rero. e ca11pri is therdore alwaysmaxts -rT = S ) as c tendstc zerc. it can be shown that the put priceis alwaysmaxtre .
,
-x
-
,
,
13.9 CUMULATIVE NORMALDISTRIBUTIONFUNCTION is in calculayingthe The only problemin implementingequations (13.2)and (13.21) cumulativenormal distributionfunctioh,Nxj. Tablesfor #(# are providedat the ed of this book. The NORMSDISTfunction calculates N(z) in Excel.A polynomial appro/mation that givessix-decimal-placeauracy is7
1 1
-
N
=
-
J l + a 212+ ajk + a4k4 + ajk5) when x p N (x)(c, N(-x) when x < ()
where k=
,
y = 0.2316419
1 + yx 0.319381530, a
c, 1 81477937 c4 = =
.7
th =
1
,
-0.356563782
=
-1.821255978
,
J5
=
1.3302t4429
7 HandbookofMatematicalFuzctioas.NewYork: DoverPublications, seeM. Abramowitzand Ir stegun, 1972,
294
CHAPT:R 13
apd
fxamplet3,6
'
r
.
The stockprice 6 months frm the erpirationof an optionis $42,the exercise Price f te optionis $40,te riskufreeinjeyst rat is 1% per afmum,and the ,
vlatilisy is 2% prr annum.nis means that S = 42, k = 4, r T = ()5 . 2 1n(42/4)140.1 + /2)x = 0.7693 k = . ln(42/4)+ (.1 ()22/2)x'.5 :=F0.6278 k= 2.2+ and Ke-rz 4o-s.t5 ?s g49 .2
.1,
=
c
:::F(t2,
.5
.5
.2
-
.
=
=
.
Hence,if the optionis a Europeancall, its value c is given by c 42#4.7693) 38k049#40.6278) =
-
If the option is a Europeanput, its value p is pivenby
Ignoringthe tipe value of money, the stick pri has to yise by $2.76for the purchaserOf the call to breakeven. Similarly,the stockpli has to fallby $2.81 for the purchaser of the put to breakeven.
STOCKOPTIONS 13.10 WARRANTSAND EMPLOYEE The exerciseof a regular call option on a company has no efect on the number of the shreq outstanding.If the writer o the option doesnot ownthe company's company's he shares, or se must buythim in the market in the usual wayand then sell themto the optionholderfor the strikepri: As explained in Chapter8, warrants and employee stock options are diferent from replar call options in that exercise leads to the companyissuingmore sharesand thensellingthemto the optio holderfor the strike price.As the strikepli isless than the market price,tilisdilutesthe interestof the existingshareholders. How should potential dilution afkct the way we value outsnding warrants and employeestock options?The answeris that it should not! Assumingmarkets are
/95
The Black-scholqs-Meyt6n Mldel
Emjlpyt: jtk ugpldjnppgoi lj-j Wartqtsk,
Ojtihs; and biltion
tC.E lt Ii ptpiij Cnsldii a cdijklk with haiesiwh Apit thr y/ket kith j yttxjrytytgu )# rjj, )it j yjjyjsjs jjg wjtj a ) jyj ali >oupt fht j , :g y o y %)k, mt' pitgjjqs jtrike price f $r% If t littlibtiitfj t 4t1rjyrpldis tf ih ; tjjzyy ysstjvtu y ayyj jj j jyjjryja l st49k jygs.tj rpployet k 4s, J' ?) ins w qti r g, 4c1ii maaiirs? te qjk Ep7 rke Qll immiijily lafti: djzleht f the t elpiowt jtoct iptipps. Ete tjtct EpfickWhllpltski dttikn cs'i t 'qE shkrilkolz, $3 il, 1. i i s ( : k jiti (t q cuitens llEttpil, jltgt yrrjrtls, 4heiplpkpf tll j?( lat ptrtMw. syjjsr thtrjtljppsi tli.gilpsjyii fpythqq ll is i tE, #, i piie 'ptijjErii lrjt ! gEij tik iiiylytrsik $30 ijtltglt il) ii p: tpdilutipp' Tii''k'efi'stlrlfi'ski r jE'ilkt nlktliilid slk' io' bith ftitihfEy # fj)'Ejl jj) jr, tu jt ytjkrjjj' ; j, rj)sjjyyjj,jjjj, j jkuk ysjj hlc shilqqy tjtj; 0, ljt'j y : Ethlyqn, tf ) yjoy ytyyjys, yyjytoyyj, yiigpEg, , j t gyyj, tjjyyy,yjj, jj
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eciint the stock price will reqect potential diiutionfroma11outstandingwarrants and employee stock options.'Tilis is explainedin BusinessSnapshot13.3.8 Considernext the situaon a companyis in when it )s cntemp lting a new issueof warrants (or nployt' stock options).We supposethat the company is interestedin calculating the cost ot the issueassumingthatthere are no compensatingbenets. We ' eah and the number 0 j new optjons assumethat the company has N shares worth contemplatedis M, with each option givingtheholderthe right to buy ohe sharef0r K. value doesnot changeas a resnlt of the The value of th company todayis NS. without issuethe shareprice will be Sz at the the issut. Supposethat warpnt warrant warrant'smaturity. This mans that (with or without thewariantissue)the totalvalue will equity k%. If thewarrant are exercised,thereis a ofthe and the warrants at timeT cashiniw fromthe strikeprice increasingthisto NSI. + MK. This value is distributed amongN + M shares, so that the shafeprke immediatelyafter exercisebecomes k
rnlis
N% + MK N+M Therefore the payos to an optionholderif the optionis exercisedis N% MK - K N+M '>
8 Analystssomdimes assume that the sum of the valuesof the warrants and the equity (rather thanjustthe valueof the equity) is lognormal.The rtsult is a Black-kholes typeof equation for the valueof thewarrant in tcrmsof the value of the warrant. See TechnicatNote3 on the author's websitefor an explanationof this model.
13 UHAPTZR
29
'
.
0r
N X+ Y ..
'
.
fv S
'l
Z-
X
.
,
This shows that te value of eachoptio is the ,vlpe ef N N+ M
call options on tht company's stock. M timeqthis.
regtgar
'therefoie
ihetotal cost of the options is
fxample33.7 A company with 1 million shares worth $4 eachis considerinjissuing2, warrantseachgivingth hol ler thetight to buyone sharewith a strikeprice of $6 in 5 years. It wants to khowthe cost of this.ne interestrate is3% per annum,and the vlatility i 3% per annum. The.companyrpafs.npdikidends.From'equatio (13k2), the plue of a sryear Eulppeancall option pn the stockis $7;04.In = and M 2().0()0.so thatthe value o.feach warrant is this case,N '1,,00 '
.
.
.
'
'*'.
.
.
,
=
:
.
1,,
1'
' '
'
'
.
'
x 7.04 5.87 + 200,000 =
x 5k87 $1.17million. r $5.87.The total cost of the warrant issueis 2,0 Assumingthe rket perceivesno heneftsfromthe warrant issue,we expect the stockprice to declinrby $1.17tp $38.83. =
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13.11 IMPLIEDVOLATILITIES .
The one parapeter in the Black-schols pricing formlas that cannot be dlrctly observedis the volatility of the stock prik I Section13.4,we (lcussed how this can bi estimated froma historyof thestok prict. In practi, tradersusually workwith Theseare the volatilitiesimpliedby option prices what are knownas impliedbolatilities. obseryedlnthe market. To illustrat hpw implied platilities are calculated, pppose that the value of a Europeancall option on a non-dividend-payingstock is 1.875Fhen h = 21, K 2, and T 0.25.The impliedkoltility is the value of c that, when sustituted r= into eqation (13.2),give c = 1.875.Unfortunately, it is not possible to invertequaw tion (13.2)so that c is exjressed as a functionof h, K, r, T, and c. However,an iterativesearch produre can be used to hndthe impliedc. For example,we can stayt This givesa valu df c equal to 1.76,whkh is too low.Becausec is by tryingo' an incresing function f c, a highervalue of c is requiied. Wecan next try a value of 0.30for o'. Tltis gives a value of.c equal to 2.10,which is too highand means that c mustliebetween0,20and0.3. Ndxt;a value of 0.15can betriedfor o'. Thisalsoprves to be too high,showingthat o' liesbetken 0.20and 0.25.Proceedingin this way, we can halvethe range for c at eachitertion and the correctvalue of o' can b: calculated to any required auracy. 9 In thisexample,theimpliedvolatilityis 0.235 or 23.5% per .
'
.
'
,
.
'
.r..
.
=
.1,
=
.2.
=
,
,
9 This method is presented for illustration.Othermore powerfulmethods, such as the Newton-Raphson footnoti4 of Chapter4). DerivaGem can be used to calculateimplied are often used in practi (see method, r . atilities. irt)l ..
/
Thq fl-cplo-Mirjpn Fisure 13.4
297
Model The #Ix izdex,January24
.
ctobri 2097.
t
3j 30 25
'
15
'
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;,
'
,
:
10 5 0 2904
.
Ql1l12m.
/0(
2005
A Similar
Pr0CCdurC
Can
b.
2007
USCd
in Conjlmdion
Vkhblnomialtreesto fnd
hplied volatilitiesfoi A:mericanoptions. ' Ipplie(t vplatilitiesaie usedto monitor the market s opinion aboutthe volatiliy of a payticglarstock. Whireakhistpricalvolatilities(seeSedion 13.4)ar bakward looking, ( volatilitie are orward looking.Tradersoften quotetheimpliedvolatilityof an implied opti n rathtr than its prict. This isconvenkntbtcuse theimpliedvolatilitytendsto be lessvariable than the option price.As will be explained in Chapter 18, the implied o'ptions used y tradersto estimateappropriate implied are vo1atllitiesof activelytraded volatilities fr otief options. .
'
'''')
The v1xIndex The most populgr index,the SPX 'I'he CBOE publishes indies of impliedtolatility. vIX, is an indexof theimpliedvolatilityof 3-day options on the S&P500calculated froma wide rangt of calls and puts.' Informationon te waytheindexis calculatedis in futurts on tht VIX starttd in 2004 and tradingin options in Stction 24.13. on theVIXstarted in 2006. A tradeinvolvingfuturesor opyionson the S&P50 is a bet .on both thefuturelevelof theS&P500and the volatilityof the S&P5. Bycontrast, a futuresor options contract on the VlX is a bet only on volatility. 0ne contract is on timesthe index.Figure13.4shows the VIXindexbetween24 and 27. l, 'trading
Example13.8 Supposethat a traderbuysan pril futurescontracton ti VIXwhen tht futurts to a 3-day S&P5 volatilityof 18.5%) and closes priceis 18.5(corresponding contract'when prke is 19.3 (corresjonding futmes the the to an S&P.500 out of 19.30/:).'I'ht trader makes a gain of $8*. volatility ID
Similarly,theVXNis an inex of thevolatility of te volatilityof the Dow Jones lnastrial Average.
10 inex NASDAQ
an theVXD is an inex of the
CHAPTER 13
298
13.12DlvlnEss .
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Up to now, we haveaspled that the stockupon which the optionis written pays llo In this setion, w: modifythe Black-scholesmodzl to take account ot diviends. dividends.We assumethat the gmount andtimingof thedividendsdurkg thelifeof an, cirtinty. For sorblifi ptions this is not an unreasonption can be piidlcted wi,th pptions it is usual t assllmethat the dividendyield able assumption.(F(y ratherthe tajh. dividendpaytnentsa!e known.Optims can then V yalued as will b describedin the next chapter) The date on whkh the dividendis pa should be to be iheex-dividenddate.Onthisdatethe stockpricedeclinedbythe amount assumed 11 ' ' the dividend. of
,.' .
,
.long-lifr
.
:'
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k
.
European Options
Eurppean optionscan be'analyad by sjumingthat the kock prke is the sllmof two a riskless compo:en tat corresponds to the knwn dividendsduring'the components: lifeof the option and a risky component. The riskless component, at any given time,is thepresentvalue of all the dividendsduringthe lifeof the (tion piscuntedfromthe ' ex-dividnd datesto the presentgt the risk-frie rate. Bythe time the opon matures, the diviens will have.beenpaid and the riskless component no lnger exist. ne Biack-scholesformpl i! threfore crct if k is equalt? the risky componentpf the of the pross fllwed the risky comjonent.lz k ri'ce and c' is the volatiYy
rill
f
stoc p
Operationally,this tneansthattheBlack-scholesformulacan beusedprovided thatthe stockpriceis reduced by the pmsentvalue o a11the dividendsduring 4helifeof the option,the discountingbeingdne fromthe ex-dividenbdatesat the'risk-freerate. As alreadymentioned, a dividendis countedas beipgdurig th life f the optiononlyif its ex-dividenddgte ours duringthe lifeof the option. '
'
fxample 13.9 Considera European calloption on a stockwhen thereare ex-dividevddatesin two months and :ve months. Thedividendon each ex-zividenddateis expected pri is $*, the exerciseprice is $*, the stockprice to be $!5. The current'share volatilit is 30% pr annllm,the risk-free rate of interestis 9% per annum, and the time to maturity is sk,months.The presentvlue of the dividendsis -.l667x.,9
0.5: .
.
-0.4167x0.09
+ 0.5:
=
.
0.9741 .
'
The option price can thereforebe calculatedfromthe Black-scholesformula, 11.For tax reasons the stock pri may go downby somewhatkss thanthecash amolmt of thedividend.To takeadunt of thisphenomenon,we need to interprdtheword dividend'inthecontext of option pricing as the reduction in the spck pri on the ex-dividenddatecaused bythedividend.nus,if a dividendof $1per shazeis anticipqtedand te share pri normallygoes dgwnby 80%of thedividendon the ex-dividenddate, the dividend should be assumed to be $0.80for the purposepof the analysis. 12In theo, tis is not quite the sameas the vlat ilit of the stochasttcproqessfollowedby te wholestock prke. ne volatilityof the risky mponent is approximatelyequal tb thevolatil of the wholestock pri value of the dividends.However, an adjustment is only D), where D is the multipliedby klk jresent estimated vclatility is calculatedafter the volatilhies when historical data. An implie,d are usmg nessary presentvalue of divideds havebeen subtrade,d fromthe stock pri and is the volatility of the risky -
component.
The Slacvscht#'-v:rjp with
k
299%
Model 40 0.9741= 39.0259,#
=
4, r
=
-
'
.
.
iq
'
.
1n(39.0259/40)'+ (.I +
,z>
.3
t.9, jy
=
0,3,and T =
'
.5:
'
J
/2)x
:3
=
,5
jgjy
=
y
.5
.
' '
.
tz
=
-
.3
.''''.''
'
o
.3*/2)
.5
y
=
-:,t14
=
.5
L
13.9or the NOXMSDISY Usingthe polynomial approximationin section functiqnin Exl givej ( .g,4jjj Nkj = j (s sy 2) '
rg1 -
e
j
(13.25)
.
for any reasonable assumptionaboutthestochasticprocess followedbythe stockprice, it can be shownthat it is alWaysgptimalto exerciseat time tn for a suciently high will tend to be satisfed when the ;nal exvalue of Stn). The inequplityin (13.25) clost fairly the dividenddateis to matmy of th option (i,e.,T tn is small) and the dividnd is large. Considernext time !?;-j, the penltimate ex-dividenddate.If the option is exercised #. If th optin is not immediatelyprior to time ln-I the investorreceivesk(!a-I) -
-
,
CHAPTEk 13
30 ''''
exerited at time ln-1, the stock pri drops to 5'(!n.ul) D,,.1 and the earliest subsequenttimeat which exerdsecould take place is 1,,. Hence.fromequation(9.5), a loweybound to the optipn pri if it is not exemistd at time ln-1 is -
'
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.
-41a
5'(ln-I)
.
It followsthat if .
'
'
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'
Dn-I
-
-
' Ke
.
)
-la-j
.
-4j
).k
-j
,9(1,,-.1)Dn-1 .uJ Ke
..j
,
n
-
n
.(ln-1)
.
K
-
.
Ds-l %r(1= e-r(i,,-la-1)j
it is not optimalto exerise immediatelyprior tp time tn-I p i < rj1
.
for any i < similarly,
,-r(k+!l.j)j
-
s, if
(j?al .
.
.
'''
it is n:t dptimalto exerdseimmediatelyplior to time lj.
approximately The inequalityin (13.26)''is to rquivalent '
zj K rrqsj
.
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jj)
..
.
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-
'
.
. .
Assumingthat K is fitly clo to cuntnt stck prke, th ipequalityis satisfkd yiet whenthe dividend on rthestock is lessthan theriskfree rte of interest..Thisis 'the
oftenthe case. We an conclde fromtis analysisthat,in many circumstans, the mok likelytime for the early exercisepf an Americancall is immediatelybeforethe final e-dividend holdsfor i = 1,2, date, ln. Furthermore, if inequality(13.26) tt and inquality (13.24)holds,we can. be rtain that earlyexerce is neyef optimal. .1
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Black'sApproximation Black sugpestsan approyimateprodure for taking accountof early exemisein call 13Th involvescalculapg, as described arlierin til siction, the prices f options. Europeanoptions that mature at timesT and n, and thep setting the Alerican price approzimationseemsto work well in mostcases.i4 equalto the greatrr of the two. .
rrhis
Ekample 13.10
Considerthi situaon in Example13.9,but supposet' at the optionis American Da 0.5, St 40, K ::F:40, r = 0.09, = 11 2/12, and tl 5/12. Since
. ratherthan European.ln this case D1
=
=
=
=
' rg1 -
:-r(t2-tl)j
=
kg(1 :-0.9x.25 ) -
g 89
=
.
.
that the option shouldpever is greater than it follows(seeinequality(13.26)) immedlately ex-dividepd date.ln addition,since befotethe hrst beexercised .5,
#gl
-
-4T-q)
e
1 =
jgtj
-
:-0.09x0.0833
)
=
g yg .
13seeF. Black, Fact and Fantasy in the Use of Optiopsj''FinancialAnalystsJoarnal, 31 (July/August 3G41,61-72. 1975): 14For an exact formula,suuested by Roll, Geske,and Whaley, for valuing mrrican calls when thereis onlyone ex-dividend date, see Tecnical Note 4 on the apthor's website. Tis involkesthe cumulative normal distributionfunction.A produre for calculaungthisfunctionis givet inTecbnical Note 5 bivariate h site ant e s we on or also t 41
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T
Blqcimcholes-Mertoi Model
3t1
thai,Fhen it is vckntlydeep i$kskthat t.5,it flloks (jtt inqqality (13.25)) ltould'be exrcijid immediatlybtfore th:e second ei111the moniy the optin ,
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ditidend date. We n0F use Black's approimation to kalue the optio. The piesent plue of the.srstdividepdis j . j eu.jttyx.g ()4:jj ::;:
.
.
k
jj
the value of the optioh, on the assllmption.t at lt expiresjustbeforete sothatx-dikidenb date, can be calculated using te Black-jcholesformlla with snal
4926= 39.5074 k = 40 r = and T = 0.4167.It is c= t . the valu:. Black's appfoximationtnvclfes taking4hegieaterof this and $3..51. of4heoptionwhe it an onlybe ekercisedat tlieend of 6 monts. Frop Examjle 13.8,we knowthat thelatter is $3.67.Black'z approximation,thereforqgivesthe value of the Amerkan call as $3.61. ' The vale of the option givenby DyrivaGemusing SsBinomial merican'' with (Noti that DerivaGip rmlires dividendsto be input in 54t time steps is $3.72. ordef in the tablr;the tile to a dividendis in the flrst olumn nd chronological of the dividendis in the second column.) Thereare two reasons for alpupt the Mode1(BM)and Black'sapproximation (BA). b iweep theTBinomial diferences timig of the earlyexercisedidsion; thesecopd concerns The rst concerns the the wa volatility is applied. The timingof the early extrcisi dtcisiontends to make BVgreater thanBA.-In BA the assumption is th' t theholderhas iodecide today whether the optionwill be exemisedafter 5 months or aftet 6 months; BM point to depenbon the stock alldWsthe decisionon early eyerise at the s-month priceat tat time:Theway in which volatilityis applied tendqto mqke A greater we assume exercise takis place after 6 month, the .than BV.In BA, a pplied to the stock price lesste pysent value of the ffst dividend; vlatility.is whenwe assume exercisetakespla er 6months,the volatilityis appliedto the of price valtle both dividends. stock less$hepmsent S = 40
.9,
.l,
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SOMMARY 4xgmining the propertks of the We starte this chapter process for stock prices in Chapter 12 The process impliesthat the p'riceof a stockat jome futurc introduced g iven its price today,is lopormal: lt Alsoimplie!thgt the continuously comtime, pounLedreturn from the stock in a period of time is normally distributed.Our aout futurestockprices increasesas we lookfuithet ahead. The standard uncertainty of deviation the logarithmof tv stockpri is proportional to thr squx rot of how farahead we are looking. T0 ejtimate thevolatilityc of a stockpri empirically,the stock price is observed at : xed intervalsof time (e.g.,everyday, every week, or everymonth). For each time period,the natural logarithmof the ratio of thestockprie at the end of thetimeperiod to the stock price at tikebeginningof the time perio' is calculated, The voltility is ekiated as ihestandard deviatin of thesenumbers dividedbythe square root of the length of the time period in years. Usually,dayswhenthe exchanges are closed are igored in measuring timefor the purposesof volatilitycalculations. The difefentialequationfor thepriceof any derivativedependenton a Stock can be 'by
.
.
302
CHAPTEF13 obtainedby creating a riskless portfolio of the optiop >nd thr stotk. Beause the deriyasiveand tht stock jficeboth dependon the sape vndrlyingspurce of unrtainty,this can lwaysbe done.ne portfoliothat is created remains rkless for onlya yq short leriodof time.However,the retuynop a rkless porfoliomust alwaysbe the risk-freeinterestrate if there are yobe po arbitrageopportunities. The expectedreturn on the stockdoes not ehter into the Black-scholesdifefential equation.Thisleadsto a useful Dsult knownas risk-neuyal valuationkTilisDsult states thatwhen valuing a delivativedependnt pn stock price, Fe tan assumethat thr worldis ritk neutral.T ismeans that wecan assumeihatthe exptctedDturn fromthe stok is the riskrfree interestrate, apd then discountexpectedpayofs at the risk-free interestrate. The Black-sholes qations for Europeancall and put opti ns can be dered by either solving their difertntial equationor by/usingrisk-neitpl valuation. Ap implied.yolgtility is the volatility tat, when used in conjunction with the BlackScho1esoptinnpricingformula;ges She'marketpriceof the option.Tradersmonitor ipplied)volatilities.The ften quote theimpliedvolatility of ap optionratherthan its impliedby thepricesof hav dvloped produres for using tht wlatilitks Prfct activelytraded ptlons to estimatevolatilitiesfor otheroptions. TheBlack-scholesresults can beexteded to cover European call nd put optionson stocks. ne procedureis to the Black-scholesformulawit the divizend-paying stck pricereduced by the presentvalue of the dividendsantlcipatedduri the lif of theoptioh,an the votility qual to the volatilityof the stock pricenet of the present valueof thesedividends. I theory,it can beoptimalto exerciseAmqricancall optionsimmediatlybeforeany ex-dividenddate.In practke, it is oftenpnly ntcssary to consider t e fma1ex-dividend date.FischerBlackhas suggestedan approxilation.Th involvessetting theAmerican of tw Europeancall ption prices.The frst calloption price equal to the rreater European all optionexpires at the sametimeas the Americancall option; the second ippediately prior to the ftial ex-dividenddate. expires '
,
.
'ey
.
'useg
FURTHERREAMIjG On
the Dlstrlbutlon of Stock Prlce Changes
Blattberg,R., and N. Gonedes, $AComparisonof te Stableand StndentDistribntions as StatisticalModelsfor StockPricrs,'' Jourzal :./' Busizess,47 (April1974):24449. Fama, E. F., Behaviorof StockMarkt Pricesi'' Jourzal :./' Busizess, 38 (Jannary 1965): ' 3G15. Kon, S. J., iModelsof StockRetnrns-A Comparison,''Jourzal :./' Finazce, 39(March1984): Slrf'he
'
'
147-65.
Richardson,M.,andT. Smith,:tATestforMnltivariteNormality i!istock Returns,''Jourzal of Busizess,66 (1993): 295-321. .
.
On
the Blck-scholes
'
nlysls
' Black,F. dFact and Fantasy inthe Use f OptionsandCorporateLiabilities,'' FizazcialAzalysts Jourzal, 31(July/August1975):36-41,61-72. Black, F. We Came Up with the Option Pricing Formula,'' Jourzal of Jbrt//f: 15, Mzagemezt, 2 (1989): G8. tEl-low
Prking of Optionsand CorporateLiabilitiesi'' Jourzal Black, F,, and M. Scholes, Political Ecozomy, 81(May/lune1973)q637-59. tlrf'he
p./''
3cj
Tbe Blacsscholes-Meton Model 7
f-fheolqy
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of Rational UptionPyidng,''BellJXPIJ/ Merton,R! C., (Sprkg 1973): 141s83. t 'teb 4
. On ulsk-Ntutralvaluatlon .
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Cox,J.C., and S.A. Ross, $$TheValuationof Optionsfpr Alternativestochastic trocesses,'' Journal of K?llzlipllEconomics 3 (197$: 145-66. 3-54 C.W., OptionPricig. A keview, J oarza/ ofFiniiat kcozomis3 (19t6): . smith, On the atei 0/ volatlllty 38 (January1965): Market Prices.''hutal 0,/ Rasiness, Fama',E. F. $$The ehavior of stock :
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34-105
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k lteturns and the WeekendE;t.'' lourzalof Ff:lacfl/ Economics,8 French,K. R, 's yoc 5543. (March 180): sstock Return Variances:The Airival of Inforniationand th Frenchj K.R., and R. Ro11 ReaC tion of Tradets.'' Joartlal of Finazcialftrtzprnks?,17(September1986):5-2. Ro11R. Juke nd Weather,''AmericanEconomit Xvk, N, 5 (Dember 1984): .
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Questions an
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Prblems (Apswerjin SolufionsMahal) '
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13 1 What dpesthe Black-sholej stock option pricij Podtl assume about the krobabi' lity abut of stock cntinuously year? the Wht doesit in d istriution the nri ,t' one assume compoundedrate of return on the stockduringthe year? 13.2.The volatility of a stock price is.3% per apnumkR. at is 1hestandard deviatlonof ie n#rcentggeprice change in 0ne tradingday? ' q: ! ,
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13.34Explainthe principle of.risk-neutral valuation. 13:4.':Calculate$heprice of a 3-mont Europeanpt option on a nm-dividendupayingstock with a strik prie of $5'wheniliecurrentstockprice is $% the risk-free interestr'ateis # 10 a per annum, and the volatility is 30% prr annum. 13.5.Whatdiferencedoesit mak to your calcutationsih Piolem 13.4if a dividend f $1.50 is expected in 1 months? 136 wliatis imptiedvolatilit How can it c:culated? 13.7.A stock price is cuient 1y $40 ssllme that the expecte return fromthe stok is 15% andthat its volatility is 25%.Whatis the probabilitydijtributionfor the rate of return period? (with continuous compounding) earned over a z-year 13.8.A stock pri followsgeometricBrownianmotionwhh an expectedreturn of 16% and a volatilityof 350/:.The current priceis $38. (a) What is the pro abilitythat a Euro'peancall option on the stock with an exercise price of $40and a maturity date in 6 monthswill be nercised?' (b) What is the p'robability that Europeanput ption on the stock with the same exerciseprice and maturity will be exercised? 13.9.Using the notation in tllis chapter, prove that a 95% confdence intervalfor S k between gp,-2/2)z-1.i6,.J (j gp,-2/2)zA1.96gJ ,
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l3.l. A portfolio manager announces that the average of the returnsrealizedin each year of the last l yers is 2% per almum. In wht respect is this stattmeut miskadng?
304
CHAPTER 13 '
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13.11. Assumethat a non-dividend-payingstcck haS an expectedreturn of p an a volatility institutionhasjuq Announced that it will trade a security of c. An innvtive snancial that pys f a dollaramountequal to ln Szat timeT, where ti denotesthe value of the stok price at timi T. (a) se risk-neutrpl valuation to calculatetlie pri of the security at timr ! ip termsof : thestockprice, at time !. Conrm ihatyour pri satishes the diferentialequation(13.16). , (b) Considera (terivative that paysofl'S( at timeT, hereSvisthe stock price at that time. 13.12k Whentikestck pri followsgtometric Brownianmotion, is an be shpwn that its price at time t (1K T) has the form ht, T)Sn ,%
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whereS is the stock priceat time ! and is a functiononly 9f t and T. (a) By substitutinj into the Black-schles-Mertonpartial diferentil equation, derive an ordiqry dlferentialequationsatised by kt, T). (b) What is the boundaryconditionfor the diferentialequationfor (!,T)2 -
(c) Showthat
ht
T) =' e ()5J,l(?l-li+r(n-' l)!(r-t
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wherer is the rist-free interesttte and c is the stock pricevolatility. stoct Qhen the 13.13. What is the price of a Euppean ca11option on a non-dividend-paying . th ripkwfreeinterestrate is 12% stockpriceis $52,thestrike priceis $50, peranvm, the volatilityis 3404per annum, and the timeto maturitv 3 mopths? 13.14.Whatis the price of Eurppean put option on non-dividend-pying jtck whyn the stockprke $69,thr strikepri is $70,the risk-freeinterestfate is 5% per annum, the volatilityis 35% per annum, and the time to maturity is 6 months? 13.15.Consideran Americancall option on a stock. The stock pri is $70,thetimeto maturity is8 months the riskrfree rate of interist lb: lxr annllm,thetxercijeprici'is $65and the volatility is 320:. A dividepdof $1 is expectedafter 3 months yd agaln after that iycan nefer be optilal to exercisethe optipnon either of the two 6 months. show dates.Use DerivaGemto calculate the priceof the option. dividend ''''''''
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The stock price 13.16.A call option n a non-dividend-payingstock has a market prke of $212. is $15,the exerise pri is $13,the time to maturity is 3 monts, and the risk-free interestrte is 5% per annum. Whatis the impliedvolatility? 13.17.Withthe notation uqed in this chapter: whatis N (x)? - Ke-rT-tts'd that sxtdj) 2) where s is the stock priceat tixe ! and ( show '
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(a) b)
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dl
=
'
LnslK) +
(r# c2/2)(T
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-
!)
d2 =
,
tr T t
(c) Calculate kl, (d) Showthat wlien
kls.
T
tr
..2
-
t
c ,N(k) xk-rtr-llxt:2) =
it followsthat
!j)T
=
'
-
and
1n(&& + (r
-
Jc yt-tjygj 'j =. -rKe
-
c
sytvjj z r
.j
wherec is the pri of a call option on a non-didend-paying stock.
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The Black-schole-Merton Model .
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@)Show that 3c/3 = Ndjj. (f) Shok that c satisfksthe Black-scholesdiferentialequation. @)Showthat c ja tises the oundav condition for a Europeancall option, i.e., that = c mlxt. K, 0) as ! T. 13.18.Shok thaj the Black-choles formulasfor call and put qptionssatisfyput-call parity. 13.19.A stockprict is currtntly $50an@tht risk-frtt iterekt rate is 5%.Usethe DerivaGem options on the stockinto a to translatethe follwing tahle of European softwaye Arethe option prkes consistentwith tale of impliebvolatilities,assumingno dividendsk underlying Black-choles? theosumptions .
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Matarity(-pnfz) Strikeprice (3)
J
6
45 50 55
7. 3.7 1.6
s.3 5.2 2.9
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12 10.5 7.5 5.1
13.20.Explaincartfully why Black'sapproachto evaluanp an Amerkancall option on a dividend-payinjstbck may give an approkimateanswerevn whenonly one dividendis anticipated.Doesthi answergiven by Black'sapproachrpdefstatr 44 oyerstate the true optipn value? Explainyour answer. . 13.21.. an Amerkancall option on a stock.The stockpHc'eis $50,thetimeto matqrity is 15monthj,thi risk-free rate of interestis 8% perannum,theexewisepri is $55,and thevolatiliiy is 25%.Dividendsof $1.50are expectedin 4 mpnths and l()months. Show th it can nevrr be optimal to exercisethe option on eithr of the two dividenddates. Calculatethe pri of tht option. 13.i2. Showthat the probabilitythat a Europen call option will be exercisedi a risk-ntutral worldis, withthe notation introducedin th chapter, Nd. Whatis n expression for the va1ue of a derivativethat pays 0fI'$100if the prke Of a stockat time T is greater thah K xconjider
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1323 Showthat
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r2r/J
x
2
could be the pri
cf a traded sicurity.
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13.24.A company has an issue of executivestock options outstanding. Shoulddilutionbe take!linto accopntwhen the options are valued?Explain youranswer.
1325 A compan's stockprice is $5 and 1 millionsharesare outstanding. Te company is call options. Option consideringgiving its employees 3 millionat-the-moneys-year willbe handledby issuingmort shares.The stock pri volatility is 250/0,the exercises risk-free rate is 5%,andthe company dozsnot pay dividends.Estimte the cost to s-year thecompanyof the em/loyee stock option issue. *
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AssignmentQuestions 13.24.A stock pri is currently$50.Asspmethat the expected return fromthe stock is 18% and it8 volatility is 300/0.What is the probabilhy distributionf0r the stock pri in 2 years? Calculatethe meap and standard deviationof the distribution.Determinvthe 95% consdenceinterval.
CHXPTER 13
306
13.27.suppose that observations(m a qockprici (indollars)at th endofeachof 15consecutive Feeksare as follows: 30.2,32.0,31.i 30 1 302 30.3,30.6,33.0,32.9,33.0,33.5,33.5,33.7,33.5, 33.2 . standard error of ymr estimate? Whatij,the Estimatethe stqk price volgtilisy. .',j. )g ' j (.. . 13.28.A tnncial institutionplansto ofl-efa securitythat pays ofl-a dollarnmoupt equ: to at time T. (a) Use risk-neutral valuation to calculati the price of the surity at time t in termsof the stock price S at time 2. Hint: Theexpectedvalue of r2 can. be calculated from 13.1.) the mean' and variance of S given in section (b) Confirmthat your price satlsfks the diferentialequation(13.16). '
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13.29,Consideran option pn a non-dividend-payingstock when the stock pri is $30,the eerciseprice t29,the riskfree interestrate is 5%, the volatility is 25% per annup, .is
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and the time to maturityis 4 months. (a) What is the price of the optionif it is a European call? (b) What is the price of the ption if it is an Americancall? (c) What is the jrice of iheoptionif it is a European put? (d) Verify that put-cll parity holds. .' 13.30.Asjumethat the stock in Proem 1.29 is due to go ex-dividend in 112months.The expecteddividendis 50 nts. (a) What is the price of the optionif it is a Europtan call? (b) What is the prie of the optionif it is a European put? (c) If the optiohis an Americancall, are thire any circulmtancesunder which it will be exerdsedearly? 13.31.Consideran Americahcall optionwheh the stock prke is $18,te exercisepri is $20, thetime to maturity is 6 months, the volatility is 3% per annum, and the risk-free interestrate is 1% per annum.Tw0equal dividendsare expectedduringthe lifeof the optionwithex-dividenddates at the end of 2 monts ad 5 months. Assumethe diyidendsare 4 cents. Use Black's approximationand the DeripGem sqftware to valuethe option. H0w high can the dividendsbe without the America option being worth more than the corresponding European option? .
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The.pltlvrs/lo:lc.Jiz.ft*
307
Model
APPENDIX PR
THE BLACK-SCHULES=MERTQNFORMULA
MF 0F '
Wewill prove the Black j cholej result by flrstprying another keyresult that will also. be useful in future chapters. .
-
Key Result If P
lognormallydistributedand te standard eviationof ln P is w, then f(max(P
j
#,
-
WVFC
dl
f(9#(#1)
=
ln (&P)/-1
=
+
KNkj
-
(13.1)
*2/2
w '
.
ln(:(y)/m -
k
=
12/2
and E denotesthe expectedvalue.
krofof key Result '
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Defmeg(P) as the probability dinsityfunctip of P. It followsthat : '
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fx
flmaxtp
K,
-
))
(P
jx
=
#)g(9JP
-
(13.2)
The Yaliale ln P is normally dtributed with standarddeviaon v?.Fromthe proper15 t iCs f the lognormaldtribution, the mean f ln P is m, whert
'
lnl'(Pj
m
=
Defmea new variable
w2/ 2
-
(13.3)
ln P m -
Q '
=
(13.4)
This variable is normally distljbutedwith a mean of zero and a standard deviation 0' f 1 () Denotethe densityfunctlon.forQ by hoj so that .
.
hot
=
l
-:2/2,
e
27:
Using equation (l3A.4) to convert the expression on the right-hand side of equation (13A2) from an integpl over P to an integralover Q,we get .
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flmaxtp
()r
K, j X
qmaxty -
K,
x
.
-
))
=
Q*+M
:::
(ln#--)/?
e
Kj hojdo
-
X
'
eQ*+mhQ)dQ K -
(1n#-/?C)/u)
(lnK-mjl
16For a proof of this,see TechnicalNote 2 on te
author's website.
hQ)dQ
(13.5)
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308
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Now
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CHAPTR
13
1 (-:1+2:y+2x)/2 e 2p
Qul+#l/j(g)=
1 e.E-(:-.u,)2+on+,j/2
=
2/
2/2
M+
=
.
e
.-.;):j/2
( ((?
eT
2,
'
2
x-bw.pc
=e
'
u))
-
Tllis means tat equao (13A.5)becomes
lmaxtp' r, p)) -
=
jX:
enq-wll
X
#,'j
.(''j'''''.-'''
wjdo
-
-
-
'jdn
(lnr-m)/,,
qnK-mvw
(13A.6)
' ,
If we defne #(# aj the protabilitytat a variablewith a mean of zero and a standard deviationof 1.0is lessthan x, the firstintepal in equation (13A.6)is 1 #((ln K mjlv u)) -
-
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Or
#((- ln K + mjlv + u( Substitutingfor m frcm equation (13A.3)leadsto
x(
1n('(9/'1 +
2
v /2j
.x(#j)
.,
Similarlythe second integl'alk equaon (13A.6)is Nkj. Equaon (13A.6),therefore,
b CCOmCS
2
qmqxty r, -
0))= e''.w /2#(J 1) KNkj -
Substitutingfor m fromeqution (l3A.3)#vesthe keyresult.
The Black-scholes-Merton Result We now cnsider a call ption on a non-dividend-payingstock maturing at timeT. The strikepriceis K, the risk-fxe rate is r, thecurxnt stock priceis and the volality is c. As shown in equatien (13.22), the call pricec is given by k,
c=e
'-rTkmaxtkr
-
r, )q
(13A.:)
whereSz is the stock price at me T a:d J' deotes the expectationin a risk-neutral world.Underthe stochastk processassumedby Black-scholes,Sz is lognormal.Also, and (13.4), from equations (13.3) Jurl = Sbe'T and te standard deviationof ln Sz is cu/. From the key result justprovtd, eqtlation (13A.7)implies c= e r
-rr
(yg
c = SNlh
rr
yj(J1; jlyyjdyij
)
..
-
Ke-rrxg
)
2
z. hlacsscholes-xtonModel
309
where
.
077,/2
JI
::t
k
=
and
+ lnl't&l/m c//
lnEjrl/m .
-
=
l-ll
cx//
This the Black-schole-Mertonresult.
=
ln(,/#) +
(r+
jljT
c//
ln(5/r)+
(r- #/2)z
cx/
1.,(,.
T
J
y .1
. . z
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.q.
g
#.g.r.k'z .7'd! s, . s. #.y, ipmczu-t-/;ur
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k.
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$X '.
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+.
'-.
w.
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.,
.d'
A,rkij iw
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--
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'$ 'n
y.'. /6 ..
z
. . : m.
Drivatlke.
$x..
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1
)'G 1 'A.' m j .
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X.k..p
Lz
.
bqveloplng
'.iw
Cquntries
Derivativeshavebecomevery impoitanttoolsin the Westernworld for trqnsferring risks from one entity to another. It is not surprising that derivativesmarkets are growingfast in many developingcountries.This chapter focuseson what is happening in Clna a.lndia. These are two countries whose economies gre expted tp plqy dominantroks in the 21st centuly China's population in 2007 is estimated to he aout 1.3 billion,zwhilethat of India ls about 1.1 billion.(Bycontrast, the population biliion.)The world's population is about of the United States is tmlyabout 6.6billion,so Chiha and India betweenthem account for about 36% of the worlz's population.India is expected to overtakeChina as the world'smost populous nation by 23. There c>n be little doubt that China and India wi11have a uge impact on the developmentof derivativesmarkts tkoughout the world in the years to come. Other countrieswill also be importantplayers.For exnmple,Brazil,the Efthmost pppulous country in the world, has been very suessful in deyelopingits dirivativesmarkets. Its premier xchahge, Bolsade Mercdorias & Futuros (www.bmf.com.bz) is highly regarded. .3
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14.1 CHINA'S MARKETS The way derivativesmarkets are regulatedplays an importantrole in theirgrore sophtimarkets to reach a cated instruments and it is nessary for a countrfs fmncial ' level of snplsticatiop beforederivativescan be successfut.There are a reasonable numberof key con:tions for the growth of derivatives.It is important for the govirnmetto set up a pplatory structurethat protects investorsfrom fraud and themfeelingsurethat thir cpntrads will be honored.However,the government leaves should not impos: too any restrictions on the waf derivativescan be used because speulators and arbitrageursare importantfor the lkuidity of the market. Theremust be a sound Enacial and legal systm within the countly Volatilityis not bad for derikativesmarkets. (Indeedwithout volality therr would be liltle intepst in mny j derivativeproducts.) But derivativesare unlikely to t rive unless the economy of the countryis reasonably stableand thereis a good payments system.Stockmarkets,bond markets,and money markets'shouldhe raonably well developed.(fter all, stocks, bonds, and money market instrumentsare the underlyings for man derivatives.) Ideallythe currency shouldbe freelyconvertible and thereshouldbe no restrktions there shouldbe on the llow of the currency in and ut o.f the country.koreover, enough swaps, bonds, and money market instrmints trading for a rkkrfpe zerocoupon yield curve to be estimated, hnal very imprtant condition is that there shouldbe enough welleducatedindividualswho understand the products and howthy can be valued. An intrlguingidea for developingcountries is the possibilityof derivativestransactions betweennaonal governments.If colmtry X exports oil to country Y and country Y exports buildingmaterials t company X, they are both subjectto risks relatingt? the prices of theirexports. It might make senseforthemto enter into a swap that efectively hes pris for severalyears into the future. This example can be extendedso that it applies to groups of countriesthat tradewith each other.l .
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SUMMkRY The economies of manydevelopingcolmtriesare growingvery fast.Indeed,the phrase country'' may no longerbe an appropriate Fay of describingthemin a decadeor two.AlthoughthetradingOf derivativesoasionally getsout of control (see Chapter34),therecan be no question thatderivativesmarketshaveplayed an important role in allowing risks to be managedin developedcountries.nere is no reason to supposethat the samewillnot in thefuturebe truein everycountryof theworld. Sekeral developingcountries have laid th foundationsfor a mature derivativesmarket by creatingappropriate' legal,fmancial,and regulatory frnmeworks.In a fewdecades,it is likel that the derivativesmarkets of countries likeChina,India,and Brazilwill be as important,if not more important,than those of the Unite Statesand WesternEurow pean countr ies Edveloping
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1
idea,which has beznproposed by Rolmrt Merton,is of ccurse not justappropriate for developing colmtries. 'fkis
DeyivativesMaykets in Deretl/Alg
31s
Countyies
FURTHERREADING Dtrivatiyes Marketin India:Development,Reguktion,and Futures Ahuja,N.L., Prospects,''InternationalResearc Jourzalp.ff?2lrlc: and f'tmtmfc', 2 (26): 153-62. Jourzal ofDerivatives,4 (Fal11996): Braga, B. s., DerivativesMarketsin Brazil: n Overviewy'' Sscommodity
63-7,
.
. Markets,''in: Cinq's Knfmflf Markets..An JAr',y Derivatives Mnqer-xu, M.Y., china's Guideto Sb@ t e Markets Frk (s.N. Neftciand M.Y. Mnager-xk eds.). New York: AcademicPress,17.
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Opiioqs on
stockIq
and
le:
urtencie. '
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'Optlons on stock indicesand currencies were introducedin chapter 8. gTIS chapter anb feviewssome of the ways discssej themin more ditil. It explaipshowtheywprk chapter, valuationresults seond of used. In the half the in Chapter 13 they can be the are extended to cover Eurpean optionj n a stock payihg a knwn dividendyield.It is then argued that.both stock indls and currencies are anaiojous to stocks paying dividendyields. This enables the results for options on a stock paying a dividendyield to be applied to tliesetypesof options as well.
15.1 OPTIONSON STOCKINDICCS of the indiclstrackthe Severalexchangestrade options On stock idices. some moveOf the market ment as a whole. Oters are based On tht performance of a particular computer technology,oil and gas, transportation,br telecoms).Amongthe [email protected]., index options tradd On the ChicagoBoard Of OptionsExchange are Americanand European options on the s&P 1 (OEXand XEO),Europen options on the s&P 5 (DJX),and Euiopear (7PX),European optios on the DowJons lndustfial'Average optionson the Nasdaq 1 (NDX).ln Chapter8, we explained that the CBOE trades LEAPSand fkx options on individualstocks. It also osers theseoption product,son indices. 0ne indexoption contract is on 1 timestheindex.(Note tat the DowJonesindtx ujedfor indexoptions is 0.1 timesthe usuallyquoted DowJonesindex.)Index options aresettled in cash. Thismeansthat, on exerciseof the option, theholderof a call option receives s & x 100in cash and the writerof te option pays yhisamount in contract cash,where S is te value of the indtx at the close of trading n the day of the exercise the holder of a put option contract receives and K is the strike price. sililarly, CK 54 x 100 in cas and the writeyof the option paysthis amount in cash. '
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Portfolio Insurance Portfolio managers can use indexOptions to limittheirdownsiderisk. suppose that the of clrge well-diversifk index portfolio o f S. in todayis Consider a value an a manager whosebetais 1.. A beta of 1. impliesthat tllereturns fromtlleportfolio mirror those
317
318
CHAPTER 15 .
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fromthe index.Assumingthedividendyield flpm tht portfolio is the same as the yieldfromthe index,t percentay changes in tiw value of ye portfolio can dividend beexpected to be gpproxinptely the same as the percenge chynges in the vlue of the indtx.Each contract on the j#P 500is on 1 tiles tht index.It followsthatthe value of thrportfolio is prtected againstthe possibilityof theindexfallinsbelow:f if, for each buyson put option contract with strikepfice lkc dollarsintheportfolio, themanayer portfolio manager's K. suppose andthe valq of theindexis idwortli$5, that the l,. The pftfolio is worth 5 times.theindex.Th maliager can obtain insurance ajainst the valtle of the portfoli drpping elor S45, in the next threemonths by buinghvethree-monthput ption contracts on theindx with a strikeprice of 9. To illustratehowthe insurapceworks, onsider ih situation where the indexdrps to 88 in yhreelonths. Tht portfolio will be worth about $44,. The payos from the options wil1be 5 x (9t 880)x 1 = $1 t briging the total value of the portfolip up to the insuredvalue of S45,. .
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Whn the Portfolio's Beta Is Not 1.0 '
If the portfolio's beta (j) is not 1., j put options must b purchased for each lkc dollarsip the portfolio, where k is the current value of the index.suppose that the portfolio just considered has beta of 2. insteadof 1.. Wi coatinue to S5, assumethat the s&P 500 indexi! 1,. The number of pt options required is .
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2., x
ratherthan 5 as before.
5, l,
x 100
=
l
To calculate the appropriate strikepri, the capital assetpricing model can be used, suppqsethat the risk free rate ks12%,the dividendyield on both te indexand the is 4%, and protection is required pgainstthe value f the portfolio dropping portfolio blow 5450,0 in the n xy three months. Uder the capital asset pricing model, the excls fdurn of a jrtfolio oket the rilkzfrei ratt is assumed to etual beta expected ''%
;
.
Table 15.1 Calculation f expected value of portfolio when the indexis 1,040in threemonths and j = 2.. Value of indexin threemonths: Returnfrom change in index:
Dividens fromindex:
1,40 4/1,,
or 4% per three months 0.25x 4 1% per three tonths 4 + 1 5% per threemonths 0.25x 12 3% per three months =
Total return ftom index: Risk-fretinterestrate: Exess return fromindex over risk-free interestrate: Expected excess return from portfolio over risk-free interestrate: Expected return from portfoliok. Dividendsfrom portfolio: -Expectedintreasein varlueof prtfolio: Expected value of porifolio: '
=
=
3 3 2% per threemonths =
-
2 x 2 4% per threemonthq j + 4 7% per tkeemonths 0.25x 4 1% per three months 7 1 m 6% per threemonths $500,000 x 1.06 = $530,000 =
=
=
'
-
.
Optims (m Stocb .
.llc.
lkr /
and
encjes
319
'
;
f
;
Table 15.2 iRelationshipbetwen value of index and value of portfolio for j = 2..
alueof yfkx ?ltree mozts 1,8 1,4
,t,.
96t 92
37,
880
..
-
(3)
tree
570,00 530,000 490,000 450,000 41,
l,
..
valaefporlfolio pltzfl,
?l
jmes tlle excessreturn of te indexportfolio ver the risk-freerate. The lodel enables te expectrd value of the portfolio to be clculated for diferent values f the indexat te end of threemonths. Tablt 15.1showsthe calculationsfor the case where theindex iS 1040 In thi case te expectedvalueof te portfolio at the end of te thre months is calculationscan be carried out for other valuesof theindexat the end $53,. similar of the tree mont Te resuits are shown in Table 15.2. The strike prke for the optionstat are purchased should be the inddxlevelcorresponding to the protection levelrequired on thi portfolio. ln this case the protection leyelis $450,000 and so the 969.1 option purchased strike price 1 that tlle is for contfacts put are correct T illustratehow the insuranceworks,consider what happensif the value of the ' indexfalls to 88. As shown in Table 15.2, the value of te portfolio is then about The put options pay oll-(960 880)x 1 x l = $8,, and this k exatly $37,. Fh4t is necessary to pove the total value f the portfolio manager's position up from $370,000to te required level f .$45.l. The examplesin this sectien slloW tat tefe are two reasons why the cost of hdging increasesas the beta of p portfolio increases.More put options are required and they ave a. higherstrike price. ,
.
.
-
*
.
1s.2
tURRENCY OPTIQNS Currencyoptions are primarily tradedin te over-the-countermarket. The advantage of'this market is that largetradesare possible,with strike prkes, expiration dates,and oter featurestailoredto meet the needs of corporate treasurers.Althoug European and Americancurrency options do trade on the Philadelpla stockExchangein the United States,the xchange-tra ed maiket for these ptions is much smaller than the over-te-countermarket. An example of a European call option is a contract that givestheholderthe right to buy one millipn eurs with US dllars at an exchange rate of 1.2 US dliars per Of euro, lf the actual exchange rate at te maturity the option is 1.2500,the payof is ' pproximately I % of T500,00,or T, will be earned in dividendsover tlle next threemonts. If we want the insuredlevel of $45,0 to include dividends, we can .choose a strike price corresponding to $445,999rathet than $450,. Tls is 955.
320
CHAPTER 15 Similarly,an exmple of a European put $5,. x (1.250 l 1,, optipnis a contract that gives the holder the right to sill ten million Australian IJS dollap per Australiandollpr. dnllarsfor US dllars at an exchangerate of .2)
=
-
.7
If te actual exchange mte t the maturity of the option is .67)
$j
.6700,
the payof is
x (.7 F0r a corporation wishing to hedgea foreignexchange exposure, foreigncurrency optionsare an interestingqltenzative t forwardcontmcts. A company due to receive sterlingat a knowntimeinthefuturecan hedgeits risk bybpyingpnt optipns on sterling thlt matre at that time. The hedgingstmteg guarants that the exchange rate to thesterlinywillnot belessthanthe strikepri, wlle allowingthe cqmpany applicable acompanydueto pay tobene:t froman favorabletxchange-ratemovements.s'imilarly, calls sterling timeinthefutyrecan hedgebybuying that mature at at a known on sterling of sterlipgwill Tls thq thatthe time. hedging cost stmtegyguarantees not be greater that than a certa,in amountwhile llowingthe company to benefitfromfavobl excangea forwardcontract locksin the exchange rate fr future ratemovements. whereas an optionprovidesa type of insurance.Thisis not free.It osts nthing to transaction, entefinto a forwardtransaction,but options require a premium to be paid up front.
l
-
=
.
Range Fokwards A range forward contract is a variation on a standard forwardcontract f?r hedging toreign exchangerisk. Considera US company that knowsit will reive one million plun'dssterling in threemonths. Supposethatthethree-monthforFard exchangerate is 1. 2 dollars per pound.TVcompany could lockin this exchangel'ate for th dollars it receivesby entering into a shol't forwrd contract to sll one million pounds sterling in thr months. Tls would ensure that the amount received for the one million poundsis $l,92,. An alternativeis t buy a Europeanput option with a strike pri of rj and sell a < K. This is knownas Eurqpean all optin.rith a strikeprice Kj, where #1 < l a short range fomard contract. Thepayof is shown in Figre l5.l(a). In both cases the optionsare on one millionpunds. If the exchangergte in threemonths provesto beles than #1 the put option is exercisedand as a result the cdmpany is able to sell the one .92
,
Figure 15.1
Payofs from (a)short and (b)long range-forward contract. Payoff
Payoff
Asset . Pyktz -1
' #2
(a)
/tSSC! .
#l
K2
(b)
PFjcg
Optiom4:1 StockIndice
JA?,#
j
32
Cuyyencin
Eijgre 15.2 Exchange rate realized.wheneither (a)a shori rang-forkard contrat is usedto hedgea futur foreip currncy insowor (b)a longrapge-forwardcontractis used to hedge futureforeigncurrencyoutfow. Exchang:rat: rralized
whenrange-fomafd contrctis use,d '
g 2
'
#1
Exchangerate in markd
1)
%
millionpounds at an exchange rate of rj If tlle ekchange rate is between#1 and #2, neitheroption is exrcised and the orppany gets the currentexchang rate for the one .
millionponds. If the exchange rate is greater than #2, te call option is exercised ggainstthe company with the rsult that theone million pounds is sold at an exchange rateof #2. Theexchangerate realizedforthe one millionpounds is shownin Figure 15.2. If the company knew t was due to pay ratherthanreceiveone millionpounds in three months,it could sll a European put option with strike pri #1 and buy a European call option with strikeprice #2. Ts is knownada log range forwar contract and the payof is show i Figurt 15.1(b).If tht txchangt tatt in thrtt molths provts to beless than #k, the put option is exercise againstthe company and aj a result the company buysthe one million pounds it needs at an exchanje rate of #1. If the exchangerate is between#1 and #2, neither ojtion is exercisedand tht.companybuysthe one million pounds at te current exchange rate. If the exhange rate is greater than #2, the call option is exercised and the comjany is able So buy the one million pounds at an exchangerate of #2, The exchanpe rate paid for the on million pounds is the snme lnillion pppnds in the earlier example and is shownin as that rtceived for the one Figllfe 15.2. In practice, a range forwardbontract is set up so th>t the price of the put option equalsthe price of the call option, Thismeansthat it costs notng to set up the range fomard contract, justas it costs nothing tp sd up a regular forwardcontract. suppose that the US and Britishinterestrates are both 5%, so that the spot exchange rate is 1.9200(thesame as the forwardexchangeTate). Supposefurtherthat the exchangerate ' volatility is 140:.Wecan use DerivaGemto showthat a put with strikeprice 1.9 to sell one pound haS the same price as a call option with a strike price of 1.9413to buy one pond. (Both are worth 0.04338.)Setting#1 = 1.9 and #2 = 1.9413therefore leadsto a contract with zero cost in otlr example.
322
CHAPTER 15
As the strikepris of th# call nd put options becomecloser in a rnge fomrd tlie range fomard contract bemes a regular forFard contract. A short ranje conttact, codtract forward becomesa shortfomard cotract and a Iongrange fomard contract a lng fomard contrct. becomes '
.
.
nlvlnENn #IELDS
15.3 OPTIONS ON STQCKSPAYINGKsows
In this section t producea simple rule that epabls valuation results ?orEuropean optionson a nop-dividend-payingstock to be extended so that theyapply to European optionj on a stockpaying a knowndividendyield. Lter weshowhowthis enables s t valueoptions on stqck indis and crrencies. Dividnds cause stock pricesto redue on the ex-dividenddate by the amount of thedividendpayment. Tlle jaylent of a dividendyield at mte q thereforecauses the growthmte in the stock price to be lessthan it would otherwisebe by an amount q. 1f, with a dividendyield of q, the stock price grows from today to Sz a.t time T, then today to heq at time T. in the absence of dividendsit would grow from Alternatively,in the asen of dividendsit would grow from Se-qT today to S'E at time T. ' get the sameprobabiiitydistributionfor the stock price This argument shows that we at time T in each of the followingtwo cases: kg
kg
1. The stock strts at price % and provides diidend yield at rate q. starts at pri ke-qT no dividends. 2. The andEpay
.stock
This leadsto a simple rule. Whenvaluing a Europeanoption lastinj for time T on a stockpaying a knowndividendyield at rate q, we reduce the current stock pri from -T to he and then va 1uethe bption as thoughthe stockjays n dividends.z kg
Lower Bounds for Option Prices '
'
.
.
.
As a srstapplication of this rule, consider.theproblem of deterpiningboundsfor the of a Europeanoption on a stock paying a dividendyield at rate q. Substituting price -VT S: for S in equation (9.1), we se that a lowerboundforthe Europeancall option price, c, is given by -qT c k Se
-
yk-rr
(js
j)
.
We can also prove this directlyby considering the followingtwo portfolios: Portjblio A : one Europeanca11option plus an amgunt of cash equal to Ke-rT Portjblio B : e -qT shares with dividnds beingreinvestedin additional shares ,
To obtain a lowerbound for a Europeanput option, we can similarly replace -VT Se in equation (9.2)to get
p p Ke-rz
-
st):
-qT .
kg
by
(js a) .
2 This rule is analogous to the one developedin Section13.12for valting a Eropean option on a Stock knowncash dividends.(In that case we conclndedthat it is correct to reduce the stock price by the paying presentvalne of the dividends;in thiscase we disconnt the stock price at thedividendyield rate.)
btionstm Sck'
fngcdl an4
323
CvrrAlcfe.r
Tllis psult can also be proveddirectlyby consideringthe followipgportfolis: '
..
.
.
'
.
.
.xr
'
Portflio C : oe European put option pluse shareswith dividendson the shares reinvested in additional shares being. '
Porblio D : an mount of cashequal to K '
.
.
-rT
.
Put-call Parity #
'
.
.
'
.
.
.
Replacing by Sge-@ in equation (9.3) we qbtainput-tall parity for an option on a -stockpayinga dividnd yield at rate : kc
C
#
Ke-rT
p+
=
k-qT
(153) .
This result can also be proveddkectlyby consideringthe followingtwo portfolios: '
-rz
one European call optio plus an amount of cash equal to Kt e-qT Shares with dividendson the shaies /7pr blioC : one European put option plus lhares beingreinvestedin additional '
, Porfolio
z.l
:
Both portfolios are both woryhmaxtsy, h at timeT. They must thereforebe worth the follows.For Americ>n same today, and the put-call parity result in equation (15.3) options,the pgt-call parity relationship is (seeProbleml 5.12) kg:
-qT
-
r Kc
py
-
Kc-rT
sb -
Pricipg Fprmulas )
.
.
.
.
B repIacingk by ke-qT in the Black-scholes fonuulas,equations (13.2)and (13.21), obtain the price, c, o?a Europeyp calland the price, p, of a Europeanput on a stock payinga dividepdyield at rate q as
These results Erst derivedby Merton.3 As discussedin Chapter 13 the word dividendshoul, for the purposes of option valuation, bedefmedas the reduction in thi stockpriceon the ex-dividenddate arising fromany dividendsdeclard. If thedividend yield rate is known but not constant duringthe life of the optin, equations (15.4)
rere
3 seeR. C. Merton, tTheory of Rational Option Pricin'' Sciezce,4 (Spring1973):141-83.
,
Bell Joarzal oj' Ecozomicst/zltf Hanagemezt .
324
CHAPTER 15 are stilltru, and(11.5)
optio's life.
with q equal to tileaverageannualizeddividendyield duringthe
Differential Equationand .Risk-NeutralValuation ald (15.5) Tp prove the results in equations (15.4) more formlly, we ca either solve diferentiai equation pri the that thepptiop mst satfy or use risk-peutral valuation. Whenweincludea dividendyieldof g.in the analysisin Section13.6,the diferntial ekuation(13.16) becmes4
JJ
=+ (r t -
92J
CJ j2c jsj qjs,.b
.
.?.j
(1i.6)
2 C,T
Like equation (13.16), th does not involveany variable a*ectedby risk preferences Therefore the risk-neutral valgtion procedure describedin Siction13.7can be used. ln a risk-neutral world, the total return from thestock must be r. The dividends jrovide a return of q. The expectedgiowth rate in the stock pri must thereforebe ' q lt followsthat the risk-neutml process for the stock price is -
.
ds =
(r -
qjsdt + c'Jz
(15.7)
j
To value a derivativedqpen ent on a stock that providesa dividendyield equal to q, we .setthe expectedgrowth rate of the stockequal to r q and discountthe expectedpayof expectedgrowth rate in the stockprice is r q, the expectedstock at rate r. Wheniheer-T A similaranalysisto thatin the appendixyoChgpter13givs v T priceat tire is () the expected p ayof for a call option in a risk-neutral world as -
-
.
r-qtTv
e
g
xgj).KNLdjj
where dj and d are dehnrd as above.Discountingat rate r for time T leads to euation ' (15.). '
15.4 VALUATION OF EUROPEANSTOCK INDEX OPTIONS In valuing indexfuturesin Chapter5, we assumed that theindexcould betreatedas n aset paying a knownyield. ln vluing indexoptionj, we make similar assumptions. provide a lowerboundfor Europan index This means that equations (15.1) and (15.2) is the ptlt-call parity result for European index optipns; optibns; equation (15.3) and euations (15.4) (15.5) can be used to Value European options on an index;and thebinomialtreeapproch can be used forAmericanoptions. ln al1cases, is equal to the value of the index,c is equal to the volatilityof the index,and q is tqual to the averageannualized dividendyield on the indexduringthe lifeof the option. ka
fxample 15.1 Considera European call pptionon theS&P500thatis twomonths frommaturity. The current kalue f theindexis930,the exercise.priceis9, the risk-freeinterest rate is 8% per annum, and the volatilityof theindexis 20% per anmlm. Dividend 4 seeTecknicalNote 6 on tile author's websitefor a proof of this.
3?!
Op'tionst??1Sck Indices nd Ctz?'?ucds
jields of 0.2% and 0.3%#.re expectedin the srstmonth and the second mpnth, x'u us'tl respectively. ln tlus ' case f:u 93 K = 9 r :: 08 o. 2 and T = 2/12. The totl dividendyielddutingtheoption's lifeis0.2%+ 0.3%= 0.59/:.Tbis 3% per annum.Hece, q 0.03and ,
,
,
.
.
,
'
=
'
.
.
'
j!
ln(93$9) + (.
=
4
.
.
.
0.03+ 0.22/2)x 2/12 =
-
j.j4.
-
.2 1/12 ln(93/9)
=
+
(.8
.3
=
2/2)x j jz /
.2 -
,
0.4628
=
9.2 2/12 .l9,
Ndkj p
Nd
0.6782
=
so that the call prke, c, is givenby.equation (15.4) qs ' j yx-.8x2/12 k 90 g g 0.7069: 93 x c ,03x2/12
=
=
-
.
'
'
51. ?
.
.
One contract would cost $5,183. The cakulation of q should 'includeonly divideds whose ex-dividend date occurs duringthe life'of the option. ln the Vnittd Statesex-diyidend datestend to occur during thehrst weekof February, May, August,and November.A any given timethe correctvalue of q thereforelikelyto dependon the life of the option. This is even moretrue for some foreignindices.In Jap>n, for exmple, all compaies tend to use the sameex-divitend dates. g lf theabsolute amount of the dividend that ll be-pai thestocksunderlying ihe index(rather than the dividendyield)is assumed to be known,thebasicBlack-scholes formulacan be used with the initialstock price beingrduced by the prejent value of the dividends.Tllis is the approach recommendedin Chapter 13 for a stock paying knowndividends.Isweyer, it may be dicult to implementfor a broadlybase,dstock indexbeause it pquires a knowletlge of the dividendsexpectedon everystock pnder
lyingthe index. It idsemetimesargued that the return froma portfolioof stocks is certain to beatthe return from a bond portfolio in thelong.runwhep both havethe same ipitialvalue. lf this wereso, a lonpdated put option on the stpck portfolio where the strike price equled the future value of the bond portfoli? would not cost very much. In fact, as expensive. indicatedby BusinessSnapshot15.1,it is 'quite
.
Forward Prices Detinef'g as theforwardprice of theindexfor a contrAct with maturityT. As shown by ker-T This meanstat the equations fQr the European call price l; equation(5.3), and (15.5) can be wrhten c and the Europeanput price p in tquations (15.4) =
.
-rTNd 1 ) c = he
p
where
$ ),
#1
=
=
Ke-rTN-d a)
1n(&r)+Tlz c;T
Ke-rTsd
-
-
)
(15.:)
2
he-rTyf-h)
and J2
=
(15.9)
1n()
-
c;T
T!2
326
CHAPTER 15 ('
Buinsj 59p: jlt
llki
'
an We Guaranteethat stcksWill ai Bondsin
kup?
teEjapg
'
. .
'
,
' '
.
,
. .
7
.
Ii is ofte jal: . th/' . lf ypu am a lqnptirm inkrstoryou shguld by stocks rather tlka bppds. Cosldqr y Uj fppdmpger who is tryingto peisuadi investorsto ( fln-terp ipkesttnent pp viuiyyfrd thlt i! ipcid tp rror the buy, a: :t iof pvrclserj oser S&P :. the fupd a ii yry mght bt iipted zgogd >i liit risk-free the j f i pytlrwill kit turn n glwrnyirt p gndq as : s HktorkAll jtotks havi ipstyfformedbonzs i the United ' ixt it t yeksk oket Slateso# ) lji An? je thatthe fud mygir wold no! h rppfap rr E begivipg ) ih pra. Er q, ( y ,In falci,.ibljrgtg qf braiir ij suijrisigly eypesive. Supppse aj n ej ujjy ikx i! l,# iih.thr ividrd yirld dp teginikxis 1% annpln, te volatilky q. 1j9(15p/a an te' lzyear rijkmfreerate is 5% ?eyannum.T qfth ipdi; qlifykils, jrr jnnllm, lpu' thi kStok! udirlg thi inde st i.rjw ear mor than' 56 ptr utkfpo ( >nuk, dividendhildwj11provij e js pejyamvjm, yagjjjj gains on rthe 4% pef aimu. Thit ans thatWerequirt the,indrx q ' ' stkj ' m jt'tlqefoytlki .' )14412 ' in 1 yerj. livrlto he t lesi 1)k0@ = in,iikeindrxwill bejfeatir tkan the Sti,q Shrlteiqr iTetz : 'Apianteithqt Eiptts ipTeste hver 4hrnep 10 years iqtkereibreyequikllent to rw!'n(pp $1,t() 'ih npht y etlqtheidik fof 1k492i l yrar. Tfii is Europianput optionon the r y iz 1 K y1'411;r = 5, ipikxandrcofb gfltl flpm tquation(15.5) rith r f y 1 i/akT # 1t. anj q = 1a/n.The value f thiput optipnis 169.7.This'showsthpt of thefyndv ie yure cpptmplted b the fn langger ls orth about llte/a : hqrly nmithingtht shoyld e givrpaway! J
.
.
.
.
'
.
'
. .
.
J
.s
.
,
'lkytrr
'flpdk
.
u
.per
:
: y.
'
'
. .
'
J
.
.
.
.
.
.
'lii
.9lx.l
.
.
:
i
,
''
. . .
.
.
tn.
'
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.
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...
''
'''''.',
.
.
,
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'
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.
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.
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.
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.
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.
.
.
.
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'
The put-call parity relationship in iquation (15.j)can be written -rT c + Ke
p+ y e
=
-rT
r'
0F k + (c >-
pler'r
(15.1g)
markets, pairs of puts and callswith the If, as is not uncopmon in the'exchange-traded samestrike price are tradedactivelyfor a particular maturity date,this equation can be usedto estimate theforwardprice of tlieindexf0rthat maturity date.Oncetheforward pris of the indexfor a number of diserentpaturity dateshavebeen pbtained, the termstructure of forwardrates an beestimated,andotheroptionscan be kalued using equations(15.8) and (15.9). The advantageof this,approgchis that the dividendyield indexdoes have the not to be estipated explicitly, on .
Implied DividendYields If estimatesof the dividendyield are reqtlired(e.gbecausean Americanoption is being valued),callsand puts with the same strike price and time to maturitycan be used. From equation (15.3), -rF 1 c p + Ke ln -
q
=
-
'f
k
327
Optionstm Stock Indices and Cuyyencies t
For jartiular strikeprie and timeto mlturity,the estlmatesof q cakulated .fromthis equationarr liableto be unreliable. But when the Dsultj from many matced pairs of (t yi ld beingassumed by the callsand puts are combined, a clear pictux of the,diviend nlarketeplerges.
15.5 VALUXTIONOk ECROPEANCURFENCYOPTIQNi To valqe cprrency options, we dene h as the spot exchangerate, To be predse, h is the vlue of one ult of theforeigncrrency in Us dollars.Asexplainedin section 5.1, stockpaying yield. analogous kpowndividend The owner of foreigncurrencyis to a a foreigncurrency riciives a yield equal to the risk-free interestrate, rf, in the foreign with q replaced by rj, provideboundsfor the and (15.2),' curreny. Equations(15.1) Europea,ncall price, c, apd the Europeanput pri, p: -ryr
he
C7
-
-rT p 7 Ke
Kg-rT
s
-
g-rgT
with q replad by rj, providesthe put-call parity msultfor currency Equation(15.3), 0Ptions: '
'
-rT c + Ke
p+
=
g-rgT
v
provide the pricingformulasfor turrencyptions and (15.5) Finally,equations (15.4) is replaced by rJ : wzsy N@j) xe , 2; (j5jj; c Se
When q
.r);.
=
p
where
=
-
.
.-ryrxt-:
Ke-rr N-d
-
ke
I
)
(jsojy
ivsbjp # (?. y + g2yzly -
dL
=
cxff
'
k
=
lnts/m + t
-
ry
cxff
-
2
/2)T
.
dk
=
-
(TW
Boththe domesticinterestrate, r, and the foreigninterestrate, rf, are the rates for a T. Put and call options on a currencyare symmetricalin that a put option to maturity sellcurrency A for currency at strikepri K isthe sameas a all option to buy B with A gt strike price 1/# (seeProblem15.8). currency
fxample15.2 Considera four-monthEuropeancall option on theBritishpound. suppose that the exercisepri is 1.6, the risk-fret the curent exchange rate is l is 8% per annum, the risk-free interestrate interestrate in the United states Britain is 11% per annum, and theoptionpri is4.3cents. In tlliscase, S 1.6, = 0.3333,and 1, volatility T K = 1.6, r = 0.043.The implied 0.1 rj c calculated volatility and by trial of 20% gives an option price be A error. can of 0.0639;a volatility of 10% gvesan option prk of 0.0285;and so on. The impliedvolatility is 14.1%. .6,
.in
=
.8,
=
=
328
CHAPTER 15
Using ForWard ExchangeRates BrcAuse baks and ther Enncial ipstitutionstrgde forward contmcts on foreign rates actively,foreignexchangerates are often used for valuing options. exchange the forwardrate, &,for a maturity T is givenby From equation (5.9), (r=rJ)T
ke
Fq =
This relationship allowsequations (15.1 1)and (15.12) to be qimplifkdto '
.
-rr
c.= e
p
where J,
=
tt
=
=
(y#x(#j) sxtyy)j <x(.tj) e-rr E sgxt-gj)j
(15.13)
.
(15.14)
-
1n(1$/:)lc
2T/2
cq// ln(//#)
j
-
=
cY
dj
-
cd
These equations are the same as equations (15.8) and (15.9). As we shall see in Chapter 16, a European option on any asset can be-valued in terms of forward or and (15.14). futures contracts on the asset using etuations (15.13) Note that the maturityof the forwardor futurescontract must be the sam as the maturity of the European option.
15.6AMERICANopylss As describezin Chapter 11, binomialirs can be used to value Americanoptions on indicesand currencies. As in the case of Americanoption on a non-dividend-paying stock,the parapeter determiningthe size of up movelents u ij jet equal to ez' wherec is the volatility and ! is the lengthof time steps. The parameter determining For a non-dividendthe size of down movements, d, is set equal to 1/If, or e-'. stock, probability of paying the an up movementis ,
/=
,
,
a-d u-#
CXt
' wherea = e For options on indicesafd currencies,th formulafor p is the same, but a is defned diferently.In the case of options on n index, .
a=e
(r-)A!
whereq is th8 dividendyieldon the index.In the case of optionson a currency, J=.
(r-r/)Al
wherer/ is the foreignrisk-fr rate. Example11.1in Section11.9shows how a twosteptree can be constructed to value an optionon an index.Example 11 showshow a three-steptreecan be constructed to value an option on a currency. Furtherexamples .2
329
Options on Stock Indicet and Cuyyendes
of the use of binomialtreesto value ojtions on indicesand currencies are given in Chapter 19., . ln some circmstances it is optimal to exerciseAnerican currency options prior to Ameiican maturity.Thus, currency options are worth more than their European countrrprts. ln general, call optionson lghuinteresycurrencies and put ojtions on lowrinterestcurrencies are the most likelyto be exercisedprior iomaturity. The reason is that a lgh-interest cufreny is expectedto deprciatt nd a low-interestcurrency is d appreciate expecte.tp .
IUMMAkY The indexoptions that trade on exchangesare settled in cash. On exerciseof an index call option, the holder reives 1 timesthe amount by whichthe indexexceedsthe strikeprice. jimilarly,on exerciseof an indexput option contract, the holderyeceives 1 timesthe amount by which the strikeprice exceedstheindex.lndex options can be used f0r portfolio insurancekIf the value of the portfolio mirrors' the index,it is appropriate to buy 0ne put Option contract f0r each ls' dollarsin th portfolio, Where S is the value of th index.If the portfolio does not nrror the index,j put optioncontracts should be purchased for each ls' dnllarsin the portfolio, where jis the beta of the portfolip calculated using $h:capital ajset pricing model. The strike priceof the put options purchased shoqld reii the levelof insurancerequired. Mnst currency optiops are iradedin the over-the-countermarket. They can be used by corporate treakurersto hedge a foreignexchpnge exposure. For example, a US treasurerwho knowstht the companywill be receivingsterling at a certain corporate timeip thefuturecan hedgebybuyingput options that mature at that tile. Similarly,a US corporate treasqrerwho knowsthat the company will be ppyingsterling at certain timein the future can hedgeby buyingcall options that mature at that time.Currency optionscan also be,used to create a range fomard cntrad. Tls is a zero-cost contract giving of upside. that provides downsideprotection while up some the Th Black-scholesfoeula for valuingEuropan optios on a noh-dividend-paying stockcan be extended to cover Europeanoptions on a stock paying a knpw dividend yield.T11 e extension tan be ujed to value Europe/n options on stock inices and currencies because: 1. A stock indexis analogousto a stock paying a dividendyield. Thedividendyield is the dividendyield on the stocks that make up the indexr 2. A forelgncurrenty is analogous to a stock paying a dividendyield. The foreign interestrate plays the role of the dividendyield. risk-free '
Binomialtrees can be used to kalue Americanoptions on stock indicesan currencies.
FURTHERREADING Amin, K., and R. Rates,'' Joarnal
p.f
Jarrow. Foreip Currency Opons under Stochasticlnterest InternationalVprlcy and Finance,l (1991): 310-29. ttpricing
.
Bicer.N.. and J. C. Hull. (Spring1983):24-28.
St-fhe
Valuation of Currency Optionss''Filmial Matagement, 12
330
CHAPTER 15
!Qntlle Risk Of Stocksin tlle Long Runq'' Fizazcial Azalyst,Joarzal, 51, 3 (1995): 18.222.
Bodie, Z.
Garman, M. B., and S.W. Kolhagen sForign CurrencyOptign Values,'' Jourzal of J'?l/pr?ll/p?ll/Hozey J?l# Fizazce, 2 (Decembey !983):231-37. ' ,
Giddy, 1,H., and G. Dufey. an 49-57. Coorate J'fzllr4c',8, 3 (1995): teuses
.
buses of CurrencyOptions,'' Joarzal of Applied
Grabbe, J.0. iThe Pricing of Call and Put Options On Foreign Exchangej''Jourzal oj. .J?l/t?r?lJ&?ll/ Hozey J?l# Fizazce, 2 (Decembet1983):239-53. Jorion, P.
lpredicting
597-28. 2(1995):
Volatility in te Foreign Excange Marketr'' Jorzal oj' Finance 5,
Merton, R. C. ttTheoryof Rational OptionPricinp'' Bll Jourzal ofEcozomicsand #f/zzlye??le?l/ Sciezce,4 (Spring1973):141-83. ' '
.%
'
z
.
Manual) and Prpblems(Answers in solutins Quesfions 15.1.A portfolio is currently worth $1 zillion pd has a beta of 1.. An indexis currently standingat to. Explain how a put option on the inbexwith a strike pyiceof 700can be used to provide portfolio insurance. we knowhowto ylue options on a stockpaying a dividendyield, we knowhow 15.2. to value options on stockindicesand currencies.'' Explain this statement. dsonce
15.3. A stockindtxis currently 3, thedividendyield on theindexis 3% per annum, and the jj . risk-freeinteresirate is 8h per annum. Whatis a lowerboundfor the price of a SmOn th Europeancall option on the indexwhen the strike price is 2902 '
15.4. A currency is currently worth $0.80and has a volatility of 12*:. The doestic and foreignrisk-fre interestrates are 6% and 80:, resptikely. Use a two-stepbinomialtree t value (a)a Europeanfour-monthcall option with a ltrike price of 0.79 and (b)an ' Ameyicanfour-monthcall option with the samestrike price. fange forward contracts to hedge their foreigp 15.5.Explain how corporations can exchangerisk. :
.
.
'
'
15.6.Calculatethe value of a thr-month at-the-money Europeancall option on a stock indexwhen the index is at 250, the risk-free interest fate is lts annum, the and yield of the indexis 18% per annum, the dividend on the indexis 3% per volstility 'per
annllm.
15.7.Calculat the value of ap eight-month European put option on a currency with a strik The current exchangerate is0,52,the volatilityof the exchangeratq is 120:, priceof the domesticrisk-free interestrate is 4% per annum, and the foreigntisk-freeinterest ' 9 '' rate is 8/Cper annum. for a put option to sell one uit of currency A 15.8.Showthat the formulain equation (15.12) for currency B at strikeprice # giveste same value as equation (15.1 1)for a call option strike price units of 1/#. A K for to buy currency at currency B 15.9.A foreigncurrency is currently worth T1.5. The domesticand foreignrisk-fr interest rates are 5% and 9%, respectively.Calculatea lowerboundfor the value of a six-tnonth call option on th currencf with a strike price of $1.40 if it is (a)Europea: and (b) American. .5.
'
'
''
'
'
'
331
Optinns !l Stock Indlces tlxd Cuyyencies r'
15.1. Considera stock indexcurrthtlystandingat 250.The dividendyield on th.eindexis 4% per annum, and the tisk-freerat is 6% per annumkA three-monthEurpean call optionon theindexwith a strikeprke of 245i! currehtlyworth $1. Whatisthe value of a three-monthnut option on ihe indexwith a jirike price of 2452 '
.K
.K
.
.
,
1. An indexcurrently stands at 696and hs a volafilityof 3% per annum.The risk-free 15.1 ratt of interestis 7% per annup and the indexprovides a dividendyield of 4% per wlth a exerciseprice of 7. annum,Calculpa.the alueof a three-monthEuropean put
15.12. Showthat, if Cisthe price of an mrrican call with exerclsepli K and maturity T on a stockpaying a dividendyield of q, aild P is tlzepri of an Americanpllt on the same stockwith the sam strikepri nd exercisedte, then he .zg
-
,
.
y. < '
.
c
.,
llint: To obtain thefirsthalf
.
Porfblio : a European call o'ptionplus an amount K investedat the risk-free rate -VT Porfblio B : an Americanput option plus e of stock with dividendsbeing reitock investedin the ,x
To obtain te secnd half of the inequality,consider possiblevalues of t -rT Porfolio C : an Americancall option plus an amount Ke investedpt the riskfre rati Borblio D : 4 European put optin plus one sto'ckwith dividendsbeingreinvestedin ihe stck.) '
Showthat a Europeancall option on a currency hasthe sameprice as the corresponding 15.13,.
European put option on the currency when the fomard kri eqqals the strike pri. 15.14.Wouldyou expet the volatilityof a stock indexto be greater or-lsj thanthe vplatilityof typicalstock? Explain your answer. '
.
.
.
a
1515 Does the cost of portfolio insuran increas or decreaseas the beta of a portfolio increases?Explainyour answer. 15.16.Supposethat a portfolio is worth $6 millionand theS&P 54()is at 1,2. If thevalue of the portfolio mirrors the value of the index, what ptions should be purchased to provideprotection against the value of the portfolio fallingbelow$54 million in one year'stime? 15.17.Consideragain the situation in Problem 15.16.Supposethat the portfolio haq a beta yield on both the of 2., the risk-free interestrate is 5% per annum, and the zividend portfolioand the index is 3% per annum. What options should be prchased to provideprotection aginst the value f the portfolio fallingbelow$54 million in one year'stile? 15.18.An index currently stand at 1,5. European call and put optios with a sike pri of 1,4 and time.to maturity of six months havemarketpfis of 154.00and 34.25, respectively.The six-monthiisk-free rate is 50:. Whatis the implieddividendyield?
15.19.A iotal return index tracks the return, includingdividends,on
rtain portfolio. Explainhow you would value (a)forwardcontracts and (b)Egropeanoptions oii the a
index. 15.20.What is the put-call parity relationslp for European currency options?
332
CHAPTER 15
1521 Can an optipn on the yen-euto exchapgerate be created from two options, one on the dollar-euroexchang rate, an the other on the dollar-yenexchangerate? Explainyout
using the portfolios indicated. and (15.3) 15.22.Provethe results in equations (15.1), (15.2),
Assijnlent
Questions
15.23.The DowJoneslndustrial Averageon Jnuary 12, 27, was 12,556and the price of the March126call was $2.25.Usethe Derivasem softwareto calculate theimjliedvolatility of this option. Assumethe risk-free rate ws 5.3% and the dividen yield was 30:. The option expires on March20,2907.Estimate the pri gf a March 126 put. What is the volatility impliedby te prke you estimati for this ojiion?xtNote that options are on the DowJones index dividedby 100.) 15.24.A stock indexcurrentlystqnds at 3 and has a volatilityof 200:.The risk-free interest rateis 8% and the dividendyield on the indexis 39/:.Use a three-stepbinomialtree to valuea six-month put option on the indexwith a strike pri of 300if it is (a)European and (b) merican? ' the spot price of the Canadiandollaris US j 85 and that the Canadian 15.25.Supposethpt ollar dollar/us exchange rate has a volatilityof 4% per annum. The risk-free rates of and the Urlitedsptesare 4% nd 5% per annum, respectively. in canada interest Calculatethe valu of a European call option to buy one Canadiandollarfor US $0.85 in nine months. Use put-call parity to calulate the jrie of a European put option to sellon Canadiapdollarfor US $0.85in nine months. Whatis the pri of a call option to buy US $0.85with one Canadiandollarin nine months? 15.26.A mutual fund announces that the salaris of its fund managers will depend on the performanceof tht fund.lf te fund lses mony, the salarieswill be zero. lf the fund makesa prolh, the salaries will be proportional to the prot. Describe the salary of a How is a fundmanager motivated to behavewith this type fund manageras an of remqneration package? 15.27.Assumethat the price of currency A expressedin termsof the price of currency B follows the proess ds (0 rhjsdt + asdg , whererA is the risk-free interestpte in crrency and rB isthe risk-free interestrate in currencyB. What isthe pross followedbythe price of cqrrency B expressedin termsof currencvA? 15.28.The three-monthforward UsD/euro exchange rat: is 1.300. The exchangerate volatilityis 15%. A US company will have to pay 1 nllion euros in three months. Yheeuro and USD risk-freerates are 5% and 4%, respectivelf The company deides to use a range forwardcontract with the lowerstrike pri equal to 1.2500. (a) What should the lgher strike price be to create a zero-cost contract? (b) What position in calls and puts should the companytake? (c) Does your answer dependon the euro risk-free rate? Explain. (d) Does your answer dependon the USD risk-free rate? Explain. .
.
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The options we haveconsidered so far provide theholderwith the riglit to buyor sella rtain asset by a certain date. They are sometimestermedoptions on spot oi spot optionsbecquse,when the options are exetcised,the sqleor pmchase of the asset at the agteed-onprke takespla immediately. In tlzischapter wemoveon to consider options onfutures,also knownzbfutures ptions. In thesecontmcts, exerise of the option gives the holdera position in a futurescontract. The CommodityFuturesTradingCommissionauthorize the tradingof options approved in 1987,and futureson an experimetal basisin 1982.Permanentfrading.was popularity of with investorshas grown very fast. the contract tilfc the the In thischapter we consider howfuturisoptionswork and thediflkrences betweenthse an@ optimy options spot options. We examine howfutures a be prked using iither binomialtreesor formulassimilar to thoseproduced by Black, scholes, and Mertn lor stock options. We also explore the relativeprking of futuresoptions and spot options. .on
.
.
..
16.1 NATUREOF FUTURESOFTIONS A futuresoption isthe right, but not the obligation, to enter into a futurescontract at a certainfuturesprice by a certain date.Specically,a call futuresoption is the rkht to enter into a longfuturescontract at a certain price; a put futuresoption is the right to enter into a short futures contmct at a ertain pri. Futres ptions are genemlly American;that is, they can be exemisedany timeduringthe lifeof the contrat. If a call futres option is exercised, the holder acquires long position in the underlyingfutures contract plus a cash amount equal to the most recent settlement futures price minuq the strike pyi. If a put futures option is exercised,the holder pcquiresa short position in the underlyingfuturescontmct plus a cash amount equal to the strike price minus the most recent settlement futures price. As the following examplesshow,the efective payof from a call futuresoption is the futuresprice at the timr of exerciselessthe strikepri; the efective payof froma put futurespption is the strike price lessthe futuresprice at the time of exmise.
fxample16.1 futurescall option Supposeit is August and an investorhas one september with pound. prke of 24 cents per Onefuturescontract contmcton copper a strike '15
333
CHXPTER 16
334
that the fututes pric of copper for is on 25,000polnds of cojper. sppose currently deliveryin Septemberis 251 cents, an at the closr of trading on sdtkment) it was 25 centas.If the option is exerced, the August 14 (thelast investorreceivesa cash amount of
-
25,000x (250 240)cents = $2,500 -
plus a long positbn in a futurescontract to buy 25,000pounds of copper in Septtmber.If desireb,the pojitiop in the futures contract can be closed out immtdiately.This would leavethe investorwith the $2,500cash kyofl-plus an amount
'
25,000
(251 z50lcents$250 =
-
the change in thefuturesprice sin thelastsettlement.'I'hetotal payof refkcting #), fromexercing te option on upst 15 i8 $2,750,which equals 25,(F of pri and K pric.' the time > is futures is the strike at exerce where -
'the
fxample16.2 An investorhas one December futuresput optio on corn with a strike price of 400cents per bushel.0ne futurescontract is on 5, bushelsof corn. Suppose thatthe turrentfuturespric Of corn for deliviryin December is 380,and the mostrecent settlement price 379cents. If the option is exercised,the investor
receives a cash amount of 5,
x (4
379) nts
-
=
$1,050 .
plusa short positionin futurescontract to sell5,000bushelsof corn in becember. If desired,the position ip thefuturescontract can be csosedout. nis would leave the investorwith the $1,050cash payofl-minus a amount 5,
x (380 379) nts = $5 -
refkctingthe change in the futurs price sin the lst settlement. The net payof F), where #' isthefuturesprice at from exerciseis $l,, whichequals 5,(# the timeof exerciseand K te strike price. -
ExpirationMonths Futuresoptions are referred to bytheddivtry month of the underlyingfuturescontract .--not by the expiration month of the option. As mentioned earlier, mdst futures options are American.The expiration date of a futuresoption contract is usuallyon, or a fewdaysbefore,the earliest delive dateof the underlying futurescontract. (For example,ihi CBOT easury bond futuresoption expires on the latestFriday that precedesby at leastfwebusinessdaysthe end of the month beforethe futu:esdelivery month.) An exception is the CME mid-curveEurodollar contmct where the futures contractexpires either one or two years after the options contract. Populai contracts trazing in the Unitedstates are thoseon corn, soybeans, cotton, sugar-world,crudt oil, natural gas, gold, Treajurybonds,Treasurynotes, fwe-year Treasurynotes, 3-day federalfunds,Eurodollars,one-year and two-yearmid-curve Eurodollars,Euribor,Eurobunds,and the S&P5.
335
Futures Options
Options on Interest Rate Futures The most activly trded interestrate optiops ofered by exchangesin the UnitedStates arethose on Treasury bond futurts, Treasury notr futures,and Eurodollarfutures. A Treasu bondfuturesoption, which is tmdedon the ChicagoBoard of Trade, is an optitl tp enter a Treasury bofldfuturescontmct. As mentiond in Chapter6, one of Treasury bonds.The Tresury bnd futurescontmci is for the dlivery of $l, pric of a Trtasury bondfuturesoption is quoted as a percentageof the facevalue of the undrrlying Treasury bondsto th nearest sixty-fourthof l %. An option on Eutodollar.ftures, which is traded on the Cjj cag? u ercanto Exchange,is an option to enter into a Euiodollarfutuyescontrag. Aj explained in Chapter6, when the Eurodollar futura qu te changes h l basispoint, or %, ther is a gain or loss on a Eurodllar futurescontl-at of $15.Similarly,in the pricing of optionson Eurodollar fgtures,l basispoint represents $25. Interest rate futuyesoption contractswork in the same way as the other futures optiohscotracts discussedin thischaptr. For xample, in addition to thecash payof, the holder of a call option obtains a long position in the futurescontract when the optionis qxercised and the option writer obtains a onvsponding short pojition. The total payof fromthe call, includingthe value of thefuturesposition, is maxtf #, ),. whereF' is the futuresprice at the tiine of exerise and K is the strike price. when interestrates Ipterestrate futurespris increasewhen bondprices increas (i.e., fall). They decreasewhen bond prices decrease(i.e.,when interestrates rise). An investorwho thinks tat short-term interestrats Fill lise can speculate by bying tknks the rates will fallcan put options on Eurodollar futures,wheyeasan investorwho speculateby buyipgcall optiops lo Erodollar futures.An investorwho thinksthat lonpterm interestrates will rise can speculateby buyingput options on Treasury note futuresor Treasury bondfutures,whereas an investorwho thinksthe rates will fall can speculateby buyingcall options on theseinstruments. .l
-
fxample16.3 It is February and the futuresprice' for the June Eurodollarcontract is 93.82 (corresjondingto a 3-nionth Eurodollar interest mte of 6.18% per annum). The price of a call option on the contract with a strike price of 94.00is quoted or l.basis points. This option could be attractive to an investorwho feels as that interestl'ates are likelyto come down.Supposethat short-term interestrates do drop by about l basis points and the investorexercisesthe call when the Eurodollarfuturesprice is 94.78(con-esponding to a 3-month Eurodollarinterest of The payof 25 5.22% annum). is x (94.7894.) x l = $1,950. rate per = The cost of the contract is l x 25 $250.The investor'sproht is therefore .1,
-
.$l,7. fxample 16.4 It is August and the futures price for the December Treasury bond contract traded on the CBOT is 96-09(or961 96.28125).The yield on long-term govemmentbonds is abut. 6.4% per annum. An investorwho feelsthat this yield will ?allby Decemer might choose to buy December4 calls with a strik priceof 98.Assumethat the price of thesecalls is 1-4 (or1g 1.0625(j/(')of the principal).If long-termrates fall to 6% per annum and tlze Treasury bond =
=
'
336
CHAPTER 16 '
:
.
.
. .
' .
futurespri
futuresof
,
rises to lg-,
?1.0
the investorwill inakea net proft per $l =
of bond
92.40 1$625= 0.9375 -
Sinceollr option contract is for the purchase br sale of intruments with a face value of $1,, the investorwould make a proft of $937.50per ojtion contractbought. '
YPTIONS 16.2 REASONSFORTE POPqLXRITYOF UJUkES ' .
.
It is natural to ask why people choose to trade ptions on futuresrgthrr than options on the underlying gjset. The main reason appears tc he that a futurs contract is, in mpnycimumstanc's, more liquid and easier to tpde thali the underlying asset. Furthtrmort,a futurts prict is knpwn immediattlyfrom tradlng on the futures excange,whereasthe spot priceof the underlyingasset may not be so readilyavailablr. The market for Treasury bondfuturesis much more actiye ConsiderTreasury tonds. particular Treasury bond.Also,a Treasury bond futuresprice market the for than any is knownimmediatelyfromtradingon the ChicagoBoardof Trade. By contrast, the currentmarket price of a bondcan be obtaine only by contacting one or'more dealers. It is not surprising that investorswould rather takedeliviryof a Treasury bondfutres ' contractthan Treasury bonds, Futures on cpmmodities are also often asier to tradethan the commodities themr For example,it is much easierand more convenientto make or take.delieryof a selves. futurescontract than it is to make or tke deliveryof the cgttle themselves. livewcattle An importantpoint about a futuresoption is that exercisingit doesnqt usually lead to deliveryof the underlying asset, as in most circumstans the underlying futurts conttactis closed out,prior to deliverykFutures options am thereforenormally evenr tuallysettled in cash. This is appealing to many investors,particularly those with capital who my finilit dicult to come up with thefundsto buythe underlying limited asset when an option is exercised. Another advantage sometimes cittd for futures optionsis that futuresand futuresoptions are tradedin pits sideby sid: in the same exchange.This facilitateshedging,arbitrage, and speculation.lt also tendsto make the markets more ecient. A fnal point is tht ftures options tend to entail lower transactionscosts than spot options in mpny situations.
14.3 EUROPEANSPOT AND FUJURESOPTIONS The payofl-from a European call option with strike prite K on the spot price of an asset
is
maxtir
#,
-
t))' -
whereS;. is the spot pri at the option's maturity. The payof from a Europeancall Op tionwit the same strike price (ln the futurespri of the asset is maxtFr
-
#,
%
whereFr isthe futuresprice at the option's maturity. If thefuturescontract matures at
33)
Futues O>j?l:
the same time as the option? then FT % and the two options qre equivalent. Similarly,a Eurojean futgresput ojtion is worth the same as its spot put option countsrpartwhenthe fuiure contrct matures at the same timeas the option. Mostof thefuturesoptions thattradeare Alerican-style.However,as we shall see, it is useful io study Europeanfuturs options bcauye resultsthat are obtained can be used to value the corresponding Europeanspot options. =
.
'the
16.4 PuTucl.t
,
'
PARITY
In Chapier9 we deriveda put-call parity plationskp for Europeanstock options. We now consider a jimilar argument to derivea put-call parity relationship for European futuresoptios. CbnsiderEurbpeancall an l pt fututes options both with strikepri K and time to expiration T. Wecan form two portfolios: ,
Ke-rT Porqlio : a Europeancall futuresoption plus an amount of cash equal to Porolio B : a Europeanput futuresoption plus a long futurescontract plus an -rT where h is the futurespri amouyt of cajh eqpal to he ,
In portfoli A, th cash can beinvsted t the risk-freerate, r, and growsto K at timeT. Let FT be the futuresprice at maturity of 'the option. If FT > k, the call option in portfolioA is exercisedand portfolio A is worth l%. If FT %K, the call is not exemised and portfolio A is worth K. The value of jortfolioA at time T is therefore
maxtfz,Kj In portfolio B, the cash can lieinvestedai the risk-freerayeto grow to F at tipe T. The put option provides a paolllof maxtr &, ).The futurescontract provides a payof F.' The value of porolio B at time T is tlierefore Of F z -
-
h + (&
hj + mqxtf
=
h, )
-
=
maxt/,
Kj
Becausethe two portfolios havethe same value at timeT and European options cannot be exercisedearly, it followsthattheyare worth the same today.The value.of portfolio A todayis -rT c + Ke wherec is the price of the call futuresoption. The marking-to-marketprpcessensures thatthefuturescontract in portfolioB is worthzero today.PprtfolioB isthereforeworth p + he
-rT
wherep is the price of th put futuresoption. Hence -rT c + Ke
=
p+
he
-rT
(16.1)
The diference betweenthis put-call parity relationship and the one for a nondividend-payingstock in eqpation (9.3)is that the stock price, %,is .replacedby the . discountedfutures price, he x
.vy
.
! This analysis assumes that a futqrbs ontract is like a forward coqtrat and settled at the end of its Iife ratherthan on a day-to-day basis.
338
CHAPTER 16 '
j.
As jhowp in Section16.3,when the underlyingfgturescontract mature! at the same and spoy options'are the same. Equatio (16.1) tim as the option, Eropan' futures relayionship thereforegives a betwei tt pri of a call option on the pot price, the priceof a put option on the spot price, and thefuyurespricewhen bpth optigns mature at $hi same time as the futurescontiat. a:'
'
fxample16.5 ,
'
...
.
.
.
.
.
.
.a
.
,.
,
, .
.
,
..
Suppostthai tke price of a Europea:call optio on spot silyerfor deliveryin six mnths is $0.56per ounce whente exrcisepri is $8.50.Assnie that th: silver futurespricefordeliveryin six monthsis currently $8.00,and the risk-free interest rate f0r an invtsymentthat mtlres in jix monthsis 1% per nnuin. From a the plice of a Europeanput option on spot rtarrapgementof equation(16.1), and with exewise date as the call option is silver the same maturity ' 8.50:
x6/12
-.1
-.'
'
.
.
..
0.56 +
.
..
.;
.,
..
.
.
.
.
.
.
x6/l2
g g(k-().1
-
.
=
j 'g4 .
.
We can use equation (16.1) for spot options becausethe futures price that is paturity considerd haj the snme as th ojtion price.
F& Americanfuture options, the put-ca'ilrelationshipis (seeProblem 16.19) F
:-V
y
.,y
-
#
) and q = r: with pnd (15.5) ig
'
.
'
'
'
-
-rT
c= e p
where
tfj
=
=
e
(f'g#@j)KNdj '
LKN-dj)
+ 1n(:g/r)
-
(16.9). (t6.t0)
-
-
1$#(-))
l'lj
cvT
1n(>)/r) :2 cx/f -
=
'
'
lj
=
dj
-
cW
and c is the voltility of the futuresprke. Whenthe cost of carryand
'
athe
convenience
Aswe willdiscoverin Chapter i7,a more precisestatement of theresult is:$tAfuturesprice hasaro driftin thetraditionalrisk-neutralworld wherethe numeraireisthe money market account'' A zero-driftstochastic prowssisknownas a martingak. A fomardpriceis @martinpalein a dferent risk-neutralworld. Thisk one wherethe npmeraire is a zero-couponbondmaturing at timeT. 4 SeeTechnicalNote 7 on the anthor's websiiefor a proof of thk. 3
5 seeF. Black, 167-79.
lt-l-he
iMcing of CommodityContracts,'' Journal :./' Fizancial
'cdntlplfc',
3 (March1976):
FutIqes
343
Oyftm,
yieldare functionsonlyof time,it can be showniai the volaiilityofthe futuresprice is thesame as the volatility of the underling ssit. Note tat Black'smpdel es pot requirethe option cgntract and the futurescontractto niatuye at the snmetime. Ekmplq 16.6 Considera Eropean put futuresoption on crude oil. The timeto thebption's matqrity is4 mpnthk, the crrent futu s pri is $20,iheexercisepri is $2 , the ritk-freeinterestrate is 9% pir annllm,pd the volatility of the.future! pri ;:iz T = 4/12, c = 0.25, is 25% per annum, In this ase, F 1 2, K = 2, ?' = and lnt&/& so that gw,/=': = 2.27216 dj 2 .9,
,
M
dz = N-djj
=
clT
-0.07216
-
-
=
2
.
0.4712, N-d
=
0.5288
and the put price p is givrn by p= e
or $1.12.*
-.99x4/12
(2 . x 0.5288 -
20 x 0.4712)= 1.12
Using Black's Model Instead of Black-jcholes The resultj i Section16.3 showthat futurs optipns apd jfot options fe equivalent whn the ojtion contmct latures at the same time as th futurej contract. Equaand (16.1)thereforeprovide a q of calculating the vatue of European tions (16.9) optionson the spot pri of a asset. .
Expmple16.7 Considefa six-monthEurolean call option on the spot pri of gold, ihatis, an option to buy one oln f gld ln six months. The strikeprim is $600,the sixmonthfuturesprice of gold is $620,thetisk-freerate pf intest is 5% per annum, and the volatility of the futuresprke is 2%. The option is the same aj a sixmont Europeanoption on the six-monthfuturesprke. The value of the option is thereforegiven by equation (16.9) as e
where 4
=
-.5x.5
(jzgxtl
-
jggxgzj
1n(62/6) + 0.22x 0.5/2 = 0.3026 .2 x k5 2 x g jy2 ln(62/6) .2
-
k It is $44.19.
.
=
.1611
=
.2 x
.5
Tmders like to use Black'smodel rather than Black-schles to value European options on a wide ranje of underlying assets.The variable h in equations(16.9) and (16.10)is set equalto eltherthefuturesor thefomard prke of the underlyingassetfor a and (15.14) show contfactmaturingat the sametime as the ojtion. Equations(15.13)
344
CHAPTER 1t
Black'smodel beingused to value Europeanoptions on the Spot value of a currency. Theyavoid the need to estimate the fomignlk-free interst rate explicitly.Equations shw Black'smodelbeihgused to value Europeanoptions on the spot (15.8)and (15.$ of value an index.Theyavoidthe need to estimte the dividnd yield explicitly. As explained in Section15.4,Black'smodel can be used to imlly a term kructureof forwardratesfrop activelytradedindexoptions. TlieforWardrtes c4nthenbe used to pri otlter optlons on the index.The sameapproachcan bt ustd for other underlying ass 1
its .
16.9 AMERICANFUTURESUPTIONSvs. XMERICANSPQTOPTIONS Tradedfutures options are in prgcti usually Amerkan.Aspming thgt the risk-fre rate of interest,r, is positive, thereis plwayssome chan that it will be optimal to exercisean American futures option early. Americanfutures options ar therefore worthmore than their Europeancounterparts. It is ot generally true that an Amerkanfuturesoption is wofth the same as the dlngAmericn spot option when the futuresand opiions contractshavethe correspon 1 thai thereis a normnl marketwith futuresprices samematurity.6Suppose,f r exampe, higherthan spot prkes prior to maturity.Thisis the'casewith mst stock consistently gold, silver, low-interestturrencies, and some commdies. An Aperican call indices, futuresoption lnust be worth morethanthe corresponding Amerkanspot cpll ojtion. The reason is that in somesituationsthefuturesopn will be exercis:dearly, in which castit will proyide gzeater prot to the hlder. Similaflyjan Americanput futures opjn must be woith lessthanthe correspondingAmericanspot put option. If tei: is consistentlylowerthan spot jrkes,as isthe case an invertedmarket with futures withhigh-interestcurrenies and somecommodities,the reversemstbe trqe. American callfuturej pptions ar wgrth lessthan the corresponding Amerkan spot call option, whereasAmericanput futures options are worth more than the correspondingAmerican spot put option. The diferencesjustdejribed betweenAmerkanfuturesoptionj and Amerkan sppt optionshold tfue when the futurescontiact eipire laterthan the options contract as Fell as when the two expire at the same time.In fact, the later the futurescontract expiresthe greater tlle diferens tendto'be. '
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.
'jris
OPTIONS 16.10 FUTURES-STYLE optiozs.Theseare futurescontracts Someexchangestradewhat are tznnzdfutures-style who option. payof trader buys(se11s) Normally the from a an an option, whether on on the spot prict of an ssd or on the futuresprice of an assd, pays (reiyes)cash up front. By contrast, traderswho buy or sell a futures-styleoption post margi in the sameway that th j do on a regular futurescontract (seeChapter2). The contractis settleddaily as wit any other futurescontract and the %al settlementpri is the Payofl-flpmthe option. Just as a futurescontract a Vton what the futureprke of an 6 The spot option corresponding'' to a future,soption is dened hereas one with the samestrikepri and the sameexpiration date.
:'
Futgyes
345
.pftm,s
'
. .
.
'.
'
.
J '
'
.
assetwill be, a futures-styleoptln is p bet on what the payo froman optionwill e.7 lf intefestmtesar conjtant, the futuresprke in a futures-styleption isthe sameas the that the futures fomardpri in fomard contract op the optlon payof. This optlon paib prke would futures-'style that be fo optio isthe if pafment for the price 4 weremadein arrears.It isthereforethe value of rejular ojtion compounded fomard as the risk-free rate. and (16.1t) givesthe pric of a regular European Black'smodel in equatibns (16.9) of optionon an assetin terms the futures(orforward)prke F for a contract maturing at the same time qs the option. The futuresprice in a call.futurl-style option is 'shows
therefdre
'
ygxgj) rx(4) -
and the futurespri in a put futures-stykoption is KN-d
-
hN-djj
and (16.10). wheredj and k are as defned in equations (16.9) Thesefdrmulasare o ption option on the and futures-style for futures-style fnturej,contract correct a a on a spot value of an asset.In the flrstcase, F' is th currny futurespri for the contract underlyingthe option;in the sond case, it is the currentfuturesprke for a futures contract on the underlying asset matqringat jhe Same time as tlie optioh. The put-call parity relationsllip for a futures-styleoptions is .
p + >) = c + K '
' .
.
'
.
An Aperican futres-style option can be exercisedearly, in wch case there is an immediatefnal settlement at the option's intrinsicvalue. As it tlrnsout, it is never optimal to exercise an American futures-styleoptions on a futures contmct early becausethe futuresprice of the option is alwaysFeater than the intrinsicvalue.This type of Americanfutures-styleoptitm can ihereforebe treatedas thoughit were the correspondingEropean futures-styleoption.
SUMMXRY Futures options require deliveryof the underlyihgfuturescontmct on exercise.Whena callis exercised,theholderacquiresa longfuturesposition plus a cash amountequal to the excessof the futuresprice ovtr thestrikeprke. Simllarly,when a put is exercisedthe holderacquires a short position plus a cash nmount equal to the excessof the strike priceover thefuturesprice. Thefuturescontmct thatisdelivefedusqallyexpiresslightly
laterthan the option. A futuresprice behavesin the snme way as a stockthat prokides a dikdend yield equal to the risk-free rate, r. This means that the results prodced in hapter 15for options on a stock paying a dividendyield applyto ftures options if we replace the stockprice by the futurespri and set thi dividendykld equal to the risk-freeinterest Pricingwith Futurestzle 7 For a more detaileddiscussionof futures-styleoptions, see D. Lieu, option Margining,'' Jbarnal ofFutures Markets, 1, 4 (199),327-38.For pricingwheninterest rates are stochastic, hterest RateFutures Optionswith Futurestyle Margining.''Joarnal seeR.-R. Chenand L. Scott, ofFutures Marktsj 13, 1 (1993) 15-22). :pricing
i' i i :
l .-2
346
CHAPTER 16
rate.Pricingformulasfor Europeanfutures opti4ns were frst
produced by Fischer
Blackin 197. They assume that the futurts price is loporlally distributedat the option's expiration. If the expimtiondayesfor the option and futurescontmctsare the sme, a European futuresoption is worth exactly the'sameas the prrespondinjEirpean spot option. result is not true for Th result is often used to value Europeanpot options. larket is nonlpl, an Amirkan callfutre is worth miricanoptions. If the futures call option, whilean Amerkanput futures thecorrspondingAlerican spot than more is worth lessthan the corxspondingAmericanspotput option. If the futuresmarketis inverted,the reverse is true. '.'l'he
FURTHERREXDING @'
Back, F. tsThe Pricingof CommodityContracts,'?Joarnal of f?larlcfll Ecozomics,3 (1976): 167-79
&
.
of CommodityFutures and Optionsuqder Stochastic Hilliard, J. E., and J. Reis. valuation Yields,Interest Convenien Rates,an JumpDfuslons in fheSpot'' Jourzal ofFizatial Jrltf Azalysis, l 33, (March1998):61-86. Qaaztitative Miltersen, K.R., and E.S.Schwartz.pricingofOption o Commodity Futureswith Stochastic Term Structures of Conveniefl Yields and Interest Rates,'' Joarzal of Fizatial and QaaztitateAzalysis, 33, 1 (March1998):33-59.
and Problems(njwersin SolutionsManual) Questions betweena call Option on yen and a call option on yen futures. 16.2.Why are options on bondfuturesmore actively tradedthan options on bonds? 16.3. ttA futurespri is like a stockpayinga dividendyield.'' Whatis the dividendyield?
16.1 Explain'the difertn .
16.4.A futurespriceis currently50.At the end of six months it will be eithey 56 or 46. The risk-freeinterestrate is 6% per annum. Whatis the value of a six-month Europeancall option on the futureswith a strikeprke of 52 16.5.Howdoesthe put-callparity formulafor a futuresoption difer fromput-call parityfor n option on a non-dividnd-paying stock? 16.6.Considerp Amerkan futurescall option wherethe futurescontract nd the option contractexpire at the snmetime.Underwhat circumstans is the futuresoption worth lore than the corresponding merkan option on the underlyingasset? 16.7.Clculate the value of a fveumoth Europeanput futuresoption henthe futuresprice is $19,the strike price is $20,the rk-free interestrate is 12% per annum, and the volatilityof the fuiurespriceis 20% per annum. 16.8.Supposeyou buy a put optitm contracton Octpbergold futureswith a strikepri of $700per ounc. Each contractis for the deliveryof 1 ouns. What happensif you exercisewhe the Octoberfuturesprke is $6802 16.9.Suppoleyou sell a call option contracton Aprillivecattlefutureswith a strikepri of pounds. Whathappensif 90 cents er gound. Eachcontract is forthe deliveryof 9$ cents? the contrgct ls exerc ed whenthe futurespri ,000
347
FutuyesO>fpn5' 16.1.
Considera two-monthcall futuresoption with a strike price of 40 whe: the risk-free interest rte is 10/a ; per annum. The current futurespriceis 4) at kya,jowry jouu: wjy f?r the value of the futuresoption if it is (a)Etlroyan and (b)American? .
onsider a fotlr-monthjut futuresoption with a strikeprice of 50 whtn the risk-fr interestirateis '1% per annum. The current futuresprice is 47. Whatis a lowerhound for thevalue of the futuresoptjon if it is (a)Eurpean and (b)Amerkan? 16.12.A futuresprice is currently 60and'its yolAtility is 3%. The risk-free intirest rate is 8% perannum.Use a two-stepbinoial tre to calculatethe value of six-monthEuropean call option on the futureswlth a strikepriceof 62 If the call were American,would it everbe worth exercisingit early? 1. 16.1
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@
.
.
16.12, what does the binbmialtree give for the value of a six-monjh 16.13.In troblem Eurppealiput o/tion on futres with a strikepri of 6? If the put were merkan, would it ever be worth exercising it tarly? Verifythat the call pris calculatd in Problem 16.12and the put pris calculated ere satisfyput-call parityrelationslps. 16.14.A futuresprice is currently 25, its volatilityis 3% per annum,and the risk-free interest rate is 1% per annum. What ls the value of a'nine-month Europan call on the futures with a strike price of 26? '
.
.
16.15.A futuresprice is currently 7, its volatilityis 20% per annum, and the risk-free interest , rate is 6% per annum. Whatis the value of a hve-monthEuropeanput on the futures with a strike pri of 65? '' .'
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''
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16.16.Supposethat a one-year futuresprie is cufrently 35.A ne-yearEuropean ll optin anda one-year Europeanput ojtion on the futurej with a strike p:.jce of 94 ax yotjy prkedat 2 in the market. The fisk-free intefestrate is 10% per annum. Identify an arbitrageopportunity.
16.17. srfhe pri f an at-the-money Europeancall futuresoption alwaysequals the price of a similarat-the-money Europeanput futuresoption.''Explainwhy tilis statement is true. 16.18.Supposethat a futuresprice is currently 3. The risk-freeinterestrte is 5% pir annum. A three-monthAmericancall futures opon with a strike pri of 28 is worth 4, Calculatebovndi f0r the pri of a three-monthAmericanput futures option with a strikeprice of 28. '
.
.
16.19.Showthat, if C is the price of ap Amerkancall option on a futurescontract when the strikeprice is K andthe maturity is T, and / isthe prke of an merican put on the same futurescontract with the samestrikepri and exercisedate,then
he-rr -
g
) is the futurespri and r isthe risk-freerate. Assumethat r > and that there is no diserence betweenfomard and futures contracts. Hizt: Use an analogops approachto that indicatedfor Prohlem15.12.) 16.20.Calculatethe price of a three-monthEuropeancall option On the spot value of silver. The three-monthfuturespri is $12,the strikepri is $13,the risk-free rate is 4% and the volatility of the pri of silver is 25%. 1621 A corporatin knowsthat in threemonths it'willhave$5million to investfor % daysat LIBOR minus 50hasiskointsand wishesto ensure that the rate ohtained will be at least 6.5%.What positionin exchange-tradedoptions should it take? .
?
348
CIJAPTER 16
AssignmentQuestions '
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16.22.A futmespri is currently 4: It is knownthat at the end of threemonthsthepri will be either 35 or 45. Wat i; the value f a three-month Eurnpeancall option on the futureswith a strikepri of 42if $h9risk-fr interestrate is 7% per anmlm? 16.23.It is Febrlarf 4. Julycall options011 r futuieswith strikeprkes ot26 27 , 28 29 and 30 coqt 26.75,21.25,17.25,14.t, and 11.375,fespectely. July put optionswith thesejtrike pricis cost 8.50,13.50,19.t, 25.625,and 32.625,respectively.The optiohs m tureon June 19,the current Jly cop fptqrespri is278.75,andtherisk-freeinterest rate is 1.1%.Calculateimpliedvolatilitiesf0i the optionsusingDerivaGem.Comment on the results you get. Calculatethe impliedvolatility of soyben futurespris fromihefollowinginformation 16.24. conrning a Europeanput on soybeanfuturey: ,
,
,
'
-
Cmrentfuturespri Ererciseprice . Risk-fre rate Timeto maturity Put pri
gj
5 525 6% per annllm 5 moths 20
16.25.Calculatethe price of a six-monthEuropeanput optionon the spotvalue ofthes&P 5. The six-monthforwardpric of the lndex is 1,40, the strikepri is 1,45, the risk-free rate is 504, and the volatility ofthe indexis 15%. ''
l
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C
*he lGtek
APTE
Letsr.
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A Enancialimtitutio,that sells an option to a client in the over-the-countermarketq is facedwith tlie problil f managing its risk. If the option happensto be the same as one thatistradedon an exchange,thefnancialinstitutioncan neutraliveits exposurehy buyingon the gxchange the sameopon as it has sld. 'But whe the option has been tailoredto the needs of a client and doesnot correspond to the standardizedproducts tradedby exchanges, hed/ng the exposure is far mom dicult. In this chapter we discusssome of the alternativeapproachesto this problem. We coverwhat are commonly referred to as the dGreekletters'', or simplythe SdGreeks'' EachGreekletter measure! a diferentdipensionto th risk in an pption pojition and theaim of a traderis to mnage theGDeks so that all risks are aeptable. Th analysis presentedin this chapter is applicableto maiket makers in options on an exchange as lell as to traters working in the over-the-countermarket for fnancialinstitutions. Towardthe end of ihechapter, we willconsider the creationof options synthetically. This turns out to be very closelyrelated to the hedgipgof bptions. Creatingan option position synthetically is essntially the sam task as hedgingthe opposite option position.For example,creating a long call option synthttially is the same as hedging a shortpositionin the c411option. .
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-.
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17.1 ILLUSTRATION 1trthe nextfewsectionsweuse as an examplethe position f a Mancialinstitutionthat shares of a non-dividendhas sold foy $30,000a Europeancall option on lr payingstock.We assume that the stockpriceis $49,thestrikeprke is $50,the risk-frie interestrate is 5% per annum,the stockpri volatilityis 2% per annum, the timeto years), and the expectedDturn fromthe stock is 13%ptr maturityis 20 weeks (0.3846 annum. With our usual notation, this means that '
j
k
=
49, K = 5,
.2,
.5,
r
=
c
=
T = 0.3846, p
The Black-scholes prke of the option is about $24,.
.13
=
Thefnancialianstitution has
1 Asshownin Chapters11and 13,theerpected return isirrelevantto theprking of an option. lt is givenhere becauseit can havesome bearingon the efkctivenessof a hedgingscheme.
349
350
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CHAPTER 17 '
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thexforesoldthe option for $60,000 more thanitstheoreticalvalue, but it is facedwith the problm of heging the rkks. j NAkEDAsD tovERzD Po ITIONS 17.2
open to tmanualinstitution Well ifthestockprie i below .onestpter that YxXvpositiollt is $5 at the
isto donotung. Tus is sometimesreferred
the
10
Z.SQ
.
Works
2 Stfategy
nothinganb it makes lnstitution endof the20weeks.The ovtion thencoststheVancial works positi well call of less if yhe is exercisedbecausethe apro t $3,. A n keb i'i shares the Market p'riceprevailing Mancialinstitutionthenhas buf
l, at in 20 weeksto cuvrr the call. 'Thecost to thefmandalinstitutionis l, timesthe amount by which the stockpri exeds thestrikeprice. For exapple, if after 20weeksth9 stock Tllisis considerably Priceis $6 ? ti option costs the Vancial instution $t option. charged for the greaterthan the $3, As an alternativeto a naked position, the Enancialinstituon ca dopt a coyered posttioz. Thisinvolves uying l, sharesas soonas the option has een sold.If the ' straiegy well,but in other circumstancesit could lead to a works exerised, this is option loss7 Fr example, if the stock price dropsto $40,the fnancia'linstitution signiqcant position. This is considerably greater than the $3, lsij Sl? on its stock 3 chargedfor the optin. Neither a naked position nor a cbvered position provides a good hedge.If the assumptionsunderlying the Black-scholesformulahold, the cost to ihe fnancial institntionshould always be $240,000 on average for both approachtd? But on any 0n oasion the cost is liableto range from zero to over $1,0,00. A good hedge wouldensure that the cost is alwaysclose to $240,000. to
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17.3 A STOP-LOSSSTRATEGY 0ne interestingheging procedure that is somtimes proposed involvesa stoyloss strategy.Toillustratethebasicidea,consideran institutionthathas writtrn call option withstrikepri K to buyone unit of a stock.Thehedgingprocedureinvolvesbuyingone unit of the stock as Soon as its pricerises aboveK and sellingit as soon as its pri falls belowK. The objectiveis t holda ake positionwheneverthestockprice islissthan K and a covered position whnever the stock price is greater than K, The produre is designedto ensure that at timeT theinstitutionownsthestockifthe option closesin the not own it ifthe option closesout of the money.The strategyappears to moneyand zoes lroduce payofs that are the sameas thepayofs on the option. In thesituationillustrated in Fipre 17.1,it involvesbuyingthe siockat time 11 selling it at time l2, buyingit at time l3, sellingit at time!4, bying it at time !5, anddeliveringit at time T. ,
z A call option on a non-dividend-payingstock is a convenientexanple withwhich to developour ideas.The point.stkat will be made apply ti other typesof options and to other derivatives. 3 Put-call parity showsthattheexposurefromwriting coveredtallis the4ame as the exposurefromwriting
a naked put. 4 More precisely,the present value of the expected appropriaterisk-adjusteddiscotmtrate,sare used. .
,cost
,
is.$240,0*for both approache,sassuming tkat
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Fi! ure 1/.1
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A stop-loss strategy.
Stock Price S ,
Time :
.
11
tg
13
z
lj
14
l
As usual, we denotethe initialstockprice by k. The cost of setting up the hedge is k if k > K and zero otheme. It ems o fhoughthe total cost, Q, of initially ' writing and hedgingtLeoption is the option's intrinsicvalae: '
Q
=
maxgc
-
#, 0)
(17.1)
This is becausea11purchases and sales subsequentto timet are made at prke K. If this werein fact correct, the hedgingscheme would work perfedly in the absece of costs. Furthermoa, the cost of hedgingthe option would alwaysbe less transactions thanits Black-scholes pri. 2Thus, an investor ould earn risklesspro:ts by writing .
optionsand hedgingthem.
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There are two basic reasons' why equation (1t.1)is incoyrect.Te flrstis that the csh :owj to the hedgeroccur at diflrent timesand must be dcounted. The second thatpurhases and salescannot be made at exactly the same price K. Th second pointis critical. If we assumea rk-neutral world with zero interestrates, we can justifyignoringthe time value of money. But we cannot legitimatelyassumethat both purchasesand sales are made at the same price.If marketj are ecient, te hedger cannot know whether, Fhen the stock pri equalj #, it will continueabove or below #. As a practical matter, purchasesmust be made at a price K + 6 and salesmust be made at a pri K 6, for some small positive number 6. Thus, everypurchaseand subsequent sale involvesa cost (apartfrom transaction costs) of 26. A atural responseon the part of te hedger to .monitorprice movements more closely, so tat 6 reduced. Assllmingtat stock prices changecontinuously,6 can be made arbitrarilysmall by monitoring the stock prices closely. But as 6 is made smaller, cost per trade is ofset by the tradestend to occur more frequently.Thus, te tower -
CZAPTER 17
352
Performa of stop-loss strategy. ne perfornpncepeasu' ri is the ratio of the standarddeviationof the cosi of kriting tht ption and hedgingit to the theoyeticqlprice f the option.
Table 1 7.t
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s
s.
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t (weeks)
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5
Hedgeperforman
1.02
'4
1
0.93
0.82
.5
0.77
0,25
0.76
9.76
the expected number of trades tends to increasedfrequencyof trading.As insnity.5 A stop-loss strategs althoughsupecially attractive,doesn0t work particularly well option.If the kockprice as a hedgingscheme.Considerit ude for all out-of-the-money neverreachesthestrikepricr #, theheging schele ccsts nothing. Ifthepath ofthestock pri crosseste strikepricelevelmany times,the scllemeis quite expensive. MonteCarlo simulationcap be sed to assss the overallperfprmance of stop-loss hedging. involvesrandomly samplingpaths forthe stockpri and observingthe results of using the scheme.Table17.1showsthereslts fprth option consideredearlier. It assumesthat ' the stock price is observed at the end of time intervalsof length ht. The hedge performancemeasure is the ratio of 4hestandard deviatknof the cost of hedgingthe pptionto te lack-scholesoption price.Eachresult isbad on l, saple paths for the stock pri and has a standard etror of about 2%. A perfecthedgewould have a hedgeperformancemeastlreof zerooIp thiscas: it appears to beimpossibleto produce a valuef0rthe hedgeperfoimancemeasurebrlow0.70regardlessof howsmall ht is made. .->
,
'this
17.4 DELTAHEDGING Most traders use mol'e sophisticatedhedgingschemes than those mentioned so far. Theseinvolvecalculating measures suchas delta,gamma, and vega. In this section we lple ptayed by delta. consider the The delta(A) of an optionwas intfoducedin Chpter 11. It is desned as the rate of changeof the option price kith respect to the pri of the underlying asset. It is the slope of the curve' ihat re1ates the optionprke to the underlying asset prke. suppose Thk means that when tlkestock price that the de1taof a call option on a stockis changtsby a small amount, the optionprice changes by about 6% of that amount. Figure17.2shows the relationslp betweena call pri e and the underlying stockprice. Whenthe stock price corresponds to pointA, the optionprice corresponds to point B, and A is the slopeof the lineiydicayed.In general, '
.6.
:
w
3c
''e
3S
wherec is the price of the call optionand S is the stock price. 12.2,the expect d'number of timesa Wienerprocessequalsanyparticular valuein 5 As mentionedin section given timeintervalis inlite. a 6
'
'
Theprecisehedgingl'ule used was as follows.If the stock pri movesfrombelowK to above K in a time intervalof length Al, it isboughtat the end of theinterval.If it movesfromabove K to belowK in thetime interval,it is sold at the end of the interval;oherwise, no adion is takem '
knhetzr
353
Iuetteys Fijure 17.2
Calculation f delta.
Option
pric
Slope=
.
'
.6
=
j 1
Stock Price
l I
.
l
A
is $100.and the option price is $10. Supposethat, in Figur 17.2,th stock grice Imagine an investorwho has sold 20 call option contracts-that is, options to buy 2,000shares. The investor's position ould be hedged by buying 0.6 x 2,000 on the opon positionwould then tend to be ofset by 1200 sha'res. The gain (loss) stock position. the loss (gain) For example, if the stock price goes up by $1 on the (producinga gain of $1,200on the shares purchased), the option pri will tend to go on the options written); if the stock a lossof $1,200 up by x $1= $0.60(producing pricegoes downby $1(producing a lossof $1,200on the sharespurchased), the option pricewill tend to gt) downby $0.60(producing on the options written). a gain of $1,200 option positio of example, is In this the delta the invstor's =
1
.6
.6 x
(-2,)
-1,2
=
jhort optin position when the stock . ln other words, the investorloses1,2A. on the priceincreasesby hS. The delta of the stockis 1., so that tLelong position in 1,2 shareshas a delta of +1,2. The delta of the investor's virall position is, therifore, zero.The delta of the stock position ofsets thedeltaof the option position. A posiiidn with a delta of zero is referred to as beingdeltaneatral. It is importantto realize that, becausedeltachanges,theinvestor'sposition remains deltahedged(ordelta neutral) for onl? a relativelyshort period of time.The hedgehas to be adjustedperiodically. This is knownas rebalancing.ln our example, at the end of 3 days the stock price might increaseto $11. As indiated by Figure 17.2, an increasein the stock price leadsto an increasein delta.Supposethat delta rises from l shares would thenhaveto be purchased to 60 to 0.65.An extra 0.05x 2, maintainthe hedge, The delta-hedgingprocedux justdescribedis an exampleof dynamic edging.lt can be contrasted with static edging,where thehedgeis set up initiallyand never adjusted. Statichedgingis sometimes alsoreferredto as edge-ahd-forget.Deltais closelyrelated to the Black-scholes-Mertonanalysis.As explained in Chapter 13, Black,Scholes, and Merton showed that it is possible to set up a riskless portfolio consisting of a positionin an option on a stock and a position in the stock. Expressedin termsof A, =
,
334
CHAPER
17
the Black-scholesportfgliois - 1: option ( + : shares f thi stoc Using ollr new terminlogy, we can sy that Black and Scholejvalued options hy settingup a delta-neutralposition andarping th>fthereturn on tLepositipn shouldbe the risk-frer interestrate. .
'
.
,
Delta of Eufopean Stock Options For a Eropean call option on a non-dividend-payingsyock,it can be shown (see Problem13.17)that .
atcall) =
Ndjj
'
. .
wherek is desnedas i equatkon(13.2).The formulagivesthedeltaof a longposition in one call option. The delta f a short position in one tall option is -Nk4. Using deltahedgig for a shortposition in a Europeancalloptibn involvesmaintaining a lmg posionof Ndbj foreachoption sold.Similarly,using deltahedgingfor a longposition ina Europeancall pption' ipmlyr'spaintaininga shoytposion of Nkj sharesfor each option purchased. For a Europeanput option on a non-dividend-payingstock,deltais given by Amt)
=
Nkj
-
1
Delta is negative, whih means that a long position in a put opon shouldbe hedged with a long position in the underlying stock, and a shol.t position in a ppt opon shouldbe hedgedwith a short position in the underlying stock.Figure17.3showsthe vatiationof the deltaof a calloption and a put option with the stockprke. Figure17.4 showsthe variation of diI with the timeto maturity for in-the-money,at-the-money, and out-of-the-moneycall options. Figure 17.3
Variationof deltawith stock price for (a)a call option and
option on a non-dividend-payingstock.
Delta of
Delta of
1..
call
put
Stcckprice >
K
(a) .
sockprice
0.0
.
0.0
(b)a put
,, .
K
-1.0 (b) '
355
The GveekLettevs '
Fijure 17.4
Typical patttrns fcr variatin
.
optin.
f delta with timt to patrity for a call
Delta
jn (u mcjwy
At themoney
0ut of te mney
Time to expiration
fxample17./ Consideragainthe call optio ot a non-dividend-paingstockin Sedion 17.1 wherethe stbckprice is $49,the strikeprice is $50,the risk-free rate is 5%, the timeto maturityis20weeks(=0.3846years),andthevolatilityis20%.In thiscase, n
Delta is Nhj, or 0.522.Whenthe stockprict changes by hS, the option price changesby 0.522A,%
Dnamic Aspects of Delta Hedging Tables17.2and 17.3provide two examplesof the operation of deltahedgingfor the examplein Section1t.1. The hedgeis assumedto be djustedor rebaland weekly. The initial value of deltafor the optionbeingsold is calculated in Exnmple17.1as 0,522.This meansthat the delta of the short option position ij icitially As wrhtep, be borrowed optionis buy must the 52,200shares at $2,557,800 to soon as a price of $49.The rate of interestis 5%. An interestcost of approximately $2,500is thereforeincurredin the Erst week. The deltaof In Table17.2the stock price fallsbythe end of theftrstweek to $48.12. of option position option declinesto 0.452,so that the new delta the is the This means that 6,400pf the sharesicitiallypurchased are sold to mntain the hedge. in cash, and tht cumulative bonowingsat the end of The stratec rtalizps $302,000 During the secondweek, the stockprice redus to Week1 are redud to $2,252,300. $47.37,delta declinesagain, and so on. Towardthe end of the life of the option, it -52,2.
-45,200.
3S6
CHAPTER 17
Table 17.2 simulation of deltahed/ng. Optiop losesin the money and tost of hedgingis $263,300. Fddk
becomesapparent that the option will be exercised and the delta of the option 1.0. By Week2, therefore,the hedgerhas a fully verd posititm. The approaches receives $5million for the stock held,so that thetotal ost of writing the option heger andhedgingit is $243,300. -
Table 17.3illustratesan alternative sequen of events such thatthe option closesout of the money. .Asit becomesclear that the option will not be exercised, delta zero. By Week20 the hedgerhas a naked position and has incurredcosts ap/roaches
totaling$256,600.
Ip Tables 17.2 and 17.), the costs of hedgingthe option, when discohted to the of the period, are close to but not exatly the ssme aj the Black-scholes beginning If the hedgingschime worked perfectly, the cost of hedgingwould, priceof $240,000. afterdiscounting,be exactlyequal to the Black-scholesprke for everysimulatedstock pricepath. The reason for the variation in the cost of hedgingis that te hedgeis only once a week. As rebalancingtakes place more frequently,the rebalanced variationin tlie cost of hedgingis reduced. Of coupe, the exsmples in Tables 17.2 and 17.3are idealizedin that theyassume that the volatility is constant and there are no transactioncosts.
357
zh (7r::# Iuetteys
Table 17.3 Simulatio of dlta hedzing.Optkoncloses out of the moneyand cost of hedgingis $256,699. Fe/c
Table 17.4 shows statisticspn the peribrmance of delta hedgipgobtained frpm 1, random stock pri patli in our example. Aj ip Table 17.1,the peiforman is the ratio of the standarddeviationof te cost of heiipg the option to the measure Black-scholes pri of the ogtion. It is clear that deltahedgingis a greatimprovement overa jtop-lossstrategy.Unllkea stop-lossstrategy,the performan of a delta-hedging strater gets steadilybetter as the hedgeis monitored more frequently. '
,
performan masure is the Table 17.4 Prrforman of deltahedging. ratio of the standarddeviationof the cost of writing theoptionand hedging it to the teoretical price'ofthe option. 'rhe
Time betweenhedge
rebalancing(weeks): Performancemeasure:
5 0.43
4
2
0.39
9.26
1 ,19
.5
.14
0.25 .9
35y
CAPTER 17
belt hedgingaimsto keepthe value of the
nancial institution'sposition as clost to unchangedas.possible.Initially,the value f the written option is In the value calulated of theoptioncan be situationdepictedin Tablel7.2,the in as $414,500 option position. itsshort Week9.Thus,the nancial institutionhaslost$174,500 Its on cumulative osition, measured is in Wek by the cost, $1,442,900 9 h ihan worse p aj cas il Wttk Tht valut of tht shartsheldhas incrtastdfrom $2,557,8*to $4,171,1. The net efect of all this is that the value of the fnancial institqtion'sposition ha7 changed by oply $4,1 betweenWk (i and Week9.
s24,.
.
Where the Cost Comes From j
The delta-hedgingprocedure in T&bes 17.2and 17.3creates the equivalentof a long positionin the option. Tltis neutmlizes the short position the fmancialinstitution createdby writing the optkon. s the tbles lllustrate, deltahed/ng a short position jj generallyinvolvessellingstock just ter the pri has gone downand buyingstockjust after the price has gone up. It might be termeda buy-high,sell-lowtradingstratety! paid for the The cost of S24, comes froM the averagediferencebetweenthe stockand the price realized for it. 'price
Delta of a Portfolio The delta Of a portfolio of options or other derivativesdependenton a sincle asset Price is S is W'
Whose
S
)' where1: is the value of the portfolio. ' The delta of the portfolio can be calculatedfromthe deltasof theindividualopons in theportfolio.If a portfolio consistsof a quantity wf f opjqn.f (1%i K r),lh! delta of the portfoliois given by '''t
-
'
'
,
-
.
hi h l'-lu I =
'
izz
where Af is the delta of the fth option. The fonnula can be used to calculate the positionin the underlying asset necessaryto makethedeltaof the portfolio zero. When this positiqn has beentaken,the jortfoliois referred to as beingdeltaneatral. Supposea fnancial institutionhas the followingthree positions in options on a Stock :
call optionswith strikepri $55and an expiration date 1 A longposition in l, in 3 months.The delta of tach optionis 0.533. call options with strikepri $56and an expiration 2. A short posion in z, date in 5 months.The delt of eah option is 0.468. 3 A short positin in 50 put options with strikeprice $56and an expirationdate in 2 months. The delta of each option is #
*
N
-.52.
The deltaof the whole portfolio is lo,
x 0.533 20,0 -
x 0.468 5,0 -
x
(-.508)
-14,9
=
This meansthat the portfolio can lx made delta neutral by buyinj 14,9 shares.
359
lnhe(irdd Ivetters J .)
. .'
.'Transactions Costs '
.
..
.
'
.
''
'
..
..
Derivatives dealers usually rebalance theit positios once a day to mpintain delta neutrality.Whenthr dealerhas a smallnumber of options on a partiplar asset, thisis liabli to be prohibitivelyexpensivebecauseof the transactionscost!icprred on trades. For a largeportfolio f opons, it is moft feasible.Onlyone trad: il the undflying asset is necessaryto zero out ddta forthe wholeporjfolio.Thehedgingtransactionscostsare absorbedby the profhs on manydifefenttrades.
17.5 THETA of portfolioof options the rate of changeof the value of theportfolio The theta (@) with respect to the passagt of iimtwith all dst Dmaining tle same. Thtta is sometimes referredto as the time decay of the portfplio.For a Europeancall opon on a nonstotk, it ca be shwn fromthe Black-scholesformulathat dividend-paying sbN'djp
etcall)=
-,z
rKe
-
-
2.//
#(J2)
wherek and k are defned as in equation (13.2)'and 1
#'(x)
=
-
-z2/2
(17.2)
e
is the probability deysit functionfor a ktqnlard nrmal dtribution. F0r a Evropeanp7t option on the stock(seePrihlem 17.17), kN'djjo'
gtput)
=
=
2x/@
-/zx(.;y)
+ rKe
1 #(J2), the theta of a put exceedsthe theta of the corresponding BecauseN-kj -rT callby rKe timeis ' 1l thesi forrls, timeis measured in years. Usually,when thetais (uoted, when change value 1 portfolio in the in days, that thetais the day so measured passes cakndar day'' or witha11else remaining the same.We can measure teta either tradingday''. To obtain the thetaper calepdar day,theformulafor theta must be divitkdby 385;to obtain theta per tzadingday,it must be dividedby 252. (DerivaGem theta per calendar day.) measures =
-
.
Sdper
Stper
Example17.2 Asin Example17.1,consider a call opon on a non-dividen-paying stock where the stock price is $49,the strikepriceis $50,the rk-free rate 5%, the time to maturityis 20 weeks (=0.3846years), and the volatility i; 2%. ln this cae, and T = 0.3846. S = 49, K = 5, r = c= The option's theta is .5,
.2,
kNjdjja
The thetais tradingday.
-,z
-
2W
-4.31/365
=
=.1
rKq
#(J2)
=
-
4 yj
18per calendarday,or
.
-4.31/252
-.171
=
per
CHAPYER 17
360
Figufe 17.5 Variationof theta of a Europeancalloptionwith stockpri. '-.
*
.
.
Thqta Stockprice
0 .
nge the Qf gamma a portfolio, What is requiredis a poshion in an instrumentsuch s an option thatis not linearlydependent p the underlying assd. Supposethat a delta-netl'al portfolio has a gnmma equal io r and a tradedoption ,
has a gamma ual to rr. lf the number oftradedoptions aded to the porfolio is gr, the,gammaof the portfolio is wzrr + r
Hen, the position ip thetradedoption necessaryto make the portflio gamma neutl'al lnclvding the trade(1 optionis likelyto change the delta of the portfolio so is the position in tlte underlyingasset thenhasto be changed to paintain deltaneutrality. period of iime.As time Note that the portfolio is gamma neutral only for a inaintainedonly psition eutrality the in the tradedoption is if can be passes,gamma adjultedso that it is alwaysequal to Making a portfolio gamma neusral as well' as dlia-neutral can be regarded as a for $he hedgingerror illustratedin Figure17.7.Delta neutrality provides correction against relptivelysmall stqck pri mpves betweenrebalancing. .Gamma protection neutralityprovidesprotectionagainst larger movements in ths stock pric between hedge rebalancing. bupposethat a portfolio is delta eutral and has a gamma of The delta and gamma of a palticular traded call option are 0.62 and 1.50, -3,. The portfolio can be made gammaneutral byincludingin the portfolio a respectively. long position of 3,009 2,000 1.5 = -r/rz.
,
.jhort
-r/rr.
'
'
.
.
,
in tLecall option. However,the delta of the portfolio will then change fpm zero to 2,000x 0.62 1,240.Therefore1,240units of the udirlying asset must be soldfrom the portfolio to keep'itdelta neutral. =
Calculation of Gamma For a European call or put optin on a non-dividend-payingstoct, the gamma is #ven by r=
N'djj
kqt
wheredj is dened as in equation (13.2)and N'xj is as gi.venby eiuation (17.2).'The gammaof a long position is alwayspositiveand vaiies with k in the wayindicatedin Figure 17.9.The variation of gamma with time to maturity for out-of-the-money, at-the-money,and in-the-moneyoptions is shown ih Figure17.1. For an at-the-money option, gamma increasesas the time to paturity decreases.Short-lifeat-th-money options have very hiph gammas, which means that the value of the nption holder's positionis highlysensitive to jumpsin the stock pri. Example17.4 As in Example 17.1,consider a call ption on a non-dividend-payingstock where the stock price is $49,the strikeprice is $50,the risk-free rate is 5%,the timeto
364
CHAPTER 17 Figure 17.9 Variatio of gmma withstock pri
for an option.
Oamma
K
Stockprice
maturityis 20 wezks (=0.3846years), and the volatility is 2%. In thik cas, = K 5, r = and T 0.3846. c= k 49, e The option's gamma is N /(z1) = 0.066 .2,
.5,
=
SOCWT
When the stockpri changesby hS, thedelta of the option changesb 0.0661z% ' .
Figure 17.10 Variationof gamma with time to maturityfor a stockoption. Oamma
0ut of te mcney
At the money
/
In the money
0
Tjme to maturity
365
The GreekLetten
'
,
17.7 RESATIONSHIPBETWEENDELTA?THETA?AND GAMMA .
'
'
The price of a single derivqtivedependenton a non-dividend-payingstock must satisfy the diereniial equaon (13. 16).It followsthat the valu: f 11of a portfolio pf such derivativesalso satissesthe dierential equation '
3H 311 + TS +
*
Smce
.
.
'
i-f
37
.
@
=
jgz 2
=
-
,
it followsthat
(ij+ rk
-
2 2 3 11 = 2
S
rn
s
gn
g2n
r
,
= + 1c2k2r 2
=
2
n
(17.4)
Similarresults can be produd for other underlying assds (Ke Problem17.19). For a delta-neutralportfolio, = and
@+l2c
2,92r,
=
rn
'
..
'
This shows that, when e is largeand positive, gamm of a portfolio tendsto be large and negative, and vice versa. Tls is consistentwith the way in which Fipre 178 has be regardedas a proky for been drawn and explains why theta can to somr rytrnt gammain a delt-neutral portfolio. .
.
1i.8 vEc UP to now we haveipplkitly assumedthat the volatility of the asset underlying a derivativeis constant. In practice, volatilitieschange over time. This meansthat the valueof a derivativeis liableto change becauseof movementsin volatility as well as becauseof changes in the asset price and the passage of time. The vega of a portfolio of derivatives,V, is the rate of change of the value of the 9 portfoliowith respect to the volatility of the underlyiny asset. ' '
:
P=
..
.
l
-
jc
If the absolute value of vega is high,the portfolio's value is very sensitive to small changesin volatility. If the absolute value of vega is low, volatility changeshave relativelylittleimpacton the value of the portfolio. A position in theunderlyingassethas zerovega. However,the vega of a portfolio can be changed by adding'a position in a tradedoptio. If Visthe vega ofthe portfolio and Vr is the vega of a tradedoption, a position of -V/Vr in the tradedoptionmakes the portfolio instantaneous: yega neutral. Unfortunately,a portfolio is gamma neutral will not in generalbe vega neutral, and vice versa. lf a hedger requires a portfolioto be both gamma and vega neutral, at leasttwo tradedderivativesdependent on the underlying assetmust usuallybe uqed. 'that
9 vqa is the name given to one of the Greekltters'' in opton pridng, but itis not one of te letters in the Greekplphabet.
366
CHAPTER 17 fxample 17.5 Considera portfolio that is delta neutral,
(itip! j tliiyysis t egtikbq misyiiktq,jylrd thif, rkitlk. r liitllt; tcwklte ' mtrm'ql' tE E yk1a,> k) 'p ppigulsies jk' T lapgye opy ri t tk Atiqti,ga jk by yinj yi//ns iqjiiititz.jrjcis. g jmnq ntk j rf ptftji tltjt iijatesrjkel 4f m>njikg ict a iii t i'$ hr' khe: jhrykigrst' rt th: '#pmmias d kpyq.oitliki/q (iip's kt gfpp tki lmf'kd sie 'jprtflt
lspi,
l'fltiFtifitidt
,
,if
,j
Etyjjjjji
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k
z
;.
.
.
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.'
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tktitdg '
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vekas.c jt'ftii ind.aj s ' . . :tleltlienii, y.. .. elkpq:,'.tb.tl,ditifiy''psit',pfir'ilkkr'fith fof Cnyt,iir (trj rtlte,kr' i, k'r t r kmmaqndiyrgtjlp,tpv t tktj lf,rtttl:rpj, q, )jpll ln iiijlktitqt jg for, an (jjtjtys troy jj r tt. c.l Th.qcltise ,trrll'n i,tte ihry as th4 paiity daty is '#htq %iittin qopsiops yl pjpr/ e yr
Eip1-) J (.'. .
:
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tky hsyeliivttf, hk Jka,m'.' J
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ln Table 17.5, the greatestloss is in the lowerright corner of the table. The loss correspondsto the volatility increasingt9 12% and the exchangefate moving up to 1.6. Usually the greatest lossin a table such as 17.5 occurs at one of the orners, but this is not alwaysso. Consider,for exnmple,the situationwhere a bank's pdrtfolio consistsof a short position in a butteriyspread(seeSiction1.2). The greatestlosswill be exjeriented if the exchange rate stays where it is.
Proft or loss realized in 2 weeks under dferent scenarios($million).
Table 17.5 Volatility
F-xckmp rate
0.94 8% 1p% 12%
+12 +8 +6
0.96 +55 + +25
0.9$ +25 +17 +9
1.02
,/.p
'
1.04
1.06
-34
-8
-14
-38
-85
-18
-42
-%
. .
.
.
.
.
.
-1
+ +2 -2
.
-
.
. ..
CHAPTERF
370
17
1 7.12 ixTEN5IoN oF ForMUL5 Theformulasproduced so far for delta,theta, gamma, vega, and rho havebeenfor an optiop on non-dividend-payingstock.Table17.6showshowthey changewhen the stockpaysa continuou dividendyield at rate q. ne expressionsfor h and k area; for equations(15.4) and (15.5). Bysettingq equal to the dividendyield on an index,we obtain the Greeklettersfor Eurpean. options on indices.By settingq equal to the foreignrkk-freerate, we pbtain theGreeklettersforEuropeanoptionsQn currency.By stting q = rbwe obtainGrk lettersfor Europea optionson a futurescntract. A exceptionliesin the calculationof rho for Europeanoptionson a futuresconttact.The rh0foracall futuresoption is -cT andtherho fora Europeanput futuresoptionis -pT. In the caseof currency options,thereare two rhos correspdndingto the two interest rates. The rho cprresponding to tlie domesticinterestrate is givenby the formulain The rho corresponding to the foreign Table 17:6 (with42 as in equation(15.11:. interestrate for a Europeancall on a curreniyis rho For a Europeanput, it is
=
-T: -ryzy sd I )
rho = Te-rjs x(-Jj)
kith dk as in equation (15.11).
Dlta ef Forward Contracts instrumentsother than options. Consider Theconceptof deltacanbeapplkdto snancial shows thatthe value stock.Equ>tipn(5.5) a forwardcontract on a non-dividend-paying Ke-rT where and K is the delivery S pri T is the forward 0 f a forwardcontract is ' When the prici gf te stoik''hangesby LS with al1else contract,s jimr to m>turity. remainingte samekthevaltleofaforwardcontract on thestockalsochangesby LS. The deltaof a longforwardcontract on one share of the'stockis thereforealways 1.. This -
,
,
Table 17.6
Greekletter
Greeklettersfor options on an aset that provides a yield at rate q Calloptios
Pat option' '
.
Delta Gnmma Theta
Vega Rh0
e-6LNdj)
e-%TNd3) N'd
.
1)
#'(Jl):-iT
le-qT 1
KcW
kt '(J1)cz-T/(2/f)
- qsNkle-qT +
-
-
- kN'k)o.e-qT(2I) qsN-kle-qT + rKe-rTN-k) -
rKe-rTNd
&/W'@l):-T
SlN'dbte-qT
KTe-rTNk)
-#p-rT#(-J2) .-.
--L
371
knhe(7r::l Iuetteys
one'share; a meansthat a longforwardcontracton one share canbehedgedbyshrticp . j3 . shott forwardcontracton one share can be.hedjedbypurchasiygone shqre. fbr an assrt providing a-qCdividendyleld t mte q, eqqtion (5.7)shows that the $ fomard contract s deltais e For the (klta of a forwardcontracton a ssotk index,q is set equa1 to te difidend y'ktd on the inde! i tllisexpressipn'.For the delta of a fomard contpct, it is set equal to the foxig risk-fperate, r/. .
'
Delta of a Futurqs,Contract .
'
'
.
2
'
From equation (5.1), the futurespri fol.a contmcton a non-dividend-payingstock is :'T S whereT is the timeto maturityof the futurescontract.Thisshowsthai whenthe , the same, the futres pri pri f the. stock chnges by Ls, withall else remaininglarket rt dily, the holderof a marked Lse by futures to changes since contractsare longfuturespositlonmakesan almostimmediategain of this amount.The drlta of a futurescontract isthereforee For a futurespositionon an assetprovidinga dividend er-oI yieldat pte q, equ ti9n (5.3)shows similarlythat deltais It is interestingthat mrking to mafket makesthe deltas f futures anb forAard rates are constant and the contractsslkhtly diflrent. This is true even whenlnterest related pointis adeinBusinessSnapshot5.2.) fomard pri equals thefuturespri. (A position.Defne: Sometiniesa futurescontract is sed to achievea beltarneutral .
y
.
.
T: Maturityof futurescontract Hz : Requiredpositionin asset for deltahedging ,
'
.
Hy : Alternativerequired positionin futurescontmctj for deltahed/ng
'
If the underlying asset is a nn-divided-payinj stock, the anlysis we havt justgiven showsthat HF = wzs jg j; e g
(
.
Whenthe underlying assetpaysa dividendyield q, -jr-qjj.j
Hg
=
.e
(jg6)
x
.
.k
For a stock index,wi set q equal to the dividendyield on the lndex; for a currency, we set it equal to the foreip risk-free rate, tf, so that HF
=
-(r-rJ)zs
e
g
(jp7) .
fxample17.8 Suppse that a portfolioef currencyoptions held by a US bank can be made delta neutral with a short positionof 458,000poundssterling. Risk-freerates are hedgingusing g-month 4% in the US and 7% in the UK. Fromequation(17.7), currencyftpres rquires a short futuresposition -tm-.:)xg/lzjjs ggg '
or:468,442.Sin each futurescontractisforthepurchaseor saleof :62,500,seven woulz be shorted. (Sevenis thenearestwholenumber to 468,442/62,500.) contracts 13Tllese are hedge-and-forget scemes. sin delta is always1., of inthe stock duling the life the contzact.
no canges need to be made to theposititm
372 ..
d
,
CHAPTER 17
'
J
y,
.
;:zjy.
g'. .
INSURANCE PORTFOLIU
17.13
A portfolio managtr is often interestedin acquiringa put option .on his or her portfolid.Tilisprovidesprotection againstmarket declineswilik preservingthe potenin Sectionl 5.1) is to tial for a gain if the market does wellk0ne approach(discussed buy put optionson a market indexsuchas the S&P 5. An lternativeis to create the optionssynthetic>lly. Creatingan option syntheticallyinvolvesmaintaininga jositionin the nderlying asset(orfutpreqon the underlyl.g lsset)so that thedilt of thepositionis equalto the deltaof the pquired o?tion.Thepositionnecessaryto create an optbn syntheticallyis jt the reverse of that nessary to hedgeit. Thisis becauset e procedurefor hedgingan optioninvolvesthe creatipn of an equal and opposit option synthetically; There are two reasons why it may be more attractive for the portfoliomanager to thanto buyit in theparket. Fipt, options createthe required put optionsynthetically nbt liquidity k havethe do always to absorbthe tradrs required by managers of mar ets largefunds.Scond, fundmanagep gftenrequir strike pricesand exercisedatesthat are diferentfrm thoseavailablein exchang-traded optins marketj. 'The synthetic option can be created from tradingthe porfolio or fromtradingin indexfuiurescontracts. Wefirst examine the creatign of a put option by tradingthe FromTable 17.6the delta of a Europeanput on the portfolio is portfolio. .
-
where,with our usual notation,
h
=
e-qTfNdl
: lnt,%/m+
dj ,=
(r
)
-
-
1)
(1:.8)
y
+ c=/2)T
cxf
k is the value of the portfolio, K is the strike pri, r is the risk-free rate, q is the dividendyield on the portfolio, c is the volatilityof the portfolio,and i is the life of the option.The volatilityof the prtfolio can usually be assllmed to be its beta mes the volatilityof a well-diversifkdmarketinex. T create the put option synthetically,the fund manager should ensure that at any jiven time a proportion ,
e
-q1'
(j
.xgjj
of the stcks in theoriginalportfoliohasbeensold and theproeds ivested in riskless assets.As the vlue of the original portfoliodeclines,the delta of the put given by becomeslore negativeandthe proportion of the original portfoliosold equation(17.8) must be increased.As the value of the originalportfolioincrases, the delta of the put becomesless negative and the plvportion Qf the origihal portfolio sold mqst be decreased(i.e.,someof the orinal portfoliomust be repurchased). insmancemeans that at any given timefunds Usingthis strategy to create portfolio on whkh insuran is required and riskless are dividedbetweent'hestock assets.As the value of the stockportfolioincreases,rkkss assets are sold and ihe Positionin the stockportfolioisincreased.Asthe value of the stock portfolio declines, the position in the stockportflio is decreasedand riskless assetsare purchased.The cost of the insuran ares fromthe fact that the portfoliomanager is atwaysselling after a declinein the market and buyingafter a re in the market. /
,
'portfolio
373
Te Gre Letten Example 17.9
A jortibliois worth $9 million. To proteci apinst markeydownturnsthe managersof the portfolio requirea Gmonth European put option on th.eportfolio witha strikepri of $8tinillion.The risk-freerateis9% per annum, thedividend yieldis 30/cper annum,and thevolatilityof the portfolio is 25% per annqmkThe s&P 5 indexstands at 9. Asthe portfolio is consideredto mimicthe s&P 5 fairlyclosels one altemative,discussedin Section$5.1,is to buy l, put option contractson thes&P5 witha strikepriceof 870.Anothr Alternativeis to create the required option synthetically.In this case? S = % millin, K = 87 million, q = 0.31 c = 0.25,and T = 0.5, so that r 1#90/87)+ (.9 0.03+ 0.25?/2)0.5 .9,
=
-
Jj
.
.
.
0.25+
og.jj99
,
and the delta of the required option is e (1(J1) 1)= Thisshowsthat 32.15%pf the portfolio should be sold initiallyto match thedelta of the ieqired option. The amount of the portfoliosold must be monitored For example,if the value of the portfolio reduces to $88nllion aft#r frepuently. 1 day,the deltaof the mqttiredoption changs to 0.3679and a further4.64%of the original portfolko should he sold. Jf the value of the portfolio ipcreases and 4.28% to $92million,the delta of the required option changes to of the original poryfoli sduld be rijrchased. -0.3215
-
-0.2787
Use of Index Futures Using index futures to create pptions synthetically can be preferable to using the underlyingstocks becausethi transacon costs associatedwith tmdesin indexfutures are generally lowerthan thoseassociatedwith thr corresponding tradesin the underlyingstocks. The dollaramount of the futurescontracts shorted as a proportion of the be va1urof the portfolio should from equations (17k)and (17.8) =qk
#(T#-T) -rT*
-(r-#)z*
e
(1 x(4))
e
e
=
-
e
(1 =
N(4))
whereT* is the maturityof the futurescontmct. If the portfolio is worth l timesthe indexand each indexfutmescontract is on 2 timesthe index,the number of futures contractsshorted at any given time should be #(T*-T) -rT*
e
e
E1 #(gl))1/2 -
fxample17.1: Supposethat in the previousexample futurescpntracts on the s&P 5 maturing in 9 months are used to create the option synthetically.In this case initially T* = 0.75, l = l,, T= 2 = 250,and J1 = 0.4499,so that the number of futurescontracts shorted should be .5,
e (1 #(J1))1/42 122.96 e or 123,rounding to the nearest whole number. As time passes and the index changes,the position in futurescontracts must be adjusted. -
=
74
CHAPTER 17 :'
E.(r
l
t .. Bikilittiltsraliat
j j. q ) '' ... . jttalke)(..E t Blant:ti /.1 Ws i ('.'.:';.'.,..).-:;J tt :g,) f (. t .iujh (2E t.-:.E .' '. '. ! ' L t)'f i j '. 1Qqgi?-'. (,,.k4.4 .' . .. r,)). . E . ..L ( .. ) L' (.' ( .)'' ( (, ) ) .(.. '.. : (-, yJ., .. ' L.., ( r .. .... J ( . ) .... ( . L... q, ;.t '- )'' . ... .). . ., r. E ; .y.: y.;t .',... . . r.. ( . . ' . ( : -E ,. ( ). (J., (.. (. r,( . y. .. . . ) t ).,. Eiil,'g ; t. ., 'Jf9,ij Isiittfiat krij yt:hd' 'Ii 1 927, 1, i E et, bw 1,9; f t4) ' ylidj' rEiRjilrktiriq kjt) j't p' itibli lijtljlitiktiEtj 'tiEl)f//tkyfiyiMijijj. ) '..)
.yy ( ... t : j j' jjtjtj .Ejjj. (jjjtt'jkpay ; . jyjtrytyjjyyyyyj jtjkE jy : jtyjjjt j o l'ijjj EL i6, @ yjyjjjjby,.( q(yjjjtt.tty jj y'rtkjjyyt,y ttljik:kfjjjjijrjj.jt jj' jgt (j'; y yjjjrjj? k(j kijj; rt L:: t tyj ), (y:: yjtj, ljtli r ; ) s L ( r ( c r;g 2 ( y yjtyuj;k zk y ) gw; ccy xuk j ( jyjjjq jjtyjjj a q6 lffi y,jjjkks qty jl gzlt ut ku,,!? ytj j tu j.y j,. y' u kj ;: ju t: e rn. jlai ( u j. y sp y j jjjjs q ( p ..Ek',k, ojtyutks ijjjjtjj qfm ) y y y ya)t)(,, y( y jysy j y j ys,?
cW
.
-
Ksdzlj
and k
=
d
-
cW
and #, r, T, and c are the strike pri, interestrate, time to maturity, and volatility, respectively. (a) Provethat hN'dj) = KN'k). pri is (b) Provethat the delta of the call price with'tespt t th: I ). hvN'hle-rl' of call pri that the is (c) Prve the vega (d) Prove the formulafor the rho of a call futuresoption given in section 17.12. option and of call the futures Thedelta,gamma, theta, vrga a are sameas thosef0r a call ad h replad by 1$. Op tion n a stock paying diviends at rate q, bit q replaced i of option. of call why futures the same is not true the rho a Expln is satised for the option considered i 17.29.Use DerivaGemto check that eqnation (17.4) p' calendar daf'. The theta section17.1. Notel DerivaGem fouces a value of theta year''.) in equation (17.4) is 17.30,UsetheDerivaGer ApplicationBuilderfunctionsto reproduceTable 17.2.(lnTable17.2 the stockposition is rounded to the nearest 1 sharesr)Calculatethe gamma andtheta of the position each week. Calculatethe change in the value of the portfolio each weekand checkwhether equation (17.3) is appfoximatelysatished. Notel DerivaGemproduces a year'''.) calendar daf'. Thethetain equation (17.3) is valueof theta 'e-rfkd
'fututes
.
'r
Gper
tsper
ddper
Ssper
,
30
CHAPTER 17
AP?tNDlx AND HEDGEPARAMETERS TAYtORSER4ESEXPANSIONS ('
A Taylorseriesexpansion of the chanze in the portfolio value in a short period of time shoFs the roleplayed by diferentGretkletters.If the.volatilityof the underlying asset isassllmedto be constant, the value 1: of the portfolio is a functionof the asset price S, ' . . and time !. The Taylorseriesexpgnsin gives .
AH =
JH
LS +
i-f
JII -7
ht +
jl
21:
2
2
bs
2 ,
+ j2
11yp 9!2
F1 ystt
.j.
'
.j.
...
(17A.1)
jjyjj
whereAH and LS are the change in 17and Sin a small timeintervalA!. Deltahedgng eliminatesthe frst term on the right-handside. The second term is nonstochastic. The is of order A!) can be made zero by ensuringthat the portfoliois third term (which neutral well as gamma as delta neutrtl. Othertermsare of order higherthan Al. For a delta-neutralportfolio, theftrsttermon iheright-handside of equation(17A.1) i is ero, so that '
i
-
'
.
An
cz
8Al+
lj.sj,:
2
wen terms of order hkher than A! are ignored.Thisis equation (17.3).
Whenthe volatilityof the underlyingasset is uncertain, H is a functionof c, S, and !. Equation(17.1)then becomes AH =
JH
hS+
JH
Ac+
iH
Al+
jl 2
2H
2
LS + a
j
2
92H a
gc
A
+
..
.
whereAc is the chage in c in time A!. In this case, deltahedgingeliminatesthe frst is eliminated by making the portfolio term on the right-hand side. The second third term is nonstochastic.Thefourth termis liminatedby makinj veganeutral. the portfoliogamma neutral. Traderssometimes desne other Greeklettersto correspondto later termsin the expansion. 'term
'fhe
.
.
Volatiliiy
jmiles
Howcloseare the market prices of options to those predicted by Blak-scholes?Do traders really use Black-scholeswhen determininga price for an option? Are the probabilitydistributionsof asset prices really lognormal?This chapter answers these questions,It explainsthat tradersdo use the Black-scholesmodel-sbut not in exactly the way that Black and Scholesoriginally intended.This is becausethey allow the volatilityused to pri an opon to dependon its strikeprice 4ndtimetn maturity. A plot of 4heimpliedvolatilityof an option as a functionof ii strikeprice isknownas a volatilitysmile.Thischapter describesthe volatilityjmiles thattradersuse in equityand foreigncurrencymarkets It explainsthe relationsip betweena volatilitysmile>nd the risk-neutralprobability distributionbeingassumed for the future asset pri. It also discussehowoption tradersallowvplatilityto be a functionof option maturityand how they use volatilitySurfas as pricing tools. .
181 @
wHY THE(OLXTILITYSMISEISTHESAMEFOr CALLSANDPUTS This ection shows that theimpliedvolatilitiesof Europeancall and put options should be equal when theyhavethe same strikeprice and timeto maturity. As explained in Chapter9, put-call parity provides a relationshipbetwen thepris of Europeancall and put options when theyhavethe same strike pri and tipe to maturity.With a dividendyield on theunderlying asset of q, the relationship is p
+
-gr
he
-rz
=
C+
Ke
(jy
.
j;
As usual, c and p re the Europeancall and put price. Theyhavethe snmestrike price, K, and time to maturity, T. The variable k is the prke of the uderlying asset today, r is the risk-free interestrate for maturity T, and q is the yield on the asset. A keyfeature of the put-call parity relatiobhip is that it is based on a relatively simpleno-arbitrage argument. It doesnot require any assumptionabout the probability distributionof the asset pri in the future. It is true both whrn the asset price distributionis lognorml and when it is not lognormal. supposethat, for a particular value of the volatility, pBs and cBs are the values of European put and call options calculated using the Black-scholesmodel. suppose
381
382
CHAPTER 18
furter that pmktald cmktare the market Values of these options. Becauseput-call Parity olds'for the Black-scholesmodel, we must have
gg-vy
.gy
FBS
y
..u
.y
K#p
I
=
p
>BS
In the absen of arbitrageopportunities,put-call parity also holdsfor the market Plices so that -qT ry-rT ,
rmkt+ Se
=
Cmkt +
Subtractingthesetwo equations,we get #Bs
rmkt CBs =
-
-
(18.2)
Cmkt
Thisshowsthatthedollarprking errorwhen theBlack-scholesmodelis used to pricea Europeanput optionshouldbe exactly te snme as the dollarprking error when it is used to price a Europeancall optionwiththe same stiikepriceand time to maturity. Supposethat the impliedvolatility of the put option is 22%. Tls means that pas = pmktwhen a volatilityof 22% is used in the Black-scholejpodel. From equavoltility is used. Theimpliedvolatility of tion (18.2), it followsthat cas = cmktwhen t.11% the call is, thepfore, also 22$$.This argument showsthat the impliedvolatility of a European call option is alwaysthe same as the impliedvolatility of a Europeanput option when the two havethe samestrikeprice and maturity date.To put this another way,for a givenstrikepri and maturity, the corrict volatilityto usein conjuncon with theBlack-scholesmodel to prke a European callshopl alwaysbethe sameas that used to price a European put. Tllis meansthat the volatility smile (i.e.,the relationship betweenimpliedvoltillty and Strike pricef0r a particular maturity) isthe Same for calls and puts. It alsomeans that the volatility termstnlcture(i.e.,the relationship between impliedvolatilityand maturity for a particular strike)isthe samefor calls and puts.
fxample18.1 The value of the Australian dllar is $.6. The risk-free interestrate is 5% per annumintheUnitedStatesand 1% perannuminAustralia.Themarket price of a European call optionon theAustralian dollarwith a maturity of 1 year and a strike ' Price f $0.59is 0.0236.DerivaGemshows that iheimpliedvolatilityof the call is 14.5%.For there to be no arbitmge,the put-call paly relationship in equation (18.1) must apply with q equalto th foreip risk-free rate. Theprice p of a European put option with a strikeprke of $4.59and maturity of 1 year therefore satisfks 1 ' 1 = 0.0236 0.5% + p+ -.1x
-.5x
.6(k
DerivaGemshowsthat, when the put hasthispri, itsimplied so that p = Thisis what Fe expectfromthe analysisjustgiven. volatilityis also 14.50/0. .419.
18.2 FOREIGNCURRENCYOPTIONS The volatility smileused by tradersto pri foreigncurrency optionshas the general form shown in Figure18.1.The impliedvolatility is relatively lowfor at-the-money optios. It becomesprogressivelyhigheras an option moveseitherinto the money or out of the money.
383
Volatily Smiles
Fijire 18.1 Volatilitysmile for foreigncurrencyoptioni. Illlllii:l ()1 v atilit)r
Strikeprice .
'
->
In the appendixat the end of this chapter we showhow to dettrminethe risknentral probability distributionfor an asset price at a futuretimefromthe volatility smile given y options maturing at that time. We refer to tllis as the implied distribution.The volatilitysmile in Figure18.1corresponds to the inplied distzibgticn shownby the solid linein Figure18,2.A lopormal distributionwith the samemean and stpndard deviationas the implieddistributionis shown by the dashed line in Figure 18.2.lt can be seen that the implieddistributionhas heaviertails than the lognormaldistribution.l 'fo see that Figulvs12.1ad 18.2are consistentwith eah bther, considerfrst a deep'
'Fllisis knownas kurtos. Note that, in addition to having a heaviertail, the implieddistributionis more 'pekxr'. Both small and large movementsin the exchangerate are more likelythan.with the lognormal distribution.lntermediatemovementsare lesslikely. '
;84
CAPTER 18 '''
call option with a highqtrikeprice of #2. This option pays of only if out-of-the-money exchange the rate provesto be aboke#2. Fijure 18.2shoksthat theprobabilityof thisis higherfor the impliedprobability distributionthanfor the lognormaldistributin. We thereforeexpet the implieddistributiont? gikea relativelyhighpri for the option. A r'elatively highpri leadsto a relatltely highimpliedvolatility-and tllisis exactlywhat we observe in Figure18.1 for the option. The two fgures are thereforeconsistent with each other for high strike pris. Considernext a deep-out-of-the-money put pption witha 1owstrike pri of #1 Thisoptio pays of only if the exchangerate provesto -be below #j Figure 18.2showsthat the probabiEtyof this is also higher for implied probabilitydistributionthan for th lognormaldistribution.We thereforeexpect the implieddistribution to give a relatitely high price, and a relatively lligh implied volatility,for this option as rell.Again,th is exactlywhat we obserye in Figure18.1 '
--
-
-
,
.
.
.
.
Empirical Results Wehavejustshownthatthe volatilit smileusedbytradefsfor foreigncurrency options impliesthat theyconsider thatthelognormaldistrlbutionunderstates the probability of extrememovements in exchange rates. To test whether they are right, Table 18.1 examinesthe daily movements in 12difirent exhange ptes ovef a l-year period.z 'l'he ftrst step in the production of the tableis to cal,culatethe standard deviationof dailypercentage change in each exchangerate. The neyt stage is to note how often the actualpercentage change exceeded1standarddeviation,2 standard deviations,and so on. The snalstageis to calculatehowoften tlliswould havehappenedif the pementage changeshad been normally distributed.(Thelognormalmodel impliesthat perceptage changesare almost exactlynormally distributedover a one-day timeperiod.) Dailychanges exceed 3 standarddeviationson 1.34%of days.Thelognormalmodel predictsthatthis should happenon only 0.27%ofdays.Dailychangesexed 4, 5, and 6 and 0.03%of days,respectively.The lognormal standarddeviationson predicts model that we shouldhardlyever obsene thishappening.The tabletherefore vidence to supportthe existenceof heavytails(Figure18.2)and the volatility provides 18.1 shows you cld have smikuse' by tratkrs (Figure18.1).Businesssnapshot mademoney if you hd donethe analysisin Table 18.1ahead of the rest of the market. .29%,
.8%,
rhow
j
Table 18.1 Percentageof dayswhn dai y xchanje rate six standard moves are greater than one, two, of deviations(sD standarddeviation dailychange). .
.
.
,
=
> l SD >2 SD >3 SD >4 SD >5 SD >6 SD
Real world
Lognormalmodel
25.04 5.27
31.73
1.34
0.27
0.29
4.55 .l
.8 .
0.03
.
2 ThistableistakenfromJ. C. Hull and A. White,S'Valueat Risk WhenDaily Changesin Market Variables 1998):9-19. Are Not Normally Distributed.'' Joarnal p/Derflufel', 5, No. 3 (sprinp
385
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Reasonsfor the Smile in ForeignCurrenc Options Whyare exchangerates not lognormallydistributed?Twoof theconditions for an asset price to have a lopormal distributionare: 1. The volatilityof the asset is constant. 2. The price of the asset changes smoothlywish no jumps. In practice,neither of theseconitions ij jytisfied foyap exchangemte.The volatilityof anexchange rate is far from constant, and exihnge rates fiequntlyexhbit jumps.It turns out that the efect of both a nonconstant volatility and jllmpsis tat extreme
ovtcomesbecomemore likely. The impactof jumpsand nonconstt volatiliyydependson the optionmaturity. As the maturity of the option is increased,the percentageimpact of a nonconstant voltility on pris becomesmore pronound, but the percentageimpacton implied volatilityusually becomeslesspronounced.The perentage impactof jumpson both pricesand the impliedvolatility becomeslesspronouncedas the maturity of the option 4 result of a11thisisthat thevolatilitysmilebecomeskks pmnounced as isincreased.The option maturity increases.
18.3 EQUITYOPTIONS The volatilitysmile f0requityoptionshasbeenstudied by Rubinstein(1925, 1994)and Prior to 1987therewas no marked volatility smile. Jackwerthand Rubinstin (1996). Knce1927the volatilitysmileusedbytradersto prke equityoptions (both on individual 3
the jumpsare in response to the actions of sometimes
ntral banks.
4 When we look at suciently long-dated options,jumpstendto get out'' so that the exchange obtained when almost when the exchange indistinguishahle fromthe distrihution there is one rate are jumps changessmoothly. rate saveraged
/86
CHAPTER 18 ''
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.
sigure 18.3 Volatilitysmile forequities, Impliel
volatility
Strikeprice
.
stocksand on stock indices)has had the generalform shown in Figure 18.3. This is sometiles referred to as a volatility J/tzw. The volatility decreasesas the klike price Qption increases.The volatility usedto pli a low-strike-price (i.e.,a deep-out-of-the) call) is s gl ifkantlyhigherthan that used to pri a moneyput or a deep-in-the-money option (i.e., highrstrike-plice call). put or a deep-out-of-the-money a deep-in-the-money The volatility smilefor equity options correspondsto th impliedprobability distributiongiven by the solidlinein Figure 18.4.A lognormaldistrbutionwith the same Shown by thedottedline. It mean and standarddeviationas theimplieddistributionis and heavier rkht tail be has lefttl that theimplieddistribution lejshavy can seen a a thap the lopormal distribution. ' '
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To se that Figures18.3 and 18.4 are consistentwith eqch other, we proceed as for Figuies18.1and 18.2and consideroptins that are deep out of the money.From Figure 18.4 a dtep-out-of-the-money cll'' with a strikeprice of #2 has a lowerprice used is than when thelopormal distributionis used. This whentheimplieddistribution is becausethe ojtion pays of only if the stockprice proves to be above #2, and the of tlzisislowerforthe impliedprobabilitydistributionthapforthelognormal probability Therefore,we expectte implieddistributionto givea relativelylowpri distribution. this forthe option. A relativelylowprice leadsto a relatively1owimpliedvolatilitf option. exactlywhat observe 18.3 Consider deep-out-of-theFkure f0r in ntxt a ihe is we option pays of only if the stockpri option with a strikeprice of #1 Ylzis put moey #I. plves to be below Fipre 18.4shows that the probabilityof this is higherfor the probability dislributionthanfor te lognormaldistribution.We thereforeexpect implied theimplieddistributionto give a relativelyltigh pri, and a relatively high implied volatility,for thisoption. Again,tls is exactlywhat we observein Figure18.3. ''''
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0ne ppssible explanation for the smile in equity options conrns leverage.As a company' s equity declinesin value, thecompany's levepgeincreases.This meaps that the equity becomesmore risky and its volatility increases.As a comjny s equity increasesin value, leveragedecreases.The equity then becomesless risky and its volatilitydecreases.This argument shows that we can expect the volatility of equity to be .a decreasingfunctioi of pri and is consistent with Figtlres18.3 and 18.4. 18.2). Anotherexplanatiqn is (seeBusinesssnapshot ,
Sdcrashophobia''
THE WAYSOF CHARACTERIZING 18.4 ALTERNATIVE VOLATILITY SMILE Sofar Fe havedefnedthe volatilitysmile as the rrlationshi? betweenimpliedvolatility and strike price. The relationship dependson the cur/nt prke of the asset. For example,the lowst point of th volatility smilein Fipre 18.1 is usually close to the turrent exchangerate. If the exchangerate incrases, the volatilitysmik tendsto move to move to theleft. to t'heright; if the exchangerate decreasts,the volatilitysmiletends Similarly,in Figure 18.3, when the equityprice increases,the volatilityskew tends to
CHAPTkR 18
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moveto the right, and when te equity pricedecreases,it tendsto move to theleft.aFor thisreason te volatilitysmileisoftencalculatedas the relatiomhjp betweentheimplied and.r/s rather than astllt rtlationship bitweentht implitdvolatilityand #.' volatility The smile is thep much more stable. A refmementof tllisis to calculate the volatilitysmilt ak the relationship betweenthe and K/h, where J' is the frward prke of the asset fgl' a coniract implied at the same time a: the optios tat are onsidered.Tradersalsooftendefme tnaturing option'as an optionwhere K = J', not as an optionwhere K = S. an ne argumentfor this is that J', nQt4 k, is the explted stockprke on the option's date in a risk-neutral world maturity another approach to defmingthe volatilitysmileis as the relationsllipbetweenthe Yet volaiility and the deltaof th option(whtre deltais defned a in Chapter17). implied apply volatility smilesto optionsother sometimesmakes it possibleto 'tls approach thanEuropeanand Americancalls and puts. Whenthe approach is used, an at-themoneyoptionisthendned as a call optionwith a deltaof 0.5or a put option with a options''. Theseare referred to as so-delta deltaof 'volatility
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TERMSTRUCTUREANDVOLATILITY SUFFACES 18.5 THE VOLATILITY ln addition to a volatility slnile, tmders se a volatility term stnzcture when pricing Thismeans that the volatilityused to price an at-the-moneyoption depnds on options. functionof maturity thematurity of the option. Volatility tends to be an inreasing ( short-dated volatilities htorkally low. Thls is becausetere is then an when ar expectationthat volatilitieswill increase.Similarly,volatility tends to be a decreasing funcj.on of maturity when short-dated volatilitiesare historkallyhigh.Thisis because thereis then an expectation that volatilitieswill decrease. Volatility surfaces combine volatility smiles with the volatility term stpcture to tabulate te' volatilities appropriatefor pficing an option with any strike price and any matprity. An exampleof a volatilitysufface that might beusedftmforeigncurrency optionsis given in Table18.2.ln this case, we ajsume that the slnile is measuredas the relationshipbetWienvolatilityand K/SL. 0ne diniensionof Table18.2is K/k; the oter istimeto maturity. The main bodyof thetable shows impliedvolatilitiescalculated fromthq Black-scholes model.At any giventime,someof the entries in thetableare likelyto correspond to options for which reliablemarket data are available. The implied volatilities f0r these options are directlyfrom their market prices and enteredinto the table. The rest of calculated thetableis typkallydeterminedusing interpolation. When a new option has to be vrlued, fmancialengineers look uj the appropriate in the table. Fof exqmplejwhen valttinga g-monthoption with a ratio of volatility strikeprke to asset price of 1.05,a fnancial engineerwould intepolate between13.4 and14.0in Table 18.2to obin a volatilityof 13.7%.Thisis te volatility that would be used in the Black-scholes formulaor a blnomialtree. When valuing a 1.5-year .
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5 Research by Derman susests that this adjustmentis sometimes
in te case of exchange-traded options.seeE. Derman, SRegimesof Volatility'' Risk, April1999:55-59. 6 AsexplainedinCapter 27 whetherte futuresor forwardpri of te asset istheexpectedpri in a riskworld de#nds on exactly howthe risk-neutralworld is desned. neutral 'sticky''
,
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390
CHAPTER 18 1.
volatilityof ihe opti changes t reiect the option's smoneyness'' (i.e.,the extent to whichit isin or out of themoney).The formulasforGreeklettersgiveninChapter17are no longercorred. For example,deltaof a call option is given by '
Ccss css pcimp + S Jcimp bS
wherecss is the Blaik-scholespri of th.e option expressedas a functionof the asset pricet and theimpliedkilatilitycimp.Considertheimpact f thisformulaon thedeltaof an equity call option. Volatilityis a decreasingfunctionof K/S. This means that the impliedvolatility increasesas the asset pri incrrases,so that 3..:zc > q 3S As a result, deltais higherthan that given by the Black-scholesassumptions. In practice bankstry to ensure thai theirexposufeto the most commonly observed clpgqs in 4hrvolatiy surfneais reasonablysmall. Onetechniquefor identifyingthese changesis principal components analysis,which we discussin Chapter2. ,
' .
18.7 WHENA SINGLELARGEJUMPIS ANTICIPATED Let us n0w consider an example of how an unusual volatility smile mfjlg arise in equiiy markets.Supposethat a stock pri is currently $5 and an important news announcement duein a fewdaysis expectedeither to increasethe stot price by $8or to reduce it by $8. lThis announcement could conrn the outcome of a takeover attempt Or the vrdict in an importahtlawsult.)The probability distributionof the stock price in, say, 1 month might then consist Of a mixture of two lognormal distributions,the hrk corresponding to favorablenews, the second to unfavorable mixture-ofnews.The situation is illustratedin Figure18.5,The solid line shows 'the
Efect Of a single largejump.The sglid lineis the true d'istribution;the dashedlineis the lognormaldistribution.
Fi! tlre 18.5
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50
42
lognormaljdistributionfor the stock pri in 1 month; the dashed line shows a lognonal distributionwith the snmemean and tandarddeviationas tls distribution. The tnle probability distributionis bimodal(certainly not lognormal).0ne easyway to investigate the general efect of a bimodalstock pri distrlution is to consider the extremecase where the distributionis binomial.Tllisis what we will now d. sujpos ihatthe stockprice is currently $5 and that it is knownthat in 1monthit willbe either $42or $58.Supposefmher thatthe risk-free rateis 12%per annum.The situationis illustratedin Figure18.6.Optionscan be valued using the binomiallodel from Chapter 11. In this case a 1.16, d 9.84 a = 1.11 and p 0.5314.The resultsfrom valuing a range of diferent options are shownin Table 18.3.The srst column shows alternative strike prices; the second column shows pris qf l-month European call options; tv third column shows the prkes of one-month European ppt option pris; the fourth column showsimpliedvolatilities.(Aj shownin section 18.1, the impliedvolatility of a Europ. put option is the same as that of a European call option when they havethe same strike pri and maturity.) Figure18.7 displaysthe volatilitysmilefromTable18.3.It is actually a (theopposite of that observed volatilities currencies)with decliningas we move out of or into the money. The for volatilityimjlied froman option with a strikeprice of 50 will overpri an option with a strikepri of 44 or 56. :2
=
=
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*
.
Table 18.3
Impliedvolatilities in situation where true distribution is binomial.
Strikeprice
Callprice
($)
($)
42 44
8.42 7.37 6.31
46 48 50 52 54 56 58
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($) .
0.93 1.26 2.78 3.71
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6.50 7.42
.
.
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58.8 66.6 69.5 69.2 66.1 60.0 49.0 .
---- '' ' -
392
CHAPTER 18 Figure 18.7 Volatilitysmilefor sittiop in Tabie183. ,7
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80
.
(%)
70 60 50
30 20 10
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0 46
48
50
52
54
56
SUMMARY The Black-scoles modeland its extensionsassume that the probability distributionof theunderlying asset at any given futuretipe is lopnormal.This assumption is not the onemade by traders.Theyassume the probabilhy distributionof an eqity price has a heavkrleft tail and a lessheavyright tail than the lognonpaldistributipn.Theyalso that 4h probability distributionof an exchangerate has a heavierrighttail and assume a heavier tefitil tlia'n the lojnormaldtribution. The volatilitysmiledefines Traders use volatilitysmilesto allow fornonlognormality. therelationship betweenthe impliedvolatility of an option and its strikeprice. For equityoptions, the volatilitysmileteds to be downwardsloping.This means that outcalls tend to havehkh impliedvolatilitieswhereas puts and in-the-money of-the-money calls ad outrof-the-money in-the-moneyputs tend io havelowimpliedvolatilities.For and foreigncurrency options, the volatility smileis U-shaped. Bot gpt-of-the-money volatilities options havehigherimplied than al-the-moneyoptions. in-the-money Oftentradersalso use a volatility termstructure.The impliedvolatilityof an option thendepends on its life. When volatilhy snles and volatilhy tenh structuresare theyproduce a volatilitysurface.Thisdesnesimpliedvolatilityas a function combined, of both the strike price and the time to maturity. ,
FURTHER READING Bakshi,G., C. ao, and Z. Chen. tfEmpiricalPerformance of lternativeOption Pricing Models,''Jourzal ofFizazce, 52,No. 5 (December1997):2004--.19. Bates,D. S. $Post-'87 Crash Fears in the S&P Futures Marketj'' Jourzal 0./ Ecometrics, 94 (January/Febnary 20):
181-238.
(' Volatility Smiles
393
of Volatility,'' Risk, April 1999:55-59. Derman,E. SsRegimes
Ederington,L.H.?and W. Guan. lWhyAreThose OptionsSnlingj'' Journal d.fDerfvclive',1, 2 (22): 9-34. Probability Distributionsfrom Option Jackwerth,J. C., and M. Rubinstein.S'Recoveripg Priesyi' Journal of Fisance, 51 (December1996):1611-31.
'
Lauterbach,B., and P. Schultz. Warrantq;An EmpirkalStudyof the Black-stholes Modeland Its Alternatiies,''Jounal ofFisance, 4, N?. 4 (September199): 1181-121. Spricing
Tenn Stocture of Volatility Impliedby Foreign Exchange Xu, X., and S.J. Taylor. XNJ/JJJJ, 57-74. Options,''Journal ofFinazcialJrl# Quaztitative 29 (1994): ssne
and Problems (Answersin SolutionsManual) Questions '
.
18.1 What volatility smileis likelyto be bservedwhen: (a) Bothtails of the stock price distributionare lessheavythan thoseof th.lognormal '''. ' : distribution? (b) The right tail is htavier, and the left tail is lessheavyythan that of a lognormal distribution? .
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' ' 18.2. What ylatility smile is obse/ved for equities? 18,3.What volatility smile is likelyto be caused byjumpsin the underlyingasset price? Is the pattern likelyto be tnprepronogncd for a z-yearption than fr a 3-month optio? 18.4.A European call and put optionhavethe same trikeprice an timt to maturity. The call hs an impliedvolatitity Uf 3% and the put has an impliedwlatility f 2501.What trades would you do? 18.5.Explaincarefull why a distribptin with a heavierlefttail and lessheavyright tail than t e lognormal distributiopgivesrise to a downwardsloping volatilitysmile. of a Europeancall is $3. and i: pri givenbyBlack-scholesmodel 18.6.The market yrice with a volatllity of 30% is $3.50,The price given by this Black-scholesmodel fr a Eurpean put optin with the same strikeprlce and me t maturity is $1.. What shuld the market price of the put option be?Explainthe reasons fr your answer.
18.7.Explainwhat is meant by
Sscrashphobia''.
18.8.A stock prke is currently $2. Tmrrow, news is expected to be annund that will eitherincreasethe price by $5 or decreasethe pri by $5.What are the problems in usingBlack-scholesto value l-mnth ptins on the stock? *18.9,What volatility smileis likelyto be observed for 6-mnth ptions when the vlatility is uncertainand psitively correlated to the stockprice? 18.1. What problems d you think would be encuntered in testinga stock option prking modelempirically?
.
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394
CHAPTER 18
tat a ntral bank'spolicyis to allow an exchangerate to uctuate between 18.11. suppose 0.97an 1.03.Fhat patterp of implkd volatilities for options on the exchangerate would you expectto see?
18.12.Optiontraderssometimesrefer tp dep-out-of-the-poneyoptions as beingoptions on Volatility. Why o you think theydo this? '
18,13.A Europpan alloption on a certainstok has a strikeprke of $30,a timeto maturity of
1 year, and an impliedvolatilityof 3%. A Europeanput option on the sape stock has a strikeprketof $30,a timeto maturity of 1 year, and an impliedvolatility f 33%. What iS the arbitrage opportunity open to a trader?Doesthe arbitrage wofk only when the lognormalassumptim underlying Black-scholesholds?Explaincarefullythe reasons for your answer.
18.14.Supposethat the result of a major lawsuitafecting a companyis due to be announced :is tomorrow.The company'sstock prke currently $60.If thi ruling is favorableto the company,the stock price is expectedto jumpto $75.If it is unfavorable, the stock is expectedto jumpto $50.What is the risk-neutral probabiliiy of a favorableruling? Assume that the volatility of the company's stock will be 25% for 6 months after the rulingif the rulihg is favorableand 40%if it is unfavorable. UseDerivaGemto alculate the relationship betweenimplied volatility and stfike pri for f-month European options on the company today. The companydoes not pay dividends.Assllmethat thef-month risk-frre ryte is6%. Considercall options with strikeprices of $30,$40,$5, $60,$70,and $80. The volatility of the exchange'rate is quoted as 18.15.An exchange rate is urrently countries and ili interest rates the two are the same. Using the lognormal 12% estimate the probability that thr exchange rate in 3 months will be (a)less assumption, #0 and than (c)beywen0.7500and 0.8, (d)between (b)between and (f)greater than Basedon .80 and (e)between0.8500an# market for exchange rates, which of theje the volatility smileusually observe in the which and would you expect to be top high? ixpect be 1ow wouldyo to too estimates 18.1. A stock priceis $40.X f-monthEuropen call option on the stock with a strikepri of call option on the stockwith $3 has an impliedvolatilityof 350/:.A GmonthEuropean 28B/:. The6-monthrisk-free rate is 5% a strike priceof $5phas an impliedvolatilityof .8.
.70
.
.7500,
,
.9,
.85,
.9.
and o dividends are expected.Explainwhythe twoimpliedvolatilitiesare dferent. Use DerivaGemto calculatethe prices of the two options. Use put-call parity to calculate the prkes of f-month Europeanput options with strike prics of $3 and $50.Use DerivaGemto calculate the impliedvolatilitiesof thesetwo put options. 18.17. S'TheBlack-scholes model is used bytradersas an interpolationtool.'' Discussthis view. '
'
18.18.UsingTable 18.2,calculate the ipplied volatility a trgder would use for an 8-month option with K/S 1.4. =
AssijnmentQuestions *
.
18.19.A company's stock is selling for $4.The company has no outstanding debt. Analysts and there are considerthe liquidationvalue of the companyto be at least $300,000 See? sharesoutstanding. What volatility smilewould you expectto l,
395
VolatilitySmilet
18.20.A company is currently awaiting the outcope of a major lawsuit:This is expected to be knownwithin 1 month. The stock pri is currertl $2. If the outcome is positike, the Stock Pri is expeded to b $24at the end Of 1 mih. 11the outcope is negative,it is CXPCCtCd to be $18at thistime.The l-monthrisk-freeinterestrate ij 8% pr annum. Is th risk-neutrl probability of a positive outcome? What (a) (b) What are the-values of l-month call options with strikepris o? $19$2 $21$22 ,
,
and$23? (c) Use DerivaGemto calculate a votatility smilefr l-monthcail options. (d) Verifyihatte sape volatilitysmile is obtained for l-month put options. '
'
...
.
,
,
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18.21.A futuresprice is currently $40.ne risk-freeinterestrate is 5%.Somenews is expected tmorrow that will cause the volgtilityover the ndt 3 months to be either lFts or 3%. There is a 6% chance of the hrst outcome and a 4% chance of the secondoutcome. Use DerivaGemto calculate a volatility smilefor 3-monthoptions. 18.22.Data for a number of fofeigncurrencies are provided on the >uthor's website: http://uw.rotmv.utoronto.ca/whu Choosea currency and use the (iata to producea tahle similar to Table 18.1. 18.23.Data for a number of stock indicesare providedon the author's website: http://uw.rotmo.utoronto.ca/whu downmovelent happens Choosean indexand test wether a:three-standard-deviation more often than a three-standard-deviatinup movement. 18.24.Considera European call and a European put withthe same strike price and time to maturity.ShoF that they chang in mlueby the same amount when th volatility increasesfrom a levelcl to a neWlevel c2 wiihin a short period of time. llitltk Use Put-call parity.)
J
18.25.An' exchange' rate is currently 1. and the impliedvolatllities of 6-month European. options with strike prices 1., 1.1, 1.2, 1.3 are 130:, 12%, 11%, 1%, 11%, 12%, 13%. The domesticand foreignrisk-fre: rates are both 2;5t)$.Calculatethe impliedprobability distribufionusing an approachsimilarto that used for Example 18A.1 in the appendix to this chapter. Compareit with the implied.distfibutionwhere a11the impliedvolatilitiesare 11 18.26.Using Table 18.2, calculate the impliedvolatilit a trader would use for an 1l-month option with KJSL 0.98. .7,
The price of a Europeancall optioa on an assetwith strike pri
.J,
J
.
,
K and maturity T is
givenby
'
-,r
.
c
=
(j r $=K
e
.sitsltsy
constant),S is the assetpri at time T, and g is whre r is the interst rate (assumed risk-neutral the probabilitydensityfunctionof Sy. Diferentiatingon with respect to K gives X c= -rz gtsrl ds r -bK ;z ,
-e
.r
.
Diferentiatingagain with respect io K gives 2
9c
-rr
(p
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This shows that the probability densityfunctiong is gvenby
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j2 C
rz
e
=
r
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This result, which is from Breedenand Litzenberger(19 ), allowsrisk-neutr prb abilitydistributionsto be estimatedfromvolatility smiles.gsuppose that cl c2, :nd c3 call options with strikepricesof of prices T-year K the European , #, and K + are respectively. Assuming is small, an estimateof g(#) is ,
-
,
C1 + erz
C3 2
-
2C2
For another way of understanding thisformula,supposeyou set up a butter spread #, and K + and maturityT. This means that you buy a call with strike prices K with strike price K buy a call withstrike pri K + and sell two calls with strike of The value #. prke your position is c! + c3 2c2.ne value of the position can also be calculated by integratingthe payof ovtr the risk-neutral probabilitydistribution, gtr), and discountingat the risk-freemte.The payof is shownin FigureI8A.I. Since is small, wecan assume that gtizl = gKj in the wholeof the range K ts < Sy < K + where the payof is nonzero. The area under the in Figure 18A.1 is 2 of payof small) The value the is thereforee-rT g(#g2 It .5 x 74x t (whenis followsthat j2 C1 + c3 2c2 e-rzj(*) -
of State-contingent ClaimsImplicit in OptionPricesj''
VolatilitySmiles '
91
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Payoff
25
5
kr
#+5
K
#4
whichleadsdirectlyto g(Kj
=
cl + c3 err 2
-
2c2
(18.1)
ample 18.1 Supposethat the price of a non-dividend-payingstockis $1, the rk-free interest rate is 3%, and the impliedvolatilities of three-monthEuropeanoptions with strikepricesof $6,$7,$8,$9,$1, $11,$12,$13,$l4 @re3%, 29%, 28%,270/:, 260:, 25%,24%,23%,220:, respectively.Oneway f applying the above results is as fllows. Aswmethat gtkrl is constant betweenST 6 and ST = 7, constant betweenST = 7 and ST 8, and s? 0n. Desne: =
=
g(Sz)
=
gj f0r 6
ST 14 is therefore0.0015.)Althoughnot obvious fromFigure 18A.2,the implieddistribls tion does have heavierleft tail and less heavy right tail than a lognormal distribution.For the lognotmaldistributionbasedon a singlevolatility of 26%, with . the prbabilty of a stock price between$6 and $7 is 0.0022(compared 0.0057in Figure 18At2)and the probability of a stock price between$1j and with 0.0113in Figure 1A.2). $14is 0.0141(compared
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Control Variate Tpchnique A technlqueknown as the control variate tecbniqaecan improvethe accuracy of the pricingof an Amerkanoption.E Thisinvolvesusing the sametreeto calculte both the valueof theAmericanoption,A, and the value of the correspondingEuropean option, Js. We also calculate the Black-schqles pri of the Eutpean option,As.The error give by the tree in th pricingof the Epropean option is assumedequal to thpt given by the tree in the pricing of thi Americanoption. Thisgivesthe estimate of the price of the Americanoption as
J
+
As Js -
To illustmte this approach, Figure 19.1 values the option in Figure 19.3 on the assumptionthat it is European. The pli obtainedis $4.32.Fromthe Black-schoks tS formula,the true' European prke of the option $4 2. The estimateof theAmerkan pri in Figure 19.3is $4.49.The control variate stimate of the Americanprke, therefore,is .
4.49+ 4.08 4.32 4.25 -
=
A good estimateof theAmericanprice,calculatedusing 1 stepq,is 4.278.The control varite approach does,therefore,produce a considerableimprovement over the basic tree estimate of 4.49in this case. The control vayiate techniquein eect involves using the tree to calculate the diflkren betweenthe European and the Americanpri rather than the American pri itself.We givb a fuither application cf tlt control variate techniquewhen we discussMonte Carlosimulationlaterin the chapter. 7 Another problem is that forlong-datedoptions,S%is signifcantlylessthanS and volatilityestimatescan .be Very high. S See J. Hull and A. White, Use of the ControlVariate Technique in OptionPrkin'' Jonal of Azalys, 23 (Septemr 1988):237-51. Fisazcial azd Quaztitative 'l'he
414
CHAPTER 19
Tree, as produced by DerivGem, for Europeanversion of option in Fipre 19.3.At eachnodei the upper number isthe stock price,and thelowernqmber is the option price.
Figure 19.10
Ateath ncte: Uppervalue= Undorlying AssetPrice = OptionPrice Lowervalue Shating indicateswhere optionis exercised Strikeprice= 50 Disddt faitor per btep = 0.917 Time step,dt = 0.0833 years, 30,42 ays Growthfactor er step,a = 1.9t84 of up move,p 0.8073 Probability Up stepsize,u 1.1224 Downstepsize,d 0.8909 62.99 0.64 56.12 2.11 50.00 50,00 3.67 4,32 44.55 6.69 ' 39.69 9.86
89.07 0.00
.
79,15 0.00
=
70.70 0.00
=
=
56,12 1.30
70.70
. . 62.79 . 0,00
...0.0
.
.
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50.00 2.66
M.54
44.55 6.18
t
(?,)l5i
39.69
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35.36 .
1t4
31.50 18.08 28.07
..
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0.0000
0.0833
0.1667
0.2500
0.3333
.;j;t:!4. (1ti.. .,
0.4167
PROCEDUREjFORCONiTRUCTINGTREES 19.4 ALTERNATIVE The Cox,Ross, and Rubinsteinapproach is not the only way of buildinga binomial and (19.2), tree. Instead of imposingthe assumption a 1/J on equations (19.1) we = when of solution order higher equations A the than Al are terms to can set p ignoredis then =
.5.
(r-t?-c2/2)A!+c
u=e
d
=
etr-'3-cz/zlz-c4-f
415
Bic NumeyicalPrtlc:uAw :
This allows trees with p to be bullt for options on stocks, indices,foreign exchapge,and futures. Tls alternativetree-buildingprocedure hasthe advantageover the Cox,Ross,and Rubinsteinapproach that the probabilitiesare always rejrdkjs of the value of c or the nllmber of time steps.9 Its disadvantageis that it is not as straightfomard to calculatedelta, gamma, and rho ffomthe tree becausethe tree is no longercentered at the initial stock price. .5
=
.5
fxample19.6 Considera g-monthAmericancall option on the Cnadian ollar. The current rate is 0.79, the strike price is 0.7950,the Us risk-free itemst rate is exchange 6%per annum, the Canadianrisk-free interestrate is le; .per gnnum, and the Figure 19.11 inomialtreeforAmericancalloptipn on theCanadiandollar.At each node, upper number is Spot exchqngerate and lowernupber is option price.All probbilities are .5.
At each node: Asset Price Uppervalue Underlying Lowervalue OptionPrice Shadingindicateswherecpticnis exercised =
=
Strikeprice 0.795 Discountfactor per step = 0.9851 Timestep,dt 0.2500 years, 91.25 days =
=
Probpbility of upmove,p 0.5000 =
0.8136 i, y y E
'
k
0.8056 0,7978 0.0052
,;
0.7817 0.0000
0.7740 0.0000
0.7900 0.0026
0.7510 0,0000
0.7665 0.0000 0.7437 0.000
0.7216 0.0000 NodeTime: 0,0000
0,5000
0,2500
0.7500
9 whentime steps are so large that o' < lv' the Cox,Ross, and .Rubinsteintree gives nqative The alternativeprocedure describedheredoesnot have that rawback. ' Probabilities.
((r -
1,
416
CHAPTER 19
volatility of the exchangerate is 4% per annum. In this case, = 0.79, and T = 0.75.We dividethe life of K 1 0.795,r = c= ry = the option into 3-monthperiods forthe purposes of constrcting thetree, so that ht = 0.25.We.setthe probabilities on eah branchto 0.5and '
.6,
Kg
.4,
.1,
'
a
=
d
=
(.6-.1-.16/2).25+.@X
e
.
=
%
(.6-.1-.16#N.25-.4
e
=
1.0098. .
gyg?
The tree for the exchangerati is shownin Fipre 19.11.The tr givesthe value of the ojtion. as $.26. '
Trinmial Trees Trinomialtreescan be used as an alternativeto binomkltrees.The generalformof the treeis as shownin Figure19.12.Supposethat pu, pm,and h are the probabilitiesof up, middle,and downmovementsat eachnode and ht isthelengthof th: timestep.For an assetpaying divideds at a rate q, parnmeter values that match the mean and stanbard deviationof price changes when terms of higherorderthan ht arr ignoredare jj
'
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=
-
ht r 12c2
-
q
-
c2
-
2
+
-
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=
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,
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,
p.
=
-
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u
Pu
-,
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q
-
J
-
2
+
1 6
-
haq
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ha
=
4:2
,oIf
ha S0
Sn
0 hdl s 0d 2
hdq
.' ''
417
Banc NumeyicalPyocduyes
Calculationsfor a tlinbmialtree are analogousto thosefor a binomialtree. Wewnrk frim the end of the tree to the beginning.At each noik we calculate the value of exercisingand the value of coptinuing. The vale of continuig is '''''''''
-r2(
e
'
'''''''''
puj u y pmjg + yhj
here J., A and fd are the values of the option at the subsequent up, middle,and downnodes, repectively. The trinomialtrdeapproach provesto be equivalentto tLe 19.8. eplicitfnite diflbren method, wlch will be describedin section W
,
Figlewskiand Gao haveproposedan enhanmtnt of the trinopial tree method, whichthey call the adaptiye mes model. ln this, a high-resolution(small-l)treeis 10 When valuing a regular American gfafted onto a low-resolution.(largerl)tr. option,high resolution is most useful for the parts of the tree close to the strike pii at the end th life of the option. ,of
PbkAMETERS 19.5 TIME-DEPENDENT Up to now we have assllmed that r, q, rg, and c are cohstants. ln practi, they are usuallyassumed to be timedependent.The values of thesevariables betweentimest 11 and t + l are assumed to be equal-totheirforwardvalues. To make r and q (orry) a functionof timein a Cox-Ross-Rubinstein binomialtree, WC SCt
S
m
jggpggjjyt
(jj
.
jjj
for nodes at time 1, where J(l)is the fomard interestrate betweentimest and t + l and g(l) is the forward kalue of q betweenthese times.This does not change the gi 0 inetry of the tree bcause u and d do not12 epend on a. ne probabilitieson the branchesemanating from nodes at time t are: (.f(l)-g(l))A!
p 1-p=
=
d
u-d u-e
(19.12)
(/l)-:(l)1Al.
u-d
The rest of the way that we use the tree is the same as before, expt that when discountingbetwn timest and t + l we use ftj. Makingc a functionof timein a binomialtreeis more challenging.0ne approach is to makethelengthsof timestep inverselyproportionalto thevarian rate. Thevaluesof u that c(l) istheyolatility and d are then alwaysthe same and the treerombines. suppose foya maturity t so that c(l) 2t is the cumulativevarian by time 1. Defne P = c(T)2T whereT isthe lifeof the tree,and let ti be the end of the fth timestep. lf thereis a total ,
10sees.Figlewskiand B. Gao, t'The AdaptiveMeshModel: ANewApproachto Ecient OptionPricingj'' 313-51. Jourzal of Fizatial Zconomics,53 (1999): ll The forwarddividendyield and forwardvariance rate are calculatedin the same way as the forward interestrate. t'lYevariance rate is the square of the volatility) 12For a suciently large number of lime steps, theseprobabilitiesare alwayspositive.
418
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of N timesteps,we chooseti to satisfyc(li)2 i iV(N The variance bdweentimesti-l and ti is then VlN fora11 prodtlre canbe used to latch timrWitha trinomialtr&, a generalizd tree-building dejendentinterestrates apd volatilities(se TechnicalNote9 on the author'swebsite). =
.
.
SIMULATION 19.6 MONTECARLO We now explainMonte Carlo simulation,a quite difkrent apprpachfor valuing derivativesfrombinomialtrees.BusinessSnapshot19.1illustratesthe random sqmpling idea underlying MonteCarlosimulationby showinghowa simple.Exel prograp can be constructedto estimate z. Figure 19.13
Calculationof z by throwingdarts.
419
Basic NumeyicalPyoeduyes - .- .-
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Table 19.1 Samplespreadsheetcalculationsin
'
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C
0.6% 0.520
4
:
1 2 3 . :
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1
0.198
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When used to value an option, Monte Carlo simulationuses the risk-neutral result. We sampk paths to obtainthe expectedpayofl-in a risk-neutralworld valuation andthendiscountts payof at the risk-freerate. Consideya derivativedependenton a singlemarket varble S that provides a jayofl-at time T. Assumingthat interestrates areconstant,we can value the derivativeas follows:
1. Samplea random path for t in a risk-neutral world. 2. Calculatethe pamfl- fromthe derivative. the derivative 3. Repeatsteps 1 and 2 to get many samplevalues of the payofl-trom in a risk-neutral world. 4. Calculatethe mean of the samplepayofli to get an estimateof the expectedpayof in a risk-neutral world. t 5. Discountthisexpectedpayof at the risk-freerate to get an estimateof the value cf the derivative. Supposethat the process followedby the underlyingmarket variale in a risk-neutral worldis
dv
vdt
=
+ o'vdz
(19.13)
wheredt is a Wiener process, ; is the expectedreturn in a risk-neutralworld, and c is tht volatility.13To simulate the path followedby S, we can dividethe life of the into N short intervalsof lengthAl and approximate equation (19.13) derivative as St + Al)
-
St)
=
Al + cktll6;';k41)
(19.14)
whereSt) denots the value of S at time t, 6 is a random sample from a normal with mean zero and standarddeviationof 1.. Thisenablesthe value of S distribution at time ht to be calculated fromthe initial value of S, the value at time2 ht to be from te value at time ht, and so on. An illustrationof the procedureis in calculated Section12.3, 0ne simulationtrialinvolvesconstructinga completepath for S using N randomsamplesfrom a normal distribution.
13If S is the price of a non-dividend-payingstock thenJ1, r, if it is an exchange'rate then;= r r/, and so on. Note that the volatility is the same.in a risk-neutral world as i: the real world, as illustratedin section 11.7. ' =
-
!
420
CAPTER
.
19
In practice,it is usuallymre accuratet simulatelnS rather than S. Frm It's lemmathe process follwed by ln S is 2
#1nS .
s that
;-
=
ln st + t) ln st) = -
r'equivalently st +
,)
st)
-
c dt + o'dz 2
(19.15)
-
(; ) sJ
t+
-
expgj
v)
-
' . ..
!+
ln.t)
-
=
'?.y
;-
.
ov47
ovz-j
(19.1$
Thisequatin is used t cnstruct a path fr S. Wrking with ln S rather than S gives mre nezturacy. Als, if ; and c are then
lns')
. .
nstant,
T + cf/f
is tnle f r a11T 14 It fllws that .
sT)
=
.()
expgj
-j-jr
-
+
cf/fj
(19.17)
This equatin can be used t valae erivativesthat provide a nnstandard pays at time T. As shown in Businesssnapsht19.2, it can als be used t check the Blackscholesfrmulas. The key advantage f Mnte Carl simulatin is that it can be used when the pays dependson the path fllwed by the undrrlying variable S as well as when it depends only n the fnal value f Si (Fr example, it can be used when payofs depend n the averagevalue f S.) Payfs can ccur at severaltimesduringthe life f the derivativerather than all at the end. Any stchastic prcess fr S can be accmmdated. As will be shwn shrtly, the prcedure can als be extended to accmmdate situatins where the payf frm the derivativedepends n several underlyingmarket variables. The drawbacks f Mnte Cafl simulatin are that it is computationally vtry timt cnsuming and cannot easily hanple situationswhere there are early exercise pprtunities. 15 '
Derivatives Dependent en More than One Market Variable Considerthe situatin where the payf from a derivativedepeds n n variables bi (1 i n). Dehnesi as the vlatility f %, zas theexpectedgrwth fate f bi in a risk16 neutral wrld, and pfj as the instantaneus crrelatin between f and j. As in the single-variablecase, the life f the derivativemust be dividedint N subintervals f is exactly true only'in te lint as l tendsto zero. !4By contrast, equation (19.14) 26, a numer of researcers ave suggestedwa'ysMonteCarlosimulationcan 15As discussedin chapter options. value Amerkan to extended 16
Notethat si.
j,
and pa are not necessarilyconstant; theymay deynd on the gf .
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.. .
. .
.,, lutti nx! i9 fm Thi?i ilyprsint vlue oiwih:po#fk jjjj.k i j( . jjt jiiilpfty f 4heSpriadshet t the lt n... lTltisis # $yjt .(yjjtj g gprratjyu.yjjj. iy:kieE(B1:b1) E4$ a'B'tjj Ex.1.) be :192 >j $ .D#(1:Bt)! glstlwkt prkiltl',Eij : f tth fille L te '12 (khih is i;8 in thrrspi: p1tfrm sikll ei i tt Blqtcitoli tlii4.'j.ke tio. Tli olp lx pott ff 9p r 'thifijtilte. 'Ekalple 1.8, B13 a Ee j:d to siqq tL: cufcf f '
.
.
.
2
.
j
. .
.
w
.
.
j.j
.
:
',
.
.
length ht. The discreteversion of the pross f0r 0i is then
bit + ht)
-
bitj
=
A! + sibitji.vh
ttititj
(19.18)
whereki is a random samplefrom a standard normal disiribution The coecient of correlationbetweenq and 6j is pii (1% ; k %n). 0ne simulation trialinvolve obtaining N samples of the i (1%i %nj from a multivariate standaldizednormal distribution.Theseare substituted into equation (19.18) to produ simulatedpathsfor each bi, enabling value for the derivativeto be calculated. thereby a sample .
*
.. . . .. . -
Table 19.2
5
.
Monte Carlo simulation to checkBlack-scholes
/1
1 l 3 4
..
.. . .
k5.95 54.49 50.09
B
C
D
4.38
50
k
K 50
.9
47,46 44.93
1000 68.27 101 1002 Mean: 1003 SD:
k
0.2239
17.82 4.98 7.68
E
F
t7
r
c
T
:2
BS price 4.817
.5
.118
.3
.5
CHAPTER 19
422
Generating te RandomSamplesfrom Norml Distributions An apprzpate samplefrom a univariate standardizednolpal distributioncan be obtainedfromthe formula '
'
.
12
E, =
J-l
-
f
6
(19.1p)
j=1
Where the Ri (1K i
K 12)are indmendentrandm numbers between and 1, and 6 is
tht required sapple from(, 1).ThisApproximation is sajisfactoryfor most purposes. An alternativeapproach in Excelis t use =NOkMSINVIRAND) as in Business Snapshot19.2. When two correlatedsamplesel and 62 from stanzard normal bistributions are Independentsamples and from required'an appropriate procedure is as follows. a unvariate standardizeb normal distributionare obtned as just described.The requiredsamplesel and 62 are then calculated as follws: .xl
.x2
,
f1=
.:71
62= pxl +
1
.x2
-
p
wherep is the coecient of correlation. More generally, considerthe situation where we require n correlated samples from normaldistributionswith the correlationbetweensnmplei and sample j beingpij. We flrst sample n indeprndentvarkbles xi (1 i n), from univariate standardized normaldistributions.ne required samples,Ei (1 i n), are thendehnedgs follows: 6j
=
Jjlll
62= a2T.xj + a22.:2 63= a3l.xl + a32.x2+ q33.x3 and so on. We choose the coecients aij so that the cprrlations and varians are correct. This can be done step by step as follows.St alj 1',chopse a21 so that 2 = = p2I; choose 2 + a22 (2I(lI 1. choosea3! so that (qlall a22 so that a21 pql,'choose a2 a2 + a2 = 1. and that that h + + 33 31 32 33 a31121 a32a22 p32,c so on.17 ' oose a so a32so Tls procedure is knownas the Cboleskydecomposition. =
=
,
=
,
,
Number of TriAls The accuracyof the result givenby MonteCarlo simulationdependson the number of tiials. It is usual to calculate the standard deviationas well as the mean of the discountedpayofs given b.ythe simulationtrials. Denote the mean by Jz and the standard (1eviation by The variable Jz is the simulaon's estimate of the value of th derivative.The standard error of the estimate is .
where M is the number of trials.A 95% conden
intervalfor the price
J
of the
17If the equations for the a'3 do not havereal solutions, tlle aoumed correlation structure i3 intenmlly inconsistent will bediscusxdfurter in Section21.7. -1113
Basic Numeyical fWcdlv?w
42
erivativeis thereforegivenby p
.
1.961
j
si-randomsequence(alsocalkd a kw-discrepahcyequenc) is a sequence'of representativesamplrsfrom a probability dt ribution.zoDescriptions of 4he use of 21 ' quasi-randomsequencesappear in Brotherton-Ratclife,and Presset. aI. Quasi-random sequencescan havethe desirableproperty that theylead to the standard error f an estimatebeingproportional to 1/# rather than 1/UV,where M is the samplesize. sampling similar to stratifkdsampling. The objedivr is to sample Quasi-random representativevalues for the underlyingvariables. In stratifed sampling,it is assumed jginples will be tken. A quasi-randol sampling that we knowin dkanceho iy schemeis more fkxible.Te samplesare takenin sucha way that we are always in'' gaps btween existingsamples.At eackstage of the simulation, the sampledpoints are roughly evenlyspaced throughoutthe probability space. Figure19.14showspoints generated in twO dimensionsusing a procedure suggested by Sobol It can be seen that suessive points do tend to 511in the gaps left by ' Previouspoints. '
dslling
.
METHODS 19.8 FINITEDIFFERENCE
'. ,
.
Finitediflkrencemethods value a derivativeby solvingthedifkrentialequation that the derivativesatkes. The diferential equation is convertedinto a set of diflkrence equations,and the diflkrenceequations are solved iteratively. To illustratethe approach, we consider howit might be used to value an American put option on a stotk paying a diyidendyield of q. The difkrentialequation that the option must satisfyis, from equation (15.6),
X+ (r -
!
qjsj'' J
2
2
!
s
c
J
s
=
rf
(19.21)
Supposethat the lifeof the option is T. Wedivideth into N equally spacedintervals of length A! T/#. A total of N + 1 timesare thereforeconsidered =
, 20The term
A!, 2 Al,
.
.
.
,.
T
aasi-razdom
is a misnomer.A quasi-ranom seuence is totally eterministic. 21 seeR. Brotherton-Ratclifk, ttMonte CarloMotorin'' Rk, December1994:53-58,.W.H. Press, A. s.
Teukolsky,W.T. Vetterling,and B.P. Flannery, NamericalRecipesf?l C.. Tbe:rl of Sckntsc Cbpyltf#lg, 2nd edn. CambridgeUniversityPress, 1992. 8&112. A see 1.M. sobol USS.RCompatationalMatbematicsJ?IJMatbematicalPbysks, 7, 4 (1967): procedure is in W.H. Press, s.A. Teukolsky,W.T. Vetterling.and B.P. Flannery, descriptionof sobol's ,
NamericalReces i?lC.' Tbe :rl of S.cientsc Compating,2nd edn. CambridgeUniversityPress, 1992.
428
CHAPTER 19 Fijure 19.14 First 1024points of a '
.
,
.
.
sqqen. sobol'
Points1 to 128
Points129to 512
Pointq513 to 10N
Points1 to 1024.
supposethat is a stockprice suciently ltighthat, when it is reached, the put has and consider a total of M + 1 equally virtuallyno value. We defmeLS = spacedstock pris: hS, 2 hS, , kmax
kmax/v
.
.
.
,
kmax
The level is chosen so th>t one of tese is the current stockprice. The time points and stock price points defme a grid consting of a toyal of CM+ 1)(# + 1) points, as shownin Figure19.15.The (f,j4 point on the grid is the pointthat corresponds to timei Al &nd stockprice j Ak. Wewill use the variable h j to denotethe value of the optin at the (i,j4 point. kmax
Implicit Finite DifferenceMthod For an interior point i, j) on the grid, j'IS can be ajproximated as
CJ
=
'S
,.sl
-
fi'.i
A,S
(19.22)
429
Basc Numeyal Prtlcdt&?w 0r
as
,./ J,.sl -
=
'S
(19.23)
k
Equation(19.22) is knownas the/rwcr# dferezce approximatioz;equation (19.23) is tfy/rac: approxlapproximatioz. We symmetrical backward use a mpre knownas the averaging : by the two mation '
3J 'S
.,y+1
hjj-L
-
=
(j, x) .
25'
For f/t, we will use a forwarddiflrenceapproximationjo tht thevalueat timei ! is related the value at time i + 1) (: 'to
JJ
=
+1,./
-
i,j
(19.25)
l
t
The backwarddiferen approximationfor f!S at the i, j4 point is givn by The backwarddiferenceat th (, j + 1) point is iquation(19.23j.
h.jq-j -
,./
XS
'
'
. . . .. . . . - - . .
..
'
..
Figuie 19.1 5 Gfid for snitediferenceapproach. Stockprice,S Jmu
@
@
@
@
@
@
@
@
@
@
@
#
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
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@
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@
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@
@
@
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@
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@
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@
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@
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@
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@
@
@
@
@
.
@
@
@
2M
@
@
@
@
@
@
.
@
@
@
M
@
@
@
@
@
@
4
@
@
@
e .
=
e
2
z
2
2
:
Time,t
0 0
z
z
T
r
.
CHXPTER 19
430 diferen approximationfor J j'js
llite
Hencea
o J*J s1
J,./ h,. Jf,./-1
-
../+1
at the (f,jj point is
=
=
gs
-
hv
k
0r
J
2
J j
s1
'
+j
,./+1
=
'
.2
,./-l ,
k2
y
(19.26)
.
and (19.26) substitutingequations(19.24), ito thediferentialequation(19.21) (19.25), = noting S and that j gives k
-
+1,./ + (r l
.gs
)
-
for j
=
1,2,
Jtsl
,./
.
.
M
,
.
,sl
=
1
,
.
.
,
.
N
+
,./+l
+ Ljzgsz 2
21k
1and i
-
-
J,.s1 2Jf,./ g rjj -
k
2
-J j ! 2 aj 1(r 2!+ bj = 1 + c 2j =
Cj ::F
-
1(r 2
The valut of the put at timt T is maxtf
Hence,
j
-
%,j = maxtf
-
j
y
1. Rearmngingterms,we obtain
-
(19.27)
+ b)'h-).+ h-).+3.= +1,./ al.h,).-j
where
,
1c2j2j
-
2
r! l
1c2y.2!
-
2
&, 0), where Sr isthe stock price ai time T.
-
k,
), j
=
,
1,
.
.
.
,
M
(19.28)
The value of the put option when the stock price is zero is K. Hence, 1
= .= g K, i (),1,
.
.
.
,
N
We assumethat the put option lsworth zero when S = h,u
=
,
i=
,
1,
.
.
!
,
(19.29) kmax,
so that
N.
(19.39)
and (19.3) ene the value of the put option alongthe Equations(19.28), (19.29), S= threeedgesof the grid in Figure 19.15,where S and l T. lt remains to to anive at the value of J at al1other points. First the points use equation(19.27) l are tackled.Equation(19.27) with i N 1 gives correspondingto time T =
,
=
kmax,
=
-
ajJx-l,pl+ h Jx-1,/+ cj Jxr1,./+lJx,/
-
(19.31)
M 1. The right-hand sides of these equationsare known from for j = 1,2, Furthermore, from equations (19.29) and (19.3(p, equation(19.28). -
.
.
.
,
Jx-1,()K =
.&-l,M
=
()
(19.32) (19.33)
431
Bas NumeyicalPyoceduyes
fiihetefre M 1 simltneous equationsthat Equatilis (19.31) can be solvedforthe /N-1,2, JN-1,v-l 23Afterthi! has beendene,eachvalue M 1 qnknowns: Jx-!,1 jf compared wit ; s, iarjy exexjx at tjme K j 5'. j's-b,j< g of %-3,jis %-L,j is set equal to K j 5'. The nodes corresponding to T t is optimpl >nd time T i t are handledin a similar way,and so on. Eventually,J 1, J,2,J.3, J,v-1are obtained. 0ne of theseis the option prkt of interist. Tht control variate techniquecan be used in conjunctionwith 6nite diferene The samegrid is used to value an option similarto the one under consideramethods. tionbut for which An analyticvaluation is available.Equation(19.2)is the used. -
-
.
,
.
,
,
.
..j
-
-
-
-
.
.
.
,
fxample J9.J 0 Table 19.4 shows the result of usig tht implidt nite diferen meihodas just describedfor prking the Amerkanput option in Example19.1.Valuesof 2, 1, and 5 were chosen for M, N, an. S, resptively. nus, the option prict is at $5Stock pri intprvalsbetween$0and $100and at half-monthtime evaluated throughoutthe life of the option. The option price given by the grid is intervals of the correspondingEuropeanoptin as $4.07.The snmegrid gives the yrice price bythe Black-scholesformulais $4.08,ne glven $3.91.Yhetnle European controlvariate estimateof the Americanprice is iherefore
4.07+ 4.08 3.91 $4.14 =
-
ExplicltFipite DifferenceMethed Th implicithnitediferencemethod hasthe advantageof beingvery robuq. It always convergesto the solution of thediferentialequation as LS ahd l approach ztro.24One of the disadvantagesof the implicitfmitedifkrencemethod is thatM 1 simultaneous equationshaveto bejolvedin order to calculatethe h,jfpm the h.+3,j, ne method can 2 be simplied if the valuej of fls and 32f/'S at point(f,j) on the grid art assumej to and (19.26) then Vcome be the same as at point (f+ 1, j4.Equations(19.24) -
CJ Jf+1,./+1+I,./-1 -
=
25'
'S
32J
+1
=
+
./+1
1
Jf+1,.s12+1,./ -
.
,2
35'2 The diflkrenceequation is .
Z'+1,j.Jf,.f + (r !
+1,/+1
-
-
q)j gs
-
/i+1..s1
lS 2c + .t
2
.2
)
,
2
+ +1,s1 +1,./+1 2
-
2+1,y =
l
rfi,j
23 can e used to express /N-j,2 in nis doesn0t inVolve inVertinp a matrix. The j 1 equation in (19.31) termsof fx-j,1',tbej 2 equation,whencomined
inwhy it is necessary to calculatesix values of the option in each simulation trial variable techniqe are used. Whenbth the control yariate and the qntithetic when the implicit nite diferece 19.17.Explain how equations (19.27) to (19.3) methodis beingused to evaluate an Ainericancall option on a currency. E
.change
19.18.An imerican put optionon a non-dividend-paing jtock ha! 4 mopthsto maturity. The exerciseprke is $21,the stock price is $20,the risk-free rate of interestis 1% per anmlm, and the volatility is 30% per annum' Use the explkit version of the fiite diference approach to value the option. .Usestok price intervalsof $4 and time intervalsof 1 month. 19.19.The spot prke of copper is $0.60per pound.Supposetht thefuturespris (dollars per pound)are as follows: .
3 months 6 months 9 months 12 months
0.59 0.57 0.54 .5
The volatilityof the price of copper is 40% per annum and the lisk-free
l'ate
is 6% per
Bas
.NArldrl'cg!
441
Pyoceduyes
wit an anum. Use a bipomialtree to value an Ameria call op'tion on price gnd ption matrity of Diyidethe 1 lifeof time the of to $9.60 a yer. eercise ppstpcting the tree. Hint:As explaid lbt four j-ponthpirios for the purposes pf ,copper
16.7,the fiureq in sectipn
neutral world.)
'
.
price of a variable is itj expectidfuture price in a risk'
.
'
19.2. Use the binomialtree in Problem19.19to whtre x is the price of copper. '
'
q
.
'value
'
a securitythat pqys ofl-jh in 1.year
and S x afect the estimates of When do the boundary conditions for S 19.21. di#ren method? exnlkit nricesin the qnite ' derivative How wuld you use the antithetic variable method to improvethe estimate of the 19.22. -#
=
.
.K
.
.'K
''.
'
'
.
'
.
Europan optionin Businessjnaphot19.2and Tabk 19.22 bondthat has a fae value of $25and can be 19.23.A companyhas issueda 3-yearconvertible exchangedfor two of the company's shares ai any tile. Yhe company can cll theisspe, eqal pri,e version, frcing is Feater than or to $18.Assuing that wen the share the company willfore cpnversion at the earliest opprtunity, what are the boupdgry would usi fnite diferen conditionsfor the pri of the convertible?Describehow methodsto value the convertible assuping consnt interestrates. Assumethereis no risk of the company defaulting. 'you
'
'
,
19.24.Provide formulasthat can be used for otaiping three random samplesfrom standard nprmaldtributions when the correlatioc betweensample and samplej is pi,j.
AssignmentQuestions '
s
19.25. n Americanput optionto sella Swissfrancfordollarshas a Strike priceof $9.80and a of 1 year. The Swissfrancs volatility is 1 /o,the dollr interestrate ij iinieto turity 6%,theSwissfrancinterestrate is3%, and the current exchangerate is sea threo value optionfrom thedeltaof the the option.Estimate binqmial treeto your tree. step 19.26.A l-year Alerican call option on silverfutureshas an exerciseprice o $9.. The t future price is $8.5, thr risk-free rate of interestis 12%per annum, and thr curren of the futurespri is 25% per annum.UsetheDerivaGemsoftwprewith four kolatility time steps to estimate the value of the option.Displaythe tree and verify that 3-month and penultimatenodes are correct. UseDerivaGemto value prices at the snal option the theEuropean version of the option.Use tke control variate thnique to improveyour of the price of the Ameficanoption. estimate .81.
.
(
19.27.A GmonthAmericancall ption on a stock ls .expcted to pay dividendsof $1per share at the end of the sicond month and the ffth month. The cufrent stok pri is $30,the exerdsepriceis $34,the risk-freeinterestrate ls 1% per annum, and the volatilityofthe part of the stockprice that will not be used to pay thedividendsis 30%per annum. Use the DerivaGemsoftwarewith thelifeof theoptiondividedint sixtimestepsto estimate the value of the option.Compareyour answerwith that givenbyBlack'sapproximation (seeSection13.1$. '
19.28.The current valueof the Britishpound is $1.60and the volatility of the pound/dollar exchangerate is 15%per annum. AnAmericancall optionhaS an exerciseprice f $1.62 and a time to maturity of 1 year. The risk-fr rates of interestin the UnitedStatesand
442
CHAPTER 19
the United Kingdol are 6% per ammm and 9% jer annut, rispectikely. Use the explicithnittdiferen meiod to'valueth ption. Considerexshange rates at intervals '
.
of 0.20betweep and 2.40and timeiptervalsof 3 months. 19.29.Answr tlkefollowiilgquestigpj,conrned with the altrrnative prodtlps fty construct; , 19.4: ingtfeesin section that the inomi?,1mpel i:.section 19,4is exactly cosistent with the mean (a)show andvaance of the chang in the logarithm'of th stockpri ip time h(. 19.4 is coistent
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.
(
,'
;
.,f,,
--,
. . ., ,...,,. :, rr g
j
y ,.
,:
..
:,,)
-j
-
.,.
y,
.
?
.
yg
y,t,,
,
,
j.:y
, .,.,.
,-.
.,
; .; t yj (t .t. .. tjy(t,,y)t,:j, r(,#t:. y
.y
,
jjz
,..
:
j)
ysjj
..
y , ,:.j.. (.,y;,-,j,:..y,g .
.,r
.
q
,
,y
.,j
.
.? 1-'.21.;1.14.1k 'k-1'1k'J'' '$..'
,y
t c, y
,
-
yj
;
.
,
,
,
,
..
.,.
,-y
., ,
y. :,j),,y.,( ,y . r g ,
y ,,
.:
.
y
j: ;.:. , . y.
(
.. ,
of the portfolio over the next 5 days.VaR is illustratedfor the tuation where the normally distributd in Figure2.1. changein the value of theprtfolio isapproximately VqR is an attmctive measurebecauseit is easy to understand. ln essen, it asks the simple question $$Howbad can thinjs get?'' This is the queytilm a1: senior managefswant answered.They are very cpmfortqblewith the idea of compresjing . a11the'Greeklettersfor a11the markt variables unzrlying portfoliointo single a a number. . If we acceptthatit is useful to havea singk number to describethe risk of a pertfolio, reseawhers have an inierestingquestion is whether VaR is the best alteinative. some Figure 20.1 Calculationof VaRfromthe probabilitydistributionof thechangein the portfoliovalue;confdence levelis X%. Gainsin portflio value are positive;losses negative. 1* q' are 1*
''.' 5* .':':' ''
(100-X)%
(:1.
.
61 ()y;,(-.-!.,,-lj 't.E ('l.:.-. -.!. ( '
.
E'
()(: ,. :
'
-
-
.)t
(
... ...
.
:
.
.
.
.)
VaR loss
. .
.
Gain(loss)over N days
45
'klueat Rsk j.
Fijure 20.2 Alterngtivesituationto Fipre 2.1. VaR is the sae, bt th potential .z
lossi8lrger.
(l
-
X)%
y., iE i:-,.-@f. .. .)li)'?ri $1-.. ; .
'
- ,
. .. .
-.
' .
. . . .- . - .
.
- . .. ....
.
.
Gai
VG loss
k'
' ,
(loss)cver Ndzys '
.
Argued that VaR may tmpt traders to hoosea portfolio with a return distribution Similar to that in Figpre20.2.The portfolioj Figures2.1 and 20.2havethe Snme VaR,but the portfolio in Figure20.2is much rijkier brcausepotentiallossesare much 'i
larger. A measqre that deals with the probkm we hqvejust mentioned is expectedsort' ' tt ' bad an thingsget? exjected shortfall fall.l Wherea VaR asks' the question I'Jpw asks ttlf . things do get bad, how much an the cqmpypy expect to 'lose?''Expcted shrtfAll js the expected lossduring an N-day period chdltional that n outcme in #)% left tail of the distributionours. For exnmple, with X = 99 and the (l N 1, the expected shorsfallis the average amount the company losesovera l-day period when the loss is in the 1% tAil of te distributipn. . . In spiteof its weakpesses,VaR (nptexpectd shortfall)is the most popular measure WeFill theieforedevotemost of the (.- of risk nmong both replators and risk inanagers. rest of this chapter to howit can be measured. ,,
,
.
.
.
-
=
The Time Horizon VaR has two parameters: the timehorizon N, measured in days,and tie confidencr levelZ. In prgctice, analysts almost invariablyset N l i tlle firstinstan. Tilisis becausethere is not enough data to istimatedirectlythe behaviorof markeivariables usual assvmption is overperiods of timelongerthan 1 day. =
ne
y-dayvaR
=
l-dayvaR x
4#
nis frmula is exactlytruewhen the changesin the valueof thepoftflio on sucssive dayshaveindependentidenticalnormal distributionswith mean zero. In other casesit is an approkimation. BusinessSnapshot2.1 expfains that replators requirea bank's capital for market risk to be at leastthree tipesthe l-day 99% VaR. Giventhe way a l-day Vak is clculated, this minimumcapital levelis 3 x l = 9.49timeste l-day 99%VaR.. '
-
'
Q
.
. ,
l This measure,whichisalso knownas C-VaRor tailloss,was suggestedbyP. Artzner,F. Delbaen,J.-M.Eber, Measuresof Riskj'' Mthematical fncnce, 9 (1999): 293-28.Thex autors defme and D. Heath, coherent goodriskmeasureshould haveand showthattEestandardvak measuredoe,snot have certainpropertiesthata all of them.
446
CHAPTER20
20.7 HISTORICALSIMULATIQN Historica1simulation is one popular way of esiigting VaR.It involvesusing past data that VaR in a ery directway as a guide to what mighthappenin the future.suppose portfolio using calculated covdence l-day timehorizon,a*99% be for level, to is a a of market and501days data. Tht.flrst sttp is to identifytht mriablesasectig the equity ii will pris, be iterest tpically exchange r ates, These portfoo. rates,and so on. in tese markd variables ovr the lost iecent Data isthen collected oli the movements today 51 days.This provides5 alternativesnarios for what can happenVtween chapges values ofyll variables in th8 S cenario1iswhrrethe percentage andtomorrow. arethe sameas they were betweenDay0 and Day 1, snrio 2 is where they are the snmeas they were betweenDay 1 and Day2, and so on. For each snario, the dollar changein the value of the prtfolio betweentoday and tomorrowis calculated. Tis desnes a probahilitydistributionfor dailychanges in the value of the portfolio.ne sfth-worstdaily change is the flrstperntile of thi distribution.The egimate of VaR is thi loss at this flrst prcentile point. Assuming that the last 51 days are a good ttide to what could happendulingthe nexi dly, the companyis 9j% certajn that it will not take a loss zreaterthan the VaR estimate. The histrial simulationtnethodologyis illustpted in' Tbles 20.1 and 20.2. Table 2.1 shows observations: on mafket variables over the last 51 dys. The the ap taken at someparticular point i time duringthe day (usually observations closeof tfading).We denoteth flrst day for wllich #ta is avilable as Day the secondas Day 1, and so on. Today Day 5; tolorrow iqDay 51. T&ble20.2 shows the valuis f the market variables tomorrowif their jerceniage changesbetweentod>y>pdtoporpw are the sameas theywere betweenDay i 1 and Day i for 1 K i K 5. The i'st row in Table20.2showsthe values of market variables tomorrowassuming theiy perntage changes betweeptnday ad tomorroware the sme as theywerebetweenDay0 and Day1,the second row showsthe values of market variablestomorrowassumingtheirperceniagechangis betweenDay1 and Day2 our, and so n. The 5 rows in Table20.2are the 5() snarios considered. Desne vias the value ot a marketvariable on Dayi and supposethattodayisDay m. ,
-
k'
Table 20.1 Day
1 2
3 :
498 499 5
Data forVaRbtorical simlation calculation. Market variable1
Market variable2
variable n
20.33
0.1132 159 0.1162 0.1184
65.37 64.91 65.02 64.94
20.78 21.44 20.97 :
25.72 25.75 25.85
.1
!
.'
0.1312 0.1323 0.1343
Markt
:.
62.22 61.99 62.10
p' *' a'
447
Value t Risk
Scenariosgenerate for tomorlpw(Day51) uslng.ta in Tahle2.1.
Table 20.2
Market Jbrt/f
Scenario Market Markqt number variable 1 vrkk ) '
,
'
26.42 '
1
-
73.12
62.21 6199
1294
.
.
.
.
.
0.1354 0.1363
25.88 25.95
'
=0.56
.
.
.
.
.
.
499 500
0.21
'
-.j8
,
:
21.71
:
.
.
.
- .-- ..
6t.46
0.1375 0.1746 0.136
.
26.67 25.28
varile n --
valae Cang in vl/p: millionsj (J milliotsj
'$
61.87 61.21
.
'
.
.
23.63 22.87'
'fhefth scenario assllmesthat the value of the.markei variale lv
. .
'
0.13 .-0.63
'
,
x.
.
. .
tomorrw will be '
.
h i 1 -
In ur example, m 5. For the flrst variable, the value today,ty, is 25.85. lso t;g 20.33and 171 20.78.It followsthat the value of the srstmarket variahle in the rt scenario is ' =
=
=
20.78 = 25.85 x )g,??
26.42 '
,
The penultimate column of Table20.2shows the value f the portfolio tomorrw for ech of the 500scenarioskWesuppose the vlut ofthe portfolip todayis $23.50 million. This leads to the numbers in the nal column for the change in the value between today and tomorrowfor all the difkrent scenarios.For Scenari 1the change in value for Scenario2.it is in our example is +$21,, and so We are interestedin the l-perntik point of the distrihutin of changes in the portfoliovalu. Becausethereare a total of 500scenariosin Thle 20.2wecn estimate number in the nal colllmnof thetablq.Alternativly,wecan use this as the sfth-worst what o f isknownas extremevala teory to smoth thenpmbersintheleft thetechniques tail of thedistributionin an attempt to obtain a pore arate stimaie of the 1% point 0f the distribution.zAs menone in the previous sectipn,ihe NudayVaR for a 99% timesthe l-day VR. con dencelevelis calulated as U-# Eachday the VaR estimatein gr examplewould be updated using the most recent 51 daysof ata. Consider,for example,what happenson Day51. Newvalues for a11 the market variables becomeavailableand are used to calculate a hew value for our portfplio.3 The procedure we haveoutlinedis employedto calculate a newVaR using data on thi mrket variables fromDay 1 to Day51. tTh gives 51 osrvations on the percentage changes in market variables; the Day- values of the market variahles are no longerused.) Similarly,on Day 502,data fromDay2 to Day 52 are used to determineVaR, and so on. -$38,,
.
,
'
2
P. Embrechts,C. Kluppelbergjand T. Mikosch. se,e
odelizgExtremalEyeztsfor l'rllrtzce ld J'tzce. New York:Springer,199%A.J. McNeil,''Extreme ValueTheory for Risk Managers,''in lztertal lp#efg azd CXD II. London, Risk Books, 1999,and availablefromwww.math.ethz.ck/rwmcneil. 3 Ncte
tat the portfolio's compition mayhavecanp
betwq Day 5 and Day 51.
CHAPTkR 20
448 20.3 MODEL-BUILDINGAPPROACH
The main alternativeto historicalsimulation is the model-buildingajjroach. Bfore ettig into the details of the approach, it is appropriateto mention one ijsue conrned with the units for measuringvolatility.
Daily Volatilities In ption pricing timi is usually measured in jears, and the volatility of an asset is usually quoted as a per year''. Whenusing the modtl-buildingapproach to . calculateVaR,timeis usually measuredin daysand te volatilityof an assetis usually tvolatility quotgd s a pe: day''. rlatinship betweenthe vlatility peryearused in optiol pricingand the isthe What vqlatilityper day used in VaRcakulatins? Ltt us deEptcsar as the volatilityper year of a certain asset and cday as theequivalentvolatilityper dgyof the asset. Assuming252 givesthe standarddeviationof the continuously trding daysin a year, equation(13.2) compoundedreturn on the asset in 1 yearas eit er cyearor cday 252 It followstht '
'
Ssvolatility .
'
.
J
ear
=
Qay 252
0r .
gyor
.
ggay cc
252
so that daily volatility is about 6% of annual volatility. As pinted out in Section 13.4, cay is approximaiely equal t the standard deviation f the percentage change in the asset pri i one day. For the purposes of calculating V4R we assllme exact equality.Wedefne the dailyvolatilityof an asset prke (or'apy othe: variable) as equal to the standarddeviatio of the percentage changein one day. Ourdiscussionin the next fewsectionsassltmesthatestimattsof dailyvolatilitiesand correlationsare available. Chapter21discussejhowthe estimates can be produd. .
Single-AssetCse apprach in a verj simple ConsidtrhowVaR is calculated using the model-building situationwhere the poytflio consistsof a positionin a singlestock:$1 million in shares of Microsoft.We sppposethat N = l and X 99,so that we are iterested in the loss levl over 10daysthat we are 99%confdent willnot beexcded. Initially,we consider a l-day timehorizon. Asspmethat the volatility of Microsoftis 2% per day (orrtsponding to about 32% million, the standarddeviationof daily per year). Becausethe sizeof thepositionis $1 changesin the value of the position is 2% of $1 million, r $200,000, approach to assumethat theexpectedchange in It is customary in themodel-building market variable considered is ztro. Th is not strictlytrue, but it a overthetimeperiod is a reasonable assllmption. The expectedchange in the pri of a market variable over a short timeperiodis ge:erally smallwhencompartd with the standard dviation of the change. Suppose,for exakple, that Microsofthas an expectrd return of 20% per whereas annum.Ove:a l-day period, the expectedrtturn is 0.20/252,or about stapdard expected l-day period,tht the deviationof the return is2%. Overa return is ='z
.0%,
vlueat
449
isk '
0.08x 1, or pbout aboat6.394.
.8%,
v
.
return is 2,/15, or
the standarddeyiaiion f te whereas
Sofar, we haveestablishedthat the change in th value if the portflio of Micrcsoft sharesover a l-day period has a slndard deviationof $20,400and (atleastapproxiFromthe ma tely)a mean of zero. Weassllmethpt the change normallydistributed.4 This tens tht there is a 1% iables.at the end of this bo t #(- 2 33)= that a normlly d'istributedvariable will (kcreae i value by more than ptobability 2.33standarddeviations.Equiyalently,it means that we are99% rtain that a normally distributedvariable will not derease in value by more than 2.73standard deviations. The l-dqy99%VaRfor oui portfolio consistingof a $1 milliopposition in Microloft .1.
.
,
is
.
ZWYWC
233 x 200 *
1
$t66
=
'
'
1
,
As discussedearlier. the S-day VaR is calculated as U# timesthe leda VAR.The l-day 99/k VaRfor Microsoftis therefore . ,/i5
. .
466 1
x
$1,473,821
=
Considernext a portfolio consistingof a $5millionposition in AT&T,and supposethe dailyvolatilityof AT&Tis 1% (approximately 16%pr year). A similar c>lculatiohto that for Microsoftshows that the standarddeviatio of the chanje in the value of the portfolioin 1 dayis 5, 5,j x ' Assupingthe change is hormally distribuyed,the l-day99% Valllis :'
'
'
.
.
'
.1
=i
''
.
,
'5, '
'
and the l-day 99% VaRis
' .
.
x 2.33= $116,500 ,/f'
x 116,500
$3681405
=
Two-Assd Case Nw consider a portfolio consisting of both $t nillion of Mitrosoft shares and $5 millionof AT&Tshares.We supposetat the returns on the two shareshave a A standardresult in statistics biariate normal distributin with a correlation of tellsus that, if two variables and F havestandard deviationsequal to cz >nd cy with the coecient of corre'lationbetweenthemequalto p, the standarddeviatn of + F .3.
is #venby
.
cy-j.y =
2
cy2 + cy +
2/rkcy
.
To apply this result, we set X equal to the change in the value of the position in Microsoftover a l-ay period and F equal to the change in the value of the position in AT&Tovera l-day period, so that cy = 2,
and cy
=
5,
4 To be consistet with the opiion pricinz assumption in Chapterl3, we could assume that the price of Microsoft is lognormaltomorrow.Because l day is such a short prriod of time, this is almost indistinguishable hom the assnmptin we do make-that the thangeip the stock price betweentodayand tomorroW
is normal.
UHAPTER 20
450
The standard deviationof the chahge in the value ot the portfolio consistig of both stocksover a l-day period is therefore
2,2
2
+ 5,
.3
+2x
x 5,
x 2,
=
220,227
The mean change is assumedto be zero and thr change is normally dtributed. l-day99% VaR is theftfore
sothe
x 2.33.= $113,129 220,227
'
.
.
The l-day 99%.VaRis l timesthis,
'
..
$1,622,657.
.0r
'
.
.
.
The Benefitspf Diversification ln the examplewehavejustconsldered: 1. The l-day 99%VaRf0r the portfolio Microsoftsharesis $1,473,621. 2. The l-day 99%VaR f0r the portfolio of AT&Tshares is $368,405. 3. Tht lo-day 89%VaR'for tht portfoEoof both Microsoftand AT&Tshares is $1622657 .of
$
$
*
.
.
The amount
= (1,473,621 + 368,405) 1,622,657 $219,369 -
the bene ts of diversihcat represents ion. If Microsoftand AT&T were perfectly of portfolio both Micrqsoftapd AT&Twould equalthe the V?.Rfor the correlated,
VaR for the Microsoftportfolio plus the VaR for the AT&Tportfolio. Ltss than away''.s perfect orrelation leadsto some of the risk being ''liversifed
20.4 LINEARMODEL ,.
..j
The examples we havejustconsidered am simple illustmtlonsof the use of the linear modelfr calculating VaR.Supposethat a portfolio worth P consists of n assets with an amount ai beinginvestedin asset i (1%i %n). Wedefne if as the lturn on asset i in 1 day. 14followsthat tlie dollar change in the valueof the investmintin asset i in and 1 day is ai
m
p
=
l-lafxf
(zg.1)
i=l
whereXP is the dollarchange in the valueof the whole portfolio in 1 day. In the exampleconsideredin the previoussection, $1 million was investrdin theflrst asset (Microsoft)and $5 millionwas investedin the second ajset (AT&T),so that (in millions of dollars)al = 1, a2 = 5, and XP
=
lxl
+ 5.:2
If we assume that the xi in equation (2.1)are multivariatenormal, XP is normally
5 Harry Markowit was one of the fu'stresearchersto study the beneftsof diversifcation to a portfolio He was awardeda Nobel prize for thisresearchin 1999.SeeH, Markowitz, manager. selection,'' Stportfolio
Journal ofFinance, 7, 1 tMazch 1952):77-91.
451
Value t Risk
need to cal.culate nly the mean and distributed.To calculate VR, We therefore stapdarddeviationof hP. We assume, as discussedin the previoussection,that the expectedvalue of each hxi is zero. Thisimpliesthat the mean of LP is ziro. To calculate the standarddeviationof hP, we defineo.ias the dailyvolatilityof the th assetapd pij as the coecient of correlationbetweenreturns on asseti and assetj. Tllis means that o'i is the standard deviationof Lxi, nd j is the coecient of correlation beiweenAxf and hxj. The varian of hP, whkh we will denoteby c2,, is givenby .
'n
?j
c2,
pqaiajaib.
=
izz:1
1
zl-zzz
This equation can alsobe written
2 oy
n
2 2
ai tn
=
n
+2
(20.2)
pijaiajioj. fcrl )
ircl
.:i
The standarddeviationof the changt overN daysis cy/#, and the 99% VaR for an N-day tl'mehorizoniq 2.33,,/#. ,2 = In the exnmple considertd in 4he previoussection, = and m noted, al that and already 5, As a2 so p12 .2,
.3.
.1,
,1
=
=
=
12 0.022+ 52x = x
,12
+ 2x l x 5x
.3
x 0.02x
.1
=
0.0485
and oy = 0.220.Tllisis the standarddeviationof the change in theportfoliovalue per day (in millions bf dollan). ne l-day 99% VaR is 2.33x 0.220x & $1.623million.This agreeswith thi calculation in the previoussection. =
Handling Intvrest Rates It is out of the questionin the mdel-buildingapproachto defmea separatemarket variablefor everysipglebond price or interestrate to hich a company is exposed. Some simplifkatloqsaye necessary when the model-building approachls used. 0ne possibilityis to assumethayonly parallelshiftsin the yield curve our. It is then necessaryto desne only one market variable: the sizeof thr parallelshift.The changes in the value of a bondportfoliocan thenbe calculatedusing the durationrelationship LP
=
.-DFA:
whereP is the value of the portfolio,LP is the its change in P in one day, D is the modifkddurationof the portfolio,and Ay is the parallelshiftin 1 day. This approach does not usually give enough auracy. Te procedureusually followedis to chdose as market variablesth pricesof zero-couponbondswithstandard mturities: 1month,3 months, 6 months,1 year, 2 years, 5 years, 7 years, 1 years, and frominstrumentsin the 30 years. For the purposesof calculating VaR,the cash sows ourring portfolio are mapped into cash Qows on the standard matmity dates. Considera $1millin positionin a Treasuv bondlastingl years that pays coupon and 1.2 years, and the principalis of 6% semiannually.Couponsare paid in regarded as a $30,000 1.2 This bond is, the firstinstance paid in therefore,in years. positionin a position in zero-coupon bond plus a $30,000 zeromillion plus positionin a 1.2-yearzero-coupon bond. The a $1.03 coupon bond .2
.2,
.z-year
.7,
.7-year
452
CHAPTER 20
' .
.
.
.
.
.
.
.
j
.
.
.
..
.
,
positionin th 0.2-year bondisthen replnemdby an eqttivalentpositionin l-month and 3-month zero-coupon bonds; the positionin the 0.7-year bond is replaced by an equivaknt positionin Gmonth and l-year zero-coupon bonds; and the positionin bond is replad by an equivalent position in l-year and z-yerzero'' tht couponbpnds.The result isthat thepositknin the 1.2-yearoupon-bearingbondis for VaR.puposes regarded as a position in zero-coupon bonds havingmaturitiesof 1 month,3 mnths, 6 months, 1 year, and 2 years. Oneway of doingit isexplainedin the Tilisprocedure isknownas casb-fow??lclArlg. appendixat the endof this chpter. Notethat cashziowmapping is nt neqsary when the historicalsimulation approachis used. Tilisis becausethe complete termstructure of interestrates can be calculated for eKh of the snarios considrred. '1.2-j,r
Applicationsof the
al-inear
Model
rne simplest application'of the linear model is to a portfolio with no derivatives consistingof positns in stocks, bonds,foreignexchange,and commodities. .Inthis case, the change in the value of the portfolio is linearlydependenton the percentage chages in the prices of the assetscomprisingtheportfolio.Notethat,for the purposes of VaR calculations,a11asset pricts aremeasuredin thedomesticcurrency.Themarket variablesconsidered by a largebankin theUnitedStatesare thereforelikelyto include the value of the Nikkei225indexpeasured in dohrs, the price of a lo-yearsterling zero-couponbond measured in dollars,an2 so on. An exnmple of a dtrivativi that can be handledby the linear modelis a fomard contractto buy a foreigncurrency. Supposethe contract maturesat tile T. It can be regardedas the exchange of a foreip zero-coupon bond maturing at time T for a domesticzero-coupon bond maturing at time T. For the purposes of calculatingVaR, the forward cmtract is therefor Sreatedas a long position in the fofeignbond combinedwith a short positionin thedomesticbond. Each bond cn be handledusing a cash-iow mappingprocedure. Considernext an interestrate swap.As txplaine in Chapter7, this can be regarded flxed-ratebond is a as tht exchange of a Qoaing-ratebondfor a flxed-ratebond. bond is worth par just after the nixt regulr coupon-bearing bond. The Goating-rate regarded paymentdate.It can be as a zero-coupon bond with a maturitydate equal to the next paymentdate.The interestrate swapthereforereduces to a portfolio of long and short positions in bondsand can behandledusing'a cash-fbw mpppinjprocedure. 'l'he
The LinearModel and Options We now conder how we might try to use the linearmodelwhen tlere are optiqps. Considerflrsta portfolio consistingof optionson a singlestock whose current price is in the way describedin Chapter17) S.Supposethat thedeltaof the position (calculated t5 value of theportfoliowith S, it isapproximately ij 6Since isthe rate of changeof the .
p =
:
6 Normally we denotethedeltaand gammbof a portfolioby A and r. In thissectionand thenext, weuse the lowercase drtekletters t and y to avoid overworkingA.
.
.r :
,. ,
453
Value at Risk Of
LP
LS
=
(2.3)
whereLS is thedollarchangein thestockprice in l dayand LP as usual, the-dollar chage i the portfolioin 1day.Dehne Ax as the pementagechangein the stock pri in 1 day, so that LS = ,
x
It followsthat an approximaterelationshipbeyweenLP and Ax is LP
St Ax
=
When we havea positionin severalunderlyingmarketvariables that includesoptions, Dlationship betweenLP )nd theLxi similarly. we can derivean approximate linear Thisrelationshipis ,1
LP
J-lsib
=
.
xf
.
=1
(z9.d)
whereS is the value of the fth market variableand ; isthe delta of the pertfolio with respectto the fth marketvariable. Thiscorrespondsto equation (2.1): ,1
LP
qi
=
fcrl
withai of LP.
=
Axj
(2.5)
..
hh. Equation(2.2)can thereforebe used to calculatethestandarddeviation
Example20.1 A portfolio consists of pptions n Microsoftand AT&T.The options on Microsofthavea delta of 1,000,and the options on AT&Thavea delta of 20,0. The Microsoftshare prke is $120,and the AT&Tshare prke is $30.From equation (2.4),it is approximatelytrue that LP
=
12 x 1,
x Ax1+ 30 x 20,000 k 1x2
0r
LP
12,Ax1
=
+ 6,lx2
whereAx1 and 1x2 are the returns fromMicrosoftandAT&Tin 1dayand LP is the resuliant change in the value of theportfolio.(Theportfoliois assumedto be equifalentto an investmentof $120,000 in Microsft and $6, in AT&T.) volatilityof 2% and volality of Micrsoft is Assumingthat the daily the daily the standard AT&Tis 1% and the correlation betweenthe dailychangesis deviationof LP (inthousandsof dollars)is .3,
(12 x
+ (60 x () 1)2+ 2 x 12 x 0.02x 6() x
.2)
sinci 1(-1.65)
.
.5,
=
the s-day 95%VaRis 1.65x
.1
.3
x
=
7.099
Ujx 7,099 $26,193. =
454 20. .5
CHAPTER 20
Qunkylc
MonEt
When a portfolio includesoptions, the linearmodel is an appmximation.It does not uke ncrount if the gamm of the poitfolio. Asdiscussedin Chapter17,deltais desned as the rate of chapge of the portfolio value with sped to an undeflyingmarket variableand gamma is defmedas the fate of change of the delta with respect to the mlrket variable. Gamma measures the curvature of the relationship betwen the poftfoliovalue and an undeflyingmafketvariable. Figufe 20,3shows the impactof a nonzero gamma on the probability dtribution of the value of the portfolio. Whengnmmais psitive, the probability distributiontendsto be positively skewed; when gamma is nigative, it tends to be negatively skiwed. Fipres 20.4and 20.5illustratethe feason fof this result. Figme20.4showsthe relationshipbetweepthe value of a longcall option and the price of the undelying asset.A long The spreshowsthat, callis an example of an option position with positive price of theunderlyingasset at theind of l day whenthe probability distribution forthe is normal, the pfobability distributionfor the option pri is positively skewed.7 Fipre 20.5 shows the relationship betweenthe value of a shoft call position and the priceof the underlyingasset. A shortcallposition has a negativegamma. In th case,we seethat a normal dtribution forthe price ofthe underlyig assetat the end of l daygets mpped into a negativelyskeweddistributionfor the value of the option positin. The VaR for a portfolio is clitically dependenton the left tail of the probability distributionof the portfolio value.For examgle,when the coodence levelused is99%, theVaRisthe value in thelefttailbelowwhlch thereis only l% of the distribution.As indicatedin Fipres 20.3(a)and 20.4,a positive jamma porfolio tendsto havea less heavyleft tail than the nornpl dtribution. If the dtribution of F is normal, the calculatedVaRtendsto be too high.similarly, as indicatedin Fipres 20.3(b)and 20.5, a ngative gamma portfolio tendsto havea heavierleft tail than the normal dtribution. If the distributionof AF is nolpal, the calculated VaRtendsto be t0o low. For a more aurate estimateof VaRthn that given bythelinearmodel, both delta and gamma masures can be used to relate AF to the xi. Considera portfolio .gamma.
' ,
Figure 20.3 Probability distributionfor value' of portfolio: (b) negative gamma.
(a)positive gamma;
.-,--,,---(a)
7 As mentioned in footnote4, we can use the normal distriution in VaR calculations. distriution
(b)
as an approximation to the lognormal
,455
Value qt Risk Fijnre 20.4
Tranjlationof normal probability distributionfor ajset into probability for value of a long call on asset. distribution Valueof longcall
I
1 l
I I l
' j
..
.
,
.I
j
1 l
I I
I
l
I I I I
Figure
I 1
I I ' l
l 1
I j
I I
I I
I 1 l 1 1 I 1
I
j
.
I
Untkrlylng asset
.
00.5 Translationof normal probability distributionfor asset into ppbability
distributio for value of a short call on asset. Valueof shortcall
I j
'
lI '
I I
I I
1 I
l l I I
l
.
l l
Untkrlymg asset ,
I I
I 1
I
I I . I
l
I
l
I I
456
CHAPTER 20 and y al.ethedelta and gamma dependenton a single asset whose priceis S. suppose appendk of the portfolio.From the to Chapter17,the equation
h;
x
+ )2y( ;y2
,
is an improvementover the approximationin equation (2:3).i setting z= reduce:this to
'
--
,y
S x + 1y(#2 2
=
(20.6)
Moregenel-alj for a portfoliowith n underlyingmarket variables,with eachinstrllment in the portfollobeingdependenton only one of the marketvariables, equation(2k6) becomes n
,
=
J-lsth xj + f=1
n
Ls? 2
,
yf(xj)2
f=1
whereSi is the valt!eof the fth market variable, and h and yi aly the delt! and gamma ofthe'portfoliowith respect to the th market variable.Whenindividualinstrllmentsin theportfoliomay be dependenton mort than one market variable, this equation takes themore genel'alform '
n
F
=
f=l
whereyij is a
Sscross
n
hh
xj +
n
! Sihyij xj xj 2 j=1
(20.7)
-
=1
gamma'' dtfinedas &P SiJk
?ij =
Equation (2.7)is not as easyto work with as equation (2.5),but it can be used to expansion moments for hP. A result in statktics knownas theCornish-zFisher calculate be used to estimate percentilesof the probabilitydistributionfromthe moments.g
can
20.6 MONTECARLOSIMULATION f-
u
.
,
approach can be As an alternative to theproceduredescribedso far,themodel-building implementedusing MonteCarlo simulationto generate theprobabilitydistributionfor
8 ne Taylor serits expansion in the appendix to Chapter 17 suggeststhe approximation LP
=
( A!15 LS 1 1y(A5j
2 whenterms of higher order than A!.are ipored. In pradice, the (')A! term is so small that it is usually imortd. 9 the use of seeTechnical Note l on the author's websitefor detailsof the calciation of moments and Ehplj Cornish-fisher expansions. When there is a singk underlying variable, Ehpj 5'2!2c2..F and F(AJ'3) 4.5:4!2y2 + 1.875J6/16, whereS isthevalue of thevarkble And c is volatility. ApplicationE in the DerivaGemApplicationBuilderimplementsthe Cornishits daily sampk Fisher expansionmethod for thiscase. .5k2yc2,
=
,75,dc4,
=
.-
.
=
4!i
Value at Risk
hP. Suppos we wish to calculatea l-day VaR for a portfolio. The procedure is
1. Valueth portfolio today in the usual way using the currentvalues of market variables. 2. Sampleo fromthe mgltivarite pormal probability distributionof the Axj.10 3. Usethe values f the Axf that are sampledto determinethe value of each matket at the end of one day. variable 4. Revaluethe poytfplio at the end of the dayin the usual way. 5. Sttbtractthe value calculatedin Step l fro the value in Step4 to determinea '
.
samplehP.
6. RepeatSteps2 to 5 many timesto buildup a probability distributio for hP. The VaR is calculatedas the appropriateperntile of the probability distributionof hP. Suppose,for exnmple,that we calculate5, diferent samplevalues of Apin the wayjustdescribed. l-day 99%VaRisthe value of LP for.the 5th worst outcme; the l-day VaR95%is the value of LP for ihe25th worst outcome;and so on.11The N-day VaR is usually assumedto be the l-dy VaR multiplied by /V.12 The drawbackof Monte Carlo snulation is that it tends to be slowbecausea mkht onsist of hundredsof thousans of ' complete portfolio (which companys revalued instruments) diferent haj to be many times.13One way of speeding things up is to assumethat eqqation (2.7)describesthe relationshipbeteen LP andthe hxi W can then jumpstraightfromStep2 to Step in the MonteCarlo simulation and avoidthe nee for a complete revaluationofthe portflio. Thisissometimesreferred to as the partial simulatioz approac. 'fhe
.
20.8 COM4ARISONOF APROACHES Wehavediscussedtwo methods for estimatingVaR:thehistoricalsimulationapproach and the model-building approach.Theadvantagesof the model-buildingapproachare that results can be produced very quickly and it can easily be used in conjunctionwith volatilityupdatingschemessuchp,sthosewewill describein the next chapier, Themain disadvantageofthe model-buildingapproachisthatit assgmesthatthemarket variables have a multivariate normal distribution.In practice, dailychanges in market varkbles oftenhavedistributionsthat are quite diferent'fromnormal(see,e.g.,Table18.1). ne historkal simulafion approachhas the advantagetha:thistoricaldata determine thejointprobability distributionof the marketvariabl. It also avoidsthe need for mapping (see Problem2.2). Themain disazvantagesof historicalsimulation cash-iow 10Oneway of doingso is given in Chapterl9. 1l Asin the case of historicalsimulation, extremevaluetheorycan be used to
'smooth
thetails'' so that
betterestimatesof extremeperntiks are obuine. 12This is only approximatelytrue when the portfolio inclues options, butit the assumptionthat is mae inpracti for most VaR calculationmethos.
13An approach for limitingthe pumber of portfolio revaluationsis proped in F. Jamshidianand Y. Zhu 43-67. scenariosimulationmoel: theoryan methoolor,'' Fizazce azd Stocbastics,1 (1997),
CHAPTZR 20
458
are thatit is14 omputationally slowan doesnot easilyallowvolatilityujdting schemes to be'used. Onedisadvantageof the model-buildingapproach isthat it tendsto giye poor results for low-deltaportfolios(steProblem 2.21).
20.8 STRESSTESTINGAND PACk TESTING In addition to calculating VaR, many companies carry out what is known as stress testing.This involvesestimating how a company's portfoo would have performed under someof the ntost extrememarket movesseen in the last 1 to 20 years. For exapple, to test the impact of an extyememovementin US equity pris, a might sd the percentagechanges in a1lmarket variables equal to those on company the s&P 5 moved by 22.3standarddeviations).If this is Qctober19, 1987(when the to be too exeme, the company ght choose January3, 1988(when considered S&P 5 moved by 6,8standazddeviations).To test the efect of extrememovementsin UK interestrates, the companylnight set the percentagechanges in a11market variables l-year bond yiels moved by 7.7 standard equalto thdse on pril 10, 1992(when
deviations).
The Snarios used in stresstestingare alsosometimesgenerated by senior management.Onetechniquesometimesused is to askseniormanagement to mt periodically to developextreme snarios that might our given the current qpd 'sbminstorm''
econpmkenvironment and global uncertainties. jtresstestingcan be considered s a way of takinginto accountextremeevents tht do occur from time to time but are Wrtuallyimpossibleaording to the probability
daily move in a distributionsassumd for mprket variables. A s-standard-deviation such variable market is one extreme event. Under the assllmptionof a normal distribution,it happens about nce every 7,099 years, but, in practice,it is not dailymoveon or twke every1 years. uncommonto see a s-standard-deviation calcul-ating metho d d for VaR, an importantreality check ij baci Whateverthe use testing.lt ivolves testinghowwelltheVaRestimateswould haveperformedin the past. Supposethat we are calculatinga l-day99%VaR.Backtestingwould involvelooking at how often the loss in a day exceededthe l-day 99% VaR that would have.been calculatrdfor that day. lf tls happenedon about 1% of the days, we can feel reaonably comfortable wit,h the methodology for calculating V>R.lf it happened 7% of days,the methodology is suspect. . on, say,
20.9 PRINCIPAL COMPONENTSANALYSIS One approachto handlingihe risk aring from'groups of hkhly correlated market variablesis principalcomponrnts analys. Tls takeshistoricaldata on movementsin the market variables nd attemptsto efme a set of components or factorsthat explain the movements. :4 For a way of adapting thehistoricalsimulationapproach to incorporatevolatilityupdating, see 7. Hull 'andA. Wite. lncorporating volatilityupdating into the Mstoricl simulationmethod for value-at-iisk'' Joarnal ofRisk 1, No. 1 (1998): 5-19.
459
Valueat Rk Tahle 20.3
Factorloadingsf0r US
KI
PC?
N2
data. treasmy P'
'CJ
J'C4
J'C7
PCS
J'C9
nC10 '.
3/ 6m 12m 2y 3y 4y 5y 7y 1y 30y
0.21 0.26 0.32 0.35 0.36 0.36 0.36 0.34 0.31 0.25
0.50 0.23
-0.57
-0.49
-0.32
0.47
-.2
.1
.1
.
.
-9.37
-.52
-0.58
-0k37
-0.39
6.70
-.4
.1
-0.23
-.2
-:4
-.1
0.
=.5
.1 .
.2
-.3
0.14 0.17 0.27
-.12
0.33
0.15 0.28 0.46
59 0.24
.
.16
0.56
.
-.6
-.34
-.18
-0,5
-.9
--.
0.55
.15
-0.1
0.99 0.13 0.03
.
-.1
-.12
.
-.12
.1
0.12
-.79
-0.63
0.08
.14
-.4
.3
i 0.07
9.17 0.27 9.25
-0.38
-.1
0.01 .3j
-.14
-.8
0.71
7-0.26
0.48
-.54
-0.68
. -0.23
-20.63
0.52
-.9
0.52 0.26
-.13
The approachis best illustratedwith an example, The market variables we will considerare l Us Treasury ratqs with maturities ltween 3 months and 30 years. Tables 2.j and 20.4showsresults prouced by Fryefor thesemarket variables using 1,543dailyobservationsbtween 1989and 1995.15The hrst column in Table 20.3shows the maturities of the rates that were considered.The remaining 1 columns in the table show the l factors (orprincipal components) deslbing the rate moves. The first factor,shgwnin the colllmn labeledPCI, correspondsto a roughlyparallel slliftin the yield curver When there is one unit of that factor, the 3-monfhrate increasesby 0.21 basis pints, the rate increasesby 0.26basis points, and.so on. The colllmn labeledPC2. lt corresponds to a secondfactor is shownin the or of the yield curv. Ratesbetwn 3 months and 2 years move in 0ne direction;rates betwen 3 years and 39 years move in the other difecti.on.The tilird of theyield curve. Ratesat the short end and longend factorcorisponds to a of the yield curve move in one direction;ratesin the middle move in the othef direction. The interestrate move fo: a particular factorisknownzbfactor loadizg.In the example, the hrst factor' s 1oadingfor the three-monthrate ii Becausethere are l ratesand 1 factors,the interestrate changes observed on any givenday can always be expressedas a Enearsllm.of the factorsby sglving a set of l simultaneous equations. The quantity of a particular factor in the interestrate canges on a particular dayis knoe as thefactorscore for that day. The importanceof a factoris measured bythe standarddeviationof its factorscore. The standard deviationsof thefactorscres in 0ur exampleare shownin Table 20.4and -month
ttwist''
Sssttepening''
Ssbowing''
'.21.16
Tahle 20.4 PCI
17.49
points). Standarddeviationof factorscores(basis
'CJ .5
'CJ
PCI
3.1
2.17
'CJ 1.97 ,
15See J. Frye,
PC6
'C7
PC8
'C9
1.69
1.27
1.24
0.80
.
K10 '
0.79
of Risk:Findinp VAR tkouph Factor-Based Interest Rate Snariosh'' in MR.' UnderstandingJrl# ApplyingValaeat Rk, pp. 275-22.Lcndon: RiskKlicaons, 1997. tprincipals
16The factorloadinps havethe property thattk sum of theirsquares for each factoris 1..
460
CHAPTER 20
in Table20 4 are e factots are listedin order of their lmportanci. ne measuredin basijpoints.A quantity f theflrstfactorequalto one standprd deviation, therefore,correspondsto the 3-monthmtemoving by 0.21x 17.49= 3.67basispoints, the Gmonth rate loving by 0.26k 17.49= 4.55basispoints, ad so 0n, Th technicaldetailsof howthe facyorsare determlntdare not coveredhere,It is sucient f0r us to note that the factors are chosen so that the factor scores are uncorrelated.F0r instan, in our example,the flrstfactor score (amount of parallel of twist) across te shift) is uncorrelated with the secondfactor score (amount 1,543days. The variances of the factor scores (i.e.,the squares of the standard deations) havethe propertythat theyadd u? to the total varian of the data. From Table 20.4,the total vrian of the originaldata (i.e.,sum of the variance of the observationson the3-monthrate, the variance of theobservationson the Gmonthrate, ' and so 0n) is 2 52 12 = 367.9 17.49 + 6 + 3 + + t
'uibers
h
.
.792
.
.
.
.
.
.
From thisit can be seen that the ftrstfactoraccountsfor 17.492/ 3679 = 831%of the 2+ 6 5i)/j67.9 varian in the originaldata;theflrsttw0 factorsneztount for (17.49 93.1%of the variancd in the data;the third factoraccounts for a further 2.8%of the variance.Thisshowsmost of the risk in interestl'ate moves is accounted for by the frst tw0 or three factors.It suggeststhat we can relate the risks in a portfolioof interest rate dejendent instrumentsto movements in thesefactorsinsteadof considering a11 1 interest rates. The thfee most importantfactorsfrom Table 20.3 are plotted in 17 Figure20.6. .
.
=
Filure 20.6 The three most importantfactorsdrivingyield curve movements. '/'
'
Factorloading
0.6 ll
04
f
.
.
.
#
*
.
I
.
; gs
:
lI l
'
I !. I
4
w
# ..
;
;
*
e
;
:
#'
e
.-
.
.
Matunty
.
10
15
20
25
tyarsl
30
:
t
' -t
l l
1. f
z
e
#
,'
l
'
#
**
I
-0.4
.
..p
.,
1 l
-0.2
.
-*
..-
.
I I
0
-**
-***
1
0 #2
--*
-**
j
e
z. x
-
.....
-
-
-
P01 p;g N3
. l
-0.6
17Similarresults to thosedescribedhere,in resped of the nature of te factorsand the amount of the total risktheyaccount for, are obhined when a prindpal componentsanalysisis used to explainthemovementsin almostany yield curve in any country. '
461
Vale at Rk
UsingPrincipal ComponqntsAnalysisto Clculate VaR illustratehw a principal componentsanalysiscan beusedto calculateVaR,consider a portfoliowith tlieexposuresto interestratemovesshowninTable20.5.A l-basis-point changein the l-yearrate causes the portfolio value to increaseby $1 million, a l-basispointchange in the z-year. rate causesit to increaseby $4million,and soon. Supposethe frst twofactorsare usedto model rate moves.(Asmintined above,thiscaptures 93.1% of the uncertainty in l'ate move) Usingthe data in Table2.3, the exposre to tht Erst factor(measured in millions of dollarsper factorscorebasispoint)is
T
1 x 0.32+ 4 x 0.35 8.x 0.36 7 x 0.36+ 2 x 0.36= -
-.8
-
and the exposureto the secondfactoris l x
(-.32) + 4 x (-.1)
8x
=
.2 -
7 x 0.14+ 2 x
.17
-4.40
=
in basispoints).The change in Supposethat fj and are the factorscores(measured the portfoliovalue is, t a goodapproximation,given by ./2
: '
-.8
=
-
4.40./2
,
Thefactr scoresare uncorrelated and havethestandarddeviationsgivenin Table20.4. The standafddeviationof LP is therefore
0.082x 17.492+ 4.402x 6.052 26.66 =
Htnct, tht l-day 99% VaR is 26.66x 2.33= 62.12.Nte that the dat ih Table20.5 are suchthat thereis very little exposureto the flrstfactorand signiscant xposure to the secon factor. Using only one factor would sipcantly understate VaR (see Problem2.13). The duration-basedmethod for handlinginterestrates, mentionedin Section20.4,would also signiscantly understateVaRas it considersonly parallelshifts in the yield curve. A principal. components analysis can in theorybe usedfor marketvarbles other tha interestrates. Supposetht a flancial institutionhas expcsuresto a number of diflrent stock indices.A principalcomponents analysiscan be used to identifyfactors describingmovements in the indicesand the most importantof thesecan be used to replacethe market indicesin a VaR analysis. How efective a principalcomponents aalysis is for a group of market variables dependson how cbsely correlated theyare. As explained earlierin the chapter, VaRis usually calculatedby relating the actual changesin a portfolio to percentage changes in marketvariables(the xj). For a VaR calculation,it may thereforebe most appropriate to carry out a principalcomponents analysison prrcentage changes in market variables rather than actualchanges. Table 20.5 Changein portfolio value for a l-basis-point rate move($millions). l-year rate +1
z-year rcf: +4
3-year rate -8
4-year rate -7
yyear rate +2
462
CHAPTER 20
SUMMARY A value at risk @aR) calculatipn is aimedat making a statement of the form:$Weare X pewent rt atn t,hat we will npt losemore than P dollarsin the next N days.''The variableJ? is the #aR, 7% ij the conli.dencelevel,and N dap is the timehorizon. One approach to calculating VaR is htrkal simulation.This involvescreating a consisting of the daily movements in 411market variables over a perid of database 'rhe time. flrst simulation trial assumesthat the percentagechangesin each market variableare the snme as those on the flrstday covered by the database;the second trial pssumesthat the perntage changes are the same as those on the simulation stcondday; and 8o on. The change in the portfoliovalue, hP, is calcukted for each trial, and the VaR is calculated as the appropriateperntile of the simulation distributionof LP. probability An alternativeis the model-building approach.This is relativelystraightfomard if twoassllmptionscan be made:
1. The change in the value of theportfolio(A') is linearlydependenton percentage hangesin marketvariables. 2. The percentagechanges in tarket variables are mtivariate normally distributed. u Theprobabilitydistributionof LP isthen normal, and t ere are analyticfrmulas for relatingthe stand>rd deviationof LP to the volatilitks and correlations of the underlyingmarket variables. The VaR can be calculated from well-knownproyrties of the normaldistribution. ' When a portfolioincludesoptions, LP is not linearlyrelated to the perntage in market variables. From knowledgeof the gamma of the portfolio,we can changes relationhip betweenLP and perntage changes in derivean approximat tuadratic marketvaiiables, MonteCarlo simlatiop ca thenbe used to estimate VaR. In the next chapter we discusshowvolatilitiesand correlations can be estimated and
monitoyed.
FURTHERREADING ArtznerP., F. Delbaen, J.-M. Eber, an D. Heath. 203-28. fxtmcc,9 (1999):
Beder, T. $VaR:seductive 12-24. but Dangerous,'' FizancialAnalystsJtmrxi, 51, 5 (1995): Boudoukh, J., k Richar(1son, and R Whitelaw.
S''l'he
.
.
Best of Both Worls,'' Risk May 1998: ,
64-67.
Dow, K. BeyozdPbll at Risk: Te NewSciezceVAf.k Managemezt.NewYork:Wiley,1998. Due, D., and J. Pan. iiAnOveniew of Value at Risk'' Jozrnal 0./. Deriyatiyes,4, 3 (spring 1997):7-49. Embrechts, P., C. Kluppelberg,and T. Mikosch, ModelizgExtremal 1997. Finance.NewYork: springer,
'iu/.
forIzsarazce and
of Risk: FindingVXR throuzh Factor-Based Interest Rate snarios''in IJAR..Uzderstazdingand z@/yfplg qe at Risk, pp. 27548. London: Ksk Publications,1997.
Frye,J.
dsprincipals
463
Value at lbi
of Value-at-Risk ModelsUsing HistoricalDztzrEcozomic Policy Hendricks,D. SsEvaluation York, 2 (Apfil1996):39-69. Revw, FederalReserveBank of New Hopper,G. at Risk: A New Methodologyf0r MeasuringPortfolio Risk'' Basizess Aevkw, FederalRestrveBank of Philadelphia,Jl/August 1996:19-29. Ssvalue
Hua P., ad P. Wilmott, Coursesr''Risk, June 1997:64-67. Hull,J. C., ald A. White.ttYalueat Risk WhenDailyChangesin MarketVariables Are N0t . Normally Distributed,''Joarzal p/ferivl/flel', 5 (Spring1998):9-19. Volatility Updting into the HistoricalSimulation Hull, J. C., and A. Wltite. /19. Metjod f0r Value at Risk,'' Joarzal of AJl, 1, 1 (1998): Jackson,P., D.J. Maud, and W. Perraudin. t ank Capitaland Value at Risk.'' Joarzal of 73-99. Derivatipes,4, 3 (Spring1997)1 ssscenario and Methodologs''Fizazce SimulationModd: Jamshidian,F., and Y. Zhu. 43-67. Stzcatics, 1 (1995: cll Jorion,P. Val at A'k, 3rd edn. McGraw-llill,27. dtcrash
'Talue at Risk: Implelenting a Risk MeasurementStandard,'' Marshall,C., and M. siegel. Joarzal p./ Deratiyes 4, 3 (Spring1995: 91-111. McNeil, A.J. ''ExtremeValue Theory f0r Risk Vanagersr'' in Izterzal Mbtfelfngazd C.D II, London: Risk Books, 1999, Seealso: www.math.ethz.ch/-mceil. Neftci S.N. at Risk Calculations,ExtremeEventsand Tail Estimation,''Joarital of 2 23-38. Derwates, 7, 3 (Sprinj 2): Rich, D. Gelyration VaR and Risk-AdjusiedReturn on Capitalr'' Jourzal of 51-61. Deratives, 1, 4 (Summer2): .
Ssvalue
dtsecond
Manual) tionsand Pfoblems (Answersin solutions Ques investmentin asset A and a $1, 2.1. Considera position consisting'of a $l, investmentin asset B. Assumethat the dailyvolatilitiesof both assets are 1e/rkand that Whatisthe s-day 99%VaRfor the coecient of correlation bdwn theirreturns is portfolio? the 20.2. Describethree ways of handlinginstrumentsthat are dependenton interestrates when the model-building approachis used to calculate VaR. How would you handlethese instrumentswhen lltorical simulation is used to calculate VaR? institutionowns a portfolio of options on the US dollar-sterlingexchange 20.3. A snancial Uf the portfolio is 56.0.The current exchnge rate is 1.5. The Derivean delta rate. portfoiio kalpe and the 4PProximatelinear relationship betweenthe change in the volatility of exchange exchange chaqge daily lf the in the the rate. rate is percentage .7%, estimate the l-day 99%VaR. 20.4. Supposeyou knowthat the gammaof theportfli in thepreviousquestionis 16.2.How does tllis change your estimate of te relationship betweenthe change in the portfolio valueand the percentage change in the exchangerate? that the dailychange in the value of a portfolio is, to a good approximation, 20.5. suppose linearlydependent on two factofs,calculted froma principat componentsanplysis.'I'he delta f a portfolio with respect to the firstfactoris 6 and the deltawith respect to the The standard deviationsof the factor are 20 and 8, respectively. secondfactor is What is the 5-day90% VaR? .3.
-4.
464
CHAPTER 20
20.6. Supposethat a cpmpany has a portfolio consistingof positions in stocks,bonds,freign exchange,and commgdities. Asspme thas there are n derivatives.Explain the assumptions underlying (ajthe linearmodel and (b)the llistoricalsimulationmodel for calculatingVaR. 20.7. Explainhowan interestrate swapismappedinto a portfolio of zero-coupon bondsiith standrd maturities for the purposes of a VaR calculation. 20.8. Explainthe diferencebetweenvalue at risk and expectedshortfall. provide only apjroximate estimates of VaR for a 20.9.Explainwhy the linear nodel containing options. portfolio 2.10. Verifythat the 0.3-yearzero-coupon bond in the cash-iow mapping examplein the position in a 3-monthbond and a to this chapter is mapped into : $37,397 appendix $11,793positionin a f-monthbond. 2.1 1. Supposethat the s-year expressedwithannu'al rate is 6%, the 7zyearrate is 7% (both conpounding),the dailyvolatility of a s-year bondis and the daily zero-cupon Tht correlation betweendailyreturns volatilityof a 7-yeafzero-coupon bondis cash of iow Ma'pa $1, reived at time 6.5 yars into a on the two bgndsis bond and a position in a 7-yearbond using the approath in the positionin a s-year cash iws in $ and 7 years are equivalentto the 6.5-yearcash iow? appendix.What 20.12.Some tile ago a company enteredinto a forward contract to buy f 1 million for $1.5 million. The contract now ha8 6 months to maturity. The daily volatility of a f-monthzero-coupon sterlingbond (when its price is translatedto dollais)is 0.06%a'nd The correlation the daily volatility of a Gmonth zero-coupon dollar btmd is Thecrrent exchangerate is 1.53.Calculate betweenreturns fromthe two bondsis the standarddeviationof the change in thedollarvalue of theforwardcontract in 1day. Whatisthe l-day 99%VaR?Assumethatthef-monthinterestrate in bothsterlingand dollarsis 5% per annum with continous compounding. 20.13.The text calculates a VaRestimatefor the examplein Table20.5assumingtwo factors. How dpesthe estimatechange if you assllme (a)one factor and (b)threefactors. and the 2.l4. A bank has a portfolio pf options an asset. Thedeltaof the optionsis ntlmbers price be interpreted. is T he Explin howthese is 20 and asset can gamma its volatility is 1% per day.AdaptSampleApplicationE in theDerivaGemApplication Buildersoftware to calculate VaR. is 7-2 per 1% change in the 20.15.Supposethat in Problem20.14the vega of the kortfolio annual volatility. Derivea model relating the change in the portfolio value in 1 day to delta,gamma, and vega. Explainwithout doingdetiled calculationsh0wyou would use the model to calculate a VaRestimate. 'can
'
.5%,
.58%.
.6.
.5%.
.8.
-3
.n
-5.
AssignmentQuestions 20.16.A company has a positionin bonds worth $6 million. ne modifed durationof the portfoliois 5.2 years. Assumethat only parallel Shis in the yield curve ca take place yield is measured in and that the standarddeviationof the daily yield chanpe (whep percent)is Use the duration model to estimatethe z-day9% VaR for the careilly the weaknessesof this approach to calculatingVaR.Explain Portfolio.Explain two alternativesthat give more auracy. .9.
t65
valueat Risk
in gold apd a $5*,0(% Consider a position consistingof a $300,000 20.17% tnvestment investpent in silver.Supposethat te daily volatilijies of thse two assets are 1.8% and 1.2%,respectively,and thatthecoecient of corrilationhetweentheirreturns is What is tht lo-day 97.5%VaR for the portfolio? By howmuchdoes diversihcation reducethe VaR? 20.18.Considera portfolio of options on a sigle asset, Suyyosethat thedelta of the yrtfolio is 12,the vlue of the ajset is sl, and thedailyvolatlhtyof the assetis 2%. Estlmatete l-dy 95% VaR for the pprtfolio frpmthe delta.Supposenext that the gamma of tht portfoliois Derivea quadratk relationship hetweenthe chang ip the portfolio valueand the perntage hage in the nderlyingassetpri il one day.How.wuld ' you use this in a MonteCarlosimulatitm? hondand 3-yearhopd, as well a,sa short 20.19.A companyhas a long position in a z-year positionin a s-year hond. Eachhond has a principal of $1 and pays a 5% coupon annually.Calculatethe company'sexpospre to the l-yeai, z-year, 3-year,4-year,and s-yearrates. Usethe data in Tahles20.3 and 20.4 to calculatea z-day95%VaR on the assuinption that rase changesayeexplined hy (a)one factor,@)twofactors,and (c)three factors.Assumethat the zero-coupon yield curveis fat at 5%. 20.20.A hank has wlten a call option on one stock >nd put optitm on ancther stock. For the frst option the stock prke is 5, the strike prie is 51, the volatility is 28% per annum,and the time to matmity is 9 months. For the seond option the stock prke is 2, the strike prke is 19,the volatility is 25% per annum,and the time to maturity is 1 year. Neither stock pays a dividend,the rijk-free rate is 6% per annum, and the Calculatea l-day 99%VaR: correlationhetweenstockprke returns s (a) Using only delts (h) Usingthe partial simulationapproach (c) Usingthe full simulation approach 20.21.A common complait of risk managers isthatthe model-bilding approach(either linar well Testwhathappenswhendelta quadratic)does whendeltaisclose work not to zero. or is close to zero Y using SgmpleApplicationE in the DerivaGem pplication Builder software.(Youcan do thisby expen'mentingwith diserentoptionpositions andadjusng the position in the underlying to give a deltaof zero.lExplainte results you get. .6.
-2.6.
.
.4.
'
466
CHAPTER 20
APPENDIZ''
CbSH-FI.OW MXPPING In thisappendixwe explain oneprodure formpppingcash Qws to stahdard maturity dates.We will illustratethe produre by considering a simple exampleof a portfolio of a long positionin a single Tmasurybond with a principalof $1million consisting in 0.8years. We suppose tht thebondprovidesa coupon of 10%perannum maturing Thismeans thatthebondprqvidescoupon paymentsof $50,000 s emianpually. payahle in years an@ years. It also providesa principalpment of $1 miljon in zeroyears.The Treasu!'ybond can thereforebe regarded as a positionin a princlpal with of a nd positionin bond bond $50,000 zero-coupon a a a coupon witha principalof $1,50,. Thepositionin the0.3-yearzero-couponbondis mapped into an equivalentposition in 3-monthand 6-monthzero-couponbonds.Theptsitionin the zero-coupon p ositionin 6-monthand bondis mappedinto an equivalent l-yearzero-coupon bonds. coupo-bearing bond is, for VaR The result is that the position in the regarded as a positionin zero-couponbondshayingmatmities of 3 months, purposes, months, and 1 year. 6 .3
.8
.8
.3-year
.s-year
'
.s-year
.8-year
The Mappinj Prpcedure Considerthe $1,5, thpt will be reived in years. We supposethat zero rates, dailybond price volatilities, and rrelatiop betweenbond returns are s showpin 't ab1e2A.1. Theflrststageis to interpolatebetweentheGmonthrate of 6.0%andthe l-yearrate ' of 7.0% to obtain a for a11 rate of 6.6%.(Annualcompounding ij ltived cash fbw to be in 0.8 years is rates.)Thepresentvalue of the $1,050,000
.8
'
.
.
.8-year
'assumed
1,5,
= 997,662
1.06t8
We also interpolatebetweenthe 0.1% volatility for the volatilityfor the l-yeai bond to get a 0.16%yolatility fr the -month
.8-year
bond and the 0.2% bond,
Table 20A.1 Data to illustratemappingprocedure.
J-?npplf
Matarityl
Zero rate (%with annual compounding): Bondpricevolatility (%per day):
q-mozth bohd
Correlatiozbetweezdailyrdfpr?lj .
3-monthbond (-monthbond l-yearbond
5.50 0.06
. . -- .
'
1. .9
0.6
6-mozth
l-year
0.10
7. 0.20
6-moztb
l-year
prl#
boad
.
..
- . .9
.
1. .7
.7
1.
467
Value at Rk
Table 0:.2
The cash-iowmapping result.
Sl,
$1,5,
receed
in .J years Positionin 3-month ond ($): Positionin f-monthbond ($): Positionin l-yearbond ($):
receid
in
37,397 11,793
Total
.#
years
37,397 331,382 678,074
319,589 678,074
Supposewe allocatea of the presentvalpe to the Gmonthbond and 1 a of the and matching variances, we presentvalue to the,l-year bond. Usingequation (2k2). -
obtain
2 0012( + 0.0016 =
.
22(1 a)1 + 2 x
.7
x 0.0()2a(1 a)
.001
x
-
.
-
Thisis a quadratic equation that can be solvedin the usualway to give a = 9.320337. nis means that 32.9337% f the value shouldbe allocatedto a Gmonthzero-couppn bondand 67.9663%ofthe value shouldbeallocatedto a l-yearzerocoupon bond.The is thereforereplaced by a f-monthbond worth .s-yearbond worth $997,667 and a l-yearbond worth
This cash-iow mapping schemehas te advantagethatit preservesbpth the value ad the variance of the cash :ow, Also,it can be shown thatthe weightsassignedto the two adjacentzero-coupon bonds are alwayspositive. For the $50,000cash ;ow received at time years, we can carry out similar calculations(seeProblem2.1). lt turs out that the present value of the cash ;ow is $49 189 lt can be mappedinto a position worth $37397in a 3-monthbond and a in a (-monthbond. Positionworth $11,793 The results of the calculationsare summarizedin Table2A.2. The couponin a 3-monthbond, a position bearingbondis mapped into a position wcrth $37,397 in a l-yearbond. worth $331,382 in a f-monthbond, an a position worth $678,074 equation correlations in Table2.1, givesthe variance Usingthevolatilitiesand (20.2) bond with n = 3, aj = 37,397,a2 = 331,382, of the change in the price of the = = = 678,074, and p11 pj3 p23= c2 c3 = cl a3 This variance is 2,628,518.The standal.d deviationof the change in the price of the bondis therefore 2,628,518= 1,621.3.Becausewe are assumingthat the bondis the oly instrumentin the portfolio, the l-day 99%VaRis .3
,
,
.
.s-year
,s-year
.f,
.1,
.9,
.2,
=
or about $11,950.
x& 1621.3
x 2.33= 11,946
.6,
=
.7.
'
Ektimating Volatiliiies and Corrlations '
.
.
In this chapter we expkinhowhistpricaldata can be used to produce estimattsof the currentand futurelevelsof volatilitiesand correlations. Thecliapter is relevant both to the calculation of value at risk using the model-buildingapproach and to the val,uation of derivatives.When calculating value at risk, we are most interestedin the current leveljof volatilitiesand correlations becausewe are assessingpossible changes i the valueof a portfolio overa tery short pehod of time.When valuingderivatives,forecysts of volatilitiesand correlations over thewholelifeof the derivativtare usually required. ne chapter considers modelswith imposingnames such as exponentia weighted and movingaverage (EWMA),autorepessive conditional heteroscedajticity (ARCH), ) . (UARCH), autoregressive conditional heterscedasticity generalized The distmctive featureof the models is that they recognke that volatilitiesand correlations are not constant.During some periods, a particular yolatilityor cormlation may be platively low,whereas duringother periods it tnay e relaiivtlyhigh.Themodels attempt to keep track of the variations in the volatilityor correlation thrpughtime. .
.
21.1 ESTIMATINGVOLATILITY DeEneoj as the volatility of a market variable on day n, as estimated at the end of day n 1. The square of the volatility, a, on day n is the variazcerate. Wedescribed 13.. suppose the standard approach to estimating o', fromhistrical data in section that the value of the marketvariable at the end of day i is h. Thevariable u is dened theend of day i l and as the continuousl compounded return duringday i (between of i): end the day Si = ln ui Si-b -
-
An unbiased estimate of the variance rate per day, observationson the ui is 2 c?i =
m
using the most recent m
M
.) -
g2a,
(ua-j uj -
-
1
2
(21.1)
f=1
469
470
CHAPTER 21
where17 is the mean of the Iljs:
.
J
=
-
??l
j
us-f f=1
For the purposesof monitoringdaily vplatility, the formulain equation (21. 1) is usuallychanged in a nllmber of ways: 1. ui is defned as the perntage change in1 the market variable betweenthe end of daf i 1and the end of day i, so that: -
Si
u=
i
-
,f-l
(21.2)
sf-l
2. is assllmed to be zero.2 3. m 1 is replaced by m.3 -
the estimatesthat are calculated, bt Thesethr changesmake very littlediferen varkn allow simplify formulafor the the rate to they us to qto
'
.;
V,1
M
1 =
-
u'?l-f
(21.3)
f=1
4 whereui is given by equation(21.2).
WeightingSchemes 2 2 2 un-2, givesequal weight to u,-l, Equation(21.3). un-v,. our objectiveis to estimate the current levelof volatility,jn. It thereforemakes sense to give moreweight to recent data. A modelthat doesthis ks .
n
=J-laf
.
.
,
(z1.4)
-j
f=1
g' *
.
The variable ai isthe amount of weightgivento the observation i daysago. The a's are positive.If we chooe them so. that qi < aj when i > j, less weight is given to older observatios. The weights must sum to unity, so that '
a.I l'zzz1
=
1
-
' This is consistent with the point made in 29.3 about the way that volatilityis dened for the section of calculations. VaR purposes 2 As explaine,din 29.3, this assumption usually has very littleefl'ectoc estimatesof te variance section becausetheexpectedchangeic a variablein one dayis very smallwhencomparedwit thestandard deviation of chuges. 3 Replacing m 1 by m moves us from a!l unbiasid estimate of the variance to a maximum likelihoo estimate.Maximumlikelihoodestimatesare discusxdlaterin the chapter. 4 Notethat the Il's in thischapter play the same role as the Ax's in Chapter2. Both are dailypercentage changesin market variabks. In thecase of the Il's, 9hesubscriptscount observationsmade on diserentdays on the same market variable. In the case of the A.x's,theycount observationsmade on the same day oc dferent markd variables.ne use of subscriptsfor a is gmilarlydiferentltween the twochapters. In this chapter,the subscriptsrefer to days;in Chapter29 theyreferred to market variables. -
'
4l1
Estimating Volatilitiesand Ctvrlaltm, ' .
.
An CXtensionof th fdeain equation (2l is to assumethat thereis a lonprun averae variance and that this should be givenjome weiht. Thisleadsto te model that takesthe form .4)
'rate
'
m
c7,
ai g-fn
)?yz+
=
(21.5)
f=1
where Pz is the lonprun variance rate and y is the weight assignedto Fz. Sincethe weightsmust sl'm to unity, it followsthat
lz-l-l''lf =.1
f=1
..
model,It was hrst suggestedby Engle.sTVestimateof Thisis knownas an the variance is based on a long-runaveragi variance and m observations.The oldr n observation,the kss weight it is givtn. Dtining (I) = yFz, themodd in tquation (21.5) can be written ARCHIN)
#1
K
=
+
J-laf
:,2-3
(21.6)
i=1
In the nexttw0 sections we discusstwo importantapproaches to monitoring volatility and (21.5). usingthe ideasin equations(21.4)
WEIGHTEDMOVINGAVERAGE 21.2 THEEXPONENTIALLY MODEL The exponentiallyweightedmoving average(EWMA)modl is a partkular caje of ihe modelin equation(21.4) wherethe wekhts ai decreaseexpbnentia as we move back throughiime.Speciscally,aj+l = aj, where is a constant between and 1. It turns out that this Aeightingscheme leadsto a particularlysimpk formulafor updatingvplatilityestimates. The formulais n-1 2 + (1 ),z2 (217) n = cn-l The estimat, c,,, of the volatilityof a variable fordayn (maeat the end ofdayn 1)is : cplculatedfromcs-l (theestlmatethat was made at the end of dayn 2 of the volatility ftk day n 1) apd If,,-1 (themost recent dailypercentagechange in ihevariable). corresponds to weightsthat decreaseexponenTo understand why equation(21.7) 2 to get tially,we substitute for ok-l -
.
-
-
-
y2n- 2j + (1 2 gc2 n- j # (1
Gn
=
)Ifs2.j
-
-
Of )(R2
= (1
n- !
-
n
+
:2 n-
2)
+
2c2
n-2
Substitutin in a similar wayfor /.-2gives
2 (1
Gn
=
-
)(R2
n-1
+
:2
n-2
+ ),2:2 n-3 ) +
3c2
n-3
5 stt R. Enjk 4'AutoregressiveConditional Hdtrosdasticity witb Estimatts of tl Varian 987-1*8. Inqation,'' Ecosometrica,59 (1982):
of UK
472
CHAPTER 21
Continuingin thisway gives m
= (1 ) f=1 -
?1
i-1Id2 ?l-f ' .
+
kM ?1.m
-
For largem, the termk's-. is sllciently small to beignored,so that equatign(21.7) with ai = (1 ) f-l The weights for the ai declineat is the same as equaon (21.4) mte as we move backthroughtime.Eachweight is timesthe previous weight. -
.
fxample2/./ the volatilityestimatedfor a market variable for day n 1 Supposethat is mayketvarkble increased y 2%. Tllis is 1% per day,and duringday n l the:2-1 0.12 = 0.022= = nd Equation (21.7) n meansthat n-1 = .9,
-
-
.4.
.l
gives
o7, =
.9
.l
x
+
.4
.1
x
.13
=
The estimateof the volatility,h, for day n is therefore or 1.14%,per 2 2 is cn-j day.Note that theexpectedvalue of Idn-1 l. In thisexample,the , or 2 is reater than the expectedvalue, and realizedvalue of IG-1 g as result our z had beenlessthan . volatilityestimate increases.If the realized value of In-1 its expectedvalue, our estimateof the volatilitywould havedecreased. .13,
.
The EWMAapproachhasthe attractive featurethat relativelylittledata need to be stored.At any given time, nly the current estimateof the varian rate and the most recentobservation on the value of the market variable need be remembered.When a newobservation on the marketvarible is obtained,a newdailypercentagechange is is used to update the estimateof the varian rate. The calculatedand equation(21.7) o1destimateof the variance mte and the old value of the market variable c thenbe discarded. The EWMAapproachis designedt trackchanges in.thevolatility. Supposethereis R2uI a big move in the market variab1e on day n 1, so that , is largr. From equation(21 this cpuses $heestimateof the current volatility to move upward. The value of governs how responsive the estimateof the dly volatility is to the most recentdailypercentage change. A lowvalue of leadsto great deal of weightbeing 2 when c, is calculated. In this case, the estimatesprodud for the givento the p,-I volatilityon sucssive days are themselvesilighlyvolatile. A high value of (i.e.,a value clse to 1.) produs estimatesof the daily vclatiltty that respond relatively slowlyto new infrmation providedby the dailypercentagechange. The RiskMetrksdatabase,wllich wasoriginallycreated by J.P. Morganand made publiclyavailqblein 1994,uses the EWMAmodel with = 0.94f0r updating daily volatilityestimatesin its RkMetrics database.Thecompany foundthat, acrossa range of diferent market variables, this value of l givesforecastsof the varian rate that comecljest to th realized vaiian rate.6 ne realized varian rate on. a partkular day was calculated as an equallywekhted averageof the I, on the subsquent 25days (s Problem21.17). -
.7)
t'
'
6
.
.
'
1995.We +11explain an altemative (maximum S J. P. Morgan, RkMetrics Monitor,Fcurth Quarter, likelihocd)apprcach tc estimatingpjrameters later in the'chapter.
ktimatng
''fz'plcllf/:.
473
and Cprr:lc/tm.
21.3. THEGARCHII?1) MODEL We now niove oa to dcuss what is knownas the GARCHII,1)model,proposed by Bollersleyin 1986.7The diferen betweenthe GARCHII,1) modd and the EFMX and equaon (21.5). (11.4) to the diferen betweenrqultign Jn modelis cs2 varin well GARCHII,1), is calculatedfrci a tcng-run rate, Ps, as average as 1 )is equationfor GARCHII, The and us-l. fromc-l .analogus
yPs + au2s-l + j-l
2
Gn
=
(11.8)
istheweightassiped to us-l, l and jis theweight unity, it follows that must sum to
Fhere p istheweightassiped to Ps, a c2,-1. the weights
assipedto
sin
/+J+/=1
The EWMA model is a particulat case of GARCHII,I) whert y = j a = 1 , and j = 1)'' in GARCHII,1)indicatesthat , is based on the lost recent observaThe tion of and the most rent estimate of the variance rate. The more general GARCH/, modelcalculates , from the most rent p observationson al and ' 8 the most recent q estimates of the varian rate. GARCHII,1) is by far the lost popularof the GARCHmodels. 1) model can also be written Sdting (z) pps, the GARCHIi, -
.
$$41:
.
.
=
.
s
=
+a&2+ b
*
,-1
,-
(21.9)
l
Tls is the form of the model that is usnlly used for the purposesof estimntingthe ' parameters. Once ), a, and j havebeenestimated,wecan calculatey a; 1 a j. The long-termvariance Ps can thenbe calculatedas /y.jbr a stableGARCHII,1)process we require a + j < 1. Othemise the weight applied to th lonpterm variance is negative. -
-
Example 21.2
Supposethat a GARCHII,1) model is estimatedfrom (kily dqta as .2
R
+
=
0.13R2,-1
+ 0.86,-,
This corresponds to a = Because j = 0.86, and (t) = = = = ()) yps, it followsthat Because y 1 a j, it followsthat y Ps = In other words, the long-nmaverage varian per dayimpliedby = Th correspondsto a volatilityof (1.0002 themodelis or 1.4%, day. per .13,
.()($2.
.1.
-
-
.2.
.2.
.14,
'
.
y see T. Bollerslev, eGeneralize,dAtoregressive ConditionalHeterosceasticity,''JoarnalPJLnosetics, 397-27. 31 (1986): 8 Other GAR/H motkls havebeenpropose thatincorporateasymmdric ntws. nex moels are.tksipe epends sip of models appropriate the for equities than Arguably, t he that ok are more on so w-l GARCHII, 1).As mentionedin Chapter 18,te volatilityof an tquhy's price tendsto beinversely related to the price so that a negative w-l should have a iggerefl'ecton ca than te same positive un-l. For a and discussionof models for handlingasymmetricntws, sie D. Nelson, Heteroscedastkity AssetReturns: A NewApproach,'' Lnometrica, 59(l99):347-79;R. F. Engk and V. Ng, Measuringand 174178. Testing the Impact of News on Vol,atility,''Journal nJFizate, 48 (1993): .
%conditional
!74
CHAPTER 21
thatthe estimate of the volatilityqn dayn 1is 1.6%per day,so that suppose 0.0162 aln-1 and that on day n 1 the market yariable decreased 12 2 0.001. Then 1%, by so that un-j -
.256,
=
=
-
=
c2 =
=
.
-1L
.13
,z
+ 0.86x
.l
y
.256
=
The newestimateof the volatility is therefqre PCr ay.
k23516
0.t0023516 .153,
or 1.53%,'
=
'
d
The Weijhts Substitutingfor c-j in equation(21.9) #ves 0r
g2,,-2
for substituting
2 h =
au2-I
=
+
=
+ j.)+
+ jtt,l+ au2-I
(w2.-2
+ j-zl
+ aju2-2 + ,2(r22
gives + jto + j
2
2
2
2 2
+ aua-l + @uq-2 + aj uJ)
54.44 38.39 32.85 31.61 24.47
1 In the United States, the claimmade y a ondholder is the ond's facevalueplus anled interest.
492
CHAPTER22 It foundthat the followingrelationshipprovides a good t to the data:z
an d 26.
Recoveryrate
59.1 8.356x Defaultrate
=
-
wherethe recvery rate isthe veragefecoveryiate on senioyunsecuredbondsin a year measuredas percentageand the befault rate is the corporatedefaultrate in the vear ed percentage. masur as a This relationship means that a bd year for te defaultrate is usually doublybad ple, when the averagedefault it is accompaniedby a 1owrecoveryl'ate. For exam. because the expectedrecoveryrate is relativelyhighat 58.39/:.When raiein a year is only relatively high at 3%, the expted recoveryrate is only 34.0%. thedefaultrati is '
'
.1%,
f ROM BOND PRICES DEFAULTPROBABILITIES 22.4 ESTIMATING '
.'
.
,
The probability f defaultfor a companycan be estimatedfromthe prices of bondsit has issued.The usual assumptionisthat the only reason a coporate bond sells for less than a similarrisk-free bond is the possibilityof default.3 Considerfrst an approximatecalculation.Supposethat a bond yields 200 basis points more than a similartisk-fr bond and that the expted recoveryl'ate in the eventof a defptlltis4%. Theholderof a corporatibondmust beexpectingto lose2 year) fromdefaults.Giventhe recoveryl'ate of 4%, yhisleads basispoints (or2% to an estimateof the probability of a defaultper year conditionalon po eailier default of or 3.3j%. In general, 'per
.4),
.2/(1
-.
-
J
.
(22.2)
--
1-;
mte) per year, s is the spread of the wherei, is the average defaultintensity(hazard and risk-free R is the expeted recpveryl'ate. yield the bond rte, over corporate
A More Exact Calulation For a more exact calculation, supposethat the corporate bond we havebeen considersemiannually)and thatthe inglastsfor 5 years, provides a coupon 6% per annum(paid yield on the corporate bond is 7% per annum (withcontinuous compounding). The yield on a similarrisk-free bond is 5% (withcontinuouscompounding).The jields implythat the pri of the corporate bondis 95.34and the pri of the risk-free bond is 14.9. The expected lossfromdefaultover the s-year life of the bondis therefore probability of defaultper year (assumed 104.09 95.34,or $8.75.Supposethat the in this simple example to be the sameeach year) is Q.Table22.3 talculates the expected lossfromdefaultin termsof Q'on theassumptionthatdefaultscan happenat times -
.5,
2 and RecoveryRatesof CorporateBond seeD. T. Hamilton, P. Varma, s.Ou, and R, Cantor,tDefault corielation is also January 24. The Rl of the regressionis Issuers,''Moo's Investor's srvices, identifedand discussedin E. 1.Altman,B. Bra, A. Resti,and A. sironi, Link etweenDefault and RecoveryRates:Implicationsfor CreditRisk Models and Procyclicality'' Working Paper, New York .6.
'rhe
Sl-he
Univezsity,23.
3 This assumption is not perfed. In practicr the price of a cormrate bondis afected by its liquidity.The lowertheliquidity,the lower the price. '
493
CreditRisk .
.
...- - -.- - ..--
J
' .-.- - . .
.
Calculationof loss frm default on a bond in termj of the default principal Probabilitis per year, Q.kotional sl.
Table 22.3
'
=
Time
ear .5 1.5
2.5 3.5 4;5
Defaalt
Recovery Risk-free Loss gen Dcoant 'F of expected probability ('plt?ltnf(.9 valae (.9 defaalt(J) factor loss(.9
Q Q Q Q Q
.
106.73 105.97 105.17 104.34 103.46
4 4 4 4
.
..
'
66.73 65.97 65.17 64.34 63.46
0.975) 0.9277 0.8825 0.8395 0.7985
...
Total
'''
65.08: 61.20: 57.57: 54.01: 50.67: 288.48: '
.
.
1.5,2.5, 3.5,and 4.5 years mmediately beforecoupon payment dates).Rk-free rates continuous compounding). for a11maturities are aumed to be 5% (with To illustratethe calcplations, consider tlle 3.5-yearrow in Table22.3. Th expected usiug forwardinterestrates value of the corporate bond at time 3.5 years (calculated and assllmingno possibilityof efault) is -.5x
3+
3:-.5x.5 + 3e
1.
+ j.g3:-.5x
1.5 =
jgj
.
34
Giventhe desnition of recoveryrates in the previoqssection, the amountrecoveredif there a default k 4, so that the loss givep defaultis 194.34 4, or $64.34.The presnt value of tlt lossi 54.1. The expected loss therefore54.91Q. Thetotal exjected lossis 288.48:. Settingthis equal to 8.75,we obtain a value for Q of 8.75/288.48,or 3.3%. The calculations we have given assumethat the default probabilityis the sme in iach year and that defaultstakeplae at justone timeduring the year. We can exted the cqlculationsto assume that defaultscan takeplacemore frequently.Also,instradof assuminga cpnstant uncgnditionalprobabilityof defaultwe rate) or assume a partkulgr pattern for canassumea constant defaultintensity(hazard the variation of default prohabilities with time.Withseveralbonds we can estimate srveralparameters describingthe term jtructure of defaultprobabilities.Suppose,for example,we havebods maturing in 3,5,7, and l years.Wecould use thefirstbond to estimatea defaultprobability per yeaf for thefirst3 years, the second bond to estimate dfault ppbability per year for years 4 and 5, the third bond to estimate a (kfault probabilityfor years 6 and 7, and the fourthbond to estimatea defaultprobability for years8, 9, and l (seeProblems22.15 and 22.29). Th approachis analogous to the bootstrap proceure in Section4.5for calculating a zero-coupon yield urve. -
the Risk-FreeRate Akey suewhen bond prices are used to estimatedefaultprobabilitiesisthe meaningof the spread s is the the terms risk-free rate'' and rijk-free bond''. ln equation (22.2), yield yield r isk-free of bond.ln Table22.3, on a similar overthe excess the corporate bond the rk-free value of thebond must be calculatedusing the risk-freediscountrate. The benchmarkrisk-freerate thatis usuallyused in quoting corpomte bondyields theyield on similar Treasurybonds.(For example, a bond trader mkht quote the yield on a particularco:porate bond as beinga spread of 250 basispoints over Treasurks)
4F4
CHAPTER 22 Asdiscussedin Section4.1,tradersusually use LlBoR/swapratej as proxies for riskfree rates when valuing derivatives.Tradersalo oftenuse LlBoR/swaprates as riskfree rates when calculating defult probabilities. F0r exnmpl, when they determine defaultprobabilities frombond prices, the spreads in equation (22.2) is the spreadof the bind yield cver the LIBOR/swaprate. Also,the rijk-fpe discountrates used in the calculationsin Table22.3are LlBoR/swapzero rates. willbediscussedin the next chapter) can be used to imply Creit defaultswaps(which .the risk-free rate assllmedby tradeys.The impliedrate appearsto be approzmately equal to the LlBoR/swap late minus 1 basis oints on average.4This estimate is plausible.As explained in Section7.5, the credit lisk in a swaprate isthecreditrisk from makinga seriesof fimonth loansto A-rated countrparties and 1()basispoints is a reasonble defaultridk.preminm for a A-rated f-monthinstrument.
Asset Swaps In practic, traders often use asset swap spreads as a way of extracting default probabilitiesfrom bond prices. This is becauseasset swap sprea'dsprnvide a direct estimateof the pread f bond yields overthe LIBOR/swapcurve. To explain how asset swaps work, nqider the situationwhere an asset swap sPread for a particular bond is quoted as 15 basis points. There are three situations: .possible
1. Thebondsellsforitspar valueof 1. Theswaptheninvolvesone side(companyA) B)paying LIBORplus Payingthe cupon on thebgndan the other side(cotnpany 150basis points. Note that it is the promised copons tht are exchanged. The exchangestake pla regardlessof whether th bonddefaults. 2. Thebondsellsbelowits par value, say,for 95.The swapisthen stnctured so that, in additionto the coupons, company A pays $5per tlo of notignal principal at the outset. Company2 pays LIBORplus 15 basispoints. 3. The undrlying bond sellsabovepar, say,fol'18. The swlp is theystructuredso ' that, in additionto LIBORplus 15 basispoints, company B makes a payment of $8 pir $100of principal at the outset. CompanyA pays the coupons. The eflkctof a11th is that the present valueof the assetswap spreadiqthe Amountby whichthe price of th corporate bond is exeded by the price of a similarrisk-free bond where the risk-free rate is assumedto be given by the LlBoR/swapcurve (see Problem22.14). exnmplein Table22.3where the LlBoR/swapzero curve is ;at . Consideragainthe at 50:. Supposethat insteadof knowing$e bond's pri we know that the asset swap spreadis 15 basis points. This meansthat the amountby which the value of the risk-free bond exeds the value of the corporate bon is tht present value of 15 basis points per year for 5 yeais.Assumingsenziannualpayments, this ij $6.55per $100of princ pal Te total loss in Table 22.3 wouldin this case be set equal to $6.55.This means tat the default probability per year, Q, would be 6.55/288.48, or 2.270: .
'''
'
,
'i
.
.
4 Stt J. Hull,M. Prtd%cu, and A. Wbite, Rtlationshipbttween CreditDefault Swa/ Spreas, Bond Yields, and Credit Rating Xnnounments,''Jourzql d.J'Banking fzxl' Fizazce,28 (Novemer 24): 2789-2811. llrfht
495
Crdit Risk
PROBABILITY ESTIMATES 22.5 COMPARISON OF DEFAULT The default probabilities estimated from storial dpta are much less than those derivedfrombond prices. Table22.4illustratesth. 5It shows for companiesthat start with a particular rating, the ayerage annual defaultintensityover 7 years cplculated from (a)historicaldata and (b)bond prices. Thecalculationof defaultintensies fromhistoricaldata arebasedon equation (22.1) with t = 7 and Table22.1.Fmm equation (22.1), ,
k7) ...-I:lnElp(7j =
-
where (l)is the averagedefaultintensity(orhazard rate) by time t and :41)is the cupulativeprobability of defaultby time 1. The values of :47)are takendrectlyfrom Table22.1.Consider,for example,an A-ratedcompany. The value of :47)is 0.00759. The average 7-yeardefaultintensityij tVrefore
(7)
=
-J1n(.99241) .11
=
or 11% The calculations of averagedefaultintensities'frombond pris are basedon equatipn (22.2) and bond yieldspublished byMerrillLynch.The resultsshown are avepges betweenDecember 1996and October2007.Therecoveryl'ate is assllmedto be4% and, for the reasonsdiscussedin te previoussection, the risk-ffeeinterestpte is assumedto be the 7-yearswap rate minus l basispoints.For exnmple, for A-ratedbonds,the averageMerri11Lynchyield was5.9930:. The averageswp ratewas 5.398% so thatthe averagerisk-freerate was 5.2890:. Thisgiveste averge 7-yeardefaultprobability as .
.
.
,
i
0.05993 0.05298 = 1 -
.116
.4
-
or 1.160:. Table22.4showsthat tlieratio of thedefaultprobabilitybackedout frombondprkes grde to thedefaultprobabilitycalculatedfromhistorkaldataisveryhighforinvestment companiesand tendj to declineas a company's crediyrating declines.The diference betweenthe two defaultprobabilitiestendsto increaseas the credit rating declines. Table 22.4
averagedefaultintensities(%per annum). seven-year
Ratizg
Historical defaalt Defaalt iztezsity Ratio Dlhrezce
iztezsity Aaa
Xa A Baa
Ba B
C&aand lower
0.04 0.05 1 0.43 2.16 6.10 13.07 .1
frombozds 0.60 0.74 1.16
16.7 14.6 10.5
0.56 0.68 1.04
5.
1.71
2.2 1.3
2.54
7.97 18.16
1.4
i.13
4.67
1.98 5.50
..
5 The,results in Tables22.4and 22.5 are updates of the result.sin J. Hull,M.Predescu,and A.White,SlBond and Risk Premiums,'' Joarnal of CreditA'k, l 2 (Spring25): 53-6. Prices,Default Probabilities, ,
Table22.5providesanother way oflookingat theseresults.It showsthe excessreturn (Sti11 assumed to be the 7-yeazswap rate minus 1 basispoints) overthe risk-free rate earnedby investorsin bonds with diferept edit rating. Consideragain an A-rated bond.The averagespreadover Treasuriesis 112hasispcints.Oftls, 42basispoints are accountedfor by the average spread between7-yearTreasuyiesand our proxy for the risk-freerate. A spread of 7 basispoints is necessal'yto cover expected defaults.(T11is equalsthe real-world probability of defaultfrot Table22.4 multiplied by 0.6 to qllow expecteddefaultshavebeentakeninto for recoveries.)Ikis leavesan excessreturn (after points. aount) cf 63 basis Tables 22.4 and 22,5 show that a large percentage diferenc betweendefault probabilityestimatestranslatesinto a small(butsipicant) excessreturn on the bond. For Aaa-ratedbondsthe ratio of the twodefaultprobabilities is 16.8,but the expected excessreturn is only 32basispoints. Theexcessreturn tendsto increas as credit quality 6 declines. The excess return in Tabk 22.5 dces not remain constant through time. Credit spreads,and thereforeexcess returns,were high in 21, 2002,and the frst half of 2003.After that they were fairlylow until the credit cnmch in July 2007when they 23.3 for a discssiop ofthe year 2007 startedto increaserakidly. (seeBusinesssnapshbt creditcnmch.) .
Real-Worldvs. Risk-NeutralProbabilities The default probabilities impliedfrom bond yields are risk-neutral probabilities of default. To explain why th is so, consider the calculationsof default probabilities in Table 22.3. The calculations assume that expected default losses can be discountedat the risk-free rate. The risk-neutral valuation principle showsthat this is a valid procedure providingthe expected losses are calculated in a riskneutral world.This meansthat the defaultprobability Q in Table22.3mut be a risk-neutral Probability. By contrast, the default probabilities impliedfrom lstorical data are real-world k defaultprobabilhies (sometimes also calledpysical probabilitle. The expectedexcess return in Table 22.5arisesdirectlyfromthe dference betweenreal-wrld and riski The results f0r B-rated bondsin Tbles 22.4and 22.5 nm counter to the overall pattern.
kii
CreditRk
neutraldefaultprobabilities.If therewere o expectedexss rdurn, thenthe real-world and risk-neutral defaultprobabilitiesFould be the same, and vi vqrsa. Why do we see such big diferencesbetweenreal-world and risk-neutral default As we havejustarled, this is the sameas askipg why corporate bond probabilities? trad.ers earn more than the risk-free pte on avrage, One reason for the results is that corporatebo ds are rela kel?illiquiband the returnson bondsare higherthan theywoulb otherwisebe to compensate for this. But thisis small part of what is going0. It explainsperhaps25basispointspf the exess . Dturn ain Table 22.5.Anotherpossiblereason for the results is that the subjective default probabilitiesofbondtradersare muchhigherthanthe iose gye in Table 22.1. Bondtradersmay be alloV1% fordepressionscenaripsmuchwprse than anytlng seen during the 1970to 2003priod. However,it is dicult to see howthis can explain a largepart of the excessreturn that is observed. Byfar the most importantreason forthe results in Tables 22.4and22.5isthat bonds do nt defaultindependentlyof each other. There are periodsof time when default mtesar very lowand perids of timeWhentheyare very high.Eviden for this can be obtainedby lookingat the defaultrates in diferent years. Moody'sstatisticsshow that between1970ad 2006thedefaultmte per year ranged froma lowof 0.090% in 1979to variation rise to The in default high gives of q/ in 3.81 21. rates year-tyear a away) risk risk and bond tmders earn an systematic (i.e., that cannot be diversifkd rijk. The variqtion in (kfault rates from year to expected for bearingthe return excess conditions and it may be becausea defaultby yearmay be becauseof overalleconomk Dsulting in defaultsby other ompanies. (The latteris company has a ripple efect one Dferred to by researchers as credit contagion.) addition In to the systematicrisk we havejusttalkedabut themis nonsystematic(or risk associatedwith each bond.If wewere talkingabout stocks, we would idiosyncratic) argut that ilwtstorscan divtrsifytht nonsystematicrisk by choosinga portfolioof, say, 30 stocks.They should not thereforedemanda rk premiumforbearingnonsystematic risk. For bonds,the arguments are not so clear-cut. Bond returns are highlyskewed with limitedupside. (For example, on an individualbond,theremightbe a 99.75% chanceof a 7% return in a year, and a 0.25%cpce of a return in the year, the and corresponding flrstoutcome to no default the srcondto default.)This type of risk away''.' is dicult to It Fould require tens of thousandsof diferent bonds. In practi, many bond portfoliosare far from fullydiversihed.As a result, bond tradersmay earn an extrareturn for bering nonsystematicrisk as well as for bearing the systematicrisk mentioned in the previous paragraph. At this stage it is natural to ask whether we shoulduse real-world or risk-neutral defaultprobabilitiesin the analysisof creditrisk. The answerdependson the purpose of the analysis.Whenvaluingcredit derivativesor estimatingtheimpactof defaultrisk on the pricingof instruments,risk-neutral defaultprobabilitiesshould be used. This is becausethr analysis calculatesthe presentvalue of expected futlye cash Qowsand ,
-60%
Sdiversify
1 In addition to producingTable22.1,wllichisbasedon the 1979to 2996period,Moodyrsprodus a similar tablebasedUn the 1929to 2996period. When thistableis used, historkal defaultintensitiesfpr from4 to basispoints; te Aa gradebondsin'Table22.4 lise somewat. ne Aaadefaultintensityincreases increasesfrom5 to 22 basispoints; te A increasesfrom11to 29 basispoints;theBa' increaes from43 to 73 basispoints. Howrver, the noninvestmentgrade historicaldefaultintensitksdecline. 8 credit spread puzzlq'' BIS Qzarterly Acvfcw,5 tDecember seeJ. D. Amatoand E. M. Remolona, 23): 5143. 'investment
t'l'he
498
HAPTER 22 almostinvariably(implicitly or explicitly)involvesusing risk-neutralvaluation, When carying out scenario analysesto calculatepotential futurelossesfromdefaults,realworlddefaultprobabilities shouldbe used.
DEFAULTk'kd/jlLttltj 22.6 USINCEQUIT PRICESTO ESTIMATE Whenweuse a table such as Table22.1to estimat a company's feal-worldpfobability of default,we are relying on th company'scredit rating. Unfortunately,creditratings are revisedrelativelyinfrequently.Th has1edsomeanalyststo argue that equityprices can provid more ulsto-date informationfor estimating defaultprobabilities. In 1974,Mertonproposeda model where a company'sequity is an option on the has one zero-couponbond assetsof the company.9 suppose, for simplicity,that a 517)1 outstandinganLthat te bond matures at time T. Dene: Pg: Valueof company'sassets today Pz: Valueof company'sassets t ime T fg : Valueof company's equity today
'z : Valueof company'sequity at time T D: Debt repayment due at time T ctmstant) cy : Volatilityof assets(assumed c, : Instantaneousvolatilityof equity, If Pz < D, it is (atleastin theory)rational for the company to defaulton the debt at time T, The kalue of the equity is then zero.If Pz > D, the company shouldmake the debt repayment at time T and the value of the equity at this timeis Pz D. Mertn's model,therefore,givesthe value of the srm'sequity at time T as -
'z = maxtpz
-
D,
)
Thisshowsthat the equityis a calloptionon the value of the assets with a strikt prke equalto the repayment requiredon thedebt.TheBlack-scholesformulagivesthe value of the equity today as E = P Nd 1) De-rTNd 2)
(22.3)
-
where dj =
lny/o
+
(r+ 4/2)r
cpx/T
and
k
=
:1
-
cy
yj
The value of the debt todayis Js Eb. The risk-neutral probability thatthe company will defaulton thedebtis N-k). To calculateth, we require P and cy. Neitherof theseare directlyobservable. However, if the company is publicly traded,we can observeE. Th meansthat equation (22.3) providesone conditionthat must be satised by Pc and cy. We can also estimate c, fromhistoricaldata or options.Fromlt's lemma, -
cziEi = 9
'E JP
.--cypg
seeR. Merton onthe Pricinz of CqrporateDebt: The Risk Structureof Interest Rate''
449-79. Ffncncc,29 (1974):
Joarnal of
499
CreditRisk Of
csk
(22.4)
#(#l)cyy
=
This provides another equaiin that must be satissed by Ji and o'v. Equations (22.3) provide a pair of simultaneousequationsthat can be solvedfor F and av.1 and (22.4) Example22.1
.
The value of a company's equity is $3million and the vdatility of the equity is 8%. Thedbt that will haveto be paid in 1 year is $1 million. Therisk-freerate :::i T = 1, and b 1. is 5n/(, per annum.In tiliscase E = 3, c, = r = yields F 12.40and cy F= 0.2123.The parand (22.4) soivingequations (22.3) probability 1.148, of is that defaultis #(-#2) = 0.127,or the so ameterh is F 12.7%.The market value of the debt1(k-.5X1 k, or 9.49.The present value of = 9.51.Theexpectedloss promised payment on thedebt.is the on the no-defaultvalue. about 1.2% C0mis therefore 9.40)/9.51, f its debt (9.51 or paringthis with the probability of defaultgivesthe expeded recoveryin the event of a defaultas (12.7 1.2)/12.7, or about 91% of the debt'sno-defaultvalue. .8,
.5,
=
-
-
-
ThebasicMertonmoel we havejustpresentedhasbeenextendedin a number of ways. For example, one version of the model >ssumesthat p defaultoccurs wheneverthe valye of the assets flls belowa barrkr level. How well do the defaultprobabilitiesproduced byMerton'smodeland its extensions correspond to actual default experkpce? The answer is that Merton'smodel and its extensionsproduce a good ranking of defaultprobabities (risk-neutral or real-world). Thismans that a monotonic transformationcan be used to nvert the probabilityof defaultoutput fromMerton's mode1into a good estimateof eiter the reat-world()r 11 risk-neutraldefaultprobability.
22.7 CREDITRISKIN DERIVATIVES TRANSACTIONS The credit exposure on a derivativestransactionis more comp'licatedthan that on a loan. This is becaus the claim that will be made in the event of a defaultis more uncertain.Considera fnancialinstitutionthat has one derivativscontrad outstanding with a counterparty. Threepossible situations can be distinpished: 1. Contractis alwpysa liabilityto the fnancialinstitution 2. Contractis always an asset to the fmancialinstitution institution 3. Contractcan becomeeither an asset or a liabilityto the snancial An xample of a derivativescontract in the flrstcategoryis a short optionposition; an examplein the second category is a long option position; an example in the third categoryis a forwardcontract. -
10Tc sclve twontylinear equaticns cf thefcrm Ijx, yj and G(x,y) tlleSolverroutine in.Excelcan ' 2 2 be asked tc flnd the values cf and y tat minimlze(fk, y)) + lG(x,y)) 11 Mccy's KMV prcvides a servi that transfcrmsa defanltprcbabilityprcducedb Mertcn's mcdel intc it refers tc as an EDF, short fcr expectd difault frmuepcy). a real-wcrld default prcbability (which Creditthadesuse Mertcn's mcdel tc estimatecrdit spreads,which are clcselyltnkedto risk-neutraldefault probabilities. =
=
,
.
.x
.
500
CHAPTER 22 Derivativesin theflrstcategory haveno credit iisk to theEnancialinstitution.If the counterpartygoes bankrupt,there111be'no loss.Thederivativeis one of thecounterparty' s assets.It is likelyto be retakned closedout or sold to a third party. The result is no loss (orgain) to the Mancialinstitution. Derivativesin the second category alwayshavecredit risk to theEnancialinstitution. lf the counterpart. goes bankrupt,a lossis likelyto be exjerknced. The derivativeis one of the counterparty's liabilities,The fmancialinstitutionhas to make a claim againstthe assetsof the counterparty and may reive somepercentage of th value of the derivative.(Tpically, a claim arisingfroma derivativestfansactionis unsecured and junior.) Derivativesin the tllirdcategory mayor may not havecredit risk. If the counterparty defaultswhen the value of the derivativeis positive to thefnancialinstitution,a claim willbemadeagainst the assets of the counterparty and a lossislikelyto be experienced. lf the counterparty dtfaultswhtn tht value is negativt to the fmancialinstitution,no lossis made becausethe derivativeis retained,closed out, or sold to a third party.lz ,
,
Adjusting DerivAtives' Valeations for Counterpart DefAqlt Risk institution(orend-user of derivatives)adjust the value of a How shguld a snancial to allow for counterparty credit risk? Considera derivativethat lasts until derivative and has a value of J todayassuming no defaults.Let us suppose that defaults timeT = cantake pla at timesl1, l2, tn,where tn T, and that the valu of the derivative no defaults)at timeti k Defne the risk-n:utral to theEnancialinstitutipn(aisllming of lj time as qi and the expected recvery rateas R.13 ility defaultat probab The exposure at timelf is the Enancialinstitution'spptential loss.This is maxt, ). Asjumethat the expected recovery in the event of a defaultis R timesthe exposure. Assllmealso that the recoveryrate and the probabilityof defaultare independentof the alueof the derivatiye.The risk-neutral expected lossfromdefaultat time ti is .
.
.
,
..
tl(l -
Alkmaxt,
)!
J denotesexpectedvalue in a risk-neutral world. Tang where being the ost of zefaults ui Pj 2=1
present values leadsto
(2).5)
whereui equals 4j(1 Rj and vi is the value today of an instrumentthat pays of the exposureon the derivativeunder considefation at time lj. Consider again the three categories of derivativesmentioned earlier. The Erst institution)is easy the derivativeis always a liabilityto the snancial category(where and with. of alwaysnegative value expected deal The is the total lss from to so defaultsgivenby eqution (22.5) is always zero. The Enancialinstitutionneeds to of adjustments for the make no cost defaults.(0fcourse, the counterparty may want to take aount of the possibility of the nancial institutiondefaultingin its own pricing.) -
.
12Note that a company usuallydefaultsbecauseof a (kterioration ofthe value of any one transaction.
inits overallfmancialhealth,not because
22.4. 13The probability of defaultcould be calculatedfrombond pris in the way (kscribed in section
501
CveditRisk
the derivativeis alkays an asset to the fnancial For the second category (where institution), is aways positive. nis meansthan the expressionmaxt, ) always that the only payof from te derivave is at time T, the end of its equals suppose life.In this case, J must be the pmset value of so that vi = J for at1 i. The for the present value df the ost of defaultsbomes in equatio (22.5) expression .
.
,
'
' ., .
'
.
.
'
.
'
'
'
.
.
'
#
t(l
J '
')
-
f 1' ' Z2z
If Jl is the actul value of the derivative (afteraltowing for possible detaults),it followsthat
Jl J =
-
J
)7t(1 ') -
=
i=1
J
1
-
V
f(1
-
Rj
(22.6)
f=1
ne part icu1arinstrumentthat fallsinto the sond category we are considering is an unsecuredzero-couponbondtht jromises $1at timeT and isissuedbythecounterparty in thederivativestransaction.DefmeBbas the value of thebond assumingllo possibility of defaultand B3as the actual value of thebond.If we make the simplifyingassllmption thatthe recoveryon thebondas a pernt of its no-defaultvalue isthesnpe as that on the derivative,then = Bb Bb% 1
jt
-
1 R)
(22.7)
-
f=1
and (22.7), From equations (22.6)
:#
j#
.-t
=
(22.j)
-
B
J
(
*
If y is the yield on a risk-free zero-coupon bond maturing at time T and y is the yield bond issuedby the counterpartythat maturesat time T, then on a zero-coupon -?T -yT = B3 gives and B e e so that equation (22.8) .
=
,
Jl J =
-(y.-y)r
(22.9)
This showsthat any derivativepromisinga payof at timeT can be valued byincreasing the discountl'ate that is applied to theexpectedpayof in a risk-neutralworld fromthe risk-freerati y to the risky rate y#. fxample 22.2 over-ihe-counter option sold by company X with a value, Considera z-year that z-year assumingno possibilityof defalt, of $3.suppos zero-couponbonds issuedby the company X havea yield thatis 1.5%greaterthan a similarrisk-free zero-couponbond.Tht value of the option is
3:
-.15x2
=
) 9j .
r $2.91. For the third category of derivatives,the sip of h is unrtain. 'Fhevariable vi is a call option on with a strike price of zero. 0ne way of calculatingvi is to simulate the approximate underlyingmarket variables over the life of te derivative.sometimes clculations possible and Problems22.17 analytic 22.18). . are (see,e.2.,
!'J .1 ' .2'
.
!02
CHAPTER 22 Theanalyseswe hve presented assumethatthe probability of defaultis independent of thq value of the derivative'.This is likelyto be a reasonable approximationin when the derivativeis a smallpart of the portfolio of the counterparty cimumstances or when the counterparty is usinj the derivativefr hedgng purposes. When a transadionfor speculativepurposes wants to enter into a largt defivatives counterparty nancial Whenthe transaction institutionshouldbe ho a larje negativevalue wary. a forthe counterpafty (anda largepositivevalue forthefmancialinstitutin), thechan may be much higherthap'whentht situatio is ofcounterparty declaringhankruptcy the other way round. institutionuse the term right-wayrisk to describethe Tradersworking for a snancial where situatin a counterparty is mst likelyto defaultwhen the Enancialinstitution has zero, or very little, exposure.ney use the term wrtzpway risk to desibe the situationwhere tht counterparty is most likelyto defaultwhen theEnancialinstitution has a big exposure.
22.8 CREDITRISKMITIGATION Ip many instancesthe analysiswe justhave presented overstates the credit risk in a derivativestransaction.Thisis becausethere are a number of clausts that derivatives dealersincludein their contracts to mitigate credit risk.
Netting A clause that hasbecomestandardin over-the-counierderivativescontractsisknownas netting.This statesthat, if a company defaultson 0ne contract it has with a counterparty, it must defaulton a11outstanding contracts withthe counterpady. Netting has been suessfully tested in the courts in most jurisdictions. It can credit rk reduce institution.Consider,for example, a substantially for a snancial Enancialinstitutionthat hasthreecontracts outstanding with a particular counterparty. million to the The contracts are worth +$1 million,+$3 nllion, and snancial and into diculties institution.Supposethe counterparty runs defaults on its snancial outstanding obligations. To tlie counterparty the three contzacts have values of -$1 million, million, and 4-$25 million, resptively. Without netting, the would default counterparty on the lirst two contracts and retain the third for a loss institutionof $4 million. Withnetting, it is compelledto defaulton al1 to the snanclal three contracts for a loss to the Enancia1institutionof $15mi11ion.14 Supposea Enancialinstitutionhs portfolio of N derivativescontracts with a particularcounterparty. Supposethat the no-default value of the jth contract is %.and the amount recoveredin the eventof defaultis the recove rate timestllis no default value.Withoutnetting,the Enancialinstitutionloses -$25
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millionto thefnancialihstitutioninsteadof Note thatifthethirdcontract were worth million, institution. the counterparty would choose not to default and therewould he no lossto the snancial .
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Withoutnetting, itj lossis the payo- frop a portfolioof call ojtions on the cntmcts whereeachoption haj a strike price of aro. Withnetting, it isthe payof froma single option on the portfolioof contzactswith a strike pri of zero. Thevalue of an option ona portfoliois never gfeater than, and is often considerablylepsthan,the valueof the correspondingportfolio of options. The analysispresentedin the previoussectioncan beextendedso thatequation(22.5) givesthe presentvalue of the expectedlossfromall contracts with a counterpartywhen netng agreements are in place.Thisis acltievedby redening vi in theeqmtionas the presentvalue of a derivativethat paysofl-the exposureat timeti on the portfolioof all contractswith a counterparty. A challenging task for a Vancial institutionwhen consideringwhether it should enter into 9,new derivativescontrad with a counterparty is to calculatethein/mental in the way esect on expectedcredit losses.Tiliscan be doneby using equation(22.5) just describedto calculate expected defaultcosts with and without the contract. It is interestingto note that, becauseof netting, the incrementalefect of a new contract on exjected default losses can be ngative. This happenswhtn the value of the new contractis negativelycorrelated with the value of existingcontmcts. .
Collateralization Another clause frequentlyused to mitigate credit risks is knownas collateralizatioz. institutionhave entered into a mlmber of Supposethat a company and a snancial derivativescontracts. A typicalcollateralkation agreemvntspecises that th contracts be valued periodically.If the total value of the contracts to the nancial institutionis abovea specifkd thresholdlevel,the agreelent requiresthe cumulativecollatemlposte by the company to equal the diflkrencebetwn the value of the contracts to the snancial institutionand the thresholdlevel.If, after the collateral has beenposted, the value of the contracts movesin favorof the companyso thatthediferen between value of the contract to the fmancialinstitutionand the thresholdlevelis lessthanthe total margin already posted, the companycan reclm margin, In the event of a default institutioncan seizethe collateral.If the company'does y the compapy, the snancial not post collateral as required, the fmancialinstitutio'ncap close out the contmcts. Suppose,forexample,that thethresholdlevelforthe companyis $1 millionand the contracts are marked to market daily for the purposesof collateralization.If on a particulardaythe value of the contracts to the nancial institutionrisesfrom$9million to $10.5million, it can ask for $0.5millionof collateral.If the next daythe valueof $he contractsrises furtherto $11.4millionit can ask for a further$0.9millionof collaterl. If the valueof the contracts fallsto $10.9millionon thefollowingday,thecompanycan ask fOr $05 million of the collateral to be reblrned. Notethat thethreshold($1million in tis case) can be regarded as a lineof credit that the nancial institutionis prepartd to grantto the company. in cash The margin must be depositedby the cmpany with the nancial lnstitution nezteptable securitiessuch as bonds.The securies are subject to a or in the form of .
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CHAPTER 22
discountknownas a aircut aphliedto theirmarketvalue for the purposesof margin calculations.Interestis normally paid on cash. If the collateralization agreemeptis a two-wayagreementa tkeshoid will also be specifedfor th: fnancial institution.The cotpany can thenask the fmancialinstitution to post collattral when tht value of tht outstandinp cohtracts to the company exceedsthe threshold. Collateralkationagreementsprovidea yreatdealof prottion agginstthe possibility of default (justas the margin accounts discussedin Chapter2 provideprotectionfor peoplewho trade futures on an exchange). However,the thresholdamount is not subjectto protection.Futthermore,even when the thresholdis zzr, the protectionis nottotal. Tllisis because,when a company getsinto fnancialdiculties, it is likelyto stopresponding to requests to postcollateral. Bythe timethe cotmterparty exercisesits rightto closeout contracts, their value may havemoved furtherin its fapr. '
.
Downjrade Trijjers Another credit risk mitigation techniqueused by a fnancialinstitutionis knownas a downgradetrigger. Thisis a clause statingthat if the creditmting of the counterparty fallsbelowa certain level,sayBaa,thefnancialintitution as theoption to close out a derivativescontract at its market value. Downgradetriggersdo not provideprotectionfroma bigjumpin a ompany's credit rating (forexnmple, from A to default).Also, doFngade tlkgers work wtll only if relativelylittle use is made of them. If a company has many downgradetrkgers outstandingWith its counterparties, they are liableto ppvide littl protectionto the countemarties(seeBusinessSnapshot22.1). '
22.9 DEFAULTCORRELATIOM The term defaultcorrelation is used to describethe tendencyfor two compaies to ' defaultat about the sametime.Thereare a nllmber of reasons why defaultcorrelation exists.Companiesin the same industryor the same geographic region tend to be afected similarlyby external events and as a result may experienctfnancialdiculties at the same time. Economic condions genemlly cause average default rates to be higherin someyears than in other years. A defaultby one companymay cause a default by another-the credit contagion efkct. Default correlation means that credit risk cannot be completely diversifedaway>nd is the major xason why risk-neutl'aldefault probabilitiesare greater than real-worlddefaultprobabilities(seeSection22.5). Defaultcorrelation is importantin the determinationof probabilitydistributions?or defaultlossesfrom a portfolioof exposures to dferent counterparties. Two types of dtfault correlation modds that havt beensuggtsttdby researchers are referred to as structuralmodels. reducedformmodels Reducedform models assume that the defaultintensitiesfor diferent companies followstochasticprosses and are correlated with macroonomi variables.When the defaultintensityf0r company A is highthereis a tendencyfor the defaultintensity for company B to be high. This induces a default correlationbetweenthe two companies. Reduqedform models are mathematically attractiveand reot the tendencyfor 'and
SUMMARY The ptbability that a companywill defaultduringa partkular perbd of timein the futurecan be estimateb fromhistrical data, bnd pris, or equitypris. ne default probabilitiescalculated frombnd prices are risk-neutral prbabilities,whereas thse calculatedfrom histrical data are real-world probabilities.Real-wrld prbahilities shouldbe used for scenario analysisand the calculatin of edit #R. Rbk-neutral probabilitiesshould be used fr aluing cret-sensitive instnlments.Msk-neutral defaultprobabilities are often sipifkantly higherthan real-worlddefaultprbabilities.
'g
5t2
CHAPTER 22
The expected loss experienced from a couterparty defaultis reduced by what is knpwnas nqtng. This a clause in most contracts written by a fmancialinstitution statingthat, if a counteparty defaults on one contract it ha3 with the nancial institption,it must defaulton a1lcontract it has withthe nancial institution.Loses requires anddwngrade triggers.Collateralization are also redud by collateralizmtion the counteparty to post collateraland a downgradetriggergivesa nancial institution the option to closeout a cotmd if the creditratingof a counterparty fallsbelowa speciedlevel. CreditVaR can be dened similarlyto the way VaRis defmedfor market risk. One approachto calculatingit is the Gaussiancopulamodelof tim'e to default.Thisis used by regulators in thecalculationof capitalfor credit risk. Anotherpopularapproachfor calculatingcyeditVaR is CreditMdrks.Tls uses a Gaussian pula ndel for credit ratingchanges. .
FURTHERREADING Altman,E. 1., tMeasuring CorporateBond Mortality and Performancq'' Journal ofFinance, 44 (1989):902-22. Due, D., and K. Singleton,tlMbdelingTerm Structuresof DefaultableBondsr'' ,Rcvfcwof Financial Studies,12 (1999): 687-720. Fingr, C.;C, $$AComparisonof StocilasticDefault Rate Modelsj'' RiskMetrics Journal, 1 (Nkember 2):
49-73.
Hull,J., M. Predescu,and A. Wllite,SsRelationshipbetweenCreditDefaultSwapSpreads,Bopd #ields,and CreditRating Announcements,'' Journal of Bqnkingazd Finance,28 (Nbvember
Bond Valuation and the Term Stnlctureof Credit Littennan, R., and T. Iben, J ournal Management, Spring1991:52-64. (IJJb?SI/Nf: Spreadsj'' the Pricing of CorporateDebt: The Risk Strnctareof Interest Rates,'' Merton, R.C., tson
449-70. Journal ofFinance, 29 (1974):
and Time to Maturity,''Journal ofknancial and Rodri>ez, R. J., 'Def>ult Risk, Yieldspreads, 1-17. Analysis,23 1 Quantitative (1988): l Vasik, O., sLoanPorttblio Value,'' Risk(Dember 22),
160-62.
and Problems(nswers in te solutions Manual) Quedions 22.1.Thespreadbdweenthe yield on a 3-yearcorporatebondandthe yield on similarriskfreebond is 50 basispoints.The recovery rate is 32%. Estimate the averagedefault intensity per year over the 3-yearperiod. bondissuedbythe 22.2. Supposethat in Problem22.1thespreadbetweenthe yield on a s-year rk-free and yield bondis 60basispoints.Assumethe the on a similar slme company
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513
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recotery rate of 3%. Estimatethe averagedefaultintensityprr year ver thes-year . same period.Whatdo your results indicateabout the averagedefult intensityin years4and5?
defaultprobabilitiesf0r (a)calculating 22.3. Shouldresearchers use real-world or lk-neutral credit value at risk and (b)adjusting the price of a derivativefor defaults? 22.4. How are recovery rates usually dened? 22.5. Explain the diference betweenan unconditional default probabilitydensityand a defaultintensity. 22.6. Verify(a)that the numbers in the secondcolumnof Table aze onsistentwith the numbersin Table22.1and (b)that the numbers in the fourthcolumn of Table22.5are consistentwith the numbers in Table22.4and a recovery rate of 4%. 22.7, Dscribe h0wnetting works. A bank already has one transactionwith a counterparty on its books.Explainwhy a new transaciionby a bank with a counterpartycan havethe esect of increasingor reducing the bank's credi! exposure t the counterparty, is the same in therealworldandthe 22.8, Supposethat the measure j,,(T) in equatitm (22.9) risk-neutralworld. Is the sametrue of the Gaussiancoplla measure,p in a collateralizationagreement.A companyofers to post 22.9. Whatis meant by a its 0wn equity as collateral. How would yu respqnd? 22.10.Expln the diserencebetweenthe Gaussiancopula model for the timeto defaultand CreditMetricsas far as thefollowingare conrned: (a)thedefmitionof p creditlossand ' (b)the way in whichdefaultcorrelation is modeltd. .22k4
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period is 22.11. jupposethat the prohability of tompany A defaultihgduring a z-year lf the and the probability of company B defaultingbqringthis perio is what of correlation copula c orrelation is the binomial Gaussian is measure default measure? .2
.15.
.3,
22.12,Supposethat the LlBoR/swapcurve is;at at 6% withcontinuouscomppundingand a s-yearbond Fith a coupon of 5% (paidsenannually) sells for 9.. Howwouldan asset swap on the bond be structured?What is the asset swap spreadthat wouldbe calculatedin this situation? '
22.13.Showthat the value of a coupon-bearing corporate bondis the sum of the valuesof its constuent zero-coupon bonds when the amount claimed in the event of defaultis the no-defaultvale of the bond, but that thisis not so when the claim amount is the fa value of the bond plus accrued interest. and payable semiannually 22.14.A 4-yearcorporate bond provides a coupon of 4% per year has a yield of 5% expressedwith continuous compoundlng. The risk-freeyieldcurve is ;at at 3% with continuous compounding. ssumethat defauttscan takepla at the end beforea coupon or principal payment) and thatthe rovery of each year (immediately risk-neutral defaultprobability on the assumptionthatit the rate is 3%. Estimatethe same each year. 22.15.A company has issued3- and s-year bons with coupon of 4% per annllmpayable yields with clmtinuouscompounding)are 4.5% the bonds annually.The on (expressed and 4.:58/:,respectively.Rk-free rates are 3.5% with continuouscompoundingfor a11 maturities.Therecoveryrate is4%. Defaultscan take place halfwaytlrough each'year. The risk-neutral defaultrates per year are Q1for years 1 to 3 and :2 for years4 and 5. EstimateQ1and Q2.
514
CHAPTER 22
22.16.Supposeth>t a fnancial institutionhas entred into a swap dependenton the sterling interest rate with counterparty X and an exactly olletting swapwith countefpartf Y. Whichof the followingstatementsare true and which are false? (a) ne total piesentvlue of the cost of defaultsis the sum of the presentvalue of the cost of defaultson the contract with X plusthe presentvalue of the cost of defaults on the contract with Y. (b) The expected exposure in 1 year n contracts ij the sum of the exjected with expected and X th the coytract exposureon the contract wlth Y. exposureon (c) The 95% upper codence limit for the exposure in 1 year gn both contracts is the sllm of the 95% upper confdence limit for the exposure in 1 year on t'he limitfor the exppsure in 1 year on contractwith X and the 95% upper colden the contract with Y. abth
Explainyour answers. 22.17.A company enters into a l-year forwardcoptract to sell$1 forAUD15. The contract is initiallyat the money. In other words, the forwardexchange rate is 1.5. The l-year dollar risk-free rate of interestis 5% per annum.The l-year dollar rate of interestat whichthe counttrparty cap borrowis 6% per annum.The exchange rate volatility is 12% per annum. Estimatethe presentvalue of the cost of defaultson the contract. Assumethat defaultsare recopized only at the end of the life of the contract; 22.18.Supposethat in Problem 22.17,the 6-monthforward rate is also 1.50 and the f-month dollar risk-free interestrate is 5% per annum,Supposefurtherthat the f-monthdollar rate of inteyestat which the counterparty can borrowis 5.5% per annum.Estimate the preserftvalur of the cost of defaultsassumingthat defaultscan occur ither t the f-month point ot at the l-year point? (lfa defalt oczurs at the f-month point, the company'spotentiallossis the market value of the contract.) 22.19.'A longforwardcpntract subjectto credit lk is a copbination of a shortpositionin a no-default put and a long positionin a call subject to credit risk.'' Explain th statement. 22.20.Explai why the creditexposure on a mntched pair of forwardcontracts resembles straddle. 22,2). Explain why theimpactof credit risk ona matched pair of interestrate swapstendsto be lessthan that on a matched pair cf currency swaps. 22.22. sWhena bank is negotiating currency swaps,it shouldtry to ensure that it is receiving the lowerinterestrate currency from a company with a low credit lk.'' Explain why. put-callparityhold when thereis defaultrk? Explainyour answer. 22.24.Supposethat in an qssetswapB is the maiket pri of the bond per dollarof principal, B%is t'hedefault-freevalue of thebondper dollarof principal,and P isthe presentvalue of the asset swapspreadper dollarof principal.Showthat P = B% B. 22.25.Showthat under Merton'smodel in Section22.6 the credit spread on a T-year zero-rT coupon bond is 1n(#(i) + N-djjlLjlT where L = De Iy 22.26.Supposethat tlie spreadbetwernthe yield on a 3-yearzero-coupon riskless bondand a 3-yearzero-coupop bond issuedby a corporation is 1%. By how much does BlackScholesoverstatethe value of a 3-yearEuropeanoptionsoldby the orporation. 22.27.Give an example of (a)right-way risk and (b)wrong-wayrisk.
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AssijnmentQuestions 27.22.Supposea 3-yearcorporate bond provides a coupon of 7% per year payable semiwith semiannualcompounding).The yields annuallyand has a yield of 5% (expressed with semnnual maturities risk-free a1l bonds is 4% per annum (expressed for on compounding).Aslumethat defaultscan takepla every6 months (immedtely before a coupon paymeny) and the recovery rate is 45%. Estimatethe defaultprobabilities assuming(a)that the unconditional defaultprobabilitiesare the sameon eachpssible defaultdate and (b)that the defaultprobabilitiesconditional on no earlierdefaultare the sameon each possibk defaultdate. bonds outstanding,each providinga coupon of 8% per 22.29.A company has 1- and z-year yields yeai j yablt'lnually. The withcontinuouscompounon thebonds(ixpressed ing) are 6.0% and 6.6%, respectively.Rk-free ratts are 4.5%for al1maturities.The recvery rate is 350/:.Defaultscan take?la halfwaytkugh each year. Estimatethe risk-neutraldefaultrate each year. 22.30.Explaincarefully the distinctionbetweenreal-worldand rk-neutral defaultprobabilities.Whichis higher?A bank enters into a credit derivativewhereit apees to pay $1p at the end of 1 year if a certain company's credit rating fallsfrom to Baaor lower duringthe year. The l-year risk-free rate is 50:. UsingTable 22.6,estimatea value the derivative.What assllmptions are yoc makingzDo theytendto overstateor understate the value of the dtrivative. 22.31.The value of a company's eqljty is $4millionand the volatilityof itasequityk 6%, ne debtthat will haveto be repaid in 2 years is 15million.The risk-freeinterestrate is6% pr annum. Use Merton's model to esmate the expeded loss from default,the probabilityof default, and the recovtry rate in the event of default.Explainwhy functionin Exl can be Merton'smodel gives a high recovery rate. Hint : The solver question, 1.) usedfor th as indicatd in footnote 22.32.Supposethat a bankhas a total of $1 millionof exposuresof a rtaintype.The l-year probabilityof default averages1% and the recovery rate averages4%. ne copula Estimatethe99.5%l-year credit VaR. correlationparameter is .for
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total returp swaps.It explains creditindis, basketcreditdefaultswaps,asset-backed debtobligations.It expands on thematerialin Chapter22 securities,and collateralized to showhow the Gaussiancopulamodel of defaultcorrelationcan be used to value tranchesof collateralizeddebt oblkations.
23.1 CREDITDEFAULT5WAP5 The most popular crdit derivativeis a credit defaultswap (CDS).This is a contract that providesinsuran againstthe risk of a default by particular company.The companyis knownas the reference entity and a defaultby the compgnyis knownas a by the credit eyent. The buyerof the insumnceobtains the rkht to sellbonds kssued the insurance companyfor theirfacevaluewhen a crediteventours and the sellerof 1. valuewhen The credit bondsfor theirface buy total face event the a occurs. agreesto value of the bonds that can be sold is known as the creditdefault swap's zotiozal principal. The buyerof the CDSmakesperiodic payments to the selleruntil theend of the life of theCDS or until a creditevent occurs.These payments are typicallymade in arrears everyquarter, everyhalfyear, or everyyear. The settlementin thi eventof a default involveseither physical deliveryof the bonds or a cas payment. n examplewill help to illustratehowa typicaldealis structured.Supposethat two creditdefault swap on March 1, 2009.Assumethat the parties enter into a s-year notionalprincipal is $100million and the buyeragreesto pay 90basispoints annually for protection againstdefaultby the referenceentity. '
1 ne fa value (orpar value) of a coupon-bearingbon is the principal amount that theissuerrepays at maturityif it &es not default.
519
CyeditDE'?r//l''?x.r Figqfe 23.1
Crepitdefaultswap.
Default
'
9 basispcints ptr 7ear
.
.
.
prctectici btler
Ixfault
,
?rotecticn
,
Seller
Paymenyif dvfaultby
rfrtnce #ntity .:
*
.
TheCDSis shownin Figure23.1 lf the referenceentity doesnot default(i.e., thereis paytd credit event), reives eachof and thebuyer pays $900,000 on March1of no no the years 21, 211, 2012,2013,and2014.lf thereis a credit event, a substanal payof is likely.Supposethat the buyefnotifes the seller of a credit event on June 1, 2912 (a quarter of the way into thefourthyear). If the contmnt specisesphysicalsettlement, the buyerhas the right to sell bons issuedby the referen enty with a f'a valueof $100million for $1 million.If the conyrat requirescash settlement, an independent calculationajent will poll dealersto determinethe mid-maket value of the cheapest deliverablebond a predesignatednumber of daysafterthe crebit th suppose bondis worth $35per $100of fa value. The cash paytd woulbbe $65million. The regulpr quarterls semiannual, or almual paymnts fromthe buyerof protection to the seller of protection cease when thereis a credit event. However,becausethese paymentsare madein arrears,a snalaccrual payment bythebuyeris usuallyrequire, In our example,where thereis a defaulton June 1, 2012,thebuyerwould be require to pay to thesellerthe amountof the annual paymentaccruedbetweenMarch1, 2012,an but no furtherpayments would be required, June 1, 2012(approximately $225,000), The total amount paid per yeaf, as a percent of the notional plincipal, to buy largebanks are marketmakersin protectionis known as the CDS spread. several credit defaultswapon a the credit defaultswap market.Whenquoting n a new s-year ofer 26 basij points. This company,a market maker mkht bid 25 basispoints an thai protection paying maker market buy by is prepared 25 basispoints the to means per year (i.e.,2,5% of the principi per year) and to sellprotection for 260'basispoints per year (i.e.,2.6% of the plincipal per year). Manydiferent companies and countlies are referenceentities for the C9S contracts that trade. Undr the mst popular arrangement, payments are made quarterly in arrears.Contractswith matulitie of 5 yearsare most popular, but other matulities such as 1, 2, 3, 7, and l years are not uncommon. Usuallycntracts mature on one of te 2, and'December followingstandarddates:Mrch 2, June2, september 2. ne esect ofthisisthat the actual timeto maturityof a contract when it isinitiatedis closeto, but not necessarilythe sameas, the numberof years to maturity that is spised. suppose you call a dealeron November15,29, to buy 5-yearprotection on a company.The contfactwould probably lastuntil December2, 214. Yourfrst payment wouldbedue on December2, 29, and wuld qual a amount coveringtheNovember15,2009,to December2, 29, period.z A keyaspectof a CDScontract isthedesnitionof default. In contracts on European referenceentities restructuringistypicallyincludedas a creit event,whereas in optractson NorthAmericanreferen entities it is lmt. .
.event.
'
;
If the time to theflrststandard date is less than 1 mcnth, thenthefint payment istypicallymatk on the paynwnt date; otherwiseit is made on theflrststandard payment date. second .st-mdard
520
CMAPTER 23 ' .
,
Credit Default SWapsand BondYields that an investor A CDs can be used to hedgea position in a corporatebond. suppose yielding value face bond 7% for its and buys a s-year corporate at th same per yar CDs to byy protection againstthe issuer of the bond time enters into a s-year defaulting.suppose that th: CDs spreadis 20 basis points, or 29:, per annllm.The efect of the CDs is to convert the corporatebond to a rk-free bond (at least the investorearns 5% per year approximatly).If the bond issuerdoes not netted against spreadis whenthe CDs corporatebond yield. If the bond does defaultthe investorearns 5% up to the time of the default.Under the terms of the Ds, the investoris then able to exchangethe bond for it$facavalue. Thisfncavalue can be investedat te rk-free rate for the remainder of the j years. The n-year CDs spread shouldbe approximatelyequalto the exss of te par yield on an n-year corporate bond over the par yield on An n-year rk-free bond. If it is markedlylessthan this,an investor an earn morethan the rijk-free rate by buyingthe corpotatebppd and buyingprotection. If it is markedly greater than th, an inkestor canborrowat lessthan the risk-free fat: by shortingthe corporate bopd and jell ng CDs protection, Theseare not perfect arbitrages.But theyare closeto perfect and do anb bond yields. CDs givea good guide to the relationship betweenCDs spreads spreadscan be ujed to imply the risk-free rates used by market participants. As 124 the averageimpliedrisk-freerate appearsto be ppproximately in section discussed LlBoR/swap rate minus10bas points. equalto the 'deiult
.the
t
.
.
,
The Cheapest-to-Delivernond 7
.
As explained in section 22.3,the l'ecovery rate n a bondis dehnedas the value of the bond immedtely afterdefaultas a percent of fa value. nis means that the payof from a CDs is f41 Rj, where L is the notionalprincipal and R is the recoveryrate. Usuallya CDSspeces tat a numberof diferentbondscan bedeliveredin the event of a default.ne bondstypicallyhavethe same seniority, but theymaynot sell for the aftera defalt. 3Thisgivestheholderof a CDS sameperntagT of fa value immediately a cheapest-to-deliverbond option. Whena defaulthappensthe buyerof proteciion (or the calculation agent in the event f cash settlement)will revirwalternative deliverable bondsand choose for deliverythe one that can be purchasep lost heaply. -
'
13.2 VALUATIONOF CREDITDEFAULTSWAPS Mid-marketCDS spreadson individualreferenceentities(i.e.,the average of the bid andofer CDSspreadsquotedby brokers)can be calculated fromdefaultprobability esbmates.We will illustratehow this is done with a simple exnmple. Supposethat the probabilityof a refexn entitydefaultingduringa year conditioal on no earlier defaultis 29:. Table23.1showssurvivalprpbabilities and unconditional default probabilities (i.e.,defaultprobabih'ties as seen at time zero) foi each of the 3 There are a Anmbe: of reasons fo: th. Theclaim thatis made in theevent of a (kfault is tpically equal to thebond'sfacevalue plus accrued interest.Bondswith highaccrued interestat thetimeof defaulttherefore tend to have gher prices immeiatelyafter default. Also the market may judgethat in the event of a of the companysome bondholderswill farebette:than olhers. reorganization
521
CreditDeyivatives Tahle 23.1
Unconditionaldefault probabilisies and Survial
probabilities. Defaalt probability
Time
year
Sl//vivcf
probability
0.9800 0.9604 0.9412 0.9224 0.1039
.2
1 2 3 4 5
0.0196 0.0192 0.0188 0.0184
of a defaultduringthe srstyear is andthe.probability 5 years. The grobability ihe until of the end the srstyear is 0.98,Theppbability of a referenceentlty will survive :196 and the probabilityof defaultdurig the secondyear is survival x 0.98= The probabilityof default untilthe end of the secondyear is 0.98x 0.98 = 0.192, and is the third 9.9604 dring so on. year 0.02x We will assume that defaults alwys happen halfwaythrougha year and that gn the credit defyultswap are made onct a year, at the end of each year. payments We also assume that the risk-free (LIBOR) interestrate is 5% prr annumwith continuouscompounding and the recovery rate is 4%. There are three parts to the Theseare shown in Tables23.2,23.3, and 23.4. calculation. Table23.2 shows4hecalculationof the present value of the expected?aymentsmade on the CDs assuming that payments are madeat the rate of s per yearandthe notional principl is $1.For exmple, thereis a 0.9412jrobabilitythat te tld pament of s is and its present value is expected payment is threfore 0.9412J made. The -05X3 0.9412J: = 811&.Thetotal present value of the expted paymentsis 4.0704:. Table23.3showsthecalculationof the present value of theexpectedpayos qssuming a notional principal of $1.As mentionedearlier, we are assumingthatdefaultsalways probabity of a payos happenhalfwaythrouqha year. For exnmple,thereis a the throughthe thlrd Giventhat is4% the expectedpayos recoveryrate halfway year. 15. 'I'he present value of the expted payos is at this time is 0.0192x x 1 preselh value of the txpeded payofs is $.511. -'5x2.5 12 The total .115c .(2
.2
.?64.
=
.
.192
.6
.1
=
=
Tahle 23.2
Pament
.
.
Calculationof the present value of expecte paments. =
s per annum.
Time
Probility
Expected
Discoant
er 1 2 3 4 5
of Jqrvfvcl
payment
factor
0.9800k? 0.9604:
0.9512 0.9048 0.8607 0.8187 0.7788
Total
0.9800 0.9604 0.9412 0.9224 0.939
0.9412J 0.9224J
0.903%
'F of epecte payment 0.9322J 0.8690J .811J
0.7552: .74J
4.0704.
12
CHAPTER 23 -------
.
-
'
'
.
'
Y2il# 23..3 Calultion of the present value f expected payf. . . principal Notional $1. =
iscoast 'F of expected
Probability Recoyery Expeted Jcyp/. (3) of defaalt rate
Time
year
.2
5 1.5 2.5 3.5 4.5
0.0196 0.0192 0.0188 0.0184
factor
0.0118
.4
.4
.115
.4
.113
.4
.11l
yuytF(3)
0.9753 0.9277 0.8825 0.8395 0.7985
.12
.4
.117
.19
.12
0.0095 0.0088
Total
.511
is a snalstepTable23.4cnsiders the ccrual payment made inthe event f a default. Fdr example,there a 0.0192probability that therewill be a Enalaccrualpayment halfwaythrugh the third year. Tht aertrual payment is .5J. The expectedaccrual Its preseny value is payment at this time i? therefre 0.0192x 0.5: = ..85J. = The value expected ttal f the pmsent accrualpayments 0.0096,:-0./x2.5 0.0426:. is From Tables 23.2and 23.4,the present value of the expected payments is .9&.
4.0704:+ 0.0426:= 4.1130: FromTable23.3,the presnt value of the expectedpayf is gives 4.113(*
.51
1. Equating the two
.511
=
spread for the s-year deal we haw considered pints 124 r per year. Th example is to illustratethe calculationmethodolgy. In pmctice, we are likelyto fmdthat designed calculations aremoreextensivethanthoseinTables23.2t 23.4bause (a)paymetltsare made often more frequentlythan on a year and (b)weare likelyt want t assume that defatllts can happenmore frequentlythan oncea year.
The mid-market os 0124 shoul be 0124timesthe principal gr J
=
.
.
.bas
.
iable 23.4 Time
Calculationof the present value of accrual payment. Expected
Probqbility p/
yearsj '
default
'
.5 1.5
2.5 3.5 4.5 Total
.()2
0.0196 .191
'
0.0188 0.0184
Discoast
accraal paymest .. .
..----.
. ..
.1()J
factor 0.9753 0.9277 0.8825 0.8395 0.7985
.98J .6J .94J .92J -.
'F of expected actraal paymest
-
..
.97J ().91J
0.0085: .t7%
0.0074J '.
.
. .
0.0426J
5j3
Cedit Deyivatives
Markinj to Market CDS At the timeit is negotiated, a CDS,likemojt Other swaps,is worth close to zero. Later for examplethe credit defaultswap it may havea positive or negativevalue. suppose, in 0ur examplehad beennegotiated some timeago for a spread of 15 basispoints, the = 0.0617 andthe presentvalue of the payments bythe liuyerwould be4.113 x The of of value value payof the wouldbe present as above. swapto the seller wouldthrtfor be 0.9617 timesthe principal. Similarlythemark1, to-marketvale of the swap to the buyerof protection would be timesthe rincipal. P .15
.511
.51
,or
k16
=
-.16
Estimatinj Default Probabilities The defaultprobabilities used to value a CDSshouldbe rsk-neutral defaultprobabilz ies not real-world default probabilities Section22.5 for a discussion f the probabilities can be estimatedfrom Risk-neutral d efault diserencebetweenthe two). bond prkts or assetswaps as explainedin Chaptr 22. An alternativeis to nply them fromCDS quotes. The latterapproachis similarto the pradic in options markets of implyingvolatilities fromthe prkes of actiyelytradedoptions. Sujposewe change the examplein Tahles 23.2,23.3and 23.4 so that we do not know the defaultprobabilities. lnsteadwe knowthatthe md-market CDSCpread for a newly issueds-yerCDSis l basispoints per year. Wt can reverseen/neer ouccalculations to conclude that the implieddefaultprobabilit per year (conditional on no earlkr defau1t)is 1.61%pey year.4 l
(see
.
BinaryCredit Default Swaps A binary credit default swap is structured similarlyto a replar credit defaultswap except that the payof is a flxeddollar nmount. Sukposethat, in the example we consideredin Tables 23.1 to 23.4, the payos is $1 insteadof 1 R dollarsand the swapspreadis s. Tables 23.1, 23.2and 23.4 are the same, but Table 23.3 is replad by -
Table 23.5
Calculationof tht present value of expectedpayof from a binary credit defaultswap. Principal = $1. '
Time yearsj 0.5 1.5 2.5 3.5 4.5
Probability of defatdt .2
.196
0.0192 0.0188
9.0184
Expected
Dcpgrlf
Juyp.ff($)
factor
0.0200 9.0196 0.9192 9.0188
0.9753 0.9277 0.2825 0.8395 0.7985
.184
I'V ofexpected Juyp/./'($)
0.0195 .182
0.0170 0.0158 0.0147
Total
9.0852 . .
-
.
'
.
. .. . -
.
.
. ..
4 Idea we would like to estimate a diserentdefault probability f0r each year insteadof a singk dtfault intensity.We could do thisif we had spreads foc 1-,2-, 3-, 4-, and s-TearCDs swapsor bondprices.
i24
CHAPTER 23 Table 23.5.The CDs spreadfor a newbinaryCDs is givenby4.113J the CDs spread,s, is or 27 basispoints. .27,
=
0.0852,so that
'
.
.
'
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.
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'
How Importapt is the RecoveryRate? Whetherwe use CDs spreadsor bondpricesto estimate defaultprobabilitiesw need n estimateof the recoveryrate. However,providedthat we use the samerecoveryrate esiimating risk-neutral defaultprobabilitiesand (b)valuing a CDs, the value of for (&) the CDs (orthe estimateof the CDSspmad)is not very sensitiveto the recoveryrate. This is becausethe ipplkd probabilitiesof defaultare ap/roximatelyproportionalto ' 1/41 R4 and the payofs from a CDs are proportionalto 1 R. This afgument doesnot applyto the valuation of binaryCDs. Implkdprqbabilities of default are still approximatelyproportionalto 1/41 R4. However,for a binary CDs, the payoFsfromthe CDs am independentof R. If we have a CDs spreadfor both a plain vynilla CDs and a binaryCDs, wecan estimateboththe reovery rate and Problem23.26). the defaultprobability (kee -
-
-
The Fuiure of th*cDS Market The credit defaultswap market survivedthe credit crunch of 2007well. Creditdefault sw ps havebecomeimportanttoolsfor managingcreditrisk. A fnancialinstiiutioncan reduceits credit exposureto particular compahiesbybuyingprotection. lt can alsouse CD$sto diversifycredit risk. Forexam/le,if a fmancialinstitutionhas too much credit exposureto a particrlar businesssector, it can buy protection against defaultsby companiesin the sector and at the sam timeSll protectionagainstdefaultbycppanies in other unrelatedsectors. s omemarket participants thinkthe growthof theCDs market willcontinue andthat it will be as big as theinterestrate swapmarketby21. Othersax lessoptimistic.There is a potential aspnmtric informationproblem in theCDs marketthat is not present in other over-the-counterderivativesmarkets (seeBusinesssnapshot 23.2).
'23.3 CREDITINDICES Participants in creditmarkets havedevelopedindicesto trackcredit defaultswap sprtads. In 2004therewere agrments betweendiferent producers of indicesthat 1edto some consolidation.Twaimportant standardportfolios usedbyindexprovidersare:
1. CDX NA IG, a portfolio of 125investmentgrade companies in North America 2. i'rrau Europe,a portfolioof 125investmnt grade names in Europe Theseportfolios are updated on March20 and september 20 each year. Companiesthat are droppedfromthe portfolios and new investment are no longerinvestmentgrade gradecompanies afe added 5 CDX NA IG indexis quoted li'ya market laker as bid suppose that th s-year 65basispoints,ofer 66basispoints.(This is referredto as the index'spread) Roughly .
5 septemer 20, 2097, the Series 8i'rraxxEurope portfolio and the series 9 CDXNA IG portfolio were on efmed.Theseriesntlmhersinicate thatbytheend of September2007thei'l-rau Europepolfolio hadbeen update seven timesand theCDXNA IG portfolio hadbeenupated eight times.
525
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speaking,this means that a tradercan buyCDs protection on all 125cpmpaniqsinthe of prottion indexfor 66basispointsper company.Supposea traderwants$8, 125, each or $669,*0per year. x compqny. Tht tptal cost 0.($66x st, on of protection on each ofth 125 mpaniesfor a total tradercan similarlysell$2, of $650,0($per annum. When a company defaults,the protectionlmyerreceivesthe = usualCDSpapfl- and the annual paymentis redud by669,009/125 Thereis $5,280. an active market in buyin and sellingCDs indexprotectionf0r maturitiesof 3, 5, 7, and 1 years. The maturities for these types of contractson the indexare usually December20 and June 2. (Th means that a contractadually lastshetween 434and 514years.) Roughlyspeaking,theindex the averageof theCDsspreadson the companies in the underlying portfolio.6 The prece way in which the contract worksis a littlemore complicatedthanhasjust is specifkd. ?riceis been described.For each indexand each maturity a calculatedfromthe quoted indexspreadusing the followingprodure: rfhe
dss-year''
Stcoupon''
1. ssumea recoveryrate of 40% an four paymentsper year,madein arrears. rate) fromthe quoted spreadfortheindex.Th 2. Implya defaultintensity(hazard 23.2. n iterativesearchis usedto involvescalculationssimilar to thosein section defaultintensitythat determinethe leadsto the uoted spread. 6 More precisely, te index is slihtly lowerthan the averae of te credit efault swap preads for the companksin the portfolio. To understanl the reason for this consider a portfolio nsistin of two coppnnies,one with a spread of 1, basispoints an the oter with a spreadof l basispoints.To lmy protedionon the compnnieswoul cost slkhtly lessthan55 basispoints per company.Mlisis becausete 1, basislointsis nd expectedto be pai for as lonpas the J basispoints at slmul terefore carryless weiht. Another complication for CDX NA IG, lmt not i-l-raxxEurope, is thatthe (knition of (kfault applicableto the intkxincludesrestructurin, whereasthe (ksnition for CDs contractsoli the lmtkrlyin companies oe! not.
UHAPTER 23
26
3. Calculatea durationD for the CDs payments.Thisis 'the number that the index spreadis multijied by to jetthe presentvalue of the spreadpayments.(In the examplein stlon 23.2,it ls 4.113.) l x D x CS C),where S is theindexspread 4. Theprice P is given by P = l and C is the coupon expressedin decinl form. '
-
-
whdna trader buysindexprotection the trader pays 1 P per $100of the total ngtional and the sellerof proteciionreceivesthis amount. gf 1 P is remaining -
-
negative,the buyer of prottion recelvesmoney ahd the selkr f protection pays money.)The buyerof protectionten paysthe coupon timesthe remaininq notional t date.(Tht remainingnotional isthe number of nam'esin the lnex that on ech paymen havenot yet defulted multiplied bytheprincipalper name.)The payof when thereis a defaultis calculated in the usual Way.This arrangement facilitatestyadingbecausethe regularquarterly paymentjmade by the buyerof protectionare independentof the indexspread at the timethe buyerentersinto the contract.
fxample2.11 supposqthat the irfraxxEuropeindexquote is 34basispointsand the coupon is 40 basis points for a cbntract lastingexactly5 years, with both quotes being expressedusing a 30/360day ount. tT1s is the usual day cpunt convntion in CDS and CDs indexmarkets.) ne equivalentactual/actual quotes are 0.345% for the indexand 0.406%for the coupon. Supposethat the yield curve is ;at at continuouslycompoundd). Assupinga rovery rlte 4% per year (actual/actual, of 40% and four payments per yepr the impliedhazard rate is 0.57179/:.ne durationis 4.447years. The price is therefore 10
-
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x 4.447x
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.46) -
=
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j
Considera contract whereprotection)is $1mi ion per name. Initiall the sellerof protectionwould pay the buyer$1,, x 125x 0.27. Thereafter,the buyet of protection would make quarterly paymentsin arrears at an annual rate of $1,, x x rl, where is the number of companies that have not defaulted.When a company defaults,the payof is calculatedin the usual way and thereis an accnlal payment fromthe buyerto the sellercalctlated at the rate of 0.406%per year on $1million. .46
23.4 CDS FORWARDSAND OPTIONS Oncethe CDS market was well establhed, it was natural for derivativesdealersto 7 trade forwardsand options on credit defaultswap spreads. A forward credit default swap is the obligation to btly or sell a particular credit default swap on a particular reference entity at a particularfuture time T. lf the referenceentity defaultsbeforetime T, the forwardcontract ases to exist.Thus a protectionon a company for bank could enter into a forwardcontmct to sell s-yrar 28 basispointsstarting in 1 year. lf the companydefaultedbeforethe l-yearpoint,the forwardcontract would cease to exist. 7 The valuation of theseinstnlmentsis discussedin J.C. Hull and A. White, 23): 4$-50. Default Swap Optionsr''Jozrzal of Dcrfpc/cl', 1, 5 (spring
-f'he
Valuation of Credit
527
CreditD:rfpalfgx.
A credit efault swap option is an ojtion to buy or sell a particular cmditdefault swapon a particularreferenceentity at a particular futuretimeT. For example,a trader potection on a compny Starting in 1 year for could negotiate the right to buy s-year CDS spread for the compny in 28 batis points.This is a call option: If the s-year points, the option will lx exercised; 1 year turns out yo be more than 28 basis The cqst of the option wouldbe jd up front. othemise it will not be protection on a company similarlyan investormight negotiate theright to sell s-year CDs spread for for 28 basispoints starting in 1year. Thisis a put ojtion. If thes-year the company in l year tufns out io be lessthan 28 basispoipts, the option will be exercised;othirwise it will not be exercised.Againte costof the option would bepaid up front.LikeCDSfomards,CDs options areusuily strudured so thattheywillcease to exist if the referenceentity defaultsbeforeoptionmaturity. .exercised.
23.5 BASKETCREDITDEFAULTSW4PS In what is referred t as a basketcredit defaaltswap thqfeam a number of ieferen An add-ap basketCDS prvides a payos when any of the referen entities entities. CDs provides a payos onlywhen the tkstdefaultoup. A default.A hrst-to-defaalt second Afw/f CDs provides a payos only when tl second defaultours. More generally,a kt-to-defaalt CDs provides a payof onlywhenthe kth defaultours. Payofs calculated in the samewayas for a regular CDs. Aftetthe plevant default has occurred, thert is a settlemrnt. swap thenterminatesand thereal'eno further paymentsby eithef party. -t0-
.are
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23.6 TOTALRETURNSWAPS *
.
A tolal retarn swapis a typeof creditderivative.It is an agmtmtnt to exchangethetotal rdurn on a bond (orany portfolio of assets)forLIBORplus a spread.Thettal return includescoupons, interest,and the gain or loss on the ajset overthelifeof the swap. agreementwith a notional principalof An exampleof a tctal return swapis a s-year $1 million to exchangethe total return on a corporatebondforLIBORplus25basis points.This is illustratedin Fipre 23.2.0n couponpament datesthe payer pays the cupons earned on an investmentof $l millionin thebond.The reiver paysinterest at a rate of LIBORplus 25basispoints on a principal of $1 million,(LIBORis set on one cupon date and paid on the next as in a plain vanih interestmte swap) At the end of the iifeof the swap thereis a payment refecting thechangein valueof ihebnd. For example, if te bcnd increasesin value b 1% overthelifeof the swap, the payer Figure 23.2 Total
return Pver
Total return swap. Total return on bond LIBOR + 25 basispoints
Total return receiver
528
CHAPTER 23
is requked to pay $1 million(= 10% of $100million)ay te end of the 5 years.
Similarly,if the bond decases in value by 15%, th ceiver is required to pay $15 million at the end of the 5 years. If the is a default on the bond, the swapis usuallyterminatedand the reiver makesa nal payment equal to the exss of $100 millionover the market value f the bond. If the notional principal is added to both sidesat theend of thelifeof the swap,the total return swapcan bt chafactrizedas follqws.The payer pays the cashiows on an inestment of $1 million in the ofpomte bond.The reivef pays thecashiows on a $100million bnd paying LIBORplus 25 bajis points. If the payerowns thi corjorate bond,the total return swapallws it to pass thecreditrisk on thebond to the receiver. If it does not own te bond,the total return swapallows it to take a short position in the bnd. Total return swapsare often used as a Enancingtool.0ne scenario that couldlead to the swap in Figure23.2is as follows.ThereceiverFants fnancingto invest$100million in the reference bond. It appraches the payer (wlchis likly to be a snancial institution)and agrees to the swap.The payer thn invests$1 millionin the bond. Thisleavesthe receiverin the sameposition as it would havebeenif it had borrowed moneyat LIBORplus 25basispoints to buythebond.The payer retains ownerslp of the bond for the lifeof the swapand fas lesscredit risk than it would havedoneif it had lent moneyto the reiver to snan thepurchaseof thebond,with thebohdbeing used as collteral for the loan.If the reiver defaultsthe payer doesnot havethe legal problemof trying to realize on the cohteral. Total return swaps are similarto repos (seeSection4.1) in that theyafe structured to minimizecredit risk when securities are beingsnand. The spread over LIBORreceivedby the payer is compensation for bearingthe rk thatthe reiver will default.The payer will losemoney if the reiver defaultsat a time whenthe feferen bond's price has declined.The spread thereforedependson the creditquality of the receiver,the credit quality of the bond suer, and the correlation the two. between Thereare a number of variations on thestandarddealwehavedescribed.Sometimes, instiadof there being a cash payment for the change in value of the bond, there is settlement where the payer exchanges the underlying asset for the notional physical the change-in-valuepayments are at the end of thelifeof the swap. Sometimes principal made periodicallyrather than a11at the end.
SECURITIES 23.7 ASSET-BACKED An asset-backedsecarity (ABS)is a securitycreated from a portfolio of loans,bonds, creditcard receivables,mortgages, auto loans,aircraftleases,or other snancial assets. royalties of unusual of music future from the sale a: pie (Even assets as as are sometimesincluded.)As an exampleof thecreationof an asset-backedsecurityconsider a bank that has made a large number of auto loans.The loans would typicallybe snonprime'' and lassed, aording to thecreditquality of theborrower,as . nonprime loans.Rther than keepingthese as Supposethere are l, might decideto sell them to a spedal purpose its bank balan sheetthe assets on vehicle(SPV), also knownas a trust or a cozdait. The SPV issuessecurities that are backedbythe cash iows of theloansand proceedsto sellthesecuritiesto investors.The Ssprime''
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,
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529
CyeditDdritwitw
credit risk of the bankthat of . arrangemtnthas tht iflbct insulatinginvestopfrontthe issuedtheloans.'rhe investts' return dependssolelyon the cash :ows fromtheloans.
The bank earns a fet for originatingand servicingthe loans.Howevtr,tht crtdit risk associatedwith the loansis passed on to the investors. Manydiferent typescf ABScan be created. Oftenthe credit risk is allocattd to a numberof tranch.es. 0ne simplearmngementinvolvinga $1 lnillion portfolioof ikbt insirllmentsand threetranchesis shownFigure23.3.Supposethatthelifeof theABSis 5% of the total fiveyears. The rst tranche,knwn as the equitytranche,snances promised and of the secondtmnche, principal is kllownas the mezzanine a return 3%; tranche,Enans 20%of the total principal and is prolnised a retnrn of 1%; thethird tranche,the seniortranche,Enances75% of the principaland is promised a retul'nof 691.Tranches receivetheir return in order of seniorityusing a set of rulej knownas a waterfall.The cash iows fromthe portfolioof assetsare Erstused to pay theinvestors in Trpnche 3 their promised return of 69:. As far as is poible, theyare then usd to profidetheinvestorsin Tranche 2 with theirpromised return of 1%. Finally,residual cashflowsfromthe portfolio are used to provide the Tranche 1investorsw1t.11 a return what when experience Consider the portfolioof of up to 3%. happens asets starts to defaultlosses.The return to theTranche1investorsis afected flrst.They earn lesst.11% 30% on their original investmentand are likelyto fail to get someof theirprincipal back. When defaults get suciently high, Tranche 2 starts to be afected; and, if defaultsare really high,Tranche3 may not get its promed return. Typically the senior trancheis rated hhh. The mezzaninetranchemight be rated BBB.The equity trancheis usually not rated and is sometimesretainedhythe creatorof the Bs. An asset-backedsecurityisthereforeaway of takinga portfolioof riskyloans with a principalof $i million and creating fromit $75million of ikA-rated debt. The SPV or conduit buysinstrumentsfromissuersand createssecuritiesfromthemin
the way we havedescribed.
Sigure 23.3 Possible stxcture for an ABS. TrMche
Asset1 Asset2 Asset3 ''
'
Rehlrn 39% =
)
l : l
1
Assetn
1
(euity) Principal:$5m
Tranche2 (mz7znlne) .
sPv
Principal:$20m Rehtrn l% =
Tranche3
(senior) Totalprincipal
$1m
principal:$75m Iujurn 6% =
50
CHAPTER 23 Fijure 23.4 The creation of an ABs CD
.
ABS Portfolio
Eqnity anche(5% Notrated
ABSCDO Eqt
Mezzanine>che BBB
Sel m
(20%
anche(75%)
yanche(5%
Mezymnineache BBB
(20%)
seniortranche(75%
AA
The ABSmezzanine ancheis repackagedwithoter similar mezzanineanche,sto formthe ABS&0
Dealershavebeen very cmative-perhaps too reative=in their use of this type of structure.Mezzaninetranchesare dicult to sell.T0 overcomethis problem, dealers have put the mezzanine tranchesfrom,say, 20 diferent asset-backedsecurities into a pew asset-backed security. (Thisis known as an ABs CDO.)They then convinced ratingagenciesto assip a A rting to the most senir trancheof the new stnlcture. A possible structure is shown in Figu 23.4. TheAAArating for the senior trancheof the ABs CD0 is reasoable if the lossesexperknd by difkrent mezzanine tranches areindependentof each other.However,if a11of the mezzaninetranchesare likelyto experknce a high lossrate at tht same time,the AA-rated trancheis quite risky and experience losses.Thishappenedin mid-27, as describedin Business snapliableto shot23.3. Investorswho boughtAx-rated traches that werecreated fromBBB-rated tranchesthat werein turn created fromsubprimemortgages foundthat their mezzanine investments Fere steeply downgradedby ratinz agencies. .
DEBTOBLIGATIONS 2j.8 COLLATERALIZED A typeof asset-backedseurlty thathasbeenpartkularlypopularis a collateralizeddebt obllkation(CD0).ln tltisthe asiets lxihgsecuritizedare bondsissuedby corporations or countries.Thedesip oftheinstrumentis similar to thatin Figure23.3 (expt thatthere are usually more than threetranches).The creator of the CD0 acquiresa portfolio of bonds.Theseare passed (m to an sPVwltichpassestheincomegeneratedbythebondsto a series of tranches.The incomefromthe bondsis frst used to provide the prolised teturnto the most senior tranche,thento the nxt most senior tranche,and so on. As is usual with asset-backed securities, the Structure is dsipe.d so that the most senior trancheis sometimesretained by the trancheis rated AA. ne most jullior(equity) arrangerof the CDO.Assumingthatthe mezzaninetrancheis rated BBB, the structure shownin Figure23.3 could be used to take a $1 millionportfolio consisting of, say, A-rated bondsand convertingit to $75llillion of AkA-ratedinstruments,$2 milli of
531
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BBB-ratedinstrumets, and $5 lnillion of unrate(1instruments.This can add yalue becausemany investorswant AA-rated instnzmentsand thereis a limitedsupply of Dgular AA-ratrd bonds.The bjetive of the originatorof theCD0 is to make money paid selling forthebonds.In many by thetranchesto invesorsfor more thanthe amount instans thebondsin a cash CD0 can betradedbythe originator of the CD0 provided that certain conditions conrning the diversiscationof the portfolio re adheredto.
Synthetic CDOs The structua we havejustdesibed is knownas a cask CDO.A long position in a corporatebond has essentiallythe same credit fisk as a short position in the correspgndingciedit ddault swap(i.e.,a creditdefaultswapwhere protecon hasbeensold). This observationhas 1edto an alternativeway of creating a CDO.Insteadof forminga portfolioof corporate bonds,the originator of theCD0 formsa portfolio consistingof short positions in edit defaultswaps. The credifrists are then passed on to tranches. A CD0 createdin this way is knownas a synthetic CDO. A Synthetic CD0 is structured so that defaultlosses on the cadit default swap portfolioare allocated to tranches.Supposethat thetotal notional principal underlying 4heportfolio of credit defaultswapsis $100million and that there are threetranches. The situation nght be as follows:
1. Tranche 1is responsiblefortheflrst$5millionof losses.As compensation forthis, it earns l 5% on the remning Tranche l principal. 2. Trahche 2 is responsible for the next $20million of losses.AS mpensation for this, it arns 100 basispoints on the remainingTranche 2 pricijal, 3. Tranche 3 is responsiblefor all lossesin exss of $25million. As compensation for this, it earns l basis-pointson the remaining Tranche3 plincipal. The principals for Tranches 1, 2, >nd 3 in this example are initially $5 lnillion, $2 million,and $75million,respectively.The notional principal in a trancheis reduced by the lossesthat are paid forby thetrancheholders.InjtiallyTranche 1 earns 15%on that after six months losses of $1 million are a prindpal of $5 million. suppose of credit default swaps.Theselosses are paid for by expeljencedby the Tranche 1 and the notioal principal of Tranche 1 is redud by $1 million, so that 15%isthen earned on $4millionrather than on $5million, If lossesexceed $5lnillion, Tranche l gets wiped out and Tranche 2 bicomesresponsiblefor losses.Whenlosses reach$7million, the notional principal in Tranche 2 is $1smillion and Tranche 2 has 8 by that time paid for lossestotaling$2million. 'portfolio
Sinjle Trance Tradinj 23.3we discussedthe portfolios of 125companies that are used to generate In section market uses these portfolios to defne standard CD0 CDX and iTraxx indis. tranches.The trading of these stapdard tranchesis knownas sizgle trabke tradizg. 'l'he
8 In practice, trancheholdersare required to post th'einitialtrapcheprincipal as collateral up front.Tllis collattralearns LIBOR. 'Whtntht trancheis responsibleforth: payos on a CDs, the money is ken out of the collateral. The recovery nm' ounts when defaultsour are usually used to retire principal on tlw most Note that the srnior tranchedoesnot losetheprincipal thatis retired.It justfailsto earn the senior.tranche. points in our example)on it. basis spread(1
CyeditDdApaffw.s
533
A singletranchetradeis an agreementwhere one sideagfeesto sellprotectbn against losseson a trancheand the other sideagreesto buythe prottion; trancheis not jpthetic CD0 that someonehas cxated hut cash Qowsarecalculatedin the part f a if it were part of sucha sfntheticCDO. sameway as In the cyje of theCDXNAIG index,theequitytranchecoverslossesbesween ()%and principal. of 3% the The secondtranche,the mezzazittetrazce, coverslossesbetween 3% nd 7%. The remaining tranchescover lossesfrom7% and 1%, l% to 15%, 15% to lFe, and 3% to 1%. In the case of the i'Trau Europeindex,theequity tranchecoverslossesbetween0% and 3%.Themezzaninetranchecoverslossesbetween 3% and 6%. The remaining tmnchej cver lossesfrom6% to 9%, 9% to 12%, 12%to 22%, and 22% to le%. CDXN IGad i'lhxx tnrope Table23.6 shows themid-mnrketquotes for s-year trancheson March28, 2007.Onthatdatethe s-year CDXNAIGindexlevelwas38hasis 24 pointsand the s-year i'rraxxEuropeindexwas basispoints. tableshowsthat (ignoringbid-ofer spreads) TraderA could buy5-yearprotecion agnst lses cn the underlyingportfolio that are i the range 7%to 1% fromTraderBfo:29.3basispoinq. supposethat the amount of protection boughtis $6million.Thisistheinitialtrance principal.Payfs fromTraderB to TraderA dependon defatlltlosseson theCDXN IG portfolio. Whilethe cumulave lossislessthan7% of theportfolioprincipalthre is no payof. As soon as the cumulativelossexceeds7% of te portfolioprincipal,papfli stgrt. If, at the end of yea: 3, the cumulave losslises fl'om7% to 8%of theportfolio principal,TraderB pays Trade:A $2millionand the trancheprindpal redus to $4 million.If, at the end f year 4, the cumulave lossincmasesfrom8% to l% ofthe portfolioprincipal, TraderB pays Trade:A an additional $4million an# te traqce principalreduces to zero. Subsequentlossesthen giverise t no paments. Payxent.s fromTraderA to TraderB are madequarterlyin arrearsat a rateof 9.293%peryea?with thisheingajplied to the remaining trance principal. Initiallythe paymentsare at the $12,189 per Nar. rateof 0.00203x 6,, Note that the equity trancheis quoted diferentlyfl'omthe othe? trapches.Te marketquote of 26..85%for CDXNA IG Peansthat the plotectionsellerreives an initialpayment f 26.85% of the trancheprincipal plus 5 basispoifltaspel'year on the Temaininp tranche principal. similarly the market quoteof 11.25%fol'ipau protection seller receivesan initialpaymentof 11.25%of te Europemeans that trancheprincipal plus 500basis points per ear on the remainingtrancieprincipal. 'l'he
'
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=
,the
Table 23.6 Five-yea:CDX NA IG acd i'TraxxEuropetranceson Marc 28, 2007.Quotes are 30/360in basispoicts xcept for 0%-3%tranche,wherete quote indicatesthe percentof the trancheprincipal that musthe paid up frontin additionto 500basispoints per year. CDXNAIG Tranche
.
0-3%
3-7%
7-10%
26.85%
103.8
20.3
i'TraxxEcrope Tranche
0-3%
Quote
3-6% 57.7
6-9%
11.25%
Quote
14.4
10-15% 10.3
15-39%
39-1*%
4.3
2.
9-12% 6.4
12-22% 22-199% 2.6
1.2
534
CHAPTER 23
RELATIONIN A B/SKETcDS AND cbo
23.9ROLEoF co
'I'he cost of protection in a kth-to-efult CDs or a trancheof a CDO is critically depepdenton defaultcorrelation. suppose that a basketof 10 referen enties is used kth-to-default C DSand that each referen entity has a risk-nutral tc defme s-year probabilityof 2% of defaultingduringthe 5 years. When the default crrelation betweenthe referenceentities is zero the binomialdistributionshows that the probabilijyof one or moredefaultsduringthe5 years is 86.74%and theprobabilityof l cr A flrst-to-defaultCDSis thereforequitevalupblewhereas a more defaultsis tenth-to-defaultCDSis worth almt notMng. the probability of one or moredefalts dlines As the default orrelationincreases probability of increases.In the limit where the default and the l or more correlationbetweenthe referen entities is perfect the probability of one or more defaultsequals the probabilityof ten or more defaultsand is 2%. This is becausein tilis extremesituation the referen entities are esseatially the same.Either theya11default probabilhy 98%), (withprobability2%) oynont of themdefault(with The valuation of a tranche of a synthetic CDO is similarlydependenton default correlation.If the correlation islow,thejunicr quitytrancheis very risky andthe senir tmnchesare very safe. As the defaultcorrelation increases,the juniortranchesbecome less risky and the selor tmnchesbecomemore risky. In the limit where the default corrqlationis perfectand the recoveryrate is zero, thetmnches are equallyrisky. ''
.34%.
'defaults
23.10
VALUATION0F A jNTHETI? CD0 supposethat the payment daieson a synthetk CD0 tmncheare at timeszj zz, z. Deve h as te expectedtrancheprincipal at timezj and p(z) as the present and z valueof $1receivedat time z. Supposethat the spreadon a parsiclar tranche(i.e.,the number of basis points paid for protection) is s per year. This spreadis paid on the plincipal. The present value of the expectedpplar spread payments remainingtr<mche on the CD0 is thereforegive by J, where ,
=
,
.
.
,
.
*
= J=1 zj
zj-jjEjvzj)
-
The expectedlossbetweentimeszj-j and zj is Lj-q the midpoint of the timeinterval(i.e., at time expectedpayoss on the CDOtrancheis
-
.5rsj
C=
Ej-j
(23.1)
h. Assumethatthelossours .5r/.
+
'j)p(.5rs1
.5zy)
+
-
ne
at
present value of the
(23.2)
J=1
The arual jaymentdue on the lossesis given by sBnwhere m .5(zj
B=
-
rs1)('j-1
'j)p(.5z-1
+ 0.5zj)
-
(23.3)
j=j
The value of the trancheto the protection buyeris C
-
J
-
sB. Te breakevenspread
535
CyeditDeyivatives ours onthe tranche of the pay'(lfl's or
when the present value of the payments equals the present value C=
J
+ sB
The breakeven spreadis therefore c
s=
(23.4)
+#
.
'
.
jhow te kif foleplayed bytheexpectedtrancheprincipal in Equations(23.1) to (23.3) principal for a the breakevenspread for a tranche.If we knowthe expected calculating trancheon a11payment dates and we lso know the zero-coupon yield curve, the breakeventranchespread cn be c>lculatedfrom equations (23.1) to (23.4). ,
Modelof Time to Default Using the Gaussian copula The one-factorGaussian copula model of time to default was introducedip Section 22.9.Supposethat there are p companies in a portfolio, ti is the time to default of te th ompany,and Qiis the unconditionalcumulativeprobability distributionof ti (i.e.,Qitj is the probabilitythat ti < l). The model assumesthat
xi and x-lgpjtlf)! where From equation (22. 11), xi
=
-
1 az Zi
aiF +
=
(13.5)
-
and zj haveindependentstandardpormal distributions. X..j EU(1 ai y -
Qit l O
=
X
(23.6)
a'
1 -
l
whereQit 1Fj istheprobabilityofthe fth company defaultingbytimet conditional on of exactlyk defaultsbytimet as Pk, 1) the value of thefactorF. Denotethe probability and the conespondingprobability conditional n F as Pk, t I F). When we flx the valueof F, the defaultprobabilities are independent.It is thiskeypoint that makes it calcuiate B, and C in equations (2).1) relativelyeajily. to (23.3) Possibleto In the standardmarket model, it is assumed that the time-to-defaultdistributionQi and the parameter ai are the same for a1lcompaniesin theportfolio.Thismeans that we can write ai a, :j(l) = :(0, and Qit I F) :(l I Fj. The standard market model is therefore ,
=
=
Xi
wherexf N-1(p(j ij becomes tion(23.6) =
=
a >-+
l
J2z.
-
(23.7)
I
and F' and Zi a:e independentnormal distributions.Equa-
Qt
l F)
#( N-lgpjj)! =
j
.p
-
JF
)
(23.8)
wherep, the copula correlation, equlls a2 Thisis equivalent to equation (22.12). In the rate) for a calculationof :(1),it is usually assumedthat the defaultintensity(hazard companyis constantand consistentwith theindexspread.Thedefaultintensitythat is assumedcan be calculatedby using the CDS valuation approachin Section23.2and .
536
CHAPTER 23 searchingfor the defpultintensitjthat givestheindexspread. Supposethat th default is Then,from equatlon (22.1), intensity .
Q(!) 1 =
'
-l
e
-
(23.9)
.
From the pfopertiesof the hinomialdistrihution,the standard marketmodel ves n!
j
,-j
F) Qt I y1 @ j)! j! Qt l (1
:(1,t I >-)
=
(23.10)
-
-
'
.
.
Supposethat the trancheunder considerationcpvers losseson the portfolio betweena and au. Th parameter a is hown as the attachmentpoint and the parameter au is knownas the detachmentpoint. Dene Jsn
ns
=
.1
,
anG
R
-
nH
tgn
=
)
R
-
whereR is the recovery rate. lso, deve mxj as the smallestintegergreaterthan x. Withoqtlossof generality,we assllmethattheinitialtrancheprincipal is 1.The tranche principalstays1 w'hilethe nllmberof defaults,.l,is lessthan mnj. lt is zero when the numberof defaultsis greater than or equalto mnuj; Othemise,thetrancheprincipal is tzs l(1 -
JJJ
-
R)(n
-
Js
Dene EjFj as the expectedtrancheprincipal at timezj conditionalon the value of the factor F. lt followsthat .(ns)-1
m(,js)-l
fy(>-) =
.
V l=
81, zy IO +
.
V
L=mfns)
81, zj I F)
Dehne (F), BFj, and CFj as the valuesof equations(23.1) to (23.3),
J
nu
-
j(j H
-
yjjn
-
u
Jt
(23.11)
B, and C conditional on F. Cimilarly to
,
M
.(F)
zj
=
zj-jjEjlbvzj)
-
jzzz 1
BFj
=
J'l ./e1
(23.12)
.
zs1)(fs1(F)
.5(z) -
,..
..
..
-
.
.
-
C(O
X1(fs1(F)
(23.13)
.
m
=
fj(F))p(.5zs1 + 9.5zj)
-
f/(F)>(.5zs1
.
+9.5zy)
(23.14)
./=l
The variable F has a standard normal distribution.To calculate the unconditional valuesof , B, and C,it is necessary to integrate (F), BFj, and CFj over a standard ormaldistribution.Oncethe unconditionalvalues havebeencalculated,thebreakevtn spreadon the tranchecan be calculated as C/( + Bj.9 'I'heintegrationisbestaomplished with a produre knownas Gaussianquadrature. 9 In the case of the equity tranche,th quote is the upfront pyment that must be made in addition to 5($basispoints per year. The breakevenupfront payment is C + B). .5(
-
!37
CreditD:i'??alf'?v: It involvesthe followingapproximation: x
j
-po g y-jts e -x/S
l=V
ry
y ugfgj
(23.15)
,-1
As M increases,accuracyincreases.The valuesof wkand F'j for difefentvaluesof M bsite.'o relativelylargevalue of M is netsary to value aregiven on the author 's we seniortranches.UsuallyM 60gives sucient aumty. =
Example 23.2
Considerthe mezzanine trancheof i'lhxx Europewhen the copulacorreltion is 0.15and the recovery rate is 4%. In tll.iscase,as = au = z = 125, nt 6.25,!nd nu = 12.5.We snppose that the termstructureof interestrates is ;at at 3:5Fa,payments are made quartrly, and te CDs spreadon theindexis 50 basis points. A alculation similarto that in jection23.2showsthat the constanthazard rate correspondingto theCDs spreadis0.830:. n extractfrom the remaining calcplations is show in Table23.7.A value of M = 60is used in equation(23. 15).The factorvalues, Ft, andtheirweights,gj, are shownin flrst segmentof the table.The expected tmncheprincipalson payment datesconditional on the factr values are calculated frm equations (23.8) 1) gnd to (23.1 shownin tht secondsegmentof the table.Thevaluesof B, and C nditional on the factor values yrecqlculatedil the last threesegmentsof the tableusing The unconditionalvaluesof B, and C a?ecalcuequations(23.12) to (23.14). BFj, and Clh over the probability tributionof F. latedby ipiegrating (>), This is done by setting gFj equal in turn to (>), Bl, an CF) in The result is = 0.1496,B = 4.2846, and C equation (23.15). ne 0.0348,or 348 basis breakeventranche spread is 0.1496/44.2846 + points.Thisis much higherthan the spreadof 57.7 basispoints for tllistfanche in Table23.6.Thisislargelybecausewe assllmeda spreadfortheindexof 50 basis points and the indexspreadon March28,2007,was only 24basispoints. .3,
.d,
.
=
,
,
.187.
=
.187)
=
Valuationof kth-te-DefaultCDS 23.5)can also be valuedusing thestandaf mafket A hth-to-default CDs (seesection modelby conditioningon thefactorF. Theconditionalprobabilitythatthe kth default happensbetweentimeszj-j and zj isthe probabilitythattherea?ek or more defaultsby timezj minus the probability thatthereafe k or more defaultsbytimezj-j Thiscanbe calculatedfrom equatins (23.8) to (23.1)as .
n
,1
Pq, q=k
'ry
IF)
,
Pq, zsl
-
=i
IF)
Defaultsbetweentimezs1 and zj can beassumedto happenat time 1 This allowsthe present valuepf paymentsand of payofli,conditionalon F, to becalculated in 23.2).Byintegrang over F, the the samewayas for regular CDs payofli(seesection unconditionalpresentvaluesof payments and payofs tan be calculated. .5zs1
.5zy.
1 The parameters wkand Fk are calculatedfromtheroots of Hennitepolynomials,For moreinformaticnon Gaussianqua/ature, see TechnicalNote 21 on the acthor's website.
538
CHAPTER 23
Table 23.7 Valuationof CD0 in Exnmple23.2: principal= 1; payments are per unit of spfad.
Weightsanl values for facters 9.1579 'lz)j F'j 0.2920 .
.
9.1579
.
-.22
9.1342
9.0969
-.66
-1.14
.
.
.
Expectelpriacipal, Q(Fj) Time
j
=
'
i
1
' = 19 j = 29
.
.
.
.
.
. .
.
.
.
j j
=
19
=
20
.
.
.
.
.
0.0014 .55
.
PV expected payment, :(#'j) 0.2457 j= 1 .
.
.
0.9887
0.8636 9.8364
0.6134 0.5648
.96
.
=
19
=
29
Total
.
.
.
.
.
.
9.2107 0.2985 4.5624
PV expectel accrual'payment,C(Fj) j= 1 .
0.0230 0.1423
9.3823
0.2457
0.2457
0.2457
:'
j j
= =
.01
19 29
.
Total
.
.
0.0007
.
0.1299 9.1185 4.0361
.
0.0043
...
.
.
.
.
..
.
.
.
.
.
.
.
*
.
.8
.9
.
.
:
!
,.
:
0.1828 0.1755 4.4089
.
.2
*.041
-
0.2051 9.2015 4.5345
*
0.0412
.211
*
:
.
0.9962 0.0074 0.0346
:
:
j j
l.
.
.ll
Total
l.
.
9.9953 0.9936
PV expected payoF, (Fj) j= 1
1
0.0026 9.0929 0.0172
0.051 .51
0.0472
fxample23.3 Considera portfolio consistingof l bonds achwiththe defaultprobabilities in Table23.1and suppose weare interestedin valuing a third-to-efaultCDS where paymentsare made annuallyin arrears. Assllmethatthecopulacorrelationis the recovery rate is 4%, and a11risk-freerates are 5%. As in Table 23.7, we consider60 diflrent factorvalues. The unconditional cumulativeprobability of each bond defaultingby years 1, 2, 3, 4, 5 is 0.9396,9,0588,0.0776, shows that, conditional on #' = 0.0961 respectively.Equation(23.8) 0.1498,9.1848,respectively. thesedefgultprobabilities are 0.0365,0.9754, probabilhy conditional of tee or more From the binomialdistribution,the ;950, defaultsby times 1, 2, 3, 4, 5 years is 0.0344, 0.1794,t.2767, respectively.The conditional probability of the third defaulthappeningduring .3,
.020,
-1.14,
,
.1134,
.48,
539
CyeditD:A'twfvx,
9.296, 9.9844,9.9974,fespectively. years 1, 2, 3, 4, 5 is therefole An analysis similarto that in Section23.2showjthat the plesentvaluesof payofs, regular payments, and ncztrual paymentsconditionalon #' = are cafried .145, 3.8344J, and 171J, wheres isthespread.Similarcalculations al'e is used to integfateove! F. out forthe other 59factorvaluesand equatin (23.15) The unconditional present values of payofs, legula! payments,and arual paylents are 9.0637,4.9543:,and .531J. 'FhebreakevepCDSspfeadis therefofe .48,
.66,
-1.14
.1
9.0637/(4.0543 +
.531)
=
9.9155
or 155basispoints.
ImpliedCorrelation In the standard market model, the recoverylate R is usuallyassllmedto bed%. Th leaves only a as an unknown palameter in the model.Equivalently,the copula 2 ' cofrelationp = a is the only unknown parameter. Th makesthe modelslmllar to Black-scholeswhere there is only one unknown parameter, ihe volatility.Mafket pafticipantslike to imply a correlation from the mafket quotes fo! tranchesknthe o) sameway that theyimplya volatility fromthe market plices options. Supposethat the values of (as,esl for sucssively mole senior tfanchesare With (For exnmple,in the case of i'RaxxEurope, h, J11, lJ1' J2l, lJ2, J3l, a = Thefeal'e 14 11 12 a3 = a5 = 9.22,% = l (u two alternative impliedcorlelations measules.One is compoasdcorrelatios.F01'a tpnche (a:-1at?J,this is the value of the correlation, p, that leadsto the spfead calculatedfromthe model beingthe sameas te spleadin the market.It foundusing an iterativesearch. The othe! is basecorrelation.Fol'a palticula! value of h ( 7 1), with the thisis the value of p that leadsto the (, (N)tlanchebeingpricedconsistently market.It is obtained using the followingsteps: .
.
.
=
,
.
.
,
.
.6,
.3,
=
.9,
.12,
..)
=
=
,
1. Calculatethe compound corfelation fo! each tfanche. 2. U$ethe compound corfeltion to calculate the plesent valueof the expectedloss on each tfancheduringthe life of the CD0 as a pernt of the initialtfanche principal.Thisis the variable we ave desnedas C above.Supposethatthe value of for the (:-1 to aq tfancheis %. 3. Calculatetlie present value of the expectedlosson the (, Jt?l tfancheas a pefcent (v-1). Cp of the total principal of the undeflyingportfolio. 'Thisis EV=1 p 4. The C-value for the (, (t?1 trancheis the value calculatedin step3 ividedby qq. Thebasecorlelation is the value of thecolrelatibnparameter,p?thatis consistent withthis Gvalue. It is foundusig an iterativesealch. -
Thepresent value of thelossas a pelcent of undellyingpoftfoliothatwouldbecalculated in step3fortheirfuxx Eufope quotes inTable23.6are showninFipfe 23.5.Theimplied correlationsfor these quotes are shown in Table23.8.ne correlationpatternsin the table aze typical of those usually bserved.The compound collelationsexhibit a ;(corlelationsmile As the tlanchebecomesmore seniol, theimpliedcorrelationfjrst decreasesand thenincreases.Thebasecorrelationsexhibita corlelationskewwherethe implkd correlaon is an incleasingfunctionof thetranchedetachmentpoint. If malket prices were consistentwith the one-facto! Gaussian pulamodel,thenthe impliedcorrelations (bothcompound and base) would be the snmefo! a1ltfanches. '
,,
.
.
54
CHAPTER 23 Fijqre 23.5 Presentvalue of expectd loss on to 1% trancheas a percentof total underlying principal for iTraxxEuropeon March28,2($7. 1.2%
Fromthe pronouncedsmiles and skewsthat are observed in prcti, marketprices are not consistent with tis model.
we can ifer that
valuinjNonstandardTranehes Wedo not need a model to value thestandardtranchesof a standard portfolio such as iTraxxEpropebecausethe spreads for thesetranchescan be observed in the market. sopetimes quotes need to be producedfor nonstandard tranches of a standard portfolio.Supposethat you need a quote for the 44% i'lau Europe tranche.One approachis to interpolatebasecorrelationsso as to estimatethebasecorfelation for the 0-4% tranche and the 04% tranche.Thesetwo base correlations allow the present value of expected loss (asa percent of the underlying portfolio principal) to be estimatedfor these tranches.The presentvalue of iheexpectedlpss for the G8% Table 23.8
i'lau Europe trancheson lmpliedcorrelations for s-year March28,27. '
trapche (asa percent of the underlying principal)can be estimated as the diserence betweenthe prsent value of expectedlossesforthe0-8%and04% tmnches.Thiscan be used to implya compoundcorrelation and a breakevenspreadfor thetranche. It is now recognizedthat tls is not thebestway to proed. betterppproachis to calculate xpected lossesfor each of the standardtranchesand produ a chartsuchas Figure23.5showingthe variation of expectedlossforthe0-.1%tranchewith Values on this chart can be interpolatedto give theexpectedlossfor the0-4%andthe0-8% tranches.ThediferencebetFeen theseexpectezlossesis a betterestimateof theexpted loss on the G8% tranchethan that obtained fromthebasecorrelationapproach. It tan be shownthat for no arbitrage the expectedlossesin Figure23.5mustincrease at a decreasingrate. If base correlations are interpolatedand then used to calculate expectedlosses,tllis no-arbitrage copditionis often not satised. (Theproblemhereis that the base correlation for the 0-X0/ trancheis a nonlinear functionof theexpected loss on the 0-.X% tranche.)The directapproachof interpolatingexpecsedlossesis thereforemuch better than the indirectapproachof interpolatingbasecorrelations. What is more, it can be done so as to ensure that the no-arbitrage conditionjust mentionedis satised. .
TO THESTANDARDMARKET 23.11 ALTERNATIVES MODEL Thissectionoutlines a number of alternativesto the one-fattor Gaussiancopulamodel that has becomethe arket standard.
Heterogeneous Model The standard market model is a homogeneousmodelin the sensethat the time-todefaultprobability distributionsare asspmed to bethe snmefora11companiesand the copulacorrelations for any pair of companies are the same.Thehomogeneity assumjgeneral relaxed model equations in and (23.6) is so that the more on can be (23.5) used.However,this model is more complicated to'implementbecauseeach mpany has a diferent probability of defaultingby any giventimeand Pk, t l >-)can no longer be calculated using the binomialformulain equation (23.1).It is necessaryto use a numericalprocedure such as that describedin ndersen et al. (23) and Hull and White (24).
Other Copulas and (23.8), The one-factor Gaussiancopula model in equations (23.7) or the more ad (23.6), is a particular model of the correlation generalmodel in equations (23.5) betweentimes to default.Many other one-factorcopula models have1en proposed. Theseincludethe Student l copula, the Claytoncopula, Archimedeancopula, and Marshall-olkincopula. Wecan alsocreate new one-factorcopulas byassumingthat F with mean ()andstandard havenonnormal distributions' and the Zi in equation (23.7) deviation1 Supposethat the cumulativeprobabilitydistributionof Zj is ( and the .
11 L. Andersen, J. sidenius, and Basu, 1Al1Your Hedjes in OneBasket'' Rk, Ncvember23; and s. see White, and Hull A. without McnteCarlcsimulation,'' C. cf a CDOand nt-to-Default swap J. Journal of Dcrflle,, 12, 2 (Winter24), 8-23. tvaluation
542
CHAPTER 23
' ' .
,
'
.
.
ctmulativeprobabilitydistributionof' xz(Fiich mighthaveto be calculatednumerically and equation (23.8) from the distributionsof F and Zf) is :. Then xj = () EQ(!f)) becomes' -1(:(1)1 a y Q(tl O = S # -j
-
'
2
1-J
Fhowthat a good st to themarketis obtained when >-and the z have Hull and white withfom dtrees of freedom.lz%eyrefeyto tllisas te doablet distributions t student COJNlJ.
Multiple Factors If, insteadof a single factor F, thereare two faciors,& and F2, themodelin equation (23.7)becomes xi = and eqation
J1
>) + J2F2+
1
Jjz -.J2
-
2
Zi
becomes (23.8) -1
P(!l >) /'2) = N ,
# (p(lj 1
-
cl y I
-
1
.c
al2
-
2 yz .
Tllis model is slower to implementthan the standard market model becauseit is necessaryto integrateover two normal distributionsinsteadof one. The model can similarlybe extended to three or more factors,but $hecomputation time increases exponenally with the nvmber of factors.
Random Factor Loadings Andersenand Sideniushavesuggestedan altepative to the model in equation (23.7) 13 whefe xi a(F)F + l alh Zi '
=
-
Thisdiflkrsfromthe standard marketmodel inthatthefactorloadinga is a functionof F. In general, a increasesas /' decreases.This meansthat in states f the world where the defaultrate islgh (i.e., statesof the worldwhere/' islow)thedefaultccrrelation is also . illgh.There is empiricalevidencesuggestingthatthisisthecase.14 AhdersenandSidenius fnd that this model fhsmarketquotesmuch betterthan the standardmarket model. 12seeJ. C. Hull and A. White, 'Valuation of a CDO and nth-to-Ddault Spap without MonteCarlo Simulation,''Joarzal ofberivatiyes, 12,2 (Winter24), 8-23. ' :3seeL.
'and
ndersen and J. sidenius, tExtensionof the Gaussian CopulaModel:Random Recovery RandomFactorLcading' Joursal of CreditAik, 1, 1 (Winter24), 29-70. '
4'befauit
and 0. Rtnault, Correlation:EmpiricalEvidenj'' orking Paper, 14see,for exzmpk, A. sevigny Standprdand Poors (22); S.R. Das, L. Freed, G. Geng,and N. Kapadia,'CorrelatedDefault Risk'' Jou?nal:.JFixed Izcome, 16 (26), 2, 7-32, J.C. Hull,M, Predescv, and A, White,$$TheValuation of
Correlation-Dependent Crdit Derivatives Using a StructuralModelj'' Working Paper, Universityof Toronto, 20059and A, Ang and J. Chen,'Asylnmetric Ccrrdationof EquityPortfolios,''Jourzal ofFizazcial fctzdrac',63 (22), 443-494. '
543
CreditDdrfrtrd,s
Thq Implled copula Model
Hull and Whiteshow howa copula cgn beimpliedfrommarket quotes.15Thesimplest versionof the model assumesthat a certain averagehazardmteappliesto all pmpanks in a poftfotio over the life of a CDO. That averagehazard rate has a kfobability distributionthat can beimpliedfromte pricingof tranches.Theprocedul'eforimplying the probability distributionis to specifya number of alternativehnzardmtesandthen searchfor probabilitiesto applyto thehazardratesso tht eachtrancheof theCD0 and the indexare prid correctly.(Theprobabilitiespust sumto 1andlr nonnegative)A smoothnesscondition is used to choose amongthealternativesolutions.Thecalculation of the impliedcopulais similarin concept to the idea,discussedin Chapter18, of calculatingan impliedprobability distrilmtionfor a stockpri fromoption prices.
DynamicMedels The models discussedso far can be characterized as staticpodels. In essence they modelthe averagedefault environment ove!-the life of the CDO.The model constructedfor a s-year CD0 is diflkrentfromthat constructedfor a 7-yearCDO,which is in turn diflkrentfrom that constnlcted for a l-year CDO. Dynamicmodels are diflkrentfrom static models in that they attemptto model the evolutionof the loss on a portfolio throughtimi. Thereare threediferent typesof dynamicmodels: '
22.6 1. Stractaral Modelsk These are similarto the models describedin section exceptthat the stochastic processesfor the asset pris of many companies are modeledsimultaneously.Whenthe asset price for a company reabhesa banier, thereis a default.The processestollowed bythe assetsarecorrelated. problem withthesetypesof models is that theyhaveto beimplementewith MonteCarlo simulationand calibration is thereforedicult. 2. ReducedFormModels: In thesrmodelsthehazar rates ofcompaniesaremodeled. In ordbr to buildin a realistk amotmt of correlation,it is necessaryto assllmethat there arejllmpsin thehazardrates. 3. T0pDownModels: Theseare modelswherethetotallosson a portfolioismodeled The models do not considerwhat happensto individualcompanies. directly. 'fhe
SUMMARY Creditderivativesenable banksand other fmancialinstitutionsto activelymanage their dedit risks. Theycan be used to tmnsfercretlitrisk fromone companyto nother and to diversifycredit risk by swappingone type bf exposure for aother. The most common credit derivativeis a credit defaultswap,Thisis a contract where one companybuysinsuranceagainstanother company defaultingon its obligations. The payof is usually the diferencebetweenthe face value of a bondissuedby the aftera default.CfeditdefanltSwaps can be scond company and itj value immediately value of analyzedby calculatingthe present the expected paypents an the present value o the expectedpayof in a risk-neutralworld. CreditDerivativesUsing an ImpliedCopulaApproach,''Joarnal pf 15seeJ. c. Hull and A. White, valuing Derivates, 14 (26), 8-28. ,
CHAPTER 23 .
' 544
A forwardcredit defaultswap is an obligation to enter into a particular creditdefault swap on a particulardate. A credit defaultswap option is the right to enter into a particularcredit defaultswap n a particular date.Bothinstruments ase to exist ifthe referenceentit? defaultsbeforethe date,A lth-to-defaultCDSis defned as a CDSthat pays fl-when the lth defaultours in a portfolio of compailies. A total return swap ij an instrumentwhere the total return on a portfolio of credit. sensitiveassets is exchnged forLIBORplus a pread.Totalreturnswapsare often used as nancingvehicles.A company wanting to purchasea portfolio of assets willapproach a fmancialinstitutionto buythe assetson itsbehalf The nancial institutionthenenter into a total return swapwiththe compay wheri it paysthe return on the assetsto the companyand reives LIBORplus a spread.Theadvantage f thistypeof arrangement is that the fmancialinstution reduces its expsure to defaultsby the company. In a collateralizeddebtobligationa number 9f dferent securitiesare created from a portfolioof cqrporate bonds or commerdalloans.Thereare rles for determipinghow creditlossesare allocated. The result of the rules is that securitieswith both very high and very lw credit ratinjs are creaied frm the portfolio. A syntheticcollateralked debt obligationcreates a similar set of securitiesfrom credit default swaps. The standard markd model for pricingboth a kth-to-defaultCDS and tranches of a for time to default.Tradersuse the CD0 is the one-fator Gaussia copula mo dt1 to implycorrelations frommarket quotes. .
'model
FURTHERREADING Andersen,L., and J. Sidenius,EEExtensions to the GaussianCopula:RandomRecoveryand RandomFactor Loadings,'' Joumal of Credit Risk, 1,No. 1 (Winter24): 29-70. Your Hedgesin0ne Basket'' Risk,November23. Andersen,L., J. Sidenius,and s.Basu, $EAl1 Das,S., CreditDerivatives:TradingJ; Mazagemeztof Credit( DefaultRisk,3rd edn. NewYork: Wiley,25.
witout MonteCarlo Hull,J. C., and A. White, ''Valuation of a CD0 and :th to Defaultswap Simulation,''Jourral of Deratives, l2, No. 2 (Winter24): 8-23, CreditDerivativesUsingan Implied CopulaApproach,'' Hull, J, C., and A. White, Jourzal of Derivatives,14, 2 (Winter26), 8-28. Laurent, J.-P., and J. Gregdry,SBasket DefaultSwaps,CD0s and Factor Copulas,''Journal of Risk, 4 (205),8-23. Li, D. X., $iOnDefaultCorrelaon: A CopulaApproach,'' Jourzalof FikedItome, Marchz: Svaluing
and Problems(nswers in SolutionsMnual) Questions 23.1. Explain the dierence betweena regilar credit defaultswapand a binarycreditdefault swap. 23.2. A credit defaultswaptequiresa semiannualpayment at the rate of 60bas points per year. The principal is $300million and the credit default swap is settledin cash. A
!i
545
CyeditDdrip/livz,s
after 4 years and 2 months, and the calculation'ajnt estimat'j thatthe of cheapest pri the dliverablebondis 4% of its facevalue shortlyafter thedefault. List the cash qowsand theirtimingfor the sellerof the credit defaultswap,
defgultocturs
23.3.Explainthe two ways a credit defaultswap can be settled. 23.4.Explainhow a cash CD0 and a syntheticCD0 are created. 23.5.Explal what a Erst-to-defaultcredit defaultSwap iS. DoesitS valueincrealeor decrease as the defaultcorrelation betweenthe companies in the basketincreases?Explain. 23.6.Explainthe iferenc betweenrisk-neutral and real-worlddefaqltprobabilities. 23.7.Explainwhy a total return swapcan be useful as a fnancingtool. 23.8. suppose that the risk-free zero curve is ;at at 7% per annum with continnous compoundingand that defaultsc!n occur hawy througbeach year in a new s-year creditdefaultswap.suppose that the recoveryr'ate is 3% and te defaultprobabilities each Narconditional on no earliet defaultis 3%. Estimte the credit defaultswap spread.Assume payments are made annually. 23.9. What is the value of the swap in Problem23.8per dollarof notional principalto the protectionbuyerif the credit defaultswapspread is 15 basispoints? 23.1. What is the credit defaultswapspreadin Problem23.8if it is a binaryCDs? 23.11.Hw does a s-yernth-to-default credit defaultswap work? Considera basketof l referenc:entities where each referen entity has a probabilityof defaung in eac year of l %. As the default correlation betweenthe referenceentities increaseswhat would you expect to happen to the value of the swap when (a)n = 1 and (b n 25. Explain your answer. 23.12.What is the formula relating the payof on a CDs to the notional prindpaland the recoveryrate? 23.13. show that the spreab for a new plainvanilla CDSshould be (1 Rj timesthe spreadfor a similarnew binaryCDs, where R is the recovvryrate. 23.14.Verifythat if the CDS spread for the example in Tales 23.1to 23.4is 1 basispoints and the probability of defaultin a year (conditional ' on no earlier default)must be 1.61%. How does the probability of default change when the recovery rate is 2% instead of 4%2 Verifythat your answer is consistent with the impliedprobabilityof defaultbeing appromately proportionalto 1/(1 Rj, where R is the rovery rate, 23.15.A company enters into a total return swapwhere it reives the return on a rporate bondpayinga coupon of 5% and pays LIBOR. Expln thediferencebdweenth and a regularswap where 5% is exchanged fr LIBOR. 23.16.Eyplainhow forward ntracts and options on credit defaultswapsare structured. position of a buyerof a credit defaultswap is similar to the position of someone 23.17, whois long a risk-free bond and short a corporate bond.'' Explainthis statement. 23.18. Why'is there a potentialasymmetrk informationproblem in credit defaultswaps? 23.19.Does valuing a CDs using real-world default probabilities rather than riskrneutral defaultprobabilities verstate or understate its value?Explain your answer. 23.20.What is the diferencebetweena total return swap and an asset swap? 23.21. suppose that in a one-facto: Gaussiancopula model the s-yea: probabilityof defaultfol' of and colzelation copula is thepaimise Calculate,fol'facto: each 125nnmes 3% is =
'
-
-
rf'he
.2.
1 !j,
...
546
APTER 23
1,and 2: (a)4hedefaultprobability conditional on thefactorvalue valuesof and (b)the probability of more than l defaultsconditional on the fagot value. and compound correlation. 23.22.Expkin the dference betweenbase correlation 23.23.In the ABs CD0 structurein Figure23.4,suppose that thereis a 12%loss on each by each of the sixtranhes shown. portfolio.'What is the perntage loss exyried 23.24.In Example23.2, whatis the tranchespread for the 9% to 12%tranche? -2,
-1,
,
AssignmentQuestions that the risk-free zero curve is :at at 6% per annun with continuous 23.25. suppose compoundingand that defaultscan or at times0.25 years, 9.75 yean, 1.25years, vanilla creditdefaultsrap witlt semiannul payments. and 1.75years in a z-earplaip Supposethat the recovery rate is 1% and theunconditionalprobabilitiesof default(as seen at time zero) are 1% at times0.25 yearsand 0.75 years, and 1.5%at times1.25 years and 1.75years. What is the credit defaultswap spread? What would the credit default spread be if the instnlmentwere a binarycreditdefaultswap? 23.26.Assumethat the defaultprobability for a company in a year, conditional on no earlier defaultsis and the recovery rate is R. The risk-free interestrate is 5% per annum. plain vanill CDS Defaultalwaysours halfwaythrougha year.The spread for a s-year wherepaymentsare made annuallyis 12 basispoint,sand the spread for a s-year binary points. made and where EstimateR annuallyis 16 basis CDs payments are 23.27.Explainhow you wouldexpectthe returns ofered on the various tranchesin a synthdic CD0 to change when the correlation betweenthe bondsin the portfolioincreases. .
23.28.Supposithat: risk-free bondis 7%. (a) The yield on a s-year (b) The yild on a s-year corporatebondissuedby company X is 9.5%. credit default swap providinginsuranceagainst company X defaulting (c) A s-year basispoint,s 15 costs per year. What arbitrageopportunity istherpin tls situationzWhatarbitrageopportunity would there be if the credit defaultspread were 3 basis points insteadof 15 basis points? Give two reasons why arbitrageopportunities such as those you identifyare lessthan perfect. 23.29.In the ABs CD0 structurein Figure23.4,supposethat there is a 2% loss on each porfolio. What is the percentage lossexperiencedby each of the sixtranchesshown? 23.30.In Example23.3, whatis the spreadfor (a)a rst-to-default CDs and (b)a second-to defaultCDS? 23.31.In Example23,2, whatis the tranchespread for the 6% to 9% trnche?
.
@.
>:
'
,
A.
kl
K'
.
'k
4. :
w
'
Jk
Exotic Options
P
'
.
j.
'
.
vg' x.lp?u
is the value at time zero of an at-the-mneyoption thatlastsfor T2 Tl Usingriskneutralvalqation, the value of thefonprd Start optio at imz zzrois -
.
-m j ca k hereL'denotesthe expted valuein a rk-neutralworld. sinte c andk are knownand ce-q't Fora non-dividend1) ker-' , the value of thefomard start option is ELs the same as the payingstock, q = pnd the value Of theforwardstartOption isexactly Of Option value a regular at-the-money withthesamelifeas theforwar start option. =
'
24.4 COM?OUND OPTIONS Compoundoptions are options on options. nereare fou?mn lyes of compound a call on a call, a put on a call, a callon a put, ad a put ona put. Compound Options havetwO strikepris and twO exercise forexample, ates.Consider, a call on a call.On the srstexercisedate, T1 theholderOf the compoundoptionisentitledto pay and reive a call option. Thecalloptiongive.stheholderthe the srststrike price, right to buythe underlying asset fOr thesecondstrikepri, Kj, onthesecondexercise date, T2.The compund option will bi exercisedon theflrstexercisedateonlyif ihe valueof the option on that dat: is greaterthanthe firststrikepri. When the usual geometricBrownianmotion assumptionis made,Eufopean-style Ound ptions can be valued analgically in termsOf integnhof the lvariate comp normaldistribution. withour usual notation, thevalueat timezeroof a Europeancall optionon a call option is Options:
,
rj
,
K2e-%M(cj, Sbe-VTZV@ I b1; T1/T2) -
,
where
Lnsbjse) +
(r
Tj/T2)
);
:-TTI
-
l #(c2)
q-jvl
-
7FI
l
=
c cr2/2)r2 ln(&/r2)(r q + c/f2 +
-
2
,
=
l
-
c
2 y-.s
The functionMc, b : p) is th cumulative ivariatenormal distrintbn functionthat the first variable will be less than a and the secon will ix kss than when the coecient of correlation betweenthe twO is p.3 The variable S* is the asset price at time TI fOr which theoptionpri at timeTl equals11 If the pctualassetprke is above v* at timeTl theflrst option will be exercised; if it is nOt aboveS', theoptionexpires .
,
worthless. With similar notation, the value Of a Europeanput on a call is
#2e-Y2X(-
2, b2 ;
-
Tl/T2) ,%:-C2M(-cj, jj; -
.
-
Tj/T2)Y e-rl' k1NL-g)
p./ 2 63-81; seeR. Geske, Valuation of Compound Optionsj''Jozrsal Fisaial f'cpnpmicl',7 (1979): M. Rubinstein, sDoubleTroublq'' Risk, Decembefl99l/January 1992:53-56. 3 f0f calculatingM. ftmction for seeTechnicalNote 5 on the author's websitefor a nllmericalyrocedure calculatingM is also on the website. t'rhe
!!0
CHAPTER 24 The value of a Eurspeancall on a put is Kje-VT2X(-c2,
'
-z;
Tjhj
he-@T2M(-aj
-
-jj
;
j
y jhj
-
:-rT1
K1#(-(k)
The value of a Europeanput on a put is ke -VT2X@j ;
Tjhj
-jj
-
j
Kge-rT2M(a 2,
-
-#z; -
Tjjhj +
y-rTl
Kj #((k)
24.5 CHOOSEROPTIONS .' .
.'
referredto as an as yoa ff/cdit option) hasthefeaturethat, A chooseropon (spmetimes specifkd period of time,the holdercan choose whether the option is a call or a fter that the time when the choi is madeis Tj The value of the chooser put. suppose Option at thistimeis maxtc,gj .
wherec is the value of the call underlying tlie option and p is the value of the put underlyingthe option. If the options underlying the chooseroption are both Europeanand havethe same strikeprice, put-catl parity can be used to provide a valuation formula.Supposethat SL is the asset pri at time T1 K isthe strikeprke, T2isthe maturhy of the pptions, and r is the risk-free intertst rate. Put-callparity impliesthat ,
maxtc,p)
=
-r(R-Tl)
maxtc, c + Ke -(R-Tl)
= c+ e
maxt,
-
sl
:-(T2-TI))
#:-(r-)(T2-TI) -
Sj )
Th showsthat the chooseroptionis a packagt ccnsistingcf : 1. A call option with strike prke K and maturity T2
2.
:-/T2-TI)
#:-?-U(T2-Tl) Put options with strike price
and mturity T1
As such,it can readily be valued. More complexchooser optionscan be dened wherethecalland the put do not have the same strike price and time to maturity.They are then not jackagesand have featuresthat are somewhatsimilarto compoundoptions. '
24.6 BARRIEROPTIONS Barrieroptionsare options where the payof dependson whether the underlying asset's price reaches a certainleyelduringa certainperiod of time. A mlmberof diferent typesofbgrrieroptionsreplarly tradein the over-the-counter market.Theyare attracti'veto somemarket participants becausethey re lessexpensive than the correspondingregular optins. Thesebarrkr optionscan be classiEedas tither kzock-out(IJ/D?;J or kzock-izoptiozs. A knock-outoption ceases to exist whn the underlyingasset pri reaches a rtain barrier;a knock-inoptioncomes into existence only when the underlying assetprke reachesa barrier. Equations(15.4) and (15.5) showthatthe values at timezelo of a regular all and put
Exotic O>Al:
5!1
option are c=
he-qTNd1)
Kg-rTNd2
)
-
p = Ke-rr N-kj
where
+ (r 1n(,%/#) -
k
)
he
1
2 q + c /2)T
..
=
.xrxt-g
-
cxf
+ (r J2 lnt,g/rl c;/
q
-
=
o7/1)r
-
h
=
-
tr
zj
call is one type of knck-out option.It.is a replar calloption that A down-and-oat ceasesto exist if the asset price reaches a rtain barrkr level.S'.ne barrierlevelis belowthe initialasset prke. Thecorresjondingknock-inoption is a dowz-azd-incall. Thisis a replar call that comes into existenceonly if the asset pri reachesthebarrier level. call at me If S islessthan or equalto the strikepri, K, the valueof a down-and-in zero is
cdi ke (S / S )2#(y) -VT
=
where
-
=
r
#-rT(S/j%)2-2N(y
2
lnlIls grj
y=
gjl
!j
q+
-
-
c;T
+ yyc, '
Becausethe value of a regular callequals thevalue of a down-andncallplus thevalue of a down-and-outcall, the value of a down-and-outcall is givenby Qo
If H k K, then Cdo
==
S
#(x1k-VT
-
Ke-rTNxj
-
=
C
-
Qi
c.Uf) )2#(yI)
-VT
S S - ke ( / g
and
cdi
where xl = )'1=
=
+
Ke-rTH(h)U-jNLyL
-
gj)
c Qc -
?f) knsb! +
c:T
jnHlkj
c;r
c
./
+ cy
An uyand-oat call is a replar call option that ceasesto existifthe assetpri reachesa barrierlevel,H, that is higherthanthe current asset price. An ap-azd-izcallis a replar call option that comes into existen onlyifthebarrieris reached.WhenS islessthan or equal to K, the value of the up-and-out call,cuc,is zero >nd thevalueof theup-andin call, cui, is c. When H is yreaterthan K,
cui k Nx l =
):-fT
-
Ke-rTNxk
+
-
c./f)
-
Sbe-qTHjsbjjkLN-yj
Ke-rTH/k)1k-1LN-y + c/fl
-
-
xt-jjlj N-yj + yjj
UHAPTER24
552 and Cu0 = C
Cuj
-
Put balde! options al'e defmedsimilarlyto callbarrieroptions. An ap-azd-oat pat is a put option that ceasesto existwhen a barrier,S, that is greater than the currentasset priceis reached. An ap-azd-iz pat is a put that comesinto existenceonly if the barrkr is reached. Whenthe barrier, S, is greater than or equal to the strike price, K, their ' Prices are -qT )1kN-y) Ke-rTHjkjnns-y = -Sbe (Hjs + + cA)
rui
and
'
.
= = .Pvo p
rui
When S is lessthan or equal to K, #uo=
-
S
#(-xj#-?T
+ Ke-'TN-x3 + cu@l
+ he-@(zf/. )2#(-yj) -
Ke-rTlykjn-lN-yk
+ cyj)
-
an(j
#ui p =
#up
-
A down-and-oat pat is a put option that ceasesto existwhen a barrkr lessthan the dowz-azd-izpat is a put option that comes into current asset price is reached.A t . existenceonly when the barrkr ls reached. Whenthe barrieris greater than the strike Price, ho = 0 and pi = p. Whenthe bafrieris lessthan the strikeprke, -%
,di
=
#(-xj):-T
+
Ke-rTs-xj
cW)4 ke-qTujkllkfsy) Ke-rTHlsVk-lLNy cW) +
-
and
pdz p =
xtyjlj Ntyj cWlj -
-
-
#di
-
A11of thesevaluations makethe upal assumption that the probability distributionfor theasset pli at a futuretimeis lopormal. An importantissuefor barriefoptions is thefrequencythat the asset pri, S?is observed for purposes of determiningwhether thebarrkr has beenreached. Theanalyticformulasgiven in thissectionassllmethat S isobserved continuouslyand sometimesthisisthe case.4Often,the termsof a contract statethat S is observed periodically; for example, once a day at 12 noon. Broadie, Glasserman,and Kou provide a way of adjusting the formulaswe havejustgiven for 5Thebarrierlevel thesituationwhere the pri of the underlyingis observeddiscretely. TIm 5/.5826, H is replaced by fo: an up-and-in or up-and-out option and by TIm for down-and-in down-and-outoption,where is the number 5:-9.5826, m a or of times the asset price is observed (sothat Tjm is the time interval between observations). Barrier options often havt quhe dferent properties from replar options. For sometimes vega is negative. Consideran up-and-out call option whenthe example, assetpce is closeto the barrierlevel.As volaiility increases,the probability that the .
4 Oneway to track whether a banir has been reached frombelow (above) is to send a limitorder to the whether the sell barrier price and order is fllled. the exchangtto see (buy)tle asset at 5 M. Broadie, P.
'
Glasserman,and s. G. Kou, vxA Continuity Corrtion fcr Discrete Barrier Options, Mtematical Finazce 7, 4 (October1997):32549. ,,
553
Exotit Options
barrier will be hit increases.As a Dsult, a volatilityincreasecan cause the pri of the barrier option to deease in thesecircumstantes.
24.7 BINARYOPJIUNS Binaryoptions are options wit,hdcontinuous payofs. A simpleexample of a binary optionis a casb-ormotbizgcall. This pays of noihing if the assetprke ends up below the strike price at time T and pays a flxednmount, Q,if it ends up above the strike Price.Ili a risk-neutml world, the probabitity of the asset pri beingabove the strike price at the maturity of an option is, with our usual notation, Nd. The kalui of a cash-opnothingcall is thereforeQe-rTNd 2). A cas-ormotizg put is defmedanalogouslyto a cash-or-nothing call. It pays ofl-Qif the assetpri is belowthe stee price and nothing if it is above the strike prke. 'Ike value of a cash-or-nothingput is -rT
N-k).
Qe
Anothertype of binaryoption is an asset-ormotizg call. Tllispays of nothing i?the underlyingasset price ends up belowthe strikeprice and pays the asset pri if it ends up above the strike price. With our usual notation, the value of an asset-or-nothingcall is ke-qTNhj. An asset-or-zotiq pat pays of nothing if the underlying asset price endsup above thr strikeprice and the asset price ifit ends up belowthe strikeprice. The value of an asset-or-nothing put is ke-qTN-dj). A regular European call option is equivalent to a long position in an asset-ornothingcall and a short position in a cash-or-nothingcall where thecashpayof in the cash-or-nothinjcall equals the stiike price. Similarly,a regular European put option is equivalent to a long position in a cash-or-nothingput and a short position in an asset-or-noting put where the cash payof on the cash-or-nothing put equals the strike price.
24.8 LOOKBACKUPTIONS The payofs fromlookbackoptions dependon the maximum or ninimum asset price reachedduringthe life of the opon. The payof from a jkatizglookbackcall is the amount that the fmalasset price exceedsthe minimum asset price acllievedduringthe life of the option. The payof fromzjloatizg lookbackpat is the nmount by which the maximumasset price achieveddurinpthe life of the option exceeds the nal asset Price. Valuation formulashave been produced for qoating lookbacks.fThe value of a ioating lookbackcall at time zero is '-qTNa l
cs ke =
6
)
-
he-qT
07 N(-cl) 2(r q) -
'
-
Imine-rT
N(c2)
c2 -
2(r q)
t%N-aj)
-
. H. Sosin,and M. Set B. Goldman, A.Gatto, Path-Demndent Options:Bu7 at theLcw, Sellat theHigh, JoarnalofFinance, 34 (December1979):1111-27.; M. Garman, Recolltion in Tranquilits'' Rk, March (1989):16-19. x
,,
q 554
CHAPTER 24
where
+ (r q + c2/2)T lntk/kmin) -
J)
=
J2
=
-'
c;T
O'Xf + (-r + q + c2/2)T lntsf/5knl
Jj
(13 =
-
c;T
2(r
Jh =
t)
-
/2)lntx'!s.n4
-
-
c
2
''
ad smin is the minimumasset price achievedto date, (lfthe lookbackhasjustbeen = originated, h.j SeeProblem24.23for the r = q case. The value of a Eoatinglookbackput is ,
kmin
h
Smzse-rr #(:1)
=
c
-
2
2(r -
where
:1
=
r2#(-3) q4e
+
-
qjN- y2)
-
s n-r ty 2)
c;/
c/f
:1
:3 =
+ (r q lntkmax/k) c;/
-
-
F2 =
2(r
+ (-r + q + c 2/ 2)z lnts'max/5'l
bj
=
-qT
he
2(
-
q
-
ljjt
-
c 2/ 2)lnt, max!k) J
2
and smax is tht maximumassd prict achkvtd to date. (If the lokback has justbeen originated,then k.j A Eoatinglookbackcallis a way that the holdercan buy tlie lmderlying asset at the lowestpriceachievedduringthelifeofthe option. Similarly,a qoatinglookbackput is a waythattheholdercan sellthe underlyingasset at thehighestpri achievedduringthe lifeof the option. kmax
=
Enmple 24. 1
Considera newly issuedioating lookbackput on a non-dividend-payingstock wherethe stockpric is 5, the stockpricevolatility is 40% per annum, the riskfyeerate is l% per annum, and the time to maturityis 3 months. In this case, 50, r = 0.1, q = 0, c = 0.4, and T = 0.25, :1 = 50, ,max b? = 0.025, and F2 = bz so that the value of the lookbackput newly is 7.79.A issuedEoatinglookbackcall on the sameStock is worth 8.04. -.25,
=
.
=
-0.225,
=
,
In a,flxedlookbackoption, a strikepri isspeced. For zjxed lookbackcall optioz, the payof is the same as a regular Europeancalloptionexpt that the nal asset prke is replacedby themaximumasset pri achievedduringthe lifeof the opon. 'For zjxed lookbackputoption,the pamfis the sameas a regular Europeanput option exceptthat thethefmalasset pri is replacedby te minimum asset prke achievedduringthelifeof is the maximumasset price the option. Dene maxtkmax, #), where and of K achkvedso far duringthe life the option is the strike price. Also,dehne p) kax
=
kmax
555
Exotic O>xs
as the value of a Eoatinglookbackput whkh lastsfor the same period as the xed lookbackcall when the actual maximumassetpri s far, Smax,is replad X SL P put-call parity typeof argumentshowsthatthe value of the xed lookbackcall optlon, c:x is given by 'by
.
csx X ke ->
=
.
.:y
-,y
Ke
-
'
similarly,if Qin= mintskn, #), thenthe value of a flydlookback ut option, pflx,i8 givenby .y
ps
=
cl
'>
Ke.v
-
he
wherecl is the vlue of a ioating lookbackcall that lastsfr the sameperiod as the is replaced by faed lookbackppt when the actualminimumasset price so far, skin, equatilmsgivenabve for ioating lpokbacks the ' Ihis showsthat can be modihed Qin. to pri flxedlokbacks. Lookbacks are appealing to investofs,but very expensiveWhen compared with regula.roptions. As with barrkr options,the value of a lookbackoption is liable to be snsitive the frequencythat the asset price is observedfor the purposes of computingthe maximum or minimum.Theformulasaboveassume that the asset pri is observedcontinuously.Broadie, Glasserman,and Kouprovidea way of adjusting the formulas we have just given fof the situation where the asset pri is observed 8 discretely. 'to
24.9 SHOUYOPTIONS A shout option is a European option where the holdercan to the writer at one timeduringits life.At the end of thelifeof the option, the option holderreceiveseither the usual payof froma European option or theintrinsicvalm thetimecf theshput, wilicheveris greater. Supposethe strikeprke is $5 andtheholderof a call shouts when asset pri isk8sthan $60,theholder the price of the underlyingasset is $60.Ifthe snal receivesa payof of $1. If it is greater than$60,the holderreceivesthe excessof the asset prke over$50. A shout option has some of the same features as a lookbackoption, but is considerablyless expecsive.It can be vlued by noting that if the holdershouts at a time r whep the asset pri is Sz th payof fromthe option is Sishout''
.at
maxt, s'r -
Sz)+ Sz
-
&
where,as usual, K is the strike pri and Sz is the assetprke at time T. The valut at time r if the holdtr shouts is thereforethe presentvalue of h K (received at time T4 strike price S. 'I'helatter can be calculated plus the value of a European optionwith using Black-scholes formulas. A shout option is valued by constucting a binomialor trinomial tree fol' the underlyingasset in the usual way. Wolkingback throughthe tree, the valu of the -
7 Thearpment was proposed byH.Y. Won?and Y.K. Kwok, and ReplenishinpPremium: sub-replication Ecient Pricin? of Multi-stateLookbacks,'' ReykwOJDerate. Rezelrc, 6 (24B),83-196. 8 M. Broadie, P. Glasserman,and G. Kou, Discreteand ContinuousPath-Dependent
optionif the holdershoutsandthe value if the holder pesnot shoutcan becalculated at each nodekThe optio's prke at the node isthe greater ofthe two.The procedurefor valuing a shout option is thereforesimilarto the procdure for valing a regul.ar Americanoption.
24.10 ASIANOPTIONS Asian options are options where the payo; depends on the averageprice of the underlyingasset during at least somepart of the life of the option. The payo; from 'l and that from an average price pat is an avtrge price call is maxt, Szvz maxt, K Szvzj,where Ssvzis the avemgevalue of the underlying asset calculated ' ver a predtertined averagingperiod. Average priceoptins are less expensivethan regularoptionsand are arguablymore appropriatethan regular options for meeting some of the needs of corpofatetreasurers.Supposethat a US corpol-ate treasurer expectsto receive a cashflowof l lnillion Australiandollarsspreadevenly overthe next year from the company'sAustraliansubsidiary.The treasureris likely to be interestedin an option that guarantees that the averageexchange rate malizedduring the year is above some level.An averageprke put option can achievethis more efectivelythan regular put optipns. Anothertype of Asianoption is an averagestrikeoption.An averagestrike call pays of maxt, Sz Szvzjand an average strike pat pays oll-maxt, Ssvz &l. Average strikeoptions can guarantee that the averagepri paid for n assetin frequenttrading it can guarantee overa period of timeis not greater than tie snalpfice.Alternatively, that the averageprice receivedfor an assetin frequenttradingovera periodof timeis not lessthan the fmalprice. If the underlying assetprke, S, is assumedto belopormally distributedand Ssvzis a geometric average of the S'b analyticformulasare availablefor valuing European averageprice options.9 This is becausethe geometric averageof a set of lognormally distributedvariables is alsolopormal. Considera newlyissuedoptin that will provide a payof at time T based on the geometric averagecalculatedbetweentne zero and time T. In a risk-neutral world, it can be shownthat the probabilitydistributionof the geometricaverge of a assetprice over a certainperiod is the sme as that of the asset price at the end of the period if the asset's expected growth rate is set equal to than r qj and its volatility is set equal to c//j (rather (r q c2/6)/2 (rather than c). The geometric averageprke option can, therefore,be treatedlike a regular and the dividendyield equal to option with the volatility set equal to -
.
-
-
-
,
-
-
-
'/,/j
r-
i r
2
-
q
-
c2
-
=
l r+ + q 2
-
When, as is nearly alwaysthe case, Asian options are dehnedin terms of arithmetic averages,exact analyticpricing formulasare not available.This is bause the distribution of the arithmetk averageof a set of lognormaldistributionsdoes not have atelylognormal analyticallytractableproperties. However,the distributionis approxim' F
seeA. Kemnaand A. Vorst, <APricingMethodfor Options Basedon AverageAssetValues,'' Jourzal t).J Bazking and Finance, 14 (March199): 113-29.
!!7
Exotic O>Al:
and this leads to a good analyticapproximationfor valuing avemgepri options. Analysts alculatethe frst two momentsof the probability distributionof the arithmeticaverag in a rk-neutml world exactlyandthenfh a lopormaldistributiocto the
moments. 10
.
,
Consiber a newly issuedAsianoption that providesa payof at timeT basedon the arithmeticavemgebetweentimezero andtimeT. Theflrstmoment, M1,and thesecond moment,M,a, of the averagein a risk-nentmlworld can be shown to be ' (r-()T
Ml
=
!.
r #)T -
Sz
and V2 =
(r
-
n . 2 2(r-()+c*17'j'2 e 2 q + c )(2r jq + -
2J2
..j.
:(r-()7'
j -
(r2
qj r2 2(r
,
)r2 (r -
.:)
+
(j2
..u
v g+
g2
whenq # r (seeProblem24.23for the q r case). By assumingthat the averageasset pri is lojnormal,an analystcan use Black's model.In equatlons(16.9) and (16.1), =
h
=
(N.1)
Ml
and J
2
=
-
(11. ,,,,,,,,,,,,,jr;...!'I;t:;!!'' l!1 . . 221
T
(24.2)
fxample24.2 Consibera newly issuedavetagepricecall option on a non-dividend-payingstock wherethe stockprice is 5, the strikeprke 5, thestockpri volatilityis4% perannum,the risk-freerate is 1% per annum, and thetimeto maturityis 1 year. = and T = 1. Iftheavemgeis a In thiscase, S 5, K %, r , o' q geometrkaverage,the option can be valued as a replar Option with thevolatility equa 1 to 4/,Z or 23.09%, and dividendyield equal to (.1 + 0.42/6)/2 or 6.339/:.The value of the option is 5.13.If the averageis an arithmeticaverage, and we frst calculate M1 = 52,59ad M1 2,922.76.From equations (24.1) with K = %, T = 1, and (24.2),h 52.59and c = 23.54%.Equation(16.9), givesthe value of the option as 5.62. r= .1,
=
.
=
=
.4,
=
,
,
=
=
.1,
The fprmulasjust given for 21 and Mj assume that the average is calculatedfrom continuousobservations on the asset pri. Theappendix to tis chapter shos lmwM1 and M2 can be obtained when the averageis calculatedfromobservationson the asset price at discretepoints in time. We can modifythe analysisto aommodate the situationwhere the option is not newlyissuedand some pricesused to determinethe avemgehaveala beecobserved. supposetht the averajing period is composed of a period of lengt !1 over which priceshavealready beenobservedand a futureperiod of lengt,h!2 (theremaininglifeof the option).Supposethat the averageasset pri duringtheflrsttimeperiod is J. ne 10seeS. M. Turnbuli and L. M. Wakeman,A Quick Algorhhmfcr PricingEurcpean AverageOpticns,'' Asalysis,26 (September1991):377-89. Journal pf Fizancialasd Qaastitatiye
558
CHAPTER 24 payof from an avemgeprke call is mak
j !1+
saw t2
!1+ !2
-
#, 0
hereSave is the averageassetpri duringthe remaining part of the averagingperiod.
W
Thisis the same as
!2
maxtlaw K
.
)
-
,
lj -F!1
where
!1 +
....$
'
t2 = j K*.= K !2 !2 K% > valued optiop in the same ay as a newly issuedAsian When can be ? the option prvided that we change the strike price from K to K* and multiplythe result by !2/4!1 + !2). When K* < the option is rtain to be exercisedand can be valued as a forward contract. The value is tl lj + tj
(21:
-rJ2
-wzj
.
K e
-
24.11 OPTIONS TO EXCHANCEONE ASSETFORANOTHER referred to as exchangeoptionsj Optionsto exchange one assetforanother(sometimes arisein various contexts. n option to buy yen with Australiandollarsis, from the point of view of a US investor,an option to exchange one foreip currency asset for anotherforeip currency asset.A stock tenderofer is an option to exchangeshares in one stock for shares in aother stock. Considera European option to give up an asset worth Uz at time T and receivein returnan asset worth Pr. The payof fromthe option is
maxtpr -
Ur, 0)
A formulafor valuing this option wasfirstproduced by Margrabe.llSupposethat the assetprices, U and P, both followgeometricBrownian motionwith volatilitiescrt and cy. Supposefurtherthat the instantaneouscorrelation betweepU and P is p, and the yieldsprovided by U and P are qv and qy, respectively.Thevalue of the option at time zero is voe-qvTsvjj -qvTsv P(): (N.3) l) whre -
h
=
1n(P0/U0)+ qv
-
t?y +
J2/2)T
,
&41
and
d
=
c2 + c2 P U
-
4
=
dj J./-z -
zpggcy
and Uc and Ptl are the values of U and P at timeszero. 11seeW. Margrabe,
lT'f'he
(March 1978):177-86.
Valueof an Option to ExchangeOne Assetfor Anotherj'' Joarsal ofFisasce, 33
559
Exot O>f0x:
This yeplt will be proved in Chapter27. It is interestin to note that eqnation (24.3) is independentof the risk-free rate r. Tls is because,as r increases,the growthrate of both asset pricesin a risk-neutral world increases,but this is exactly oflet by an incyeasein the discountrate. The variable J i! the volatllityof Vjl. Comparkonswhh equation(16.4) showthat the option pri isthe sameas the pri of I-/aEuropeancall optionson an asset worth Vjl wlien the strikepri is 1., the rkk-free interestrate is qv, and thedividendyield on the asset is t?y. MarkRpbinsteinshowsthattheAmerican 12 versionof this option can be characterizedsimilarlyfor valuationpurposes. It can be regardedas WAmericanoptions to buyan asset worth P/I7 for l when the risk-free rate is qv and thedividendyield on the asset is y. Theoption can thereforebe interest valuedas deqcribedin Chapter19usin a binomialtree. An option to obtain tbebetteror worse of twoassts can be rearded as a position in oneof the assets combined wit ap optiop to exchangeit for the other asset: .
mintrr, Pr) = Pr maxtpr -
martr, Pr) =
WY mxtpr
W,0)
-
-
W,0)
ASSETS 24.12 OPTIONS INVOLVINGSEVERAL Optionsinvolvingtwo or more risky assetsare sometimesreferredto as rainbowoptiozs. Oni example iqthe bondfuturescontract tradedon the CBOT describedin Chapter6. The party with the shrt positionis allowed to choosr betweena large mlmber of diflkrentbonds when making delivery. Probably the most popular option involvingseveralassets is a basketoptioz.This is an option wherethe payof is dependenton the value of a portfolio (orbasket)of assets.The assetsa:e usually eithef individualstocks or stock indis or currencies.A Europeanbasketoption can be valued with MonteCarlosimulation,by assumingthat the assets follow correlated geometric Brownianmotion prosses, A much faster approachis to calculate the rst two moments of the basketat the maturity of the optionin a risk-neutralworld, and then assumethat value of thebasketislopormally distributedat that time. The option can thenbe valuedusing Black'smodel with the and (24.2). The appendix to this chapter shows parametersShown in equations (24.1) howthe moments of the value of thebasketat a futuretimecan be calculatedfromthe volatilitiesof, and correlations between,the assets. Correlationsare typicallyestimated fromhistoricaldata'. '
24.13 VOLATILIiYAND VARIANCESWAPS A volatilitySwap is an agreementto exchangethe realizedvolatilityof an asset between time0 and time T for a prespecifed xed volatility.The realizedvolatilityis calculate as describedin Section13.4but with the assumptionthatthe mean dailyreturn is zero. Supposethat thereare n dailyobservationson the asset price durin the period between 12SeeM. Rubinstein,
Sone
for Anothe'
Rk, July/Aupst 1991:30-32
60
CHAPTER 24
time and time T. The realizedvolatility is
252 ln = !t)Al:
ThefunctionQKi) isthe pri of a Eurgpeanput option with strikeprice Ki if Ki < S* andthe price of a Eropean calloptionWithjtrike pri Ki if Ki > $*.WhenKi = S%, the fuctin Q(&)ik equal to the averageof the piices of a Europeancall and a Europeanput with strikeprice Ki. '
,
,
.
fxample24.3 Considera 3-monthcontract to pay therealizedvariancerate of an indexoverthe 3 months and reive a varian rate of 0.045on a principal of $l million.The rate is 4% andthedividendyielb on theindexis 1%.Thecurrentlevelpf risk-free theindix ij l2. Supposethat, fol' strike prices of 8, 850,9, 950,l,, 1,5, 1,1, 1,15, 1,2, the 3-monthlmpliedvolatities of theindexare 2904, 279/:,269/0,259:, 24%,23%, 22b:, 21%, respectively.In th case, t = 9, 289/0, l,2/.4-'1)X0.25 = K1 = 8 #2 = 850, K? 1,2, F = 1027,68and = 11.5, S* = 1,. DerivaGemshowsthat P(#1)= 2.22,QKjj = 5.22,Q(#3) = = = QK6) QK'l) :(#4) 21.27, Q(#5) 51.21, 38.94, 20.69, Q(#8)= 9.44, for all i. Hence, :(#9) = 3.57.Als0, A#f = .
the vaiue of the variance swap (inmiilionsof dollarslis From equation (24.5), = j 9 1 x (.621 0.045):-0.94x0.25 -
.
.
Valuation of a Volatility Swap To value a volatility swap,we require L'), where t' is the averapevalue of volatility betweentime() and time T. We can write
t-f
# :(- k) -
E)
=
l+
-
E).
Expandinpthe second term on the rkht-hand sidein a seriesgives
t-f =
E) -
#
1+
E
#
:)
j
j
s
2/(: -
=
-
Takingexpectations, E-
'(#) -
-
a '(#) -
-
E
vartp s '(%z
j
-
g
k) (NJ)
wherevartp) is the varian of P.The valuation of a volatilityswap thereforerequires an estimate of the variance of the averagevariance rate duringthe lifeof the contract. The value of an apreementto receivethe Tealized voiatilitybetweentime0 an timeT and pay a volatilitybf cr, with both beinpapplkd to a principal of Lvo1, is Lvol(()
-
cr1'-RT
)' 562
CHAPTER 24 Example24.4 For the situationin Example24.3,considera volatility swapwhere the realized is regeivedand a volatilityof 23%is paid on a principal pf $1 million. volatility In tilis case E% = 4.621.Supposethat the standard deviationof the averape This means that varianceover 3 months has bee estimated as = gives Equation(24.7) vartQ) .l.
.l.
ij
gogggj 1 1sx 2
.621
=
=
-
.621
0.2484
of dollars)is The valu: f the swapin (millions -9..94x9.25
1 x (0.24840.23): -
'j
=
.
g)
The VlX lndex theln fupctioncan beapproximatedbytheflrsttwottrms in a serks In equation (24.4), '
expans
j
'
0n:
2
-!
ln v*
=
-
k#
1
-
-
2
v
.
1
This meansthat the risk-neutral expectedcumulativevariance is calculated as n 2 Ar i , 1 + 21 .2 e z QKi)
..g.>-
a
'
-
ztpz
=
-
-
c.
u)
f=l
fk
(24.8)
j
Since2004the VIX volatility index (seeSection13.11) has been based on eqvaThe procedureused on any gvendayis to calculate J'(% T for options tt tion (24.8). maturities and and market immediatelyabove have below30 days. The trad.e in the 3-day risk-neutral expectedcumulqtivevarian is calculated fromthesetwo numbers usinginterpolation.Tltis is ten multiplied by 365/30and the indexis set equalto the squareroot of the result. Moredetailson the calculation can be found n: ww.cboe.com/Mcro/v/ewMte.pd
24.14 STATICOPTIONS REPLICATION If the proceduresdescribedin Chapter17 are used for hedgingexoticoptions, someare easyto handle, hut others are very dlcult becase of discontinuities(seeBusiness Snapshot24.1).For thedicult caes, a techniqueknownas statk options replicationis 14 sotetimes useful. Tltis involvessearchingfor a portfolio of activelytradedoktions that approximatelyreplkatestheexoticoption. Shortingtis positionprovidesthe hedge.ls The basic principle underlying statk options reph'catin is as fllows. If two portfoliosare worththe same on a rtain boundary,they are also worth the sameat '
.
,
14See E. Derman, D. Ergener,.and 1. Kani, 'Static Options Replicationy''Journal of Derivatives2, 4 (summer1995):78-95. 15Technical Note 22 on the author s website provides an exampleof static replkation. It' shows that the variancerate of an asset can be replicatedby a position in the asset and oubof-the-moneyoptions on the result, which kads to equation (24.4), asset. can be used to hedge Yariance swaps, '
'rhis
kxoticOptions
563
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'
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.
.
$II
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(g(yjt jj: gjjj
B,jipj! Sipdhot 1t.1) tj bti j4.Ejys, jyj yjy yyju: y )g( l '':' ':; : .' y #. 1i:t5IL'4ii::. . j',. .() :( 1.-:;ii:.' 'i(::);..p!i''I?' . q'. . '. f' .:.. . ':IE''di:!i' ' E. '. . ' ..., '',y:,.., j) y y..,'.. g '' ' !.. . (' ) . . . . .;; r. ., .j...;..j,.j.,., .(t.,..k ..,.r. . . '
a11interior'pointsof the boundaly Consideras an examplea g-monthup-and-outcall optio on a non-dividend-payingstockwherethestockprke is50,the strikepri is50, tlkebarrieris 60,the risk-free interestrate is 1% per annum,andthe volatility 30%
perannum.Supposethat fs, tj is $hevalue of theoptionat timt ! for a stockpliceof S. Anyboundaryin (S, 1) spacecan be used forthe purposes cf producingtherelkatillz portfolio.A convenient one to choose is shown in Fipre 24.1.It is defnedby S = 60 Figure j4.1
Boundarypoints used for static optins replication example. ' g
60 50
l
0.25
0.50
0.75
>
364
CHAPTER 24
and t = 0.75.The values of the up-and-out option on the bopndarya;e ven by fs, 0.75)= maxt,
J(6, 1)
-
)
5,
when S < 60
when K t K 0.75
=
Ther are many ways that these boundaryvalues can be approximatelymatched using rejular options. The natural option to match the flrst boundqryis a g-tonth European call with a strike prie of 50.ne frst component of the replicatingportfolio is thereforeone unit of this option. (Werefer to this option as option A.) One way of matching the A6, 1) boundaryis to proceed as follows: 1. bividethe life of the option int N steps of length l 2. Choosea European call option witha strike pri of 60ad matrity at time Nht (= 9 months) to mgtchthe boundaryat the (6, CN 1)Al) point 3. Choose a European call option with a strike pri of 60 and maturity at me CN 1)l to matchthe boundaryat the (6, CN 2)Al) point -
-
-
and so on. Notethat the options are chosen in sequnce so that theyhavezerovalue on te parts of the bonndarymatched by e>rlieropons.lf ne option with a strike pri bf 60that matups in 9 months has zero value on the vertical boundarythat is patcheb by opon A. The option maturing at timei l has zero value at the point (6, i Al) thpt is matched by the option maturing at time i + 1)Al for 1 K i K N 1. Supposethat A 0.25.In additionto option A, the rkplicatingportfolioconsistsof positionsin European Qptions with strikeprke 60that matur in 9, 6, and 3 months. We will refer to these as options B, C, and D, respecyively.Givenour assumptions point.OptionA about volatilityanditerest ratis, option B is worth 4.33at the (6, is worth 11.54at tllispoint.The position in option B nessary to match theboundary point is therefore OptionC is wort,h4.33at the at the (6, and point. position takenin A is worth 'I'he B (6, 0.25) at this point. opons The position in option C necessary to match the boundaryat the f6, 0.25)point is therefore4.21/4.33= 0.97. Similarclculations showthat the position in option D necessaryto match the boudary at the (6, ) point is 0.28. The portfolio chosen is summarizedin Table24.1.(SeealsoSampleApplicationF of the DerivaGemApjlicationBuilder.) lt is worth 0.73initially(i.e.,at time zero when -
=
.5)
-2.66.
-11.54/4.33
.5)
=
-4.21
The portfolio of European call options used to replicate an up-and-out option.
Table 24.1
Optiotl A B C D
Strike price
Matarity
50 60 60 60
0.75 0.75
.Pol/io?l
lu/xe
year
.5
0.25
Itlitial
1. -2.66
+6.99 -8.21
0.97 0.28
+1.78 +.17
16.This is not a requirement.If K points on theboundaryare to be matched, we can choose K options and. solvea set of K lineazequations to determinerequiredpositions in the options.
)
565
Exotic Oyfpxs '
'
''
. givenbythe analytic foimula th.eStock pri is 5). Thiscompares.Wlth forthe up. and-outcall earlierin this chapter, The replicatingportfolio is n0t exac the snmeas the up-and-outoptionbecaustit matchesthe latterat onlythreepoints ol theSecond boundary. If we use the same produre, but match at 18 ppints on the Second optionsthat mtuie evtry half month), the value of the mplicating boundary (using portfolio reduces ' to 0.38.If 1 points are matched, the value redus furtherto 0.32. T0 hedgea derivative,the portfolio that replkates its bounda conditions must be shorted.The portfolio pust be unwound when any part of the boundafyis reached. Static options replication has the advantageover delta hedgingthat it does not requirefrequtnt rebalancing. It can be used for a wide range of derivatives,The user has a great deal of fkxibilityin choosing tlie boundarythat is to be matched and the optionsthat are to be used. .31
'
SUMMARY Exoticoptions are optionswith rules governingthe payof that are more mplicated than standard options. Wehavedcussed 12difret typesof exotic options:packages, nonstandardAmericanoptions, forwardstart options, mpound options,chooser options, barrier options, binary options, lookbackoptions, shout ptions, Vian options, options to exchange one asst foi another, and options involvingSeveral aslets. We have discussedhow these can be valued using the same assumptitmsas those used to derivethe Black-scholesmodelin Chapter 13. soine can be valued analytically,but using much more complkated frmulas than thosefor fegularEuropean calls and puts, som can be handledusing analyticapproximations,an some can be valued using extensions of the nllmericalprocedures in Chapter19.We will presentmore numerical procedures for valuing exotic optionsin Chapter26. Some exotic options are easier to hedgethan the corresponding regular options; others are more decult. In general, Asian options are easiertc hedgebecausethe payof becomesprogressivelympre certain as we approachmaturity. Barrieroptios can be more dicult to hedgebecausedeltais discontinuousat the barrier.0ne approach to hedgingan exoticoption, knownas static optionsreplkation, is to :nd a portfolio of regularoptions whose value matches thevaluepf theexoticoption on some boundary. The exotic optionis hedgedby shorting this portfolio.
FURTHERREADING Volatilityand Varian: Optionsvia swaps,'' Risk,May 27, Carr, P., and R. Lee, SsRealized 7643.
Clewlow,L., and C. Strkkland,Exotic ty/ft?r..Te Stateofte Xr/. London:Thomson Busines Press, 1997. Demeter, K., E. Derman,M. Kmal, and J. Zou, dsMorethan You EverWantedto Know Joarzal ofDerivatives,6, 4 (summer, 1999),9-32. aboutVolatility swaps,'' OptionsReplkation,''Joarzal ofDerivates, 2, Derman,E., D. Ergener,and 1. Kani, 4 (Summer1995):78-95. Valuation of CompoundOptions,''Joarnalof Fizazcial fctwdpc', 7 (1979): Geske,R., Sstatic
4Gne
63-81.
!66
CHAPTER 24 Dependent Options:Buyat the Low, and M.A. Gatto,$$Pat,h Goldman,B.,H. sosin, sellat the Highj'' Journal :./' Finance,34 (December1979);1111-27.
Value of an Optionto Exchange0ne Asset for Another,'' Journal p.J Margrabe,W., Finance, 33(March1973):177-86. Milevsky,M.A.? and S.E. Posner,StAsianOptions:Tie Sumof Lognormals,and te Riciprocal Analys, 33,3 (september 1998), GammaDistnbution,'' Journal YFfncnck/and Quantitative $$ne
409-22.
.
PricingBarrier Options,''Journal ofDerivatives, 3, 2 (Winter1995):19-28. and A. V. Vijh, $$TheValuationof Path Dependent Ritchken P., L. Sankarasubrsmanian, 1202-13. Averagej'' Contractson the Managennt Science,39 (1993): '
Ritchken, P.
$$0n
Rubinstein, M., aljdE. Reiner, SsBreaking Down the Baniers,'' Risk September(1991): 28-35. Aubinstein, M., SsDoubleTroubler''Rk, December/lanuary (1991/1992): 53-56. Rubinstein, M.,
$$0ne
Rubinstein, M.,
Ssoptions
for Another,''Rk, July/August(1991): 30-32. 70273. for the Undecided,'' Risk, Aplil (1991):
Rubinstein, M., $$PyNow,ChooseLater,'' Risk, February(1991): 44-47. Rubinstein, M,,
Sssomewhere
Ovrrthe Rainbowj'' Risk, November(1991): 63-66.
Rubinstein, k., $;Twoin One, Risk May (1991): 49. ,,
Rubinstein, M., and E. Reiner, S'Unscramblingthe Binary Code,'' Rk, October1991:7/.83. on the Minimumor Maximum of Two Assets,''Journal :.JFinancial 161-85. Economics,10 (1982):
stulz, R. M.,
SGoptions
Algorithmfor Priing EuropeanAverage Turnbull, s.M., and L. M. Wakeman, $'A Quick of and Xnclpf., Journal Options,'' Knlcll Qaantitative 26 (September1991):377-89.
and Problems(Answersin SplutionsManual) Questions '
,
24.1. Explainthe diferencebetweena forwardStart option and a chooser option. 24.2. Describe the payo; frm a portfolioconsistingof a lookbackcall and a lookbackput with the same maturity. ' 24.3. Conqidera chooser option where theholderhasthe yight to choose betweena European period. Thematuritydatesand strike calland a Europeanput at anytimeduring> z-year regardless of when the choi is made. ls it ever pricesfor the calls and puts are the same optimalto makethe choice beforethe end of the z-year period?Explainyour answer. 24.4. Supposethat cl and pj are the prices of a Europeanaverageprice call and a European averagepri put with strike price K and maturity T, cz and h are the prices of a Europeanaverage strike call and Europeanaveragestrikeput with maturity T, and c? and p are the prkes of a replar Europeancall and a regular Europeanput with strike price K and matmity T. Showthat cj + c2 c3 = pl + ,2 p. 24.5. fhe text derivesa detompositionof a particulartype of chooser option inic a call maturingat time Tz and a put maturingat time TI Derivean alternativedecmposition into a call maturingat time T1and a put maturing at time T2. 24.6. Section24.6givestwoormulas for a down-and-outcall. Thehrst applies to the situation wherethe barrier,S, islessthan or equal to the sike price, K. The second appliesto the situationwhere H k K. Showthat the tw0 fcrmulasare the samewhen H = K. -
-
.
567
Exotic Opfpxs
24.7..Explainwhy a down-and-output isworthzero when thebarrieris greatr thanthe strike price 24.8. Supposethat the strike prk pf an merica cll option on a non-dividend-payingstock growsat rate g. Showthat if g is lessthqn the risk-free rate, r, it is never optimal to exercisethe call early.
.
.
249 How can the value.of a forwardstart put op'tion on a nolpdividend-payingstock be alculatedif it is agreezthat the strike price will be l% gfeater t1:/1the Stock prke at the timethe option starts?
24.10.If a stock prke followsgeoletrk Brownianmotion,what pross does (f)followwhere (l) 1s the arithmeticgverage stock pri- betwtentimezero andtimef? 24.11. Explainwhy deltahedgingis easier for Asianoptions than for regularoptions. 24.12.Cakulate the pri of a l-yer Eufopeanoption to give up 1 ounces f silver in exchangefor 1 ounce of gold. The current price! of gold and silver are $380and $4, respectively;the risk-free interest rate is 1% per annum; the volatility of each commodityprice is 200/:.,and th: correlation betweenthe two pris is Ignore storagecosts. 24.13.Is a Europeandown-anduoutoption on an assetwort the snmeas a Europeandownand-out option on the asset'sfuturej price for a futurescontract maturingat the same time s the option? .7.
24.14.Answerthe followingquestions about compound options: (a) What putscall parity relationjhip exists betweenthe pri of a Europemncall on a call and a Europeanput on a call? Showthat the formtllasgivenin the text satisfy the relationship. (b) What put-call parity relationship exists betweenth pri of a European call on a put and a Europeanput on a put? Showthat the formulas venin the text satisfy the relationship. ' 24.15.Does a ioating lookbackcall bcome more valuable or lessvaluable as we increasethe frequencywith which we observe the assetpricein calculatingthe minimllm? 24.16.Does a down-and-outcall becomemore valuable or less valuable as we increasethe frequencywith which we observe the asset price in determiningwhetherthebarrierhas been crossed? What is the answer to the same question f0r a down-and-incall? i
Emopeancll 24.17.Exjlain why a replar Europeancall option ls the sumof a down-and-out cal. call. for the American 1options? and dowp-and-inEuropean Is sametn 'a
24.18.Whatis the value of a derivativethat pays 0fl- $1 in 6 months if theS&P50 indexis and zero otherwise?Assumethat the current leveloftheindexis 960, greaterthan l, the risk-free rate is 8% per annllm,thedividendyield on theindexis3% per apnum,and the volatility of the indexis 2%. 24.19.In a 3-monthdown-and-outcall option on silverfuturesthestrikeprke is $2 per ounce and the barrieris $18.The cnrrent futuresprice is $19,the risk-freeinterestrate is 5%, and the volatilityof silverfuturesis*% pel.annum.Explain howthe option works and calculateits value. What is the value of a Ieplar call option on silverfutureswith the same terms?Whatis the value of a down-and-incall option silverfutureswith the sameterms? .on
q 568
24 CHA?TEF
2420 A new European-stylioating lookbackcall optlonon a stockindexllasa maturityof 9 months.The current levelof the indexis 4,' the risk-free rate is 6% per annum, the dividendyield on theindtxis 4% per anmlm, and the volatility of theindexis 2%, Use DerivaGemto value the option. 24.21.Estimatethe value of a new 6-month European-style average pri call ojtion on a nondividend-payingock.The initialstockprice is $30,the strikeprice is $30,the risk-free interestrate is 5%, and the stockprice volatilityis 3%. 2422 Use DerivaGemto calculate the value f: (a) A rerlar Europeancall option on a non-dividend-payingstock where the stock Pri is $50,thestrikeprice is $5, the risk-freerate is 5% per annum, the volatility is 30% and the time to maturity is one year (b) A down-and-outEuropeancall which is as in (a)with the banier at $45 (c) A down-and-inEuropean call which is as in (a)with the barrier at $45. show that the optionin (a)is worth the sum of the values of the optionsin (b)and (c). 1
2423
.
Explainajpstments that haveto be made when r = q for (a)the valuation formulasfor lookbackcall optionsin section 24.8and (b)theformulasfor Mj and #2 in section 24.1. 01-
24.13assuming that the implied 24.24.Valu the variance swap in Example14. section with pris 1,5, 1,1. 1,15, volatilitiesfor options strike 8, 850,9, 950, l,, 1,20 are 20%, 20.5%,21%,21.5%,22%,22.5%,23%,23.5%,24%, respectively. ,
Assijnment Quekions in 1 year provided that 24.25.Whatis the value ih dollarsof a derivativethat pays of f l, dollar/steriing exchangerate is gryater than 1.5 at that time?The current te exchangerate is 1.48. The dollar and sterlinginterest rates are 4% and 8% per annum, respectively.The volatility of tly exchnge rate is 12% per annum. 24.26.Consideran up-and-out barrier call option on a non-dividend-payingStock when the stockprice is 5, the strikepri is 5, the volatility is 3%, the rk-free rate is 5%,the time to paturity is 1 year, and the barrierat $80.Usi the softwareto value the option and graph the rilationship between(a)theoptionpri and thestockprice, (b)the delta and the stock price, (c)the option pri and the time to maturity,and (d)$heoption price and the volatility. Provide an intuitiveexplanation f0r the results muget. show that the delta, gamma,theta, and vq f0r an up-and-out barrier call option can be eitherpositive or negative. considers the p1eApplicationF in the DerivaGemApplicationBuildersoftware 24.27. sam' 24.13.It showsthe way a hedgecan be sutic options replkation example in section 24.13)and two ways a hedge can be constructedusing four options (asin section constnlctedusing 16 options. (a) Explain the diferen betweenthe tw0 ways a hedge can be constructed using 16options.Explainintuitivelywhy the Second methodworksbetter. (b) Improveon thefour-optionhedgeby changingTmatforthethird and fourthoptions. (c) Checkh0w well the l6-option portfolios match the delta, gamma, apd vega of the barrieroption.
i
569
ExoticOktions
24.28.Considera down-and-outcall option on a foreip currency.Theinitialexchangerate is .9, the time to maturityis 2 years, the strike pri is 1., the banier is the risk-free risk-free domestic interestrate is 50:, thr foreign interestrate is 6%, and the vplatilhyis 25% per annm. Use Derivuem to developa staticOption replication strategyinvolvingfve options. 24.29.Supposethat a stockindexis currently 9. Thedividendyieldis20/:,the risk-freeratt is 50:, and the volatility is 4%. Usethe results in the appendixto cakulatethe value of a l-year avtrage pri call where the strikepri is 900and the indexlevelis Observed at the end of each quarter for the purposesof the averaging.Comparethis Fith the pri calculatedby DerivaGemfor a l-year average pri Option wherethepri is observed continuously.Providean intuitiveexplanationfOr any diferens betweenthepris. 24.30.Use tlie DerivaGemApplkationBuildersoftwareto compare the efectivenessOf daily deltahedgingfOr (a)the option tosidered in Tables 17.2and 17.3and (b)an average pricecall with the sameparameters. Use sample ApplicationC. FOr the averagepri Option Of Option pricein 11C16, you will fnd it nessary to change the calculation the G46to 0186 and N46to N186). the payofh in cells H15 and H16, and the deltas(cells Carry out 20 MonteCarlosimulationnms fOr each Option by repeatedlypressingF9. 0n each run record the cost Of writing and hedgingthe option, the volumeof trading Over the whole 20weeksand the volumeof tradingbetweenweeks11 and 2. Cornmnt On the restllts 24.31.ln the DerivaGemApplicationBuildefSoftwaremodify SampleApplicaticnD to test the efkctivenejsof delta and gamma hedgingfOr a call On callcompoundoption On a 1,00 lmits Of a foreigncurrency wherethe exchangerate is0.67,thedomesticrisk-free rate is 5%,the foreignrisk-free rate is60:, the volatilityis 12t)$.The timeto maturity Of the flrst option is 20 weeks,and the strike price of the flfstoption is 0.015.The second option matures 40 weeks fromtoday and has a strike prke of 0.62.Explainhow you modifedthe cells. CommentOn hedgeefectiveness. rtifkates'' certifkates'' 24.32.Outperformancecertitkates (alsocalled Of Or are oflred to investorsby many European banksas a way investingin company's stock, The initialinvestmentequals the stockpri, h. lf the stock price a goesup betweentime 0 and time T, the investorgains k timesthe increaseat time T, wherek is a constantgreater than 1.. However,thestockpriceusedto calculatethe gain at time T is capped at some maximum levelM. lf the stock pri goes down,the investor'slossis equal to the decrease.The investordoesnOt reive dividends. (a) Showthat an utperformance certifkate is a package. rtifkate when (b) Calculateusing DerivaGemthe value of a one-year Outperformance the stock price is 50 euros, k 1.5, M = 70 euros, the risk-freerate is 50:, and the stockpricevolatility is 25% Dividendsequal to euros are exycted in2 months, 5 months, 2 months, and l 1 months. 24.33.Carry out the analysis in Example 24.3of section 24.13to value the variance swap On Of assumption Swap rather than 3 months. month the that the life the is 1 .
Cgnsiderfrst the problemof calculatin theYsttwomoments of the value of a basket of assts at a futuretime, T, in a risk-neutral world. The pri of each asset in the basketis ajsumed to be lognormal.DeMe: n : The number of assets S i The vale of the fth ajlet at time T17 Fi : The forwardpri of the ith assetfor a contract maturing at timeT cf: The volatilityof the fth asset betweentime zero and time T pij: Correlationbetweenreturns fromthe fth and /th asset P1 Valueof basketat time T M1: First moment of # in a risk-neutral world Mz: Secondmomentof P in a. risk-neutral world Because P = E7 Lsi)= q.,vl = LP4 and uz c, LPl), where L denotesexT I si, pectedvalue in a risk-neutral world, it followsthat n '
Fi
M1= irr1
Also, P
2
?1
n
SiSj
=
i=1
./=l
of lognormaldistributions, From the kroperties n.
ftkfkyl= Fi/'/: pjggj Hence
5
22
5
=
ic 1 j= 1
FihepijuGjT
Asian Options We now moveon to the related problemof calculating the flrst two moments of the arithmetic averae pri of an asset in a risk-neutral world when the averageis fromdiscreteobservations. Supposethat the asset pri is observed at times calculated 7).(1K i K /n). Redehnevariables as follows'. h : The value of the assetat time 6. Fi : The forwardpri of the assetfor a contract maturing at time 7). cj : The impliedvolatility for an option on the assetwith maturity T. '
17lf the ith asset is a rtain stock and there are, say, 200 sjlares of the stock in thebasket then (forthe sasset'' is defmedas 20 shares of the stock and Si is the purposesgf the flrstpart of the appendix)the ft,h valueof 2 shares of the stock. ,
571
Zxotic O#!x:
pi): Correlationbetweenreturn op assetup to timeT.andthe return on the asset up to time Tj ': Valueof the arithmdic avemge 21 : Fkst moment of P in a risk-neutralworld M2: second momentof P in a risk-neutl'alworld In this case, #)
=
-
1
.-X''
)( Fi f=l
A1s0, ? 2
E- si s ) ./
M
--.
m
ln thiscase,
M
j =
2
t
=
j
y.j
hSj
z-fz')epilo'io't
=
TT:.
It can be shown that, when i < j, pij = so that
S( - si s ) = ./
and M2
=
1 W
2
M
g2rj
2 F'i i
e
=1
cf T o'j Tj kevi 71
z'f
'pl
+z
./=l
/-1 =1
y i yj yJ,z;.
'
C H
Weatherj E:ergy, .
...
P T
.
and Insurane
Derivaiives
The most common underlying variables in drivativescontracts are stockpris, stk indis, exchangerates, interestrates, and commoditypris. Futmes, forward,Optbn, and swap contracis on thesevariableshavebeenoutstandinglysucssful, AsdiKussed in Chapter23,credit derivativeshavealsobecomevery popular in rent years.Cllapter24 showsthat one way dealershave expanded the derivativesmafket is by devebpig nonstadard (orexotic) structures fOr defmingpayofs. This chapter discussesanother way theyhaveexpandedthe market. Thisis by tradingderivativesOn new underlying variables. . The chapter examinesthe prducts thathavebeendevelopedto manageweatherrk, markets that it will talk about are in some energyprice risk,and insurane risks. cases in the early stages of their development.As they evolve there may well be sigm cant changes in both the products that are Ofered and tllewaystheyalt used. rfhe
25.1 REVIEWOF PRICINGISSUES Chapters11 and 13 explaipedthe risk-neutral vallption result.'Thisinvolvespricinga derivativeon the assumptionthat investorsare risk neutral. The expectd payosis calculatedip a rk-neutral world and thendiscounteda,tthe rk-free interestmte.ne approach gives the correct pri-not justin a risk-neutral world,but in a11Other worldsas well. An alternativeplicing approach sometimes adoptedis to use historicaldata to calutate the expected payos and then dcount this expectedpayos at the rk free rate to obtain the pri. Wewill refer to this as thehtorical data approach.Historkal data give an estimate of te expected payof in the real world. It followstat the is historicaldata approach is correct Only when the expectedpayof fromthederivative the samein both the rtal world and tht risk-ntutralworld. Section11.7shows that whenwemove fromthe real worldto the rk-neutral world, the volatilitiesof ariablesremain the spme,but theirexpectedgrowthratts are liablet change.For example, the expectedgroWthrate of a stockmarketindexdecxasesby perhaps4% or 5% when we move fromthe reai worid t the risk-neutralwerld. ne expectedgrowth rate of a variable can beassumedto bethe snrnein boththerealworld 573
374
CHAPTER 25
and the risk-neutral wprld if the variable has zero systematicrisk so that percentage changesin thevariablehavezero correlationwith stockpafket returns. Tlzismeans that the htorical data apptoach t valuing a derivativegives the right answer if al1underlyingvariables havezeto systematicrk. A commonfeatureof most of the derivatives consideredin thischapteris thatthe underlyingvariables can reasonablybe assumed to havezero systematkrisk, so that the historicaldata approach can be used.
25.2 WEATHERDERIVATIVES Manycompaniesare in the position where their performan is liableto be adversely afected by the weather.1 It makes sensefor thesecompaniesto considerhedgingthei? weatherrisk in much the samewayas theyhedgeforeignexchangeor interestrate risks. The flrst over-the-cunterweatherderivativeswere introducedi 1997..To understandhowthey work, we explain two variables: HDD; Heating degreedays CDD: Coolingdegreedays A day'sHDD is desned as HDD maxto, 65 =
-
)
and a day's CDD is defmedas CDD = maito,
-
65)
where is the averaze of the ighest and lowesttemperatureduring the day at a specifedweather station,measuredin degrs Fahxnheit. For example, if the maxFahrenheitand the imum temperatureduring a day (midnight to midnightl' is 44O Fahrenheit,then = 56. The dailyHDD is then 9 and minimumtemperatureis the dailyCDD is A typicalover-the-counterproduct is a forwardor optioncontractproviding a payos dependenton the cumulativeHDD cr CDD during a month (i.e.,the total of the HDDS CDDS for dealercouldin everydayin the month). For exnmple,a derivatives or January208 sella client a calloption on thecumulativeHDDduringFebruary2009 at the ChicagoO'Hare irport weatherstatio with a strikepri of 70 and a payment rate of $1,0 per degreeday. If the actual cllmulativeHDD is 820,the payos is $t.2 million.Contractsoften includea payment cap. If the payment cap in ou? example is $1.5million, the contract is the equialent of a bullspread.ne client has a longcall option on cumulativeHDD with a strikeprice of 70()and a shortcalloptin with a strikeprice of 850. A day's HDD is a measure of the volume of energy required for heatingduringthe day. day's CDD is a measure of the volume of enerr required fr cooling during the day. Most weather derivativecontractsare entered into by energy producers and energyconsumers. But retailers, supermarketchains, food and drink manufacturers, health servi companies,agrkultural companies?and companiesin the leisureindust? are also potential users of weather derivatives.The WeatheiRiskManagement 68O
.
1
.
.
ne US Department of Energyhas estimatedthatone-seventhof theUS ecnomy is subjectto weatherrisk.
)
575
Weathey,Eneygy,and Jx-lzrtmc:Deyilmtives ' .
.
has been formedto srve the interestsQf the weather . Association(www.wrma.ozg) risk managementindustry. ln September1999the ChicagoMercantileExchangebegantradingwether futures and Europeanoptionson weather futures.The cpntracts are pn the olmulative HDD and CDD f0r a monthobservedat a wather station.2 The contracts are settled.incash just afte/ the end of the month once the HDb and CDD are knowp.One futures contmct is on $1 timesthe cumulative HDD or CDD. The HDD and CDD are calculatedby a company, Earth SatelliteCorporation,using automateddata-colltbn CQl1iPmC2t.
The temperatureat a certain locationcan reasonably be assumedto have zero systematicrisk. lt followsfrom Section25.1 that weather derivativescan be prip, using the historicaldata approach.Consider,f0r example,the call option o the February 2009HDD at ChicagoO'Hare airport mentioned earlkr. By collting,
say, 50 yeafsof data, a probabilitydistributionf0r the HDD in Februarycap be estimated.Thisin turn can be sed to pfovide a probability dktributionf0r theoption payof. An estimate of the value of the option is the mean f this distribution discountedat the risk-free rate. lt mightbe desirableto adjust the probability istribution f0r temprature trends.F0r exnmple,a linearregressionmight show that (perhaps becauseof global warming)the HDD in Februaryis decreasingat a rate of 4 per year on average.If so, the output fromthe regressioncould be used to estimate a tredadjustedprobability distributionfor the HDD in February2009.
25.3 ENERGYDERIVATIVES Energycompanies are among the most activeand sophisticatedusers of (krivatives. Many enerr products trade in both the over-the-countermarket and on exchanges. ij ln th section, the trading of crude 0i1, natural gps, and electridtyderivatives. examined.
Crude OiI Crude oi1is one of the most importantcommoditiesin the world, with global demand amountingto about80 million barrelsdaily.Ten-yearfued-pricesupply contractshave been commonplace in the over-the-countermarketf0r many years. Theseare swaps whereoil at a fed price is exchanged f0r 0il at a ioating pri. In the 1970sthe priceof 0il was highlyvolatile.The 1973war in theMiddleEastled to a triplingof oi1prkes.The fall of the Shahof lran in thelate19708againincreamd prices.These events 1edoi1producers and users to a realization that theyneededmore sophisticatedtoolsfor managing oil-pricerisk. ln the 1980sboth the over-the-counter market and the exchange-tradedmarket developedproducts to meet this need. ln the over-the-coutermarket,virtually any derivativethat is avqilableon common stocksor stockindicesis now availablewith0il as the underlyingasset. Swaps,frward contmcts,and optionsare popular. Qmtractssometimesrequiresettlementin cash apll sometimesrequire settlemet by physicaldelivery(i.e.,by deliveryof the 0i1), 2 TheCME has introdud contracts fot l() diserentweathr stations (Atlanta,Cbicago,Cincinna, Dallas, Des Moines,IaasVegas,New York Philadelpa, Portland, and Tucson).
i76
CHAPTER 25
Exthange-trded contrcts al'e also popular.The NewYork MercantikExchange (NYMEX)and the lnternptioal PetrolellmExchange(1PE)trade a number of oil of thefuturescontractsare settledin cash; futuresand futuresoptions contracts.some others re ettledby physical delifers For examplethe Brentcrude oi1futurestraded bn the IPE has caj settlementbased on the Brentindexpri; the light sweetcrude oil futurestraded on NYMEXrequiresphysicaldelivery.In both cases the amount of oil underlying one contmct is 1,0 barrels.NYMEXalso tradespopular contractson two rened products: heatingoi1 and gasoline. In both casesone contzactis fr the deliveryof 42,000gallons.
Natural Gas The natural gas indujtry tkoughout the worldhas beengoingthrough a period of deregulationand the eliminationof government monopolies.The supplierof natural are gas is now not necessarilythe samecompany as the producerof the gas. suppliers fad with the problem of meetingdailydemand. A typical over-the-countercontract is for the deliveryof a specifed amount of natural gas at a roughly uniform rate over a one-mnth period. Forwardcontracts, options, and swapsare Available in the over-the-countermarket. Th sellerof gas is 'usuallyresponsibl for moving the gas throughpipelinesto the specihed location. rnillion Britishthennal units of NYMEXtradesa contract for the deliveryof lj naturalgas.The contrad, if not losedout, requiresphysicaldeliveryto bemadeduring thedeliverymonth at a roughlyuniform rate to a particular hubin Louisiana.ne IPE tradesa similarcontract in London.
Eledricit. Electricityis an unusual commodity becauseit cannoteasily be stored.a The maximum electr'icityin region at any moment isdeterminedbf the maximumcapacity a supp1y of of a11the electricity-produdngplants in the region. In the UnitedStatesthere are 140 knownas cohtrol areas. Demandand supply are rst matchedwithin a.control reyions area,and any excesspower is sold to other control areas.lt is tls excesspower that the wholesalemarket for electricity.The ability of one controlarea to sel constitutes powerto another dependson the transmissioncapacity of the linesbetween two areas.Transmission fromone area to another involvesa transmissioncost, charged by theokner of the line,and themare generallysome transmissionor energy losses. A major use f electrkity is forair-cpnditioning systemskAs a result the demandfor tlieretote much prke, and is greater in the summermonths thap in the its electricity, nonstorabilityof The months. electridty causesoccasionalvery largemovements winter inthe spot price.Heatwaveshavebeenknownto increasethe spot priceby as muchas for short periodsof time. 1000% Likenatural gas, eltricity hasbeengoing througha period of deregulptionand the of governmentmonopolies.Tls hasbeenaccompaniedbythe development elimination ofan electridty derivatives arket. NYMEXnow tradesa futurescontract on the price of electricity, and there is an activeover-the-counter market in forward contracts, 'the
3 Electricitypodus
with spare capacitysometime,suse it to pump watef to the top cf theirhydyotkctric plantsso that it can beysedto ploduceelectricityat a latertime.Th is theclosesttheycan gd to storing this commodity.
7
Weathey,Eneygy,and InsuyanceDerivatives
options, and swaps.:Atypicalcontract (exchange tradedor overthe counter)allowsone . Side receive of for a specifkdprice at a specihed to a specifkd number megawatt-hours locationduring a partkular month. In a 5 x 8 contract, poweris receivd for 5 daysa week (Monday to Friday) during the oflpeak period (11p.m. to 7a.m.) for the speciEedmonth.In a 5 x 16contract, poweris reived 5 daysa week duringtht onpepk period (7a.m. to 11p.m.) for the specifkdmonth. In a 7 x 24 contract, it is reteivedaround the clock every day duringthe month.Optioncntlacts haveeither dailyexerciseor monthlyexercise.In the case of dailyexercije,the option holdercan chooseon each day of the month (bygiving one day'snotice)to reive the specifkd amount of powerat the specihedstrike price.Whenthre is monthlyexercise,a single decisionon whether to receivepower forthe wholemonth at the specifkdstrikepri is made at the beginningof the month. An interestingcontract in electricityand natural gas marketsis what is knownas a swizgoption or takand-pay option.In tlliscontract a miniplm andmaximumfor the amount of powerthat must be pumhased at a certainprke by the optionholde!is Specifkd for each day durig a month and for the monthin total. The opon holdr swing) change the rate at whichthe power is purchased duringthe mohth, but can (or usuallythereis a limit on the totpl number of changes that can be made.
Modeling EnergyPrices A plauble modelfor energy and other commodity price! should incorporateboth mean reversion and volatility. 0ne possiblemodel is: #ln S =
(p(l) a ln -
.j
dt + c Jz
(25.1) .
whereS isthe energy price,and a and c are constant parameters. Thisis similar to the modelsfor interestrates describedin Chapter3. Theparameter c is te volatilitkof S, and a nteasuresthe speedwith whiih it revertsto a long-funaveragelevel.Thep(l) term capturesseasonalityand trends.Chapter33 showshowto constrtlct a trinomialtreef0r with p(l) beingestimated fromfuturespris. The parathe model in equation(25.1) estimayed and fromhistoricaldata or impliedfromderivativeprkes. be metersa c can The parameters and c are diferentfordiferentsources of energy.For cfude oil, the and the volatilityparameter c is about reversionrate parametera in equation (25.1) is about 209/:;for natural gas, a is about 1.0 an c is about40e/:;for electridty, a is typicallybetweenl and 2, whilec is 1 to 2%. The seasonalityof electrkity prices is also greater.4 .5
How an EnergyProducer canHedge Risks
ThereAre two components to the risks facingan eergy producerk0ne tltepricerisk; the other isthe volumerisk. Althoughpricesdo adjust to refect volumes,thereis a lessthan-perfectrelationship betweenthe two,and energy producershaveto takebothinto accountwhen developinga hed/ng strategy. The prke risk can be hedgedusingthe enerr derivave contracts discussedin this section. The volumerisks can be hedged usingthe weather derivativesdiscussedin the previoussection. 4 For a fullerdiscussionof the spd price behaviorof energyproducts, see D. Pilipovic,EzergyRk. New York: McGraw-llill, 1997.
578
CHAPTER 25
Defme: l': Prost for a month ': Averageenergypris f0r the month T: Relevanttemperaturevariable (HDD or CDD)for the mnth An enerjy prodacer can use historicaldata to obtain a best-st linear regression relationshipof the form 1' = a + bp + cT + E: whereE:is the error term. The energyproducercn thenhedgerisks for the month by taking a positionof in enerr forwardsor futuresand a positionof in weather forwardsor futures.The relationship can also be used to analyzethe efectiveness of alternativeojtin strategies. -.b
-c
25.4 INSURANCEDERIVATIVES Whenderivativecontracts are used for hedgingpurposes,theyhavemanyof the same characteristicsas insurancecontracts. Both typesof contracts are desigd to provide protectionagainstadverse It is n0t surprisingthat many insuran companks rvents. have subsidiariesthat trade derivativesand that many of the activities of insurance companiesare becomingvery similarto thoseof investmentbanks. Traditionallythe insufn industryhas hedgedits exposuie to catastrophic (CAT) risks suh as hurrkanes and earthquakes using a practl known as reinsuran. Reinsuran contracts can takea number of forms.Suppojethai an insuranc company has.an exposureof $100millionto earthquakes in Californiaand wants to limit tllisto is to enter intoannualreinsuran contracts that coveron a $3 million. 0ne alternative of pro rata basis70% its exposure.If Californiaearthqu>keclaims in a particularyear total $5 million, the costs to the companywould thenbeonly x $50,or $15million. Another more popular alternative, inolving lowerreinsurance premiums, is to buy a seriesof reinsurance contracts cvering what are knownas excesscost layers.The frst layermight provideindemnifkationforlossesbetween$3 millionand $4 million;the next layer might coverlossesbetween$4 millionand $5 million; and so on. Each reinsurancecontract is knownas an excess-f-loss reinsurance cntract. The reinsurer has written a bull spreadon the total losses.It is long a call option with a strikeprice equalto thelowerend of thelayerandshorta call option with a strikeprke equal to the 5 upper end of the layer. The principal providersof CAT reinsurance have traditionallybeen reinsuran companiesand Lloydssyndicates(which are unlimited liabilitysyndicatesof wealthy individuals).In rent years theindustryhas come to the conclusin that its reinsurance needs have outstrped what can be providedfrom these traditionalsources.lt ha$ searchedf0r new ways in which capitalmarkets can providereinsuran. 0ne of the eventsthat caused the industryto retnk its practiceswas Hurricane Andrewip 1992, wllichcaused about $15billion f insurancecostsin Florida.Thisexeded thetotal of relevantinsuran premiumjreciived in Floridaduringthe previous sevenyears. If .3
'
.
.
5 Reinsuran is also sometimesofered in the form of a lumpsum if a rtain losslevelis reached. ne is then writing a cash-or-notlng binarycall option on the losses. reinsurer
Weather,Eneqy, rzltfInsurance fk?r/lfrx:
579
would. have . Hmricane Andrewhad hit Miami,it is estimatedthat insuredlosses other catastrophes exceeded$4 billion.HurricaneAndrewand have1edto increases in insuran/reinsurance premiums. Exchange-traded insurancefuturescontracts havebeendevelopedby the CBOT,but have not been highlysuessful. ne over-the-countermarket has come up with a numberof productsthat are alternativesto traditionalreinsuran. The mostpopularis a CATbond. Thisis a bondissuedby a subsidiaryof an insuran company that paysa higher-than-nrmalinterestrate. In exchangefor the extra interestthe holder of the bond agreest providean exss-of-loss reinsurancecontract. Dependingon the terms of the CATbond,the interestor lrincipal(orboth)can be used to meetclaims. In the considered abovtwherean insuran company wants protectionfor California . example earthquakelossesbetFeen $3 millionand $4 million, the insuran company could issueCATbonds with a total principalof $1 million. In the event that the insurance cmpany's Californiaeatthquake lossesexeded $3 million,bondholderswould lose sme or all of theirprincipal.As an alternativetheinsuran company could cover this excesscost layer by making a much biggerbond issuewhere nly the bondholders' interestis at risk. CATbondstypicallygive a highprobabilityof an above-normal rate of interestaqd a o f Why wouldinvestorsbe interestedin such instruments? low-probability a highloss. ne ahswer is that there are no statisticallysignihcantcorrelations betweenCATrisks and market returns.6 CATbonds are iherefore an attractiveaddition to an investor's portfolio.They have no systematicrk, so that their total risk can be completely diversihedaway in a large portfolio. If a CATbond's expected return is grater than the risk-free interestrate (andtpically it is),it hasthe potential to improverisk-return trade-ofl.
SUMMARY Tltischapter has shown that when thereare risks to be managed, derivativemarkets have beenvery innovativein developingproducts to meet the needs of market participants. In the weather derivativesmarket, two measurs, HDD and CDD, haye been developedto describethe tempeptureduring a month. Theseare used to defne the payofs on both exchange-tradedand over-the-counter derivatives.No doubt, as the weatherderivativesmarketdevelops,we will see contracts on minfall,snow, and similar variablesbecomemore commoplace. In enerr markets,oil derivativeshavebeenipoftant for sometimeand playa key rolein helpingoilproducersan2 oil consumersmanage theirprke risk. Naturalgas and electrkityderivativesare relativelynew. Theybecameimportantfor risk management whenthesemarketsweredereplated and governmentmonopolies discontinued. Instlrancederivativesare now beginningto be an alternativeto traditionalreinsuranceas a way for insurancecompaniesto manage the risks of a catastrophicevent such life asa hurricapeor an eartquake. No douhtwe will see other sortsof insurance(e.g., securitized similar and market automobileinsuran) being in a way as tMs insuran
develops.
6 SeeR. H. Litzenberger,D. R. Beaglehole,and C.E. Reynolds, tAssessingCatastropheReinsran-Linked 7646. Securitks as a NewAssrt Class,''Journal ofporfolio Msagement, Winter (199$:
580
CHAPTER 25
Most weather, energy, and insuran derivativeshave te propertythat perntage in the underlying variables havenegligibk corrtlationswith matkd returns. changes TMsmeans that we can valu derivativesbycalculatingexpectedpayofs using historkal rate. ata and then discountingthe expectedpayofs at the lk-fxe
FURTHERREADING On Feather Dervatlves
'' .
.
.
Arditti,F., L. Cai, M, Cao, and R, McDonald, SsWhether to Hedge,'' Risk, Supplementon 112. WeatherRisk (1999): Co, M., and J. Wei,'sWeatherDerivativesValuation and the MarketPri of WeatherRisk'' Jburnal of Futures Markets, 24, 11 (Nlwember 24): 1065-89. Hunter,R., 'Managing Mother Natmv'' DerivativesStrateiy,February(1999). On fperp Derflfat/lfe: Pricing (IIIJRisk Management Lachna Cli wlow, L., and C Strkkland Esergy Deratis: Group,200, 71-73. PowerDerivatives,'' Rk, October(1998): Eydeland,A., and H. Geman, Joskow,P., ttElectricitySectorsin Transition,''Te EnergyJournal, 19 (1998): 25-52. ,
On Insurance Derivative: Derivaiives:A NewAssetClqssfor the Canter,M. S., J. B Cole and R. L. Sandor, Capital Markets and a New fledgingToolfor the Insuran Industry,''Jourzal of Applied 69-83. CorporateFinazce, Autumn(1997): Froot,K.A., 'The Market for CatastropheRisk:A ClinicalExnmination,''Journal ofFizazcial Economics,60 (201):529-71. of ChicagoPress, 1999. Froot, K.A., Te Fizancizgof Catastrope Risk. oniversity Sslnsurance
Litzenberger,R. H., D. R. Beaglehole,and C.E. Reynolds,dsAssessing CatastropheReinsuranceLinked Securitiesas a NewAssetClaSs,'' Jounlal of 'prt//p Mazagement, Winter(1996): 76-86. '
and Prolems Questions
Manual) (Answersin solutions
25.1.What is meant by HDD and CDD? 25.2.H0F is a typicalnatural gas forwardcontractstructured? 25.3. Distinguishbetweenthe historkal data and the risk-neutralapproach to valuing a derivativt.Undtr what circumstanced thty givt tht snme answer. 68O 25.4. Supgosethat each day duringJulythe minimum temperatureis Fahreneit and the 82O Fahrenheit.Whatis the payof from a call option on the maxlmumtemperatureis olmulativeCDD duringJuly with a strikeof 25 and a payment rate of $5, per degree-day? 25.5. Whyis the price of electricitymore volatile than that of other energysours?
581
Weathey,Eneygy,and fxx?unc: Derivatives
25.6.Why is the historicaldata approach appropriatefor pricing a weather derivatives contractand a CAT bond? and CDD can be regardep qs payofs fromoptionson temperature.''Explain th 25.7. SEHDD statement. 25.8. Supposethat you have50 yeats of temperaturedata at your disposal.Explaincarefully the analysesyou would carry out to value a forwardcontract on the cupulative CDD for a particular mopth. Volality Of the l-year forwardpriceof oil 25.9. Would to be greater than or y0u expect the lessthan the volatility of te spot pri? Explainyour answer. 25.10.What are the characteristks of an enerr sour where the pri haS a very highvolatility and a very high rate of mean reversion?Givean eyampleof such an enerr source. 25.11.H0w can an energy produr use derivytivesmarkets to hedgerisks? 25.12.Explainh0w a 5 x 8 option contract for May 2009 on eltctricity wittl daily exercise works.Explainhow a 5 x 8 option contract for May2009on electridty with monthly exercis works. Whichis worth more? 25.13.Explainhow CAT bonds work. 25.14.Considertwo bondsthat havethe samecoupon, time to maturity,and pri. Oneis a B-ratedcorporate bond.The ther is a CATbond.Ananalysisbasedon historicaldata showsthat the expected losseson the two bondsin each year of theirlifeis the same. Whichbond would you advisea portfoliomanager to buyand why? '
'
'
AssijnmentQuestion 25.15.An insurancecompany's lossesof a particular type are to a reasonable approximation normally stributedwith a mean of $150million and a standarddeviationof $5 nllion. (Assumeno diferencebetweenlossesin a risk-neutralworld andlossesin the real world.) The l-year risk-free rate is 504.Estimatethe cost of the following: (a) A contract that will pay in 1 year's time6% of theinsurancecompay's costs on a pr0 rqta basis (b) A contract that pays $1 million in 1 year's timeif lossesexed $200million
A
C H
-(r-)Tj2(1-a)
a
=
E#e
lv
(1 -
where
J
p 2(r
-
2(r.x)(a-I)z
a
-
c
,
1 a
2
=
-
j'2(1-)
1
b=
,
-
1)
=
(1 -
a) 2p
j.j
and z2(z,k v) is the cumulativeprobability that a variable with a noncentral / distributionwith noncentralityparameterp and l degreesof freedomis liss than i. 2 A produre for computing z (z, p) is provided in Technkal Note 12 on theauthor s website, The CEV model is partkularly useful for valuing exotic equity options. The parametersof the mode1can be chosen to t the prkes of plain vanilla opons as closelyas possibleby minimizing the sum of the jquared diferens betweenmodel prices and marketprices. ,
: Averagenumber of jllmpsper year : Averagejumpsizemeasured as a perntage of the asset prke :
.
u
'
The percentagejunipsize is ajllmed to bedrawnfroma probability distributionin the .
model. The probabilityof a jumpin tim pricefromthe jumpsis therefore
.
ds
T = (r
is Al. The average growt rate in the asset ne risk-neutral process for the assetprice is l
q
-
l) dt + c dz + dp
-
wheredz is a Wienerprocess, dp is thePoissonprocessgenerating the jumps,and c is the volatilityof the geometricBrownianmotion. The processesdz and dp are assumed to be independent. An importantparticular case of Merton'smodel is where thelogarithmof thesizeof the percentage jumpis normal. Assupe that the standard deviationof the normal distributionis s. Mertonshowsthat a European option pricecan thenbe written '
wherey = 1n(1+ l). This model givesrise to heavierleftand heavierright tailsthanBlack-scholes.It can be used f0r prking currencyoptions.As in the case of the CEVmodel,the model parametersare chosenby minimizingthe sum of the squared diferens betweenmodel prkes and marketprkes.
The Vafiance-GammaModel An example of a purejumpmodelthatis provingquite popular is thr yariatlce-gamma mo#:/.5 DeEnea variable g as the changeover time T in a variale that followsa gammapross with mean rate of 1 and varian rate of p. A gammaprocessis a pure jump pross where small jumpsoccur very frequetly and large jumpsour only oasionally. The probabilitydensityfor g is T/p-1
-g/p
e #() g1)r/pryryp) =
Wherer4 ) denotesthe gamma function.Thisprobabilitydensitycan be computedin Exl using the GAMMADISTI function.Theflrstarpment of the functionis g, the second is Tlv, the third is p, and the fourthis TRUE or FALSE, whereTRUE probabilitydistributionfunctionand FALSE returns the probreturnsthe tumulative .
.)
.,
.,
.,
abilitydensityfunctionwe hayejustSven. As usual, we desne Sy as the asset pri at timeT, k as the asset pri today,r as the fisk-freeinterestrate, and q as the diyidendyield. In a risk-neutral world ln Sy, under the varian-gamma model, has a probability distributionthat, conditionalon g, is normal.The conditional mean is ln k + r
)T +
-
+ bg
and the conditionalstandard deviationis
c;k where T
-1n(1
=
-
r
h
-
2
c c/2)
The variance-gammamodel has threeparameters'.p, c, antl b.6 The parameter p is the varian rate of the gamma process, c is the volatility, and b is a paraineter desning skewness.When b = 0, ln Sz is symmetric;when b < 0, it is negativelyskewed (asfr equitisl; and when b > 0, it is positivelyskewed. supposethat we are interestedin using Exl to obtain 10,0 random samples of the changein an asset pri betweentime()and time T using the variance-gammamodel. 6
P.P. seeD. B. Madan, AcvfEw,
European Finance
Carr, and E. C. Chang, s-re
Varian-Gamma Prccess and OptionPricin''
2 (1998): 79-195.
Nctethat all theseparameters are liable tc changewhen we move fromtbe real wcrld tc the risk-neutral world.This is in clmtrart to pure disusicn mcdelswhere the volatilityremains the same.
587
Mbre c'n Modelt and Numeical Prtlrctflzre
As a prelimiary, we could set cellsE1,E2,E3,E4,E5,E6, and E7tqual to F, t;, 0, c, r, q, and k, respectivly. We culd also set E8equal to by desningit as = $E$1* LN(1 $E$3* $E$2 $E$4* $E$4* $E$2/2)/$E$2 -
-
We could then proceedas follws: values for g using the GAMMAINVfunction.Set the contents of cells 1. sample
A1, A2,
.
.
.
Al
,
as
= GAMMAINVIRAND, $E$1/$E$2,$E$2) 2. For each value of g we sample a yalue z for a variable that is normally distributed withmeanogand standard deviationo.l, This can bedonebydehning ll B1as = A1 * $E$3+ SQRTIAI)* $E$4* NORMSINVIRANDI similarly. and cells B2,B3, Bl 3. The stock price xr is Svenby .
BydehningC1 as
.
.,
xr
k epltr
=
-
qjT +
+ z)
= $E$7* EXP(($E$5 $E$6)i $E$1+ B1+ $E$8) similarly,random samplesfromthedistributin of xz are andC2,C3, Cl cells. createdin these -
.
.
.
,
Figure 26.1 shows the probabilitydistributionthat is obtained using the variance0 and t; = o' = gamma model for Sz when k l, T = For comparison it also showsthedistributiongiven by geometric Brownian r q= motion when the volatility, o. is (0r2%). Althoughnot clear in Figure26.1, the variance-gamma distributionhasheaviertailsthan thelognormaldistributiongiven by geometricBrownianmotion. 0ne way of characterizingthe varian-gamma distributionis that g defnesthe rate at which informationarrives duringtime T. If g is large,a great deal of information arrivesand the sample we take from a normal distributionin step 2 abovehas a relativelylarge mean and varian. If g is small, relatively littleinformationarrives and the sample we takehas a relativelysmall mean and variance.TheparameterT isthe usualtimemeasure, and g is somtimes referredto as measuringeconomictimeor time adjustedfor the i0w of information. Semi-aalytic Europeanoption valuation formulasare providedby Madan et aI. (1998).The variance-gammamodel tendsto producea U-shapedvolatility smile.The smileis n6t nicessarily symmetrical.It is vry pronouncedfor short maturities and away''for long maturities. The model can be ftted to either equity or foreigncurrency plain vanilla optionprices. .5,
Distributionsobtained with varian-gapma Brownianmotion.
Fijure 26.1
l
l l ! l
!'
I
l l l
l l
I /
/
pross and geoletric
y! !
!
#
l l !
!
j
+
-
-
VarianceOnmma .Cemetric BDOXI
-*
Motion
1.
kA
l
N
AQ
40
80
100
12
160
1*
180
200
its g parameter.Lowvaluesof g correspond to a lowarrival rate for informationand a lowvolatility; high values of g eorrespondto a hkh nrrival mte for informaon and a high volatility. An altenmtive to the varian-gnmma mod.elis a modelwhere the processfollowed bythe volatilityvariable is specied explicitly.Supposeftrstthatthe volatilityparnmeter in the geometric Brownianmotion is a knownfunctionof time.Therisk-neutral process followedby the assetprke is then dv =
sdt+ jtjspz
(r -
(26.1)
TheBlack-scholesfonhulasafe thencorrect provided thatthe variance rate is set equal to the average variance rate duringthe life of the option (seeProblem 26.6).The varnce rate is the squareof the volatility. Supposethat during a l-year period the of a stockwill be 29%duringthe rst 6 monthsand 3% duringthe second volatility 6 months.The average variance rate is 0.5 x 0.202+ 5 x 32 .
.
=
9.965
This corresponds to a It is correct to use Black-scholesw1t11a variance rate of = volatilityof 0.965 9.255,or 25.59:. Eqgation (26.1) assumes that the instantantus volatility of ap asset is perfectly predictable.In practice, volatility varies stochastically.This has led to the development of more complexmodelswith two stochasticvariabks: the stockprice and its volatility. .65.
Mpr:
l?l
!89
Mdels and NumeyicalPrpcctflzrcs
One model that has been used by researchersis
ds T = (r
dt +
-
JJ? = c(Pt
z
dzs
(26.2)
P)J! + )P' dzv
-
(26.3)
wherea,'-%, ), and a are constants,and dzs anddzv are Wienerprocesses.I'he variable P in tllismodel isthe asset's variance rate. Thevariancerate has a driftthat pulls it backto a levelPt at rate a. Hull and Whiteshow tat, when volatility is stochastic but uncorrelatedwith the asset pri, the price of a European optionis the Black-scholespriceintegmtedovir the probability distributionof the averagevarian rati duringthe lifeof the option.; Thus a Europeap callpriceis .
.
X
-
.
--.
'
'
.
-
c(P)g(P)J
where# is the averag'e value of the variance rate, c is theBlack-scholespri xpressed as a functionof #, and g istheprobabilitydensityfunctionof # in a risk-neutralworld. This result can be used to show that Black-scholesoverpris optionsthat are at the moneyor closeto the money, and underprkes options that are deepin or deep out of the money, The model is consistentwith the pattern of impliedvolatilitiesobservedfor 18.2). currencyoptions (seesection The case where the asset price and volatility are corplated is mori complicated. Option prices can be obtained using MonteCarlo simulation. In thepartkular cas Hull and Whiteprovidea series expansion and Hestonprovidesan wherea resli.' The pttern of imjlied volatilities obtained when the volatility is analytic correlated with the asset priceis similarto that observed for equities (see negatively .5,
=
section18.3).9
Chapter 21 discusses exponentially weighted moving average (EWMA) and GARCHII, 1) models. Theseare alternativeapproaches to characterizinga stochastic model. Duan shows that it is possibleto use GARCHII, 1) as thebasisfor an volatility consistentoptionpring model.l (seeProblem21.14for the equivalenceof internally GARCHII,1) and stochasticvolatilitymodels.l' stochasticvolatilitymodels canbeEttedto thepricesof plain vanillaoptions andthen 11 used to price exotic options. For options that last lessthan a year, the impactof a in percentage stochasticvolatility on pricingis fairlysmall in absolute terms(although 7
seeJ. C. Hull and A. White,
Finazce,42 (June 1985: 281-3.
8
''l''e
Yolatilitiesk''Jozrnal of Pricingof Opiionson sset; with stochastic of the process followedby the varian rate. This result is independent
Wbite, lAn Analysisof e Biasin OptionPricingCausedby a Yolatility,'' stochastic Options Researc, 3 (1988): 27-61; s.L. Heston, 4$AClosed Form solution Adyancesin Futures for Yolatilitywith Applicationsto Bondsand C'urency Options,''Aeifew6f Financial Optionsw1t.1:stochastic 327-43. Stzdies,6, 2 (1993):
seeJ. C. Hull and
.
Jzl:
9 The reason is given in foothote3. 10
GARCH OptionPricing Model,''Hatematical Fisance,vol. 5 (1995), 13-32)and ' J.-C. Duan, cracking RISK, vol. 9 (December1996),55-59. the smile'' l 1 For an exampleof this, see J. C. Hull and W. suo, ''A MethodolorfortheAssessmentof ModelRiskand Yolaiility Implied Fupction Application Jnurzal of J'zcfa/ and Quantitatiye dac/y'l, 37, the Modelj'' its to 297-318. 22): (June 2
seeJ.-c. Duan,'
ft'f'he
590
CHAPTER 26
options).It becomesprogressively termsit can be quite largefor deep-out-of-the-money larget as the life of the optionincreases.The lmpact of a stochasticvolatility on the generallyquite performanceof delta hedgingis large.Tradersrecognize this and, as describedinChapter17,monitor theirexposureto volatihtychage bycalculatingvega.
26.3 THE IVFMODEL The parameters of the model we havediscusseds far can be chosenso that they providean approximatefit to the prices of plain vanilla options on ny given day. Financialinstitutionssometimeswant to go one stage furthe: and use a model that providesan exact lit to the prices oftheseoptions.12In 1994Dermanand Kani Dupire, and ubinstein developeda model that isdesignedto do th. It has becomeknownas (IVF)modelor the impliedtree mode1.13 the implied olat 'lityfunction It provides an exactfit to the European option prices observedon any given day, regardless of the shapeof the volaiility surface. ne risk-neutral processfor the asset price in the modelhas the form ,
dv =
Er(!)qtljsdt -
+ o.s, tjsdz
wherer(l) k the instantaneousfomard interestrate f0r a contract maturing t time t and qtj isthe dividendyield as a functionof time.Th volatilityct,, 1) is a functionof both S and ( and is chosenSo that the model prices a11European optionsconsistently withthe market. It is shown bothbyDupke and byAndersenand Brotherton-Ratclile 14 that c(,, !) can be calculated analytically:
(c(#,T)12 2 =
Jcmu/lT + /T)cmkt-+ #(r(T) i r z(9cmkt/? .z
qTjczjktlK
-
(2.4)
)
wherecmkttf, T) is the market pri of a European calloptionwith strik price K and maturity T. If a suciently largemlmber of European call prices are availablein the market,this equatio can be used to estimate the ctk, !) function.15 Andersenand Brotlzrton-ltatcliflb impkment the modtl by using equation (26.4) diferen method. Analternativeapproach, the implied togetherwith the implicitsnite treemethodology suggestedbyDermanand Kani and Rubinstein,involvesconstructing a tree for the asset pricethat is consistent witholtion pris in the market. Whenit is sedin practice the IVF model is recalibrateddailyto the prices of plain vanilla optilms.It is a tool to prie exotic options consistentlywith plain vanilla options. As dcussed in Chapter18 plain vanilla options desne the risk-neutral :2There is a practicalreason forthis.If thebankloes'not use a molel with titisproperty, thereis a danger thattradersworking for the bank willspind theirtimearbitragingthe bank'sintemalmodels. 18-2; E. Dermanand 1. Kani,'Riding on a 13Se'eB. Dupire, pridng with a Smilq''Risk,February (1994): Rubinstein, BinomialTrees'' Joarnal of Finance,49, 3 32-3% Risk, February M. smile,''. (1994): '
Slmplied
(Ju1y1994),771-818.
.
14See B.
February (1994), 18-29; L.B. G. Andersenand R. Dupke, tpricing with a Smile,'' 'k, Brotherton-Ratclife Equity OptionVolatilitySmile:An ImplicitFinite DilerenceApproachs''Joarnal Finance 1, No. 2 (Winter 1997/98):5-37. Dupire considersthe case where r and q are zero; ' of compatation Andersenan'd Brotherton-Ratcliflkconsider the more generalsituation. Grf'he
.6Somesmoothingof the observedvolatility
surface is typicallynecessav.
)'' l
i
Moye0Al Models JA?,JNumeyicalPyoceduyes
!91
Probabilitydistributionof the asset pri at a11futuretimes.lt followsthat the IVF model the risk-neutral probabilhydistributionof the assetpriceat a11futuretiins all-or-notng oryect.Tls means that options providingpayofs at justone time(e.g., pricedcorrectlyby the 1VFmodel.However, the and asset-or-nothingcptions) are modeldoes not necessarilyget the jointdistribd+ of the assetpric at two or more options such as tompotmdopiionsan'd barrier times correct Tilismeans that exotk 16 optionsmay be pricedincorrectly. 'gets
.
26.4 CONVERTIBLEBONDS We now move on to discusshowthe nuperical procedprespresentedin Chapter19can be moded to handle particular valuation problems. We start by considering convertiblebonds. Convertiblebondsare bondsissuedby a company where thehclderhastheoptionto exchangethebondsforthe copany's stockat certaintimesinthefuture.ne cozversioz ratiois the mlmber of shares of stok obtainedfor one bond (thiscan be a function theissuerhastheright tq buythemback time).Thebondsar almostalwayscallable(i.e., prices). at certain timesat a predetermined ne holderalwayshastheright to convertthe bond on it has been called. The call featureis thereforeusually a way cf forcing convrsion earlier than theholderwould otherwisechoose.Sometimestheholder'scall optionis conditional ol the pri of the company's stock beingqbove a ertain level. Creditrisk plap an importantrole in the plgation of convertibles.lf credit risk is ignpred,poor pricesare obtainedbecausethe coupons and principal paments on the bond are overvalud. lngersollprovides a way of valuing convertiblesusing a model similarto Merton' s (1974)model discussedin Section22.6.17He assllmesgeometric Brownian motion for the issuer'stotal assetsand models the company's equity, its cgnvertibledebt, and its other debt as claims contingenton the value of the assets. Creditrisk is takeninso account becausethe debtholdersget repaid in fullonlyif the valueof the assetsexceedsthe nmount owing to them. A simpler model that is widelyused in practke invlves modeling the issuer'sstock price.lt is assumed that the stockfollowsgeometricBrownianmotionexcept thatthere is a probability t that therewill be a defaultin each short period of time Lt. In the eventof a dfault the stock pri fallsto zero and thereis a recoveryon thebond.The risk-neutral defaultintensitydesnedin srction variable is 22.2. The stock price processcan be representedby varying the usual binomialtree so that at each node thereis: 'of
'the
1. A probability p. of a percentageup movementof sizeli overthe next timeperiod of length Lt 16Hull and Suo test the lVF model by assumingthat all derivativeprices are determinedby a stcchastic volatilitymodel. They foundthat the model wgrks reasnably well fcr compcund cpticns, but scmetimes of givesserious errors for barrieroptions. SeeJ. C. Hulland W. Suo,S:AMethodolorfor theAssessment Model Risk and its Applicationto tlle Implied VclatilityFunction Model,'' Jozrnal of Ffntmckl and Analys, 37, 2 (June 22): 297-318 Quantitatiye 17See J. E. Ingersoll, eAContingentClaimsValuationof ConvertibleSecurities,''Journal of Fizial Economics,4, (May1975, 289-322.
592
CHAPTER 26
;. A yrobability pd of a ptrctntage down movement f sized over the next time pend of length Al 3. A probability Al, or more icuratelyl e-l that therewill be a defaultwith the stck price movinj to zero over the next time ptriod of kngth Lt -
Parameter values, chosen to match the rst two momentsof the stockprke dtribu-
tion, are:
Pu =
a
-
-A!
de u d
h
,
-
=
u:
-A! If
.1
. -
a
M
,
d
-
=
e
p
.j,)
at
,
vm
u
er-qlst r is the risk-freerate, and iqthe diviend yield the stock. q on The lifeof the tree is set equal to the lifeof the onvertiblebond.The value of the convertibleat the fmalnodes of the tree is calculated basedon any conversionoptions that the holderhas at that time.Wethen roll backthroughthe tr. At nodes where the terms of the instrumentallowconversion we test whether conversion ij optimal.We also test whether the positionof theissqercan beimprovedby calling thebonds.If so, we assume that te bonds are called and retest whethr conversion is optimal. Thisis equivalentto settingthe value at a node equal to
herea
W
=
,
maxlmintol, Q2),Q3) thatthebondis neither converted whereQ1 isthe value givenbythe rollback (assuming called value if conversiontakesplace. call and isthe price, isthe thenode), at :3 :2 nor fxample 26.1 Considera g-monthzero-couponbondisscidby company XYZ with a face ale that it can be exchanged for two sharesof company XYZ'S of $l. suppose stockat any timeduringihe9 months. Assllme alsothat it is callable for $113at any time.Theinitialstockpri is $50,its volatility is 30%per anpum, and there e; greno dividends.Th defaultintensity is 1 per year, and a11rk-free l'ates for thatin the event of a defult thebondis worth $4 all matuiities are 5%. suppose (i.e.,the recovery rate, as it is usually defned,is 4%). Figure26.2 shows the stock pri tree that can be used to value the convertible Theupper numberat eachnode isthe whenthereare threbtimesteps (Al stockprke; the lowernumberis thepliceof the convertiblebond.The treeparametersare: (.9-.1)x.25 = j j jj9 u d = qjg = g,gy j e .25).
=
=
J
=
,
.
().()5::4).25
e
=
j jjgj .
,
gu
=
g,jjjyj
pv
=
gojygy
The probabilityof a default(i.e.,of moving to the lowestnodes on the tree is :-.1x.25 = At the thr defaultnodes the stockpriceis zero an# 1 pri is 4. the bond Considersrstthe nal nodes. At nodes G and H thebond houldbe converted and is worth twicethe stock pri. At nodes I and J the bond shpuld not be convertedand is worth 10. Movingback throughthe tree enablesthe value to be calculated at earlier nodes.Consider,for example,node E. The value, if the bond is converted, is 2 x 50 = $100.If it is not cnvtrted, thenthtre is (a)a prbabilhy 0.5167that it will move to node H, where the bond is worth l 15.19, (b)a 0.4808probability .2497.
-
593
Moye on Modelsand NumeyicalPA'tlc:dTfA'd,
Figu/e 26.2 Tiee for valuing convertible.Uppernumber at each node is stockprice;lowernumber is convertiblebondprke. D 66.34 132.69
B 57.60 115.19
A 50.00 106,93
'
E 50.00 106.36
C 43,41 101
Q 76,42 152.85 H 57.60 115.19 I 43.41
F
.20
1.00
87.68
08.61
t. .
.
J
azaj
100.00
Default 0.00 40.00
Default 0.0
Default
40.00
40;0
.0
that it will mov downto node 1,where thebondis worth l, and (c)a 0.002497 pr abilitythat it t the risk-free interestmte. An estimateof the value of the derivativeis foundby obtainingmany Sample values of the derivativein this way and clculating their mean. The main problem with Monte Carlo simulationis that the computatin time necessaryto achievethe requir:d levelof accuracycan be unacceptably igh. Also, American-stylepath-dependentdrivatives (i.e.,path-dependentderivativeswhere on sidehas exerdse Qpportuni. tks or otherdecisionsto make)calmot eas bt handltd.In this siction, we show h0wthe binomialtree methods pfesentedin Chapter19 can be extendedto cope with some path-dependentderivatives. The procedurecan handle American-stylepath-dependyntderivativesand is computationally more ecient than European-style path-dependentderivatives. MonteCarlo simulation fr For the procedureto work, two conditions must be satissed: '
p
'
1. The payof fromthe derivativemust dependon a singlefunction,L of the path followedby the underlying assett 2. It must be possibleto calculate the value of F at timez + ! fromthe value of F at time z and the value of the underlying asset at time z + !.
UsingLookbackoptions lllustration
As a srstillustfaon of the procedurye consider an Americanioating lookbackput z0 . on a non-dividend-payingstock. If exercisedat timez, thispayspf the amount optlon by which the maximum stockpricebetweentime and timez excds the curent stock and Impkmentation of Convertihle Bond Models,'' 18see,e.g., L. Andersen and D. Beums calibration Joursal n/ computatosal fncace, 7, l (Winter23/4), 1-34.These authors suggest assuming that the intensityis inversdyproportional to 5', where S is the stock price an a is a positiveconsnt. default
'
approach was suggeste in J. Hull and A. White, 'Ecient Produres for ValuingEuropean and Path-Dependent Options,''Joursal n-f.nerpcfc:', 1, 1 (Fall 1993):21-31. merican 'lnhis
20nis exsmple is used as a Erstillustrationof the generalprocedureforhandlg path depenence.For a moreecient approach to valuing merican-style lookbackoptions, see TecbnicalNote 13 on the author's website.
!9!
Moye on Model and NumeyicalP?wiv?w Figure 26.3
Tree for valing an Amerkan lookhackoption.
70.% 70.% 0.
62.99 62.99 3.36 56.12
56.12
62.99 56.12 6.87 0.
56.12 4.68 50.00
50. 50.* 5.47
A
56.12 50.* 6.12 2.66
44.55
.
44.55
56.12 50. 11.57 5.45
50.00 6.38 39.69 50.% 10,31
35.36 50.00 14.64
prke is $50,the stock price volatility is 4% per ' price. Supposethat the initial stock1()B/c risk-iee l'ate interest is per anmlm, th totallife of the optionis three annum,the stock prke months,and that movementsare representedhy a three-stepbinomialtrei. Lt = 0.08333, With our usual notation this means that k = 5, c r u 1.1224,d 0.8909,a 1.0084,and p 0.5073k The tree is shown in Figure26.3.In this case,the path function>>is the maximum stock pri so far. The top numberat each node is the stock pri. next levelof numbersat eah node shows the possible maximun stock prkes achievable on paths leading to the node. The snallevelof numbers shows the values of the derivative correspondingto each of the possiblemaximum stock prices. The values of the derivativeat the snalnodes the tree are calculatedas the maximumstock price minusthe actualstock price. Toillustratethe rollbackprcedure, supposethat we are at node A, where the stock pri is $50.The maximum stock price achievedthus far is either 56.12or 5. Considerflrstthe situation where it is equal to 5. If thereis an up movement,the maximum stock pri becomes56.12and the value of the derivativeis zero. If thereis a downmovement,the maximumstock prk qtaysat 50and the value of the derivativeis 5.45.Assumingno early exercise,the value of the derivativeat A when the maximum achieved so far is 50is, therefore, .4,
=
=
*
=
=
.1,
=
=
'l'he
'of
-0.1x9.92333 = 2.66 ( x 0.5073+ 5.45 x 0.4927):
Clearly,it is not worth exercisingat node A in thesecirumstans becausethe payof
596
CHAPTER 26
fromdoingso is zero. nodeA is 56,12givest
similqrglculation frthe situgtipnwherethe maximllmvalur at e value of te derivativeat node A, ithot earlyexrcise, to bi ' .
.
..4.1.x9.98333=
( x 0.5073+ 11.57x 0.4927):
.
j jj .
In this case, earlyexircisegives a value of 6.12'and is the optimal strategy.Rolling back throughthe tree in the way we haveindicatedgives the vlue of the merican lookbackas $5.47.
Generalization The approach justdescribedis computationallyfeasiblewhen thenumberof alternative valuesof the path fuction, F, eachnodedoes n0t @ow too fast as the number of : weused, a 1ookbackoption, presents ng problems timestrps is increased.Theexgmple becausethe number of alternativevalues for the maximum asset pli at a node in a binomialtree with rl me stepsis never greater than ?l. Luckily,the approachcan be extendedto copi with situationswhere thereare a very largenumberof diferept possiblevalues of the path functionat eachnode. The basic idea is as follows.Calculationsare carrkd out at each node for a small numberof representativevalues of F. Whenthe value of thederivativeis required fUr othr values of the path function,it is calculated fromthe kown values using interpolation. The frst stage is to work forwardthroughthe tree establhing the maximum and minimumvalues of the path functionat eacb nde. Assumipgthe valur of the path functionat time r + t dependsonly on the vale of thepath funtion at timez and the undertying variable at tiinez + A!, themaximm and minimumvalues of value of the the path functionfor the nodrs at time z + Al can be calculated in a straightforward way from those for the nodes at time z. ne second stage.is to choose reprisentative valuesof tht path functionat eachnode. Thereare a number of approaches. A simple values as the maximum value,the minimumvalue, ruleis to coose the representative and a number of other valuts that are equally spacedbetweenthe maximumand the minimum.As we roll back throughthe tree, we value the derivativefor each of the representativevalues of the path flmction. .u To illustratethe nature of the calculation, consider the problem of valuingthe averageprice call option in Example24.2 of Section24.10when the payof depends initil stock prke is 5, the strike prke is 5, on the arithmetic averagestock pri. ICB/, the risk-free interestrate is the stock pri volatility is 4(j%,and the time to maturity is 1 year, For 20 time steps, the binomialtree parametersare Al = pth functionis the u = 1.0936,d = 0.9144,p 0.5056,and 1 p = 0.4944. pri. of arithmeticaverage the stock Figare26.4shows the calculations that are carried out in one small part of the tree. NodeX isthe central nodeat time year (atthe end of thefolrth timestep). NodesY and Z are the two nodes at time0.25year that are reachablefromnode X. The stock price at node X is Forwardinductiopshowsthat the muimum averagestock plice that is achievable in machingnode X is 53.83,The minimum is 46.65.(Theinitialand final stock prices afe includedwhen calculating the average.)From node X, the tree branche!to ont of the two nodes Y andZ. At node Y, the stock price is 54.68and the boundsfor the averageare 47.99and 57.39. t node Z, the stock price is 45.72and the boundsfor the averagestockprice are 43.88and 52,48. ,at
'
'
q
'
'l'he
.5,
'l'he
=
-
.2
.
597
Moye t)zl Models and NumeyicalPyoteduyes Figure 26.4
Part of tree for valing optionon the arithmtk average. S = 54.68 AverageS O?tbnprice 7.575 47.99 8.11 51.12 8.635 54.26 Y ' 5739 9.178 .
S = 5. Opucnprice 5.642 46.65 X 5.923 4904. 6.206 51.44 6.4% :3.83
Averageu
s
=4j.:2,
AveageS Z
.43.33
price Wtion 3.430 3.750 4.079 4.416
46.75
49.61 !2.48
supposethat the representativevalues of the average are chosen to be foureqlally spacedvalues at each node. Tlt means that, at node X, averagesof46.65,49.94,51.44, and 53.83are considered. At node Y, the averages47.99,51.12,54.26,and 57.39are considered.At node Z, the averages 43.88,46.75,49.61,and 52.48are condere. Assllmethat backwardinductionhas alreadybeen used to calulate the value of the option for each of the alternativevales of the average at nodes Y an Z. Valuesare shwn in Figure26.4(e.g., at node Y whec the averageis 51.12,the valueof te option is 8.11). Considerthe calculations at node X for the case where the averagi is 51.G.If the stockpri moves up to node Y, the new averagewill be 5 x 51.44+ 54.68 = 51.98 6 The value of the derivativeat node Y for tlt averagecan be foundby interpolativg betweenthe values when the averageis 51 and when it is 54.26.It is .12
Similarly,if the stockpri moves downto node 2, the new averagewill be , 5 x 51.44+ 45.72 = 50.49 6 and by interpolationthe value of the derivativeis 4.182. The value of the derivativeat node X when the average 51,44 therefore, ,
The other values at node X are calculatedsimilarly.Oncethe values at all nodes at time0.2 year havebeen calculated, the nodesat time0.15year can be considered.
598
CHAPTER 26
The value given by thefull tre for the option at timezero is 7.17. s te numberof timesteps and the mlmberof ayeragesconsideredAteachnode isincreased,the valtle of the option convergesto the correct answer. With60timestepsand 1() averagesat each node,the yalue of the option is 5.58.The analyticaplroximation for the value of the optiot, as calculated in Example24.2,is 5.62. A key advantage of the metho describedhere is that it can handle American options.The calculatioly are as we havedescribedthem exept that we test for early exerciseat each node for each of the alternatiyevalues of the path functionat the node. g practice, the early exercisedecisionis liabletodependon both the value of the path function and the value of the underlying asset.)Considerthe merican versionof the averagepri call considemdhere.The value calculatedusing the z-step tree and four averagesayeach node 7.77;with 60 time stepsand 1 averpges, the valueis 6.17. ' The approach jtst describedcan be used in a wide mngeof diferent situations. The tFo conditions that must be satisfkd were listed at the beginningof this section. Eciency is improvedsomewhatif quadratic l'ather than linearinterpolationis used at each node. ''
''
.
26.6 BARRIERQPTIONS Chapter24 presented analytic results for standrd barrier optins. Th sectionconsiders numerkal procedures that can be used f0r banier options when there are no analyticresults. In principle, a barrkr option can be valued using the binomialand trinonal trees discussedin Chapter19.Consideran up-and-out option.A simple'approachis to value this in the sme way as a regular opiin except that, when a node abovethe barrieris encontered, the value of the option is set equal to zero. Trinomialtrs work betterthan binontialtrees,but even for them convergen is vtryslow when the simpleapproach is used. A largenumber of timestepsare required to obtain a reasonably accurate result. Th reason for tls is that the barrier being by the tree is diferentfromthe true barrier.21Dehnethe izner bairier as the assumed clojer to the center of barrkr fonmedby nodesjust on theinsideof the true barrkr (i,e., the tree) and te outer barrier as tht barrier forme by nodes just outside the true barrier (i.e.,fartheraway fromthe nter of the tree).Figure26.5 showsthe innerand outer barrierfor a trinomialtree on the assumptionthat the true barrieris horizontal. The usual tr calculations implicitlyassume that the outerbarrieris the true barrier becausethebarrierconditions arefrst used at nodeson thisbarrier.Whenthetimestep is t, the vertical spaciflgbetweenthe nodes is of order X-?This meansthat errors createdby the diferencebetweenthe true barrkr and the outer barrieralso tend to be 0 f 0r der Lt On: approach to overcomingthis pmblem is to: .
1. Calculatethe pri of the derivativeon theassumptionthattheinnerbarrieris the true barrier. '
21For
a discussionof th, se,eP.P. Boyle and S.H. Lau, RBumpinzUp Azainst te Barrier with the 1994):6-14. BinomialMethod,'' Joarsal d.f Derateh 1, 4 (sllmmer
599
More t)Al Modtl and' Numerical J%cetfll?w
Figere 26.5 Barriers assumedby trinomialtrees.
Oaterbarrkr
Innr baliier -
t
r
' .
2- Calculatethevalueof thederivativeon theassumptionthatthe outer barrierisyhe tl'uebarrier. 3. 1nterpo1ate betweenthe two prices. Anotherapproachis to espre that nodes 1ieon the banier. Supposethat the initial stockprice is S and that thebanier is at H. ln a trinoml tpe, thereare threepossible movementsin the asset's zri t each node: up by a proportinal amount u; stay the same;and downby a proportional amount d, where d = 1/u. We can alwayschoose u so that nodes lie on the barrier.The conditionthat must be satisfed by a is H = Su
N
0r
ln H = lnS + N ln u for somepositive or negative N. When discussingtrinomia1treesin Section19.4,the value suggestedfor a was e''so that ln u = (r 3Al. ln the situationconsideredhere,a good rule is to chooseln u as lose as possible to this value, consistentwith the condition given above. This means that ln H lnS ln u = ,
-
N
60
CHAPTER 26 Fijure 26 6 Treewith nodeslying n banier.
:
.
barrier
where N = int
ln H
-
ln k
c 3A!
.5
+
and int/) is the integralpart of x. probabilities p., pg, and Thislead! to a tpe of the form'shownin Figure26,6. h on the upper, middle,and lowerbranchesof the tree are chosen to match the first tw0 moments of the return, so thaj 'fhe
(r -
h
=
-
-
c2/2)At
2ln li
+
o7AI
ht
2n u)2
,
p.
::
1-
(r -
2
(1nu)
,
pu =
q
-
c2/2)f
2ln li
+
c2Aj
2(lnu)2
wherepu, p., and h are the probabilitieson the upper, middle, and lowerbranches. ;
Te Adaptlve Mes Model 'fhemethods pasented so far work reasonablywellwhen the initialassetpri is not closeto the barrier.Whenthe initialassetprke is close to a barrkr, the adaptivemesh model,whkh was introducedin Section19.4,can be used.22Theideabehindthe model is that computational eciency can be improvedby grafting a fmetree onto a coarse 22sees.Figlewskiand B. Gao, ''The AdaptiveMeshModel:A New Approachto Ecient OptionPricing,'' 313-51. JoarnalofFitancial fcpn/plk., 53 (1999): '
601
Moyetm Modelsand NumeyicalPmceduyes Figure 26.7
The adaptivemesh model used to value barrieroptions.
Z
'' z ..' z'z xxN '
,'' .@* /
'e
,''
z
N X
hx
z
z
z
z
''
z
/
/
z
/
/
/
/
/
l
/
''
'' ''
x
h .
x
z
z
N N
x N. N
x
'x N
'x N
.
h
N !
'x
.
'x
xnx
.
'x
y
x
S
N
N
z'k' / z / z / z / / ze z / z / z zz / l z z 3z / z
zz
z
xN
x
.oh
z / l ,ee '' / z /
N
'x
'x
z
zzz/ ''z' /
h h
Nv
xN
'x 'x
h y h N h x h Barrier
tree to acleve a more detailedmodeling of the asset pri in the regions of the tree wheteit is needed most. To value a barrieroption,it isusefulto havea finetreeclose to barrkrs.Figure26.7 The geometryof the tree is armnged so that nodrs li illustratesthe designof the tree. probabilities the barriers.The on branchesare chosen,as usual, to match the flrst 9n tw0 moments of the processfollowedby the underlying asset. The heavylinesin Figure26.7 are the bmnchesof the coarsetfee.The light solid line are the fmetree. We frst roll back throughthe coarse treein the usual way.We thencalculatethe value at additionalnodes using the branchesindicatedby the dottedlines.Finallywe r0ll back throughthe finetree.
ASSETS 26.7 OPTIONS ON TWO CORRELATED Another tiicky numerkal problem is that of valuing Amerkanoptionsdependenton two assets whose prkes are correlated.A number of alternativeapproaches havebeen suggested.Tls sectionwill explain threeof these.
' Variables Transforming It is relativelyeasy to construct a tree in threedimensionsto Dpreseyt the movements of two uzcorrelated variables. The procedure is as follows.First, constructa twodimensionaltree for each variable, and,then combine thesetrees into a single three-
602
CHAPTER 26
dimensionaltyee.The probabilitieson th branchesof the three-dimensional tree are the product oi-ihecorrepondij probabilities on the tw-dimensionalirees.Suppose, forexample, that the variable; am stock priesi 5'l and .SZ. Eachcan be ripresented in twodimensionsby a Cox,Rpss,and Rubinsteinbinomialtree.Assumethay SL hgs a 1 pj of probability pj of moving up by a proportional amountl1 and a probability proportional Jl by Suppose down further tha amount moving' a k has a probability h of moving up by a projortional mnunt :2 and a probabilhy 1 ,2 tree ther are ofmovingdownby a proportional amountk. In the three-dimensional node. probabilities branchesemanatingfrpm four ' eac are: ne -
Considernext the situation where neu t ra1processes are:
and
,%
t.?1)5'1dt +.c1.%dz3
-
=
Supposethat the risk-
are '
dzh (r :.2 (r =
.correlated.
k
:2)5'2dt + c25'2Jz2
-
andthe instantaneouscorrelation betweentheWiener processes,dz3 and dh, is p. This meansthat Jln (r :1 l !24dt+ c! tkl =
.1
d ln k
-
-
(r
=
qz
-
2/2)
-
dt + c2 dh
'wo new uncorrelated variables can be defmed:z3 xl
=
c2 ln 5'1+ cl
.:2 c2 ln =
.l
cl
-
ln k ln k
Thesevariables followthe processes
#a1 =
citr
-
c1 -
:x2 (cztr :1 =
-
-
j
c1/2) + cltr
2 2)
cj/
-
cjtr
-:
-
qz qz
-
-
lljdt + clc2 c22/2))dt
2(1 + pldzg
+ cjcz 2(1 p) dgs -
wheredh and dzB are uncorrelatedWiener processes. The variables xj and can be modeledusing two Separate binomialtrees.In time A!, pi of increasingby hi and a probabilhy 1 pi of decreasingby hL. xi has a jrobabih'ty The variables f and pi are chosen so that the tree givis correct valuej for the frst two momentsof the distributionof and x2. Becausetheyareuncorrelated, the two trees tree, as already described.At eth can be combined into a single three-dimensional of node the tret, ,I and k can be calculatedffom zj and using the inverse .:2
-
.xl
.:2
Derivative Using the ExplicitFinite 23This ideawas suggestedin J. Hull and A. Wllitq securitie,s Diserence Method,'' Journal of Fittancialand Quantitatiye xncly, 25 (199):27-1. Eevaluing
603
Mofe n Modelt and NumeyicalPrpcdtffz/w relationships
.
+ ' ,%= exp 1 2v) .x
xl
.:2
.
(j
an
.
s
2 xaxp
.:2 -
2c1
The procdure for rolling backthrougha threv-dimensional treeto value a derivativeis analogousto that for a two-dimensional tree.
Using a Nonretanjular Tree Rubinsteinhas suggesteda wayof buildinga three-dimensional treefor two correlted stock prics by using a nqnrectanglar arrangment of the nodes.24 From a node tsl S, whre the flrst dtock price is and the second stockprice is k, there is a 0.25.chanceof moving to each of the following: 1%
,
5'zl,
(ilR1'
where and
: c
D
s2:),
(1R1,
tSldl
Id1 =
expltr
-
l
-
d1
expltr
-
l
-
=
=expr
q,
-
=
expltr qz
=
expltr q,
=
exjltr
-
-
-
qj
i20,
' cl/2)
-
-
l
c?/2)2
-
4/2)Al+ cetp
4/2)2 -
(natp
+ c1X) -
c.lG) +
1
4j
-
-
1 p2))
-
1 p2))
+ /2)l czuVtp -
(i2J1 i%D) ,
4/2)2+cetp
-
'
,
-
-
1 p2)) -
Whenthe correlation is zero, this method is equivalentto constructingseparatetres for .%And k using tht alttrnative binomialtrtt construction mttbod in Section19.4.
Adjustig the Probabilities A third approach to buildinga three-dimensional tree for and k involvesflrst assumingno correlation and then adjusting the probabilities at each node to re:ect the correlation.25The alternative binomialtreeconstfuctionmethod for each of and k in Section19.4is used. This method hasthe propertythat a11probabilities are When the two binomial trees are combined on the aspmption that there is no correlation,the probabilities are as shoFn in Table26.1.Whenthe probbilities are adjustedtc refkct the correlation, theybecomethoseshownin Table26.2. ,1
1%
.5.
26.4 MONTECARLOSIMULATIONANDXMERICANOPTIONS MonteCarlo simulationis well suited to valuingpath-dependent options and options wherethere are many stochasticvariables. Treesand 6nite disrnce methodsare well 24 SeeM. Rubinstein, sReturn to Oz,'' Rk, November(1994): 67-70. 25This approach waS suggestedin te context of interestrate trees in j. Hljll'and A. White, tNllmerical Procedure,sfor ImplementingTerm structure ModelsII: Two-FactorModels,''Joarsal t)./Derivates, Winter (1994):3748.
6t4
CHAPTER 26
Combinationofbinomialsasjllming
Table 26.1
n0
Correlation.
k-move
ul--ppe
Up
Dowz
.
0.25 0.25
Up Dowz
0.25 0.25
'
suited to valuing merican-style options.What happenj if an option is both path dependentand American? Whathappensif n Americanoptiondependsoh several stochasticvariables? section 26.5explaineda way in which the binomialtree approach can be modifed to value path-dependeht optionsin some situations.A numberof researchershaveadopteda diferentapproachby searchingf0r a way in which Monte opti0ns.26This section eiplains Carlosimulationcan be used tp value American-style two alterhativeways of proeding. '
The Least-squaresApproach In order to value an American-styleoptionit ij necessaryto choose betweenexerising and continuing at each early exercisepoint. The valpe of exercijng is norpay y easy to provide a way of determine.A pumberof resegfchersincludingLongsta- and schwartz . dtermlning the valu of continuing when Monte Carlo jimulation is used.gy Thif approachinvolvesusing a least-squaresanalysisto dtermipe the bept-ft relayionslp betweenthe value of conning and the values of relevant variables at eah time an earlyexercisedecisionhas to be made. 'I'h approachis bestillustratedwith a numerkal We use the one in the Longstaf-schwartz pgper. example. stockthat can be considera 3-yearAmericanput option dn a ngn-dividend-paying j exercisedat the end of year 1, the end of year 2, and t e end of year 3.The risk-freerate compounded). ne current stock piice is l. and the is 6% per anmlm(continuously stlikeprice is l.1. Assumethasthe eightpaths shownin Table26.3are sampledf0r the stockprice, (This exampleisforillustrationonly; in prcticemny more paths would be .
.
Table 26.2
Combinationof binomialsassuming of p. correlation
s'l-plpye
k-move
Up
Dowz .25(l
Up
p) + p)
0.25(1+ p) 0.25(1 p)
-
.25(1
Down
-
26Tilley was theflrstresearcherto publish a solution to the problem. seeJ. A. Tilley, valuingmerican ofActaaries, Societ. 45 (1993): Options in a Path simulation Trazsactions Modely'' 83-104. P.Jte 27 see F. A. Longstaf and E. 0p tions by simulation: Valuing American A simpk Leasts.scwrtz, 2001):113-47. squles Approach,''iepfew ofknancial s'llk', 14, 1 (spring -
.(
0'
More tm Models and NumericalProcedures '
'
TAble 26.3
'
.
- - .. -
-=-
..
-
-
.
.
.
Sampltpaths for put optiontxamplt,
Pat
l, l. l. ll1. 1.00 1.
1 2
3
1
l :=F2
l=
' 1.9 1.16 1.22 0.93 1.11 0.76 0.92 0.88
1.8 1.26 1.7 0.97 1.56 0.77 0.84 1.22
1.34 1.54 1.3 0.92 1.52 0,90 1.1 1.34
l
t=
=
sampled.)If the optioncan be exercisedonlyat the 3-yearpoint,it providesa cash iow equalto its intrinsicvalue at thatpoint.Thisis shownin thelastcolumnof Table26.4. point,the option holdermpstdecide If the put optionis in tht moneyat the z-year exercise. option Table26.3showsthatthe isin the money at the z-year whetherto point f0r paths 1, 4, 6, and 7. F0r thejepaths,we assumean apprkimate relationship: .3,
'
2
,
P = a + bS + cS
ppint and y is the value of continuing, where S is the stock prke at the z-year observations on s are: 1.8, 1.7, 0.97, io point.0ur back the :ve l-year discounted 28.4 thi correspondingvalues fpr P are: FromTable 77and .2%-.6X1 .%-'6X1 and The values of a, b, and c that minimize .18d-.6x1 r '
.
,
.
.7:-.6x1
,84.
.
:
)
.,
r
5
(Pf a -
-
i=1
bSi cSi2)
2
-
where Si and Pk.are the th observation on S and P, rspectivly, are a b = 2.983and c so that the best-t relationslp is -1.813,
=
P
Tle Patb
1 2 3 4 5 6 7 8
26.4
,
*
+ 2.9835 1.813:2
-1.7
=
-
Cashfows if exercke only at the 3-yearpoint. t= 1
l
=
2
l
=
.
.
.
k
.
.
.
.
.
.
.
.
.
.
.
.
0.07 0.18 .
0.20 0.09
.0 .
3
.
-1.7,
=
06
CHAPTER 26
Cash flowyif exerciseonly possible at 2- and 3-yearpoint.
Taile 26.5 Patb
t= 2
t= 1
l 2
l
.
.
3
=
.
.() .
.
.
(),()()
3
4 5 6 7
9.13
.
.
.()
(). ().()
.
0.33 0.26
.
().
8
0.07
.()
.
.
.
..
point ofcontinuingfor paths 1,3, 4, 6,apd7 of0.0369, Thisgivesthe value at the z-ye>r
0.0461, 176,0.1520,and 0.1565,respectively.FromTable26.3the value of exercising point is 0.33,and0.26.Tltismeansthat we should exerciseat the z-year for paths 4, 6, and 7. Table26.5summarizesthe cashiows assuming exerciseat either point or the 3-yearpoint for the eight paths. the z-yeat Considernext the paths that are in the money at the l-yearpoint. Theseare pths 1, Tbte'E26.3 the values of S for the paths are 1.9, 0.93,0.76, 4, 6, 7, and 8. From respectively. From tble 26.5, the corresjonding continuation values and 0.88, .13e-'6X1 33e-0.6x1 2&-.6x1 and back to t :: l are discounted respectively.The least-squaresrelatioship is .1
.2,
.3,
.13,
.92,
.,
y
=
, .
2.038 3.335:+ 1.3565-2 -
This gives the value of continuingat the l-yearpointfor paths lx 4, 6, 7, 8 as 0.139, 0.1092,0.2866, 175,and 0.1533,respectively.FromTable24.3the value ofexercising Tltismeansthat we should exerciseat the is and0.22, spectively. 0.34, l-yearpoint for paths 4, 6, 7, and 8. Table26.6 summaiizes the cajh iows assuming that earlyexerciseis possibleat a1lthreetimes.The value of the option is determinedby discountingech cash ;ow back to time zero at the risk-free rate and calculating the .1
.1,
Table 26.6 Path
1 2 3 4 5 6 7 8
.18,
.17,
optiop. Cash sows trom l
=
1
t= 2
l=
.
.
.
.
.
.
0.17 . 0.34 18 . 0.22
. . . . . .
3
0.07 .ti . . . ().()()
67
MoyePN Models and NumeyicalPyoceduyes
mean of the results. It is + g jyy-t)a()fxl + g yjy-.6xl + g jg:-.6xl + g )p-.6x1
-.6x3
.l.(.7: 8
.
.
.
.
)
g j jjj
=
.
Brcausethis is greater than 0.10,it is ot optimal to txircise $heojtion immeiatel. This method can be extended in a number of ways.If the option ca beexerciqeat points anytimewecan approximateits va1ue by consiering a largeplmberof pxercise relatins assmed to be ltip betweenP an s can be (justas a binomialtree does).The complkated. For example we could assume that P is a cubic rather than a, lore function of S. The methpd ca be ujed where the arly exerdse ecision quadratic dependsqn setil'al state variables: A fnctionAl form for th relationsltipbetwn P and the varlablesis assllmedand te pammeters are estimatedusing theleast-squaro approach,as in the examplejustconsidertd. '
The ExerciseBoundaryParameterizationApproach j' .'
A number of researchers, such as Andersen,have proposed an alternativeapproat wheyethe early exercise boundaryis parameterized ad the opmal values of tht are determinediterativelyby sfartingat the end of the life of the optn parameters working backward.28To illustratethe approach, we coptinuewith the put optbn and and assume that the eight paths shownin Table26.3havebeensampled,In example be parameterizedby a criticalvake tls case; the early exerciseboundaryat timet can s't) S'tj. price S, time is below If the at t asset we exerciseat time1; ifit is above of of is 1.1. lf thestk pri isabove ,#(l)we do not exerciseat time1. The value = ed of option's when do the life) e not txtrce; if it is belw 1.10we t 3 (the 1.10 of consider determination the We exercise. now lessthan0.77.Theoption is not exerciseat Supposethat we choose a value of of the option at the z-year value of paths. point forthe point for the The any thez-year 18:=(6Xl ,7e-'6Xl 2(k-.6x1 ()g9:-.6x1 Cight paths is thn , = 9.77. of theseis 9.0636. S uppose qnd respectively. The average next that point for the eight paths is the) The value of the option at the z-year g9:-.6xl 18:-.6x1 -.6xl and respectively.The average g g gg g ?? g 0.7: equals 0.97,1.7, and 1.8, the avemge of theseis SimilarlyFhen point is0.1032,0.0982,0.0938,and valueof the option at the z-year *(2) This analysisshows that the optimal value of (i.e.,the one that maximizesthe average value of the option) is 0,84. (More precisely, it is optimal to choose < 0.97.)When we choose this optimal value for the value of the 0.84% 0.0659,0.1695, 0.33,0.26, optionat the z-yerpoipt forthe eightpaths is respectively. The qverage value is and < 0.76the option is not exerciseat the If We now move on tp calculate l-yearpoint for any of the paths and the value ai the optio: at the l-yearpoint is .132:-'6x1 = 0.0972.If 0,76,the value of the option for each of the eiyt ,659e-'6X1 1695:-0.06x1 9.34 aths at the l-year point is value of ojtion respectively. the is and The avemge Similarly .2&-.06x1 equals 0.88,0.92,4.93, and 1.09thr averagevalue of theoptionis0.1283, when and 0.1228,respectively.The analysisthereforeshowsthattheoptimal .1202, .*43)
,#(2).
.*42)
.
,
.
,
-
,
.
,
.
,
,
.
,#(2)
.,
.,
.
,
,
.
,
.
,
,
.
.,
,
.84,
,#(2)
.813.
.
.gf3,.respectively.
,*(2),
.*(2)
,,
.,
.,
.132.
.,
.*(1)
,*41).
'i*41)
=
/
.,
.,
4
.
j
.18.
.
)
.,
,*(1)
.1215,
28seeL. Andersen,$A SimpleApproach to thePrking of BermudanSwaptionsin theMufactor LIBOR Market Modely''Joarsal of CompatatiosalFintmce,3,2 (Winter 2):
1-32.
608
CHAPTER 26 Value Of
&*41)
is 0.88.(M0reprecisely,it is optimal to hoose0.88%
,*41)
1
resimmediatelyafterthe coupon date.The current stock price is $25and the stock price volatility is 25%.No dividendsare paid on the stock. The risk-free interest rate is 5% with continuous compounding. ne yield on bondsissud byABC 7% with continuous compoundingandthe recoveryrati is 3%. (a) Use a three-steptree to calculate the value pf the bond. (b) How much is the conversion option worth?
Moyeon Models and Numeyal Prpcipzrd.
613
.(c) What diftrtn dots it makt to tht valut of tht bond !nd the yalue pf the conversionoption if the bondis callable any timewithin $he rst i jearsfor T,l15? (d) Explainllowyour anlysis would chanjeif therewere a dividendpaymentof $1on the equity at the (-month,l8-month,and3-month points.betailedcalcvlationsare n0t required. Hizt: Use equation (22.2) to estimate the averagedefaultintensity)
.*
..
'
.
C H
P T E
'
'
.
Mafiinjls
and Measures
#
Up to now interist rates havebn assumedto be constant whin valuing options. In tls chgpter, tltis assumptknis relaxed in preparation for valuing irterest rate derivativesin Chaptefs28 to 32. The risk-neutral valuation principle statesthat a derivativecan be valued by (a)calculatingthe expected payof on the assumptionthat the expected Dturn from the underlyingassetequals the risk-freeinterestrate and (b)discountingthe expectedpayof at the risk-free interestrate. Wheninterestrates are constant, risk-neutral valuation providesa well-defmed and unambiguous valuation tool. When interest rates are stochastic,it is less clear-cut. Whatdoesit mean to assumethat the expected return Qn the underlying asset equals to the risk-freerate? Doesit mean (a)that each daythe expectedreturn isthe one-dayrisk-freerate, or (b)that each year the eipected return is period the expectedreturn isthe s-year the l-year risk-free rate, or (c)that overa s-year rate at the beginningof the period? What oes it mean to discounttxpected payofs at the risk-free ite? Can we, for exapple, discountan expectedpayof realizedin year 5 at risk-free rate? today's s-year chapter In this we explain the theoreticalunderpinnings of risk-neutral valuation wheninterestrates are stochasticand sho that there are many dirent risk-neutral worldsthat can be assumed in any givensituation.Wesrstdefmea parameter knownas the market price of risk and showthat the excessreturn over the risk-free interestrate earnedby any derivativein a shortperiod of timeislinearlyrelated jo themarketpris of risk of the underlying stochasticvariahles. Whatwe will refer to as the traditiozal rk-zeatral world assumes that all market prices of lisk are zel'o, but we will fmdthat cther assumptions bout the market price of risk are useful in some situations. Martizgales and measares are critical to a full understanding of risk neutral valuation. A martingale is a zeto-drift process. A measure is the unit in which we prices. result A key in this chapter will be the eqaalest martizgale value security measareresult. Tls states that if we use the price of a tradedsecurityas the unit of measurementthenthereis some market priceof fiskfor wlch a11securityprices follow martingales. This chaptr illustratesthe powerof the equivalentmartingalemeasureresult by uing. 16.8) to the situation where interestrates are it to extend Black's model (seesection stochasticand to value options to exchangeone asset for another. Chapter28 uses the result to understand the standardmarket models for valuing interestrate derivatives, 'stchastic
615
CHAfTER 27
616
Chapter29 uses it to value somenonstanard derivatives,and Chapter 31 uses it to developthe LIBORmarket model.
27.1 THEMARKETPRiCEOF RI5K We start by considering the propertiesof derivativesdependenton the value of a single variable0. Assumethat the process followedby 0 is d0
'
%* mdt + tdz
(27.1)
wheredzis a Wienerpross. Theparameters m and s are t'h expectedgrowth l'ate in 0 We assume that theydependonly on 0 and time 1. and the volatilityof 0, respectively. The variable0 need not bethe pri of an investmentaet, It could be something as far markets as the temperaturein the center of NewOrleans. removedfrom snancial Supposethat fj and are the prices of two derivativesdependentonly on 4 and 1. Thesecan bc options or other instnlmentsthat provide a payof equalto some function of e at some futuretime.Assumethat duringthe me period under consideration apd provide no income. Supposethat the processesfollowedby and are j
dh and
'
df
.
= Jzl dt + o'1dz , t.
--.? =
'
/12dt +
,.2
dz
ttlz'' in theseprocessesmust be where/zl, /z2, cl and 12 are functionsof e and 1. The only equation becauseiyisthe the samedz as in sour gf the uncertainty in the (27.1) pricesof and and can be related using an analys Similar to the Black-scholes The pris analysisdescribedin Section13.6.Thediscreteversionsof the processesfor h and h arc ,
.
= pzlJl + ci :2/2 t + 12./.2 = !
z z
(27.2) (27.3)
We can eliminateihe z b formingan istantaneouslyrisklessportfolio consistingof Of the secondderivative. If 17is the value of the c2./2of the lirst derivativeand Ortfolio then P -2cl
,
17 @2) =
and Substitutingfrom equaons
rl
=
(cllJ2
-
c2
-
cl
th and (27.3), (27.2)
becomes
Jzzcl/i/zlt n t/zlcz/i/z =
-
! The analysis can be extende,dto derivavesthat provide
(27.4)
'income
(seeProblem 27.7).
(27.5)
617
Martingaks and Met7xre, /
.
.
riskless,it must earn the risk-freerate. Hen, Becausethe portfolio is instantaneously AI:
rn ht
=
'
.
and (27.5) Substitutinginto this equation fromequations (27.4) gives '
p 1o'2 /z2c1= rc2 rcl -
-
or #'l
F
-
/12 -
=
c1
1*
F
(:11r.t;)
2 .
dpends cnly on theparamettrsof tht Note that tLelefbhandsidecf equation (27.6) rightuhnd sidedependsiqnly followed and the by process on the prameters of the processfollowedby Defne as the valye of each sidein equation (27.6), so that .
gl
-
r
J1
=
-'Z /12 r
=
J2
Droppingsubscripts,equation (27.6) dependent showsthatif J isthepri of a derivative onlyon 0 and t with df p dt + o' dz (27.7) -
J
then
=
p
-
r
@
(27.8)
=
The parameter isknownas the marketprice ofrisk of 9. It can bedependenton both9 and !, but it is not dependenton the nqture of thederivativeJ. 0ur analyskshowsthat, that f0r no arbitrage, p r)/c must at any given timebe the samefor a11derivatives only # !. dependtnt and on are It is worth noting that c, which we will refer to as the volatilityof J, can be either If the volatility,s, of p k positiveor negative.It isthecoecient of z in equation (27.7). positiveand J is positivelyrelated to 9 (sothat JJ/J9 ispositive),c is positive.But if J is negativelyrelated to 9, then o' is negative.The volatility of J, as it is traditionally dehned,is 1c1. The market price of risk f 9 measuresthetpde-ofl betweenrisk and return that are madefor securitiesdependenton 9. Equation(27.2) can be written -
p
-
r
=
c
(1:.9)
The variable o' can be looselyihterfretedas the quantity of p-rk present in J. 0n the right-handside of the equation,the quantityof g-rk is multiplied by the price of g-risk.The left-handsideis the expectedreturn, in excessof the risk-fr interestrate, is analogousto the capital that is required to compensate for th risk.,Equation(27.9) which pricing model, relates exptcted the asset excessreturn on a stockto its risk. This chapterwill not be conrned with the measprementof the market pri of rk. Tls willbe iscussedin Chapter33 when the evluation of real options is considered. Chapter 5 dtinpished betreen investmentassets and consumptionassets. An is boughtor sold purely for investmentpurposesby a investmentasset is an asset nllmber ofinvestors.Consllmption signifcant assets are heldprimarilyforconsumption. 'that
18
CHAPTER 27 Equation (27.8) is true fora11investmentssetsthat provide no incomeanddependonly variable b itself'happensto be such an assit, then on b. If the m-r --
J
But, in other circumstances,thisielatinship is not necessarilytrue. Example27.1 Considera derivativewhose pri is pisitively relted to the prke of oi1 and variables. Supposethat it provides an expected dependson no other stochastic returnof 12%per annup andhas a volatilitypf 2% per annum. Assumethat the risk-freeinterestrate is 8% per nnum. It followsthat the marketprice of riskof oi1is 0.12 0.08 = .2 '
-
.2
Note that oi1is a consllmption assetrather than an investmentasset, so its market by settingp eqcal to the price of risk cannot be calculated from equaop (27.8) exjected feturn froman investmentin oil and tr equalto the volatilityof :il prices.
Example27.2 Considertwo securities,both of which are positively dependenton the 9-ay interestrate. Supposethat the flrjt one has an expectedreturn of 3% per amlum anda volatilityof 20%per annum, apd thesecondone ha$a volatilityof 30%per annum.Assumethat theinstantaneousrk-free-rate of interestis % per aqnum. The market prke of interestrate risk is, using th expected feturn and volatility forthe frst security, 0.03 0.06 -
-.15
.2
=
the expted return fromthe second From a rearrangement of equation (27.9), securityis, therefore, 0.06 0.15x 0.3 0.015 or 1.5%per annum. =
-
Alternative Worlds The process followedby derivativeprice J is df
=
gfdt + o'fdz
The value of Jz depends9n the risk pyeferens of investors.In a world where the market price of risk is zero, l eqlals zero. Frol equation(27.9) Jz = r, so that the processfollowedby f is JJ = rJ dt + d'JJZ We will refer to this as the traditionalrisk-zeatral world. enable other worlds that are Other assumptins about the market price of lk, internallyconsistentto be desned.From equaon (27.9), ,
619
Martingales and Measures
tht . so
df
(r+
=
c)JJ! +
CJdz
(1:.1g)
The market prk of risk of a priable determinesthe growth mtes of a11securities dependrnt the variable. As we movefrem one marketpri of risk to another, the exjectedgrowth rates of security prkes change, bt theirvolatilitiesmmainthe snme. Tis is a general property of variables fplkwingdifusion prosses and wasillustrated in sectilm 11.7.Choosinga p>rticularmqrketprice of rk is also referredto as dehing value of th marketprke of rk correspontlsto the sreal theprobabilitymeasare. some world'' and the growth rate,sof stcurityprices that are observed in practice. .n
w
.
.
'
.
.
STATEVARIABLES 27.2 SEVERAL Supposethat n variables,
1, 2,
.
.
.
%,followstochasticprosses
,
of theform
i mi#!+ si Jzj bi =
f0r i 1,2, =
.
.
.
,
(2:.11)
n, where the dzi are Wiener processes. The parameters mi and si gze
expectedgrowth rates and volatilitiesand may be functionsof the oiand time.The appendixat the end of this chapter provides a versionof lt's lemmathat covers functionsof severalvariables. It shows that the process for the price, J, of a security that is dependenton the f hs the form
df -J
' =
pdt
oj dzi
.!-
(2:.12)
j=l
ln this eqation, ;i, is th expectedreturnfromthe Security and tn dzi isthe component of th risk of tls return attributable to %. The appndix at the end of the chapter Shows that /1
.
#' - r
--
'
j=
1.
(2:,13)
foj
where j isthe market price Of risk f0r %.Tilisequaiion relatestheexpted excessreturn isthe particular that investorsrequire on the security to the j and tn. Equatin (17.9) = case of tls tquationwhen n 1. The termkioj on the right-hand sidemeasuresthe extent that the excess return required by investorson a securityis afected hy the there is no efect; if fc'i > investors dependen of the security on %.lf kio'i= requirea higherreturn to comptnsate themf0r the risk arisingfrop f ; if kio'i< the dependene of the securityon oicauses investorsto require a lowerreturn thanwould situation occurs when the variable hastheefect of fc < otherwisebe the cse. rather risks incrtasingthe than reducing in the portfolio of a typicalinvestor. ,
,
'
'*'
,
.
'fhe
fxample 27.3 A stock price dependson threeunderlyingvariables: the price of oil, the pce of gold,andtheperformanceof a stockindex.Supposethatthemarketpris of rk respectively.suppose alsothat the cj f0r these variables are and .2,
-.1,
.4,
20
CHAPTER 27 correspondig to the three va/ables havebeenestifactorsin equation(27.12) rispictively.'The excessmturnon the stck overthe matid as and risk-freerate is .2 x 0.05 + x x 0.15= 0.06 .5,
.1,
.15,
,1
.1
.4
-
or 6.9% per annum. lf variables other than those considered afect the stock price, this msultis still true provided that the market pri of risk f0r each of thes: other variables is zer. Ekation
is closelyrelated to arbitmgeprking theory,developedby stephen (27.13)
Rossin 1976.2 n e continupusltimeversion of thi capitl asset plicingmodel (CAPM) cn be rezarded as a parcular case of the equqtbn. CAPMarpes that an investor pquires excessreturns to compensatef0r any risk that is correlatedto the risk in te return fromthe stockmarket,but requires no excessreturn f0r other risks. Risksthat ari correlated with the returnfromthe stockmrket are referxd to as systematic;other risks are referredto as zozsystematic.If CXPMis true, then j is proportional to the corrilation betweenchanges in bi and the retmn from the market. When bi is uncorrelatedwith th return fromthe market, f is zero.
27.3 MARTINGALES A martmga ' le is a zero-drift stochastic process.3A variable0 followsa martingale if its processhas the form do c dz =
wheredz is a Wienerprocess. Thevariable c may itseltbe stochasc. It can dependon b and other stochasticvariables.A martingale has the convenient property that its expectedvalue at any future timeis equal to its value today.Tls means that E(r)
h
=
whereh and r denotethe valuesof b at tiles zero and T, Dspectively. T0 understand this reult, note that over a very small time intervalthe changein 0 is normally distributedwith zero mean.The expectedchangein b overany verysmalltimeinterval is thereforezero. Thechangein betweentime and time T is the sum of its changes overmanysmalltimeintervals.It followsthat the expectedchangein b betweentime and time T must also be zero.
The EquivalentMartinjale MeasupeResult supposethat and g are thepris of tradedsecuritiesdependepton a singlesoumeof uncertainty.Assumethat the securitiesprovideno incomeduringthetimeperiodunder considerationand dene ( = J/. 4 The variable # isthe relativeprke of J with respect .f
2 seeS.A. Rss, Arbitraje Thery f Capil Asset Pricin'' Joarzal of 1976):343-62. mecember . 3 Moreformally, a sequenceof randm variablesZ9,Z! . is a martingaleif,for a11i > S'ne
'ctzpplfc
,
.
.
.
:9) Ezi 1 . m-1 whereE denotes expectation. 4 Priblem27.8 extends the analysis to situationswhere the securitiesprlwide income. .#1..1
,
m-2, ,
=
Teory, 13
,
621
Maytingales and Measuyes
of f in units of g l'at er than dollars. The security price g is referred to as the nameraire. . The eqaivalent martingale measare result shows that, when there are no arbitrage oppcrtunttks, ( is a martingak for somt choi of the marktt pri of risk. What is more,for a givennumerre security g, the snme choi f the market pri of risk . makes4 a martingale for a11securitiisJ. nis choi of the marketprice of risk is the of g. In other words, whenthe market pfice of risk is set equal to the volatility volatility ofg, the ratio fjg is a martingalefor all securitypris f. To prove this reslt, supposethat the volatilitiesof f and g are c/ gnd cj. From in a world where the market pri of risk is c;, ' equation (27.10),
to g. It tan be thoughtof as measuring the pri
+ o'ffdz df (r+ c;c/).J#! =
=
UsingIt's lemmagives Jln
dt + o'ggtk
gjg
dg (r+
f (r+ cgc/ =
o'lflljdt + CJ dz
-
dt + o.gd; dln g (r+ $/2) =
sothat
Jtln
or
J
ln
-
) (cgc/ c2//2 =
d ln-J g
-
(J/ =
2
0)
-
gljj
-
-
2
dt +
dt +
(c/
-
(c/
-
o'gjdz
o'gjtk
'
.
It's lemmacan be used to detenine the proceqsfor fjg frol the processfor 1n(/7g): (2:.14) d() tcy 0)1: dz g martingale and the equivalent martingale -
-
This shws that fjg is a proves measure result.Wewillrefer to a worldwherethe market priceof risk isthevolatilitycg of g as a risk neatral with respect to g. worldthat is forward Because J/ is a martingale in a worldthat is forwardrisk neutral with respect to it followsfmm'the result at the beginningof illissection that ,
= Ea -
g
gr
0r
f
=
g :JM g
(27.15)
gr
where Eg denotesthi expected value in a world that is forward risk neutml with respectto g.
27.4 ALTERNATIVECHOICESFOR JHE NUMERAIRE k
-
Wenow present a tplmber of exnmplesof the eqmvalentmartingale measure result.Te flrstexampleshowsthat it is consistentwith thetraditionalrisk-neutralvaluation result
22
CHAPTER 27 .
'
'
'
'
.
usedin earlierchapters.The other exalples preparethe way fr the valuation f bon,d 28. opnns,inierestrate caps, and swap options in chapter
'
Money Market Account as the Numeraire The dollarmoneymarketaount is a surity that is worth $1at timezero and earns theinstani aneous risk-freerate r at any giventime.sThi fariabler maybe stochastic.If we set g equal to the moneymarktt account, it grows at rate r so that dg = rgdt
(27.16)
The drift of g is stochastic, but the volatilityof g is aro. It followsfromthe results in Section27.3 that J/g is a martingale in a world where the market price of risk is zero. This is the world we defned earlieras the tradional risk-neutral world. From equa-
tion(27.15),
J
=
J
gk
--!.
(27.17)
#z g
'
where denotesexpectationsin the traditlonalrisk-neutralworld, '
In this case, g
=
1 and
T
fz
redus to s that equation (27.17) 0r
=
eJ,
rdt
T
l =:jjg
=
rvsl
(27.18)
#(:-Frjr)
(27.19)
where is the average value of r betweentime and time T. Tls equation showsthat one way of valuing an interestrate derivativeis to shnulate the-short-terminterestrate r in the traditionalrisk-neutralworld. Oneachtrialtheexpectedpayof is calculatedand discountedat the average value of the short rate on the snmpled path. When the short-term interestrate r is assumed to be constant,equation (27.19) reducesto = e.vzg(Jr) .
J
r the risk-neutral valuation relatipnshipused in earlier chapters. Zera-coupon Bond Price as the Numeiaire Defne Pt, T4 as the price at time l of a zero-coupon bondthat paysofl-$1 at time T. We now explorethe implicationsof settingg equal to Pt, T). Let denoteexpectations in a world that is fomard risk neutral with respect to Pt, T). Because gives gr =F PCT,T) l and g v P, T), equation (27.15) 'r
=
'
./'
=
P,
T)rr(/r)
(27.28)
5 ne moneyaount isthe limltas ! appfoacheszero of thefollowinssecurity,Forthe rst short period of timeof lenzthA!, it is invistedat theinitialLt period rate; at timeAl, it is reinvestedfor a fuler period of timeA! at the new t period rate; at time 2A!,it is azain reinvestedfor a fuler periodof time ! at thenew A! period rate; and so on. 'The money market acounts in other cuaendes are desned analosously to t:e dollarmoney market aount.
.
624
Martingalesand Meaures
and (27.19). In equation (j7.19), the . Noticethe diferencebetweenequations (27.20) equation expectations the iscounting, discountingis insid $e oyrator. In (27.20) as representedby the P, Tj term,is outside the expectationsoperator. The use of Pt, Tj as the nllmeraire thereforeconsiderablysimpves thingsfor a securitythat provides a Payof Solely at time T. Considerapy variable 9 that is not an interestrate.?z A forwardcontract op b with maturityT is dened asa contract that pays ofl-gr K at timeT, where9r isthe value 9 at time T. Desne f as the value of this forwardcontract. From equation (27.2), -
T The forwardprice, F, followsthat
P, T)ISr(9r)
=
'l -
of b is the value of K fpr whichj't equalszero. It therefore P(0, T)ESz(r)
or F=
=
F1=
'z(9r)
(27.21)
Equation(27.21) showsthat theforwardpri of any variable (expt an interestrate) is its expected future spot price in a world that forwardrisk neutral with respet to Pt, Tj. Note the diserene here betweenforward pris and futures prices. The 16.7 shoWsthat the futurespri of a variable is the expected argumentin section future spot price in the traditionalrisk-neutralwprld. Equation (27.20) showsthat any securitythat provides a payof at time T can be valuedby calculating its eipected payof in a worl that is foiwardrk neutral with respectto a bondmaturingat tim T and discountingat the risk-freerate for maturity shows that it is correct to assume that the expectedvalue of the T. Equation (27.21) underlyingvariables equal their forwardvalueswhen compung the expectedpayof.
lnterestRates when a BondPrice is the Numeraire For the next result,defmeR(t, T, T*) as theforwardinterestrate as seen at timet forthe * T. (For Period between T and T* expreed with a compounding period of i the interestrate is expressedwith semiannualcompoupding; example,if T* T if T# T = it is expressedwith quqrterlycompounding;and so onk)Theforward 1, of a zero-coupon bondlastingbetweentimesT and T%is rice time as seenat P -
.5,
-
=
.25,
-
,
%
:(f, T#) Pt, T)
A forwardinterestrate is defned diFerentlyfromthe forwardvalue of mostvariables. A forwardinterestrate is the interestl'ate impliedby the corresjonding fomard bond Price.It followsthat Pt, T*) 1 (1+ (T# TjRt, T, T#)) Pt, Tj -
6 The analysis given here doesnd apply to interestrates becauseforwardcontract.sfcr interestrates re desned iferently fromfomard contracts fcr cther variables. A fomard interestrate is the interestrate implkd by the correspondingfomardbondpri,
624
CHAPTER 27
so that
1
Rt, T, T*) =
T#
F(!, F) T Pt, T#)
-
-
1
0r
Pt, T) Pt, T*)
1
Rt, T, T y) =
T
Setting
,
-
-
Pt, T , )
T
j
J
=
rE'(l,T)
T-.
Pt, T*)1
-
and g = Plt, T*), the eqtvalent martingale measure result shows that Rlt, T, T*) is a martingalein a world tat is forwardrisk neutral with respect to Pt, T#). Thij means that :(, T, T#) = ES.LRCT,T, T#)) (27.22) where E denotesexpectations a world that is forwardrisk neutral with respect to Pt, T ). interestrate betweentipes T and T* as seen The varible R(, T, T*) is the forward T*) realized interestrate betweentimes T and T*. is the at time whereas RT, T, thereforeshowsthat theforwardinterestrate betweentimesT and T* Equation(27.22) equalsthe expected future interestraie in a world that is fomard risk neutral with respect to a zero-coupon bondmaturing t time /#. This result, when combinedwith that in equation (27.2),will be critical to an understanding of the standrd market modelfor interestrate caps in the next chapter. 'in
.
,
Annuity Factor as the Numeraire For the next application of equivalentmartingale measure arpments, consider a swap startingat a futuretime T with paymentdates.at times T, T2, TN. Dene T T. Assumethat the principal underlyingthe swap is $1 Supposethat the forwardswap rate (i.e.,the interestrate on the flxedside that makes the swaphavea value of zero) is .(!) at time ! (! T). The value of the Exedside of the swap is ,
.
.
.
,
=
.
,(!)(!) where N-1
(!)
(6.+) r,)F(!, 6.+j) -'
=
f=
Chapter7 showedthat, when theprincipalis added to the payment on thelastpayment side of the swap on the initiationdate equals the date swap,the value of the soating underlyingprincipal.It followsthat if $1is added at time Tx, the:oating sideis worth $1 at tlme T. The value of $1reived at time TN Pt, TN). ne value of $1at time Tt is Pt, Tg).The value of the ioating sideat time t is, therefore, Pt,
zg) Pt, zx) -
Equatingthe values of the flxedand ioating sidesgives
.(!)(!)
=
Pt, T)
-
'(!,
TN)
625
Martingales and M:t7wr:, ' Or
stj
Pt, T)
=
Pt, Tx)
-
(l)
(27.23)
The equifalent martingalemesyre result can be applied by setting J equal to Pt, T) Pt, Ts) and equal to (t).Tll leadsto -
stj
=
'ZE-ITII
(27.24)
in a worl thatisfomard riskneutralwith pspect to (!). whereL'&denotesexpectations Therefore,in a world thatisforwardrisk neutralwith respectto (t),theexpectedfuture swaprate is the cmrent swap rate. shows that For any security, J, the result in equation(27.15) .
J
gx' 1
tl'x (r)
=
(27.25)
will be criticgl to an This result, when combined with the resnlt in equation(27.24), understandingof the standardmarketmodelfor Epropeanswaj options in the next chapter.
FACJORS 27.5) EXTENSIONTO SEVERAL The results presented in Sections27.3and 27.4can be extended to cover the situation whentherearemanyindependentfactors.? Assumethatthereare n independentfactors and that the prosses for J and in the tmtlitionalrkk-neutl'al world are
df
=
ot,ifdzi
rfdt + .izrl
and k
o'g,ikdzi
r dt +
=
f=1
It followsfromSection27.2that other internallyconsistentworlds can be dehnedby setting n
N
df
=
kiaf,i
r+ =1
and
fdt +
d
f=1 4
n =
cJ,iJJ&
o'q,idzi
kiag,i df +
r+ f=1
=1
wherethe f (1%i %nj are the n marketpricesof risk. Oneof theseother worlds isthe real world. The defnitionof forwardrkk neutrality can be extended so that a world is fomard risk neutral with respect to where j = coj for a11i. It can be shownfromIt's lemma,using thefactthat the dzi are uncorrelated,that the pross followedby J/ in ,
1 The independen conditionis not critical.If factors are not independent theycan i)eorthozonalizei
i
CHAPTER 27
this world has zeto drift (seeProblem27.12).The rest of the results in thr' last two onward) are thereforestill tnle. sections(frm equation(27.15) '
27.6 BLACK'SMODEC REVISITED ,
Section 16.8explainedthat Black'stodel is a popular tool for pricingEuropean optionsin terms of the forwardor futuresprice of the underlying asset when interest rates are constant, We are now in a position to rela te constant interest rate assumptionand showthat Black'smodel can be used to pri Europeanoptions in terms of the forwardprice of the underlying asset when interestrates are stochastic. Considera Europeancalloptionon an asset with strikeprke K that lastsuntil me T. From equation(27.2),the option'sprict is givepby c = F(, Tlfrlmaxts'r
))
K,
-
(1:.26)
wherexr is the asset pri at time T and fr denotesexpectationsin a world that is forwardrisk netral with respect to Pt, T). DefmeF and Fr as the forwardprice of the asset at i'ime and time T for a contract maturingat tie T. BecauseSz = &, c = F(, Tlfrlmaxtf'r
-
))
K,
Assllmethat F'r islognormalin the world beingconsidered,with thestandarddeviation of 1n(&) equal to cyzf. Thiscouldbe becausethe forwardprke followsa stochastic processwith constantvolatilitya'F. ne appendix at the end of Chapter13showsthat =
-
where
fr(&)#(#1)
frEmaxt/'r#, )) k
-
##/2)
(2:.25
lnlJ2rt&l/v + CI.T
=
cF W
#2cu .lnEz,(,,)/m -
csr
c,vcf
fr(&) From equation (27.21),
where
c
=
=
llrts'rl
F(,
=
>). Hence,
KNkjj T)E>)#(#1) -
#1=
lnlfk/'l + c,2r
#2=
1n(/$/m c,2z
(2:.28)
csx// -
Similarly, j'
c,vC/
p = F(, TjfKN-kj
-
1$14-#1))
(2:.29)
wherep isthe price of a Europeanput option on the asset with strikepfice K and time ' to matunty T. Thisis .Blacks model. It applies to both investmentand consumption assetsand, as we havejustshown,is tl'uewken interestrates are stochasticprvided that F' is the forward asset pri. The variable c, can be interpretedas the (constant) () of the forwardasset price. volatility
627
Maytingalesand Measuyes '
,
27.7 OPJION TO EXCHANGEONEASSETFORANOTHER donsider next an option to exchangean investmentasset worth U for an investment that the 24.11.suppose asset worth P. This has already bn (liscussed in section volatilities t U and P are cr./and cy and the coeciet of correlation betweenthemis p; Assumesrstthat the assetsprovideno incomeand choose the numeraire securityg to gives be U. SettingJ = P in equation(27.15) P
-1
P
=
UEv
(27.39)
uT
whereEv denotesexpectationsin a world thatis forwardrisk neutral with respect to U. The variable J in equation (27.15) can be set equal to the value of the option under = consideration,so that Jr maxtpr Uz, ). lt followsthat -
maxtpr -
h
U E(/
=
Uy,
)
Ur
0f
).
v
-
' The volatilityof V!U is
sr/gmaxt--p1, -
T
jq
(27.31)
Problem 27.14),where
(s
-
=
(/2
c +c
2r uc y
=
From the appendixat the end of Chapter13,equation(27.31) becomes
f
where dL
ggsr/t.-lpp -
)x(.$) xttszlq
-
tzT
ln(y IU ) +2z/2 xf
ud
k
=
-
dWr
Substuting fromequation(27.30) gives
J
=
P#@1)
U#@2)
-
(27.32)
Thisisthe value of an optionto exchangeone asset for another when the assets provide no income. Problem 27.8showsthat, when f and g pmvideincomeat l'ate qf and qg, equation (27.15) becomes = he
qg-qTE
f
...1
g
gr
and (27.31) become This means that equations(27.30) P -L
: r? Ur
.xplz
=
e
qv
.-Y U
628
CHAPTER 27 and
J
=
pqnrcgsggmaxt-) 1,
jj
-
and equation(27.32) becomes
Jq e-qvvgxtgj) =
-
e-qvl'v
x(:2)
with dk and k beingredesned as
dj
=
ln(7/I&) + qu
t?y +d2/2)r
-
&AIY
and
k 4 =
-
W
'$
.
Tls is the result given in rquation for the valle of an option to exchange one (24.3) assetfor another.
27.8 CHANGEOF NUMERAIRE In tls section we considertheimpactof a change in numeraire on the procsj followed by a mrket variable. ln a worldthat is forwardrk neutral with respect to g, the pross followedby a trade surity J is N
n
d.f
=
cj,jcj.f j'dt +
r+ f=1
=l
cJ,fJdzi
Similarly,in a world that is fomard rk neutral with resped to anotheysecurity h, the processfollowedby J i8 n
d.f
=
?1
r+
l'dt+
c,ftyi izzz1
cJ,//lzf izzz1
.
Fhere h,i is the fth component of the volatilityof h. Theeflct of moving froma world thatis forwardrk neutral with respect to g to one of changing the numerairefromg t hj thatisforwardrisk neutral withrespectto h (i.e., isthereforeto increasethe expectedgrowth rate of the price of ahy tmdedsecurity J by /1
'
.
(cJ, i. j
-
cj,jlcj,f
j=1
Considernext a variable p thatis a functionof the prices of tradedsecuries (where p is Defne necessarily plic of fth itself). tradedsecurity the the c, f as not componentof a thevolatilityof p. From It's lemmain the appendixat the end of this hapter, it is to calculatewhat happensto thepross followedby p when thereis a changt possible in numeraire causingthe expectd growth rate of the underlying traded seculitks to change.It turns out that the expectedgrowth rate of p responds to a chapgein numerairein the samt way as theexpectedgrowth rate of the pris of tradedsecrities (seeProblem 12.6 for the Situation wherethefe is only 0ne stochasticvariable and
629
Martingaksand Measuyes .problem27.13 for the general case).It increase;by
tc f=1
a 11=
.
,
j
jlc'p
o'g
-
,
. ,
,
(27.33)
f
. .
.
,
.
.
.
. .
.
.
.
Defne 1, = hjg and cmj as the fth componet of the volatilhyof u). FromIt's lemma (seeProblem27.14), o.i ::F:c i o'g,i becomes sothat equation (27.33) -
a11
o
=
,
i
Jp,f
f=1
(27.34)
is equivalentto Wewill refer to 1, as the nameraie ratio. Equation(27.34) Jp
(27.35)
Wvo
=
where cp is the total volatility of p, cb is the total volatility of u), and p is the 8 instantaneouscorrelationbetweenchangesin p and 1,. Tllis is a surprisinglysimple Dsult. The adjustment to the expectedgrowth rate of a variable p when we change from one numeraireto another is the instantaneous iovariance betweenthe perntage changein p and the pexentage changein the numeraireratio. This result will be used when timing and quanto adjustments are consideredin Chapter29.
OF JRADITIONALVALUATIONMETHODS 27.9 GENERALIZATION Whena derivativ drpendson the values of variables at more than ne point in time,it is usually necessaryto work in thetl-aditinal fisk-neutl'al world wherethe numeraireis the moneymarket account. Technical Note 20 on the author's website considersthe situationwhere a derivative'dependson variables 0i followingthe processesin equation (27.11). It extends the material in Chapter13by producing thediserentialequation that must be satisfed by the lt shows thqttraditionalrk-neutl'al valuation methodscan beimplementedbychangingthe growth l'ate of each %frommi to mi kisi and using the short-term risk-freerate at timet as thediscountrate at time!. If 0i isthe prke of a tradedsecurity that providesno income,equation,(27.9)shws thatchanging the growth l'ate from ti to m kih is equivalent to setting the retum on the security equalto the short-term risk-free l'ate. (This is as expected.)However,the 0ineed not be the prices of traded securitiesand some may be interestl'ates. .derivative.
-
-
S
To s
this, note tat the changes
p
and ul in p an ul in a short perio of timeht are givenby vki.'-'l 4.
p ul
=
=
sincethe dzi are uncorrelate,
.
.
.
.
.
.
+
it followsthat Eikj)
Ecp,, wki.''' Ecu,j
when i # j. Also, fromthedesnitionof p, we have
=
S(p) S(!$ ppcpcw Stp When termsof higherorder tan t are irored this leads to tpl
=
pocw
=
-
Eoi
c,,f
630
CHAPTER 27
In the traitional risk-neutral world, the expectedpric of 9f at time T is its futures pricef0r a cotract maturingat me T. Whenfuturescontractson f a!e available,it is thereforepossibleto estimatethe processfollowedby bi in the traditiohalrisk-neutral .worldwithout estimating the ki explkitly.Thiswill be discussedfmtherip the context of rel optionsin Chapter33.
SUMMARY The market priceof risk of a variable deMesthetrade-ofsbetweenrisk and return for traded securities dependenton the variable. Whenthereis one underlyingvariable, a derivative'sexss return overthe risk-fee rate equalsthemarketpriceof riskmultiplied bythe variable'svolatility.Whentherealemanyunderlyingvariables,theexsj return is the sum of the market priceof risk multipliedby the Yolatility for eachvariable. A powerfultool in the valuation of derivativesis risk-neptral valuatbn. This was introducedin Chapters11and 13.Theprincipleof risk-neutralvaluation showsthat, if we assume that the world is risk neutral when valuing derivatives,we get the right answer-not just in a risk-neutral worid, but in a1l other worlds as well. In the traditionalrisk-neutral world, the market pri of risk of all variables is zero. This chapterhas extendedthe principleof risk-neutralvaluation. It has shown that, when interestrates are stochastic,there are many interestingand useful alternatives to the traditionalrkzneutral world, A martingale is a zero drift stochasticpross. Any variable following a martingale has the timplifyingpropertythat itsexpectedvalue at any futuretimeequals its value today.The equivalentmartingale measure lesult showsthat,if g is a securityprice,there is a worll in which the ratio J/g is a martinjalefor all surity pris J. It turns out that,by appropriatelychoosingthe numeraire secmy g, the valuafion of many interest rate dependentdtrivativescan be simplifed. This chapterhas used the equivalent martingalemeasure result to extendBlack's model to tllt situation where interestl'ates are stochasticand to value an option to exchangeone assetf0ranother.In Chapters28to 32,it will b useful in valuinginterest rate delivatives.
FU THER READING Baxter,M., and A. Rennii, Fizaial Cblcglxy.CambridgeUniyersity Press,1996. Cox, J. C., J. E. Ingersoll, and s.A. Ross, SAnlntertemporalGeneralEquilibrinmModelof 363-84. AssetPrices,''Ecometrica, 53 (1985): Due, D., byzamc
Garman,M.,1$AOtnral Theol'yof AssetYpluationUndtr DifusionStateProcessesr'' Working Paper 5, Universityof California,Berkeley,1976. '
Harrison, J.M., and D.M. Kreps, sMartingalesand Arbitragein MultiperiodSecmities 381-408. Marketsr''Joarzalo-ffctzpznfc Teory, 20 (1979): of Harrison, J. M., and s.R. Piiska, ttMartingllesand Stochasticlntegals in the ContinuousTradin'' Stocastic Processestml TheirApplicatiozs,11 (1981): 215-69. 'l-heol-y
631
Maytingalesand Measuyes
and' Problems(Answersin theSolutionsManual) Questiohj 27.1.H0w is the market price of risk dened f0r a
variable that is n0t the prke of an
FSSCC 27.2.Supposethat the marketpriceof rk f0r gold is zero. If the storagecosts are 1% per annum and the risk-free rate of interestis 6% per annum,rhatis theexpectedgrokth rate in the priceof gold? Assllmethat gold providesno income. 27.3.Considertwo securities both of whkh are dependenton the samematket variable. The returns fromthesecuritiesare 3%and 12%. Thevolatilityof theflrstsecurityis expected 15%.Theinstantaneousrisk-freerate is49/.Whatisthevolatilityqfthesecondsecurity? 27,4.An oil company is set up solely for the purposeof expbring for 0il in a rtain small areaof Texas.Its value dependsprimarilyon tw0 stochasticvriables: the pri of oil andthe quantityof provenoil reserves.Discusswhethertheparket pti of risk fr the secondof these two variables is likelyto be positive,negative,or zero. 27.5.Deduce tlkedferential equationfor a derivativedependenton the pris of two nondividend-payingtraded securitiesby forminga rkless portfolio consisting of the derivativeand th8 tw0 traded securities. 27.6. Supposethat an interestrate x followsthe process YVCZIXCSt
'
'
,
#.x= c/
-
x) dt + c
idz
furtherthatthe market piice of risk 'where c, xg, and c are posive consnts. suppose for is What is the pross for in the traditionalrk-neutral wrld? .x
.x
.
provides incomeat mte , equatin (27.9) becomes security provides Jz + q r c. llizt : Form a new J* that no incomeby assuming reinvested a1l in incomefrom is the xthat J.) J
27.7.Provethat, when the security f -
=
27.8.Showthat when f and g provie incomeat ptes qy andqg,respedively,equation (27.15) becomes
J() =
.......IEIIC:).--;,/!?he ql-qj y g
l;!-'-gz
*
klizt: Form new securities J# and that provide no inme by assllmingthat a11the incomefrom J is reinvestedin J and a11theincomein is reinvestedin .)
expectedfuturevalue of an interestrate in a risk-neutralworld is greater thanit is in the real world.'' Whatdoesthis statementimplyaboutthe market price of lk for (a) an interestrate and (b)a bond pri. Do y0u thinkthe statementislikelyto be true? Give teasons. 27.10.The variable S is an investmentassetprovidingincomeat rate q measuredin currencyA. It followsthe process ds Ihsdt + osdz 27.9.
ilrrhe
=
in the real world. Dening new variables as necessary,#vethe proceysfollowedby S, and the corresponding mprket priceo lk, in: (a) A world that ij the traditionalrk-neutral world f0r currency (b) A world that is the traditionalrisk-neutralworld for.currencyB (c) world that is forwardrisk neutral with respectto a zeroucouponcurrency bond maturingat time T
-
632
CHAPTER 27 (d) A worldthat is forwardrisk neutfal with respect to a zero coupon currency B bond maturinc at timt T. Explain the diferen betweenthe way a forwardinteresttate is desnedandthe way the forwardvalues of other kariablessuchas stockprices, commoditypris, and exchange rates are defned.
27.12.Provethe resnlt in Stion 27.5tht when '
n
df
an
t+
=
jcsf
i=1
dg
oy,ifdzi i=1 n
;
. =
n
fdt +
r+
fcg,f
gdt +
o'g,igdzi i=l
i=1
withthe dzi uncorrelated, f/g is a martingale for
f
=
o'g,i.
in Section27.7. (27.33) 27.14. show tha,twhen / and and are each dependenton tt Wienerprocesses,the fth 27.13.Prove equation
=
componentof the volatilityof is the ft,hcomponent of the volatilityof minus the th componentof the volatilityof Usethis to prove the result that if qv is the volatilityof Hint : U and y is the volatility of P then the volatilityof UjV is cry+ cl zpcgcy. Use the result in Footnote7.) .
-
AssijnmentQuestions *'
27.15.A security'sprice is positivelydependinton two variables: the price of copper and the txchangt rate. Supposethat ttlemarketpri of risk for thesevariables is 0.5 ytn/dollar and respectively.lf the prke of copper wereheld xed, the volatilityof the security kouldbe 8% per annnm; if the yen/dollar exchangerate wereheldflxed,the volatiliyyof 'thesecuritywould be 12%per annum. Therisk-freeinterestrate is 7% per annum. What isthe expected rate of return fromthe surity? If the twovariables are uncorrelatedwith eachother, what is thr volatility of the security? 27.16.Supposethat the pj of a zero-coupon bondmaturingat time T followsthe process dpt, T) pppt, Tjdt + c,F(l, T) .1,
'
.
.
=
'
.
and the price of a derivativedependenton the bondfollowsthe process df
=
pyJdt + cy/lz
Assumeonly one sourceof uncertainty and that J provides no income. (a) What is the fomard price F of J for a contractmaturing at time Te? (b) Whatis the pross followedby F in a world that isforwardrisk neutral with respt to Pt, T)? (c) What is the process followedby F in the traditionalriskzneutralworld.j (d) Whatis the pross followedby J in a world that isforwardrisk neutral with respect to a bond maturing at timeT*, where T%# r?Assllmethat c; isthe volatilityof tllis bond. '
'
Matingales and JJl,sfzre,
63
27.17. a variahle that is not an interestrate: (a) In what world is the futurespri of the variable a Martingale? (b) In what wrld is tlle forward price of the variable a martingale? (c) Defning variables as necessary,derivean expressionfor the diferencehetweenthe drift of the futurespiice and the drift of the forwardprice in the traditionalriskneutral world. (d) Showthat your resglt is consistentwith the points madein Stion 5.8 about the circumstanceswhep$e futuresprice is abovethe forwardpri. .consider
*634
CHAPTER 27
APPENDIX
'
,
SUURCESOF UNCERTAINTY HANDLINGMUCTIPLE Tllis ppendixextendsIt's lemmato cover situationswhere thereare multiple sours relating the exss return to of uncertainty and provzsthe result in equation(27.13) marketpricesof risk when thereare multiple sours pf uncertainty.
It's
temmafor a Fqnctionof Several Variables
It's lemma,as presentedin the appendixto Chapter12,providesthe,pross folloked by a functionof a singlestochasticvariable. Herewe pxsent a generalizedversion of It's lemmafor the pross followedby a functionof severalstochasticvariables. suppose that a function J dependson the n variables a:j xn and time !. supposefurtherthat h followsan It pross with instantaneousdrift ai and instantaneousvariance , (1%i %nj, that is, .n,
,
.
'
.
..
.
.
.
.
,
.
.
.
.
.
Jxf
Jf
=
dt + bidzi
'
(2:A.1)
wheredzi (1%i %n) is a Wiener process.Eachai and#fmay be any fnctio'n of a11the xi and !. A Taylorseries expansion of AJ gives
AJ
=
.
n 3J
V sl
.
xi
iJ xi +
n
if'
i2J
n
VV ''-j=Lizf
t + lo ''
'
Azf AJ;
xj
-'-j=1
'
V
+ 1n '-
.
''-j=1
i2J xi J
xi A! +
.
.
.
(2:A.2)
Equation(27A.1)can be discretkedas xi = ai A! + bij Al where i is a rndom samplefrom a standardizednormal distlibution.ne correlation pij betweendzi and dzj is defmedas the correlation betweenEi and j. In the appendix to Chanter 12it was arguedthat n n Ar; = bkidt lim Ale -
A
-
-
'
.
Similarly, dt 1im xi A.'r;= bbiihj ?' b
*
'
A/-y
As t 0, the srstthreetermsin the expansionof AJin equation(27A.2)are of order t. A11other terms are of hkher order. Hence, .->
B
df
=
JJ
aj
JJ + dt + )
n
f=1
/=1
2
bt
f
i=1
n
i2J i.z5 xj
yi#j pv a
for dxifromequation (27A.1) Thisisthe generalizedversionofIt's lemma.substituting gives
df
=
n f=1
JJ ai + JJ + j
J.n
t
s
2 f=1
n
j2
J
n
bibjpv a-y
bxiJ.'r.j jzz1
j.j
gj yi gxi aj
4:A.3)
35
Matingales and Apcw?w
For an alternative generalizationof It's lemma,su/pose that J dependson a single variablex and that theprocess for involvesmore than one Wienerpross: .z
bidzi
#.x= adt + f=1
In tltiscase,
JJ
/ V =
f
x+
!+
V
2J
jJ 2
2
j
x +
Jx2
2
j2y
JxJ!
xj.j
.,.
biki l
.z= cAl+
i=1
and 2
11m . xf Ale
m
M
j ij jpij dt
=
f=1 j=1
where,as before,pij is the correlation betweendzi and dzj Tllisleadsto 2
J./' 3J m JJ + + J.'ta J l z J.xz iuzj
?&
.t
Jy
=
'
.
. .
.
bbpv f j y.j
a
.
+
'J
s
m
bidgi
(27A.1)
j.j .
..y. .
Finally,conider the more gineml case whereJ dependson variablesxi (1Ki Kn) and bikJzj
dxi= ai dt + )=l
A similar analysisshows that n
J.f
j
=
f=1
a, +
xj
:
jj -
i#l
n
g2
:
m
m
:
iylyl xi ylyl -
z,
-
2=1 jc j
-
(aj
jsj
j.j
bibbjlpudr+yl f.j
jj
M
.'tk
j.j
lbibdh (27A.!)
The Return for a Security Dependent pn MultipleSaurces
af Uncertaint
section 27.1 proved a result relating return to risk when there is one sour f for the situationwhere thete uncertainty.We now prove the result in eqpation(27.13) are multiple sources of uncertainty. Supposethat there are n stochastkvariables followingWienerprocesses.Consider Stcchastic variables. n + 1tradedsecuritieswhoseprices dependcn scmecr a11cf the < De6nefj as the prke of the jth security(1%j %nn + 1).Assumethat no dividends r otherinme is paid bythe n + 1tradedsecurities.'It followsfromthe previoussection '
9 This is not restrictive.A non-dividend-payingsecmity can always e otained froma dividend-paying security y reilwestingthe dividendsin the security.
636
CHAPTER 27
that the securitiesfollowplpcesses of the form '
'
.
.
.
n
dh
pjh dt +
=
oh
dzi
(2:A.6)
f=l
and n Wienerprocesses,it is possible to form an sincethere are n + 1tradedseceurities r isklessportfolio H using the securities.Dene kj as the amountof the instantaneously .jthseculityin the portfolio,so that n+l
n
(27A.:)
kjfj
=
/=1
.
The kj must be chosen so thqt the stochasticcomponents of the returns from the secrities are eliminated. Frm equgtn (27A.6),th means that n+1
Vlilh
(27A.8)
=
:,- IL ::::
for 1 %i %n. Equation (27A.8) consists of n equations in n + 1 unknowns (11,l2, la+1).Frm linearalgebra we knowthat this set of equations alwap has a solutionwhere not a11of the kj are zero. This showsthat the lisk-free portfolio FI can alwaysbe created. The return fromthe portfolio is givenby .
.
.
,
n+1
gn
kjpjfjdt
=
j=1
The cost of settingup the pprtfolio is
n+1
Tkii j=l
the portfolio mst earn the lisk-free
If tere are no arbitrage qpportunities, rate, so that
n+1
n+1 j=1
intrest
kjpjj
kjh
r
=
(27A.9)
/=1'
0r n+l
fhlh
/=1
r)
-
=
(27A.1)
Equatns (27A.8) and (27A.1) can be regarded as + 1 homogeneouslinear equationsin the k). The kj are not a11zero. From a well-known theoremin Enear algebra,equations(27A.8)and (27A.1) can be consistentonly if tie left-handsideof equation(27A.10)is a Enearcombinatio of the left-handside of equation (27A.8). Tilis means that, f0r all j, a
fg) -
r)
=
lkioij;
f=1
(rA.11)
637
Martingalesand Measures 0r
n p) .
r
-
(271.12)
jcjy,
=
f=1
.
f0r some j (1K i Knj that alt delendent only on the state variables and time. Droppingthe j subscript,this shows that, f0r any security f dependenton the n stochasticvariables, ,
t .
n .
.
.
Jzja + Tcifdzi ,df= . '
jzj
where
.
p
-
r
=
n
V f=1
Thij proves the result in equation(27.13).
fcj
'
k:
CH
PT 1*
.
,
.1
&'
-,
Interest Rate Derivtivks: T:e Stanard Mrkt Mdelj
Interestrate derivativeq are instrumentswhosepayofli are dependentin somewayon the levelof interestrates. In the 19808and 199s, the volume of tradingin intrest rate derivativesin both the over-the-counterand exchange-tradedmarketsincreasedrapi'dly. Many new products were developedto meet particular needs of end users. A key challengef0r derivativestraders was to find good, robust yrodures f0r prking and hedgingtheseprodlcts. Interestrate derivativesare more dlcult to value thanequity and foreignexchangederivativesfor thefollowingreasons: 1. The behaviorof an individualinterestrate is more complicatedthan that of a stockprke or an exchange rate. 2. For the valuation of many products it is nessary to developa model describing the behaviorof the entire zero-coupon yield curve. 3. The volatilitiesof diflkrentpoints on iheyield curve are dferent. 4. Interestrays are used for discountingthederivativeas well as defningits payof. '
.
This chapter consid,ersthe three most popular over-the-counterinterestl'ate option products:bond options, interestrate caps/ioors, and swap options. It explains how the products work and the standard market models used to value them.
28.1 BOND OPTIONS A bond optionis an option to buy or sell a particulr bondby a paricular datef0r a particularpri. In additiopto tradin?in the over-the-countermarket, bond options are frequentlyembedded in bondswhen theyareisskedto make themmore attractiveto eitherthe issueror potential purchasers.
EmbeddedBond Options One example of a bond with an embedded bond option is a callablebozd.This is a bond that contains provisions allowing the issuingfrm to buy backthe bond at a 639
640
CHAPTER 28 predeterminedprice at certaintimesin the future.The holderof sucha bondhas sold strike price or call prke in the option is the a call option to the issuer. predeterminedprice that must be paid by the issuer to the holder. Callablebonds cannot usually be called f0r the srstfewyears of theirlife.(Tls is knownas the lockout jeriod.) After that, the call priceis usually a decreasingfunctionof time. For exnmple,in a lo-yearcallablebond, there might be no call privilges for the srst 2 ap. Afterthatpthe issuermight havethe rkht to buythe bondback at a pri of 110in years 3 and 4 of its life,at a pricr of 107.5in years 5 and 6, at a price of 106in years 7 and #, and at a priceof 103in years I and 10.The kalue of the call option is rtfkcted in the quoted yklds on bonds.Bondswith call featmesgenerallyofer bkher yieldsthan bonds with no call featums. Another type of bond with an embeddedoption is a (attable bond.This contains provisionsthat allowthe holderto demandearly redemptlonat a predeterminedprke holderqf sucha bondhas purchased a put option on at certain timesin the futum. the bond as wellas the bond itself.Becausethe put option increasesthe value of the bond to the holder,bonds with put featuresjrovideloweryields than bonds with no put features.A simpleexampleof a puttable bon is a lo-yearbond where the holder has the right to be repaid at the end of 5 years. tTls is sometimesreferred to as a retractabk bond Loap and deposit instrumentsalso often contain embedded bond options. Fol' flxed-ratedepositwith a fmancialinstitutionthat can be redeemed example,a s-year withoutpenalty at any timecontainsan Americanput optionon a bond.(Thedeposit instrumentis ?. bond that the investorhas the rkht to put back to the snancial institutionat its facevalue at any timt.lPrtpaymentprivikgts on loansand mortgages are similarly call options on bonds. Finally,a loan commitment madeby a bank or other fmandalinstitutionis a put option on a bond. Consider,for example,the situation where a bank tuotes a s-year interestrate of 5% per annumto a potential borrowerand states that the rate is good for the next 2 months. Theclienthas,in efect, obtained the right to sell a s-year bond with a 5% coupon to thefmandalinstitutionfor its facevalue any timewithin the next The option will be exercisedif ratesincrease. 2 montlis. 'l'he
'
'l'he
European Bond Options Manyover-the-counterbond (iptions and someembedded bondoptionsal'eEuropean. The assumption made in the standard market model for valuing Europeanbond optionsis that the forwardbondpricehas a constant volatility o. This allows Black's and (27.29), modelin Section27.6 to be used. In equations (27.28) c, is set equal to ca and A) is set equal to the forwardbondprice h, so that c=
8,
p
80, T)LKN-d
=
TILFBNCdL)
KNdj
-
-
(28.1)
FBN-dLjj
(28.2)
where tf1
=
1n(&/& + c,2772
caxf
and
k
=
:1
-
c, yj
withK the strikeprice of the bond optionand T its timeto maturity.
641
InteyestRate .D:z'?2t'w; The Standayd.Mt7z':fModels 5.5, f can be calculatedung the formula From section &
; P, T) yg
=
=
(28.3)
whereB is the bondpriceat timezero and l is the prejtnt value of the coupos that willbepaidduringtheEfeof the option.In thisformula,boththe spot bondpri and the forwardbond prke are cash pris rather than quoted prkes. The relationsllip betweencash and quoted hondpris is explained in Section6.1. and (28.2) shouldbe the cashstrikeprice.In ne strikeprke K in equations (28.1) choosingthe correct value for #, te kreciseterms of the option are therefore If the strikepri is dened as the cash amotmt that is exchangedfor the important. bondwhen the option is exercised,K shouldbe set equal to this strike price.If, as is quoied pric applkable when the option is morecommpn, the strikeprice is the K shouldbe set equal to thestrikeprke plus acrued interestat the expiration exercised, dateof the option. Tradersrefer to the quoted price of a bond as the cleanpriceand the cash price as the dirt. trice.
#
Example28.1 ' Consider l-month Europeaniall option on a 9.75-yearbond with a face valurof $1,. (Whenthe option matures, the bon will have 8 years and thpt the currept cash bond price is $960,the 11 mohths remaining.) suppose strikeprke is $1,, the l-month risk-freeinterestrate is 1% per annllm, and the volatility of the forwardbond pri for a cpntrad maturing in l monthd is 9% per nnum. The bond pays a coupon of 1% per year(with paments made semiannually). payments of $5 are expected in 3 months and 9 months.(Tllisineansthat the accrued interestis $25and the quoted bond price is $935.)We supposethat the 3-monthand g-monthrisk-free interestrates ate 9.0% and 9.5% per annum, resptctively. The presnt vale of the coupon paymentsis, therefore, rcoupon
5: -.25x.9 + jg:-.75x,95 Qr
9j 4j
=
.
given by $95.45.The bondforwardprice is fromequation (28.3)
&
=
.1 x9.8333
(960 95.45):
j?9 jg
=
-
.
(a) If the jtrike price isthe cash pri that would be paid forthebondon exerdse, P, T) are F'B 939.68,K l, the parameters for equaon (28.1) =
e-'.1x(1W12) =
.92,
.9,
tu
=
=
and T = 1/12.
=
ne price of the call option
is $9.49. (b) If the strike price is the quoted price that would be paid for the bond on t month's accnled interestmust be added to K becausethe maturity exercise, ofthe option is 1 month after a coupon date.Ikis produces a value for K of l, + 1 x 0.08333 1,008.33 '
=
The values for the other parameters in equation (28.1) are unchanged (i.e., = = = and T 0.8333).The price of the 939.68,P, T) f c, optionis $7.97. .92,
.9,
=
'
Figure 28.1 showshow the standarddeviationof the logarithmof a bond's price
642
CHAPTER 28 Fijure 28.1
Standrd devitionpf logarithmof bondpri at futuretimes.
standarddeviationof logarithmof bond price
r
Bond
maturiY
Time
cangesas we lookfurtherahead. Thestandard deviationis zero todaybecausethereis no uncertainty about the bond's pricetoday.It is also zero at the bond's maturity
becausewe knowthat tht bond's pri will equal its fqcevalue at matuiity.Between today and the maturity of the bond, the standard dqviationhrst increasesand then decreases. Thevolatilityc thpt should be usedwhen a EuropeanOption on thebondis valued is. Standarddeviationof logarithmof bondprke at maturity of optiop -
Tjmeto maturityOf
.
Option
Whathappenswhen, for a particularunderlyingbond,thelifeof the option isincreased? Figure 28.2 shows a typicalpattern for c, as a functionof the life of the option. In general, declinesas the lifeof the option increases. ' .
.
fijure 28.2 Variationof forwardbondprice volatility c: with life of option whe'n bond is kept sxed.
Life of Bond
maturity
cption /
,
Intere't Rate Ddrfrlffgxl.. The s'llAltflrtf
643
MavketMtl##l
YieldVolatilities The volatilitiesthat are quoted for bod' optionsare oftenyieldvolatilitiesrather than prke volatilities,Thedurationconpt, iniroducedin Chapter4, is used bythemafket to converta quted yieldvolatilityinto a prie volatility.Supposetat D isth9moded duratioh of the bond underlying the ojtion at the option maturity,as defned in Chapter 4. The relationship betweenthe change / in the forward%nd price F'B and the changeLyy in the forwardyield yy is LF 15
a; -Dyg
0r
Lh
Ay,
a; -Dyg & yy . Volatilityis a measure of the standarddeviationof percentagechangesin the value of a variable.This equation therrforesuggeststhat tt voltility of the forwardbondpri O'B used in Black'smodel can be approximatelyrelated to the volatilityof the forward bond yield o'y by O'B =
(28.4)
Dyo'y
wherey istheinitialvalue of yy. Whena yieldvolatilityis quoted fora bondoption,the implicitassumption is usually that it will be converted to a price volatilityusing equaJ and that this volatilltywill thenbe used in conjunctionwith equation(28.1) tion (28.4), to pbtain the option's price. Supposethatthebondunderlyingacall optionwill or (28.2) lodifed hav a dmationof 5 years at optio maturity, theforward eldis8%,and the forwardyield volatilityquoted by a brokeris2%. Thismeans thatthe markit pri of when the option corresponding to thebrokerquote isthe price givenbyequation(28.1) volatilityvariable the a'B is 5x x .98
.2
.8
=
or 8% per annum. Figure28.2 shows that forwardbond volatilitiesdependon the option considered. Forward yield volatilitiesas we havejustdefmedthem ale more constant.This is why tmderspreferthem. The Bond-options worksheet of the softwareDerivaGemaccompanyingthisbook can be used to price Europeanbond options using Black'smoel by selting BlackEuropeanas the Pricing Model.The user inputsa yield volatility,which is handledin the way justdescribed.The strikepricecan be the cash or quoted strikeprice. fxample28.2 Considera Europeanput option on a l-year bond wit a principal of 1. ne Thelifeof the optionis 2.25years couponis 8% per year payablesemiannually. strike price and the of the optionis 115.ne forwardyieldvolatilityis 2%. The zerocurveis:at at 5% withcontinuous componzing.DerivaGel showsthatthe quotedpriceof thebondis 122.84.Thepri oftheoptionwhenthestrikepriceis a quoted priceis $2.37.Whenth: strikeprke is a cash pri, the prici of the optionis $1.74.(Notethat Derivauem'sprkes may not exactlyagreewith manually calculated pricesbecauseDerivaGemassumes 365daysper year and rounds times to the nearest whole mlmber of days.SeeProblem28.16for the manual calculation.)
644
CHAPTER 28
28.2 INTERESTRJE CAPS ND FLOORS L
.
,
A popularinterestrate option oferd by fmancialinstitutionsin the over-the-counter market is an interest rate cap. Intefest rate caps can best be understood by hrst consiering a qoating-ratenote where the interestrate is reset periodicallyequal to LIBOR.Thetimebetweenresets isknownas the ten. Supposethe tenor is 3 months, The interestrate on the note for the rst 3 months is the initial3-month LIBORrate; theinterestrate forthe next 3 months is set equal to the 3-month LIBORrateprevailing in the market at the 3-month poini;and so on. An interestrate cap isdesiped to provideinsuran againstthe rate of intereston the qoating-ratenote rising above a certain level.This levelis known as the cap rate. Supposethat the principalamount is $1 million, the tenor ij 3 months, the lifeof the cap is 5 years, and the cap rate 4%. (Becausethe paymtnts are made quarterly,tbis ap rate is expressid withquarterlycompounding.)The cap provides insuranceagainst the intereston the :oating rate note rising above i%. For the moment we ignoreday unt sues and assume that tere isexactly0.25year' betweeneach payment date.(Wewill cover day count issuesat the end of thissection.) Supposetht on a particularDset date the 3-month LIBORinterestl'ate is 591. The ioating rate note wouldrequire 0.25x 0.05x $1,,
=
$125,000
of interestto be paid 3 months later.With a 3-month LIBORmte of 4% the interest Would be Pz'ymellt 0.25x $1, x $1,,(i r Thepayofl-dpesnot our on the reset The cap thereforeprovidesa payof of $25,000. date when the 5% is observed:it occurs 3 months later.Thisrefkcts the usualtime1ag betweenan interestrate beingobservedandthe correspondingpaymentbeingrequired. At each reset datedlring thelifeof the cap, LIBORisobserved.If LIBORiskss than 4%, thereis no payoffromthe cap threemonths later.If LIBORis greater than4%,the payof one quarter ofthe exss applkd to the principal of $1 million. Notethat caps re usually dehnedso that theinitialLIBORrate, even if it is greater than the cap rate, doesnot lead to a payof on the srstreset date.ln our exnmple,the cap lastsfor 5 years. Thereare, therefore,a total of 19reset dates(attimes0.25, 0.75, 4.75years) 5.00years). 0.75,1., and 19 potentialpayoflifromthe caps (attimes '
.4
=
.%,
.
.
.
,
.5,
.
.
.
,
The Cap as a Partfolio af lnterest Rate Optians Considera cap with a totallifeof T, a principalof L, and a cap rate of Rk. Supposethat the reset datesare !! tn and defne !n+ = T. DefmeRj as theLIBORinterestrate forthe period betweentimetk and tk+lobservedat timetk(1%l %n). Thecap leadsto a n) of payos at time lj+I (1 1,2, ,
,
=
.
.
.
,
.
.
.
,
ljmaxtRj
-
Rx,
)
(28.5)
where j = tjml tj.l Both Rj and Rx are expressedwith a compounding frequency equal to the frequencyof resets. -
0 Day count issuesare discussedat the end of this secon.
Underthe standardmarket model, the value of the caplet is Ltkp,
RxNk !k+l)E/k#(#l) -
(18.7)
where #1= k
LnljRx) + c)2jk/2
ck/ jntnlRx)
=
-
44/2 dl
.g,y('
=
hs
f'kistheforwardinterestrate at time forthe period betweentimetk and 4+1,and h is the volatilityof th forwardinterestrate. Thisis a natural extensionof Black' model. The volatility h is multiplied by J'i beause the interestrate Rk is observedat time tk, but thediscountfactorP, lk+1)refectsthefactthatthepayof is at time4+1,not time4. The value of the correspnding ioorlet is !k+1)E;r#(-J2)
Lhp,
-
(-Jl)l
(28.8)
fxamplek8.3 Considera contract that caps the LIBORinterestrate on $l, at 8% per compounding) months for 3 startingin 1 year. Thisis a almum (withquarterly capletand could be one elementof a cap. Supposethat the LlBoR/swapzero curveis :at at 7% per anmlmwith quarterlycompounding and tht volatility of the 3-monthforwardrate underlying the caplet is 20% jer annum. The continuouslycompoundedzero rate fol.a11maturities is 6.9394%.ln equatio (28.7), Fk = Rz 1, tk = 1., lk+1 = 1.25, P, !k+1) = k = 0.25,L .8,
.7,
=
,
=
f teyestRa'te D:Ff'??Jn''?x.r.'
The Stan?aydMJFk!
:-9.969394x1.25 =
and cj
.919j
647
Models
Also,
.2.
=
2x 1/2 ln(.7/,8) #1 .2 x 1 k dj ,2
.
*
-0.7677
.2
=
-
-9.5677
=
=
sothat the caplet prke is 0.25x l,
1
.8#(-.77tj
.919(.7#(-,577)
x
=
-
$5.162
.
.
.
'
t
for the pri of th caplet. Tk isbezmuse it (Nbtethat Derivaem gives$5.146 wole number rounds and ofdays) times the nearest 365days to per year ssumes each Each caplet of a cap must be valned sepamtelyusing equqion (28.7). similarly, soorlet of a floormust bevalued separatelyusingequation(28.8). 0ne approachisto use Thevolatilitiesare thenreferredto as a difereqt volatity for each caplet (orsoorlet). yolatilities. An alternativeapproach is to use the samevolatilityfor 2 thecaples spot (soorlets)comprising any particulr cap (ioor)but to vary tllisvolatilitynemording to yolatilitkkl thelifeofthe cap (;oor). Tlle Thevolatilitiesused re t11 en referredto as/lf volatilitiesquotedinthemarketare usually;at volatilities.Howevir,manytradersliketo estimatespot volatilitiesbecausethisallowsthemto identifyunderpried and overprid capletj(ioorlets). optionson Eurodollarfuturesare ve# similarto caplets Theput (ca11) (Coorlets)and the spot volatilitiesused for capletsad foorletson 3-mont LIBORare fuiumsoptions. with thosecalculatedfromthepris of Eurodollpr frequentlycompared
Spot Volatilitiesvs. Flat Vnlatilities Figure 28.3shows a typicalpqtern for spotvolalities and ;at volatilitiesas afunctionof maturity.(1nthe case of a spot volatility,the maturity is the maturity of a caplet or ioorlet; in the case of a ;at volatility,it is the paturity of a cap or ioor) ne ;at Figure 28.3
The volatility hump.
Ca? or floor
implkdvolatility
Spot vols
Rat vols
Maturity
0 Flat
volatilities can be calculatedfromspot volatiliies and vice versa
(seeProblem28.29),
648
CHAPTER 28
Typicalbrokerimplied;at volatility quotesfor US dollar caps and Qoors(%per annum).
Taile 28.1 Lf/
l year 2 yearv 3 years 4 years ! years 7 years l years
Cap
Cap
bid
pfr
18.4
Floor '
20.00 24.25 25.00 24.75 24.50 22.75 21.
73.25
24.00 23.75 23.50 21.75
70.00
Floor
bli
pfr
18. 23.75 24.59 24.25 24.00 22.00 20.25
2. 24.75 '25.50
25.25 25.00 23.00 21.25
vclayilitiesare akin to cumulativeaveragesof the spot volatilitiesand therefcreexhibit in the volatilities is usually less variability. As indicatedby Figure28.3, a peak about p observed.The ofthehumpis at the2-to 3-year oint. Thishllmpiscbserved both when the ydatilities a!e impliedfrom option pris and whe theyare calculated fromhistoricaldata.Thereis no generalagreementon the reascn fcr the existente ofthe hump.Onepossibleexplanationis as follows.Ratesat the short end of the zerocurveare controlledby ntralbgnks.Bycontrast, 2- and 3-yearinterestrates are determinedto a largeextent bytheactivitiescf traders.Thesetradersmaybecverreactingtc the changes cbserve in the short rate and causing the volayilityof theserates tc be higherthan the of short rates. Fcr maturities beyond2 to 3 yeais, the volatlity meanreversionofinterest volatilities to declin. rates,whichlsdiscussd in Chapter30,causes Brokersprovide tablesof implied;at volatilitiesfor caps and :oors. Theinstruments the quotes are sually at the money. This means ihat the cap/:oor rate underlying t he equals swap rate for a swap that hasthe same payment datesas the cap. Table28.1 typicatbrokerquctesfortheUSdollarmarket. The tenorcf the cap is 3 months shows and the cap lifevaries from1 to 10years.The data exhibitsthe type cf shllmp'' shown in Fipre 28.3. dlhump''
Theordical
for the Model Justification
The extensicn of Black's mcdel used to value a caplet can be shown tc be internally by considering a wcrld that is fomard risk neutral with respect to a zeroccnsistent couponbcnd maturing at time 4+l The analysis in Section27.4shcws that: .
1-The current value of any security is its expectedvalue at tipe
lj+1 in this wcrld
multiplkd by the pri of a zero-oupon bond maturing at time 4+1 (see equation(27.2)). 2 The expectedvalue of an interestrate lastingbetweentimestk and 4+1equals the fomard interestrate in this world (seeequation (27.22/. The rst of theseresults showsthat, with the notation intrcducedearlier, the jri cf a capletthat prcvides a payos at time tjx.lis Lhp,
RK' t+1)'l+1Emax('l -
0))
(28.9)
649
Interest Rate Derivatives: The Standard Market Models
wherefj+1 denotesexpectedvalue in a world thatisforwardrisk neutrl with respectto a zeroucoupopbond aturing at timelj+1. Whentheforwardinterestrate underlyingthe #j) is assumedt havea ccnstahtMlatility cj, Rj isloponnal in tht world ca.p(initially weare considering,with 1n(&)= cjW.Fromthe appendixat the end of Chapter13, eqtlatio: (28.9) becomes Lhp,
where dj =
:2
=
4+I)Ef+1(')#()
RKNdj
-
+c j 1nWj+1(&)/'r!
hi
lnE:j+1(Aj)/Ar!
tttll
-
=
cj/
Ji
-
cW
The secondresult impliesthat
R+l(&) h =
Togetherthe results leadto the cap pricingmodel in equation (28.7). Theyshew that we can discountat the lj+l-maturity interestmte observed in the market todayproviding we set the expected interestrate equal to the forwardinterestrate.
Use of DerivaGem The softwareDerivaGemaccompanyingthis book can be used to prie interestrate caps and ioors using Black'smodel. In the Cap-and-swap-optionworksht select Cap/Flooras theUnderlyingTpe andBlack-European as thePricingModel.Thezero curveis input using continuously compounded rates. The inputsincludethe start and end date of the period cove/ed by the cap, the ;at volatility, and the ap frequency(i.e.,the tenor).The softwarecalculatesthe pament datesby working back fromthe end of periodcoveredbythe cap to thebejinning.Theinitialcaplet/Qoorlet is assumed to cover a period of lengthbetween and 1.5 times reglar period. suplose, fot example, that the period coveredby the cap is 1.22years to 2.80years andthe settlement frequencyis quarterly. Thereare sixcapletscoveringtheperiods 2.55 to 2.80years, 2.30to 2.55years, 2.05to 2.30years, 1.80to 2,01years, 1.55to 1.80years, and 1.22to 1.55 years. 'ettlemint
.5
The Impact of Da Count Conventions R'hefooulas we have presented so far in this section do not reQectday count conventions(seesection' 6.1 for an explanation of day count conventions). suppose 'that the cap rate Rv is expressedwith an actua1/36 day count (aswould be normal in This means that the time interval j in the formulasshould be the United states). accraalfraction replacedby cj, the for the time period betwn lj and lj+I juppose, for example, that lj is May 1 and lj+1 is August1.Underactua1/36thereare 92days betweenthesepayment dates so that h = 92/360= 0.2521.The forwardrate Fj must be expressedwith an adua1/36 day count, This mtans that we mustset it by solving .
1+ akFt
=
:(9,!j)
P,
!j+I)
650
CHAPTER 28
The impact of all this )s much the same as calulating t on an actual/actual basis convertingRK fromadual/36 to actual/actual,and calculating A-jon an actual/actual basisby solving y(g,tj 1 + j/'j = ,(g, jj-yj)
28.3 EUROPEAN SWAPOPTIONS Swapoptions, or swaptiozs,are options on interestrate swapsand are anotherpopular type of interestrate ption.Theygivetheholderthe right to enter into a rtain interest rate swap at a certn time in the future. @heholder does not, of cours, have to exercisethis right.) May large fmancialinstitutionsthat ofer interest rate swp contractsto their corporate clients are also prepared to sell them swptions or buy swaptionsfromthem.Asshownin BusinessSnapshot28.2,a swaptioncan be viewedas a type of bond option. To givean exampleof howa swaptionmight be used, consider a Qoating-rateloan companythat knowsthat in 6 months it will enter into a s-year agreementand knowsthat it will Wis to swap the ioating interestpayments for flxed interest payments to convert the loan into a flxed-rateloan (seeChapter 7 for a discussionof how swapscan be used in this way). At a cost, the company could enter into a swaptiongiing it the rkht to reive (-monthLIBORand pay a rtain flxed period starting in 6 months. It theflxed rate fterest, say 8% per annum, for a s-year rate exchanged for Eoatingon a replar s-year swapin 6 monthsturns out to be less 8% choose will the than not to exercisethe swaption and kill prr anum, company enter into a swap agreeent in the usual way. However,if it turns out to be greater than 8% per annum, the company willchoose to exercisethe swaptionand willobtain a swap at more favorabletermsthan thoseavailable in the market. Swaptions,when used in the wayjustdescribed,providecompanieswitha guarantee that the fixedrate of interestthe will payon a loan at somefuturetimewitl not exed called deferredswap. somelevel.Theyare an alternative to forwardswaps(sometimes Forwa'rd swapsinvolveno up-front cost but havethe disadvantageof obligating the companyto enter into a swapagreement.Witha swaption,the companyis able to benefh from favorable interestrate movements while acquiring protection from unfavorable interest rate movements. The diferencebetweena swaption and a forward swap is analogousto the diferencebetweenan option on a foreigncurrency and a forward contracton the currency.
Valuation of European Swaptions As explained in Chapter7 thi swap rate for a pqrticular maturity at a particular me is the (mid-market) flxedrate that would be exchangedforLIBORin a newlyissuedswap with that maturity.The model usually used to value a Europeanoption on a swap assumesthat the underlyinj srap rite at the maturityof the option lopormal. Considera swaption wherethe holderhasthe right to pay a rate sx and recive LIBOR on a swap that will last n years startingin T years. We supposethat there are m paymentsper year under the swapand that the notional principal is L. Chapter7 showedthat daycount conventionsmayleadto theftxedpaments under a swapbeing slightlydifexnt on each pament date.For now we will ipre the efect of
day countconventionsand assumethat eac flxedpgymenton the swapistheflxedrate timesLlm. The impactofdaycount conventionsis nsideredat theend oftllissection. Supposethat the swap rate for an n-yearswapstartingat timeT provesto he sz. By comparingthe cash flowson a swapwherethe xedl'ate is sz to thecashqowson a swap wheretheflxedrate is sz, it can be seen yhatthepayosfromthe swaptionconsistsof a seriesof cashqowsequal to L -maxtlr sin ) -
m
The cash qows are receivedm timesper yearfor the n years of the lifeof the swap. Supposethatthe swap paymentdatesare T1,T2, T., measurd in yearsfrqmtoday. (It is approximatelyyruethat T.= T + iln.j Each cash flowis thepayof froma call option on Jr with strike pricesz. Whereasa cap is a prtfolio of options on interestl'ates, a swaptionis a singleoption on the swap rate with repeatedpayofs.Thestandardmarket modelgivesthevalpe of a swaptionwhere the hlder has the right to pay sz as .
l-l-l, i=1
.
,
,
-szNdj 7).)E(:l)
where db
=
Lnslsz) +Tlj
cxf Lnslsz) c 2T/2 -
k
=
c;g
=
#1 c -
zj
and c istheforwardswaprate at timezero calculatedas indicatedin equation(27.23), . c/f standard isthe deviationof ln Jz). isthe vlatility of thefofwardswaprate (s0that This is a natural extensionof Black'smodel.The volatilityc is multipliedby JY. The E:h80, Tj)termisthediscountfactorforthemn payofs. Defining as the value of a contractthat pays 1/,n at timesT.(1 i %mn), the valueof theswaptionbecomes f(J(jN()
where --
=
-
szNLdj
1 mn p(0
,
1 fzzc
zj)
(28.10)
652
CHAPTER 28 rate o If theswaptiongl'vestheholderthe rkht to reive a sxed pamf fromthe swaptionis the L
-maxt-r
,
)
Jr,
-
insteadof pairy it,
u
Thisis a put option on Jr. Asbefore,the payofs alt reivd at times1. (1K i K mn). The standard marketmodelgivesthe value of the swaptionas LksxN-djj
J#(-J1))
-
(28.11)
fxample 28.4 supposethat the LIBORyield curve is satrat 6% per annum with continuous compunding. Considera Swaption that givestheholderthe right io pay 6.2% in a 3-yearswap startingin 5 years. 'I'hevolatilityof the forwardswap mte is 20%. Paymentsare madesemiannuallyand the principalis $1. In this case, =
.t2(e
-.6x6
-.6x5.5
-.6x7
-9.96x6,5
y.e
+e
-.6x7.5
+e
-.6x8
+e
)
+e
2.0035
=
A rate of 6% per annum with continuous cmpounding translatesinto 6.09% with semiannualcompounding. It followsthat, in tll example, Jc = T = 5, and c 0.2, So that sx = .69,
.62,
=
1n(.69/:62) + dj 0.2,1
.2
=
2x 5/2
=
0.1836 and
k
=
dj 0.215 =
-0.2636
-
1), the value of the swapton is From equation (28. '
.
1 x 2.0035x
..
(k69 x #4.1836)
0.062x #4-0.263$)
-
=
2.07
or $2.07.(T1s is in agreementwith the prke given by DerivaGem.)
Broker Quotes Brokersprovide tablesof impliedvolatilitiesfor European swaptins (i.e.,values of c 1)are used). Theinstruments impliedby market priceswhen equations (28.1)and (28.1 usually underlyingthe quotes are at the money.This meansthat th strike swap rate equalsthe forwardswaprate. Table22.2 showstypicalbrokerquotes provided for the . .
.....-.u-
'.
.. .
.
.-
.'
......
.
Table 28.2
Typicalbrokerquotes for Us European swaptions (mid-marketvolatilitiespernt peyannum). Swaplengtb year
Expiratios
1mOn th 3 months 6 months 1year 2 years 3 years 4 years 5 years
Us dollarmarket.Thelifeof the option isshcwn n the vertical scale.Thisvaries from 1 montht 5 years. The life of the underlyingswapat the matufity of the optio is shownon the horizontalscale.This varies from 1 io l years. The volatilitiesin the l-yearcolumnof the tableexhbit a humpsimilarto thatdiscussedfor capsearlier. s wemoveio the columnscorrespondingto options on longet-livedswaps,the hump but it becomeslesspronounced. persists
Mdel Thepretical Justificatiofor the swapt.ion The extension of Black'smodel used for swaptionscan be shownto be internally consistentby considering a world that is forwar risk neutml with respect to the annuity The analysisin section 27.4showsthat: .
1.
The currentvalue of any securityisthe currentvalue of theannuitymultipliedby the expectedvalue of securhy Pliceat time'T Valueof the annuity at timeT '
in this wotld (seeequation (27.25/. 2. The expected value of the swaprate at time T in this world equals the forward swaprate (seeequation (27.24)). '
The frst result shows that the value of the swaptionis
tklmaxtlr
)J
sK,
-
From the appendixat the end of Chapter13, thisis
f,E(.z)x(4)
-
sxsdj
where dj =
+ lnEktlzl/lr!
T/2
1n()(Jz)/JrJ c;J
T/ 2
c;T
-
k
=
=
tf:
-
y:j
o'
The Second result shows that k(Jr) equals s. Takentogether,the resllts lep to the swapoption pricing formulain equation (28.1).Theyshow that interestrates can br treatedas constantforthe purposesof discountingprovidedthattheexpectedswapraie is set equal to the forwarzswaprate. .
:
The Impact of Day Connt Conventions The aboveformulastan be mademorepreciseby consideringday count conveptions. The flxedrat for the swapunderlying the swapoption is expressedwith a day count conventionsuch as actual/365 or 30/360. suppose that T = / and that, for the applicableday count convention, the accrualfmctioncorresponding to the timeperipd between71.-1and T is ai. (For example,if Tf-l correspnds to March 1 and T. correspcndst September1 an the day count is actua1/365,ai = 184/365= .541.)
64
CHAPTER28 The formulasthat havebeenpresentedarethencorrect with the nnuity factor being
defned as
A.) c:Ji=qcja, 1 .by
As'indicated
the forwardswap rate J is givenby solving equation(27.23)
'
=
P, T)
-
8,
T.)
28.4 GENERALIZATIONS Wehavepresented threediflrent versionsof Black's model: one for bond options, one for caps, and one for swapoptions. Each of themodels.isinternallyconsistent, but they are nt consistent with each other. For exampli,when futtlrebond pris are lognormal,future zerp rates and swapiates are not lognormal;when futurezero rates are lognormal,futurebond pris and Swap rayesare not lopormal. The results can be generalizedas fllows: 1. Considerany instpment that provides a payof at time T dmendenton th.e value f a bond observed at timeT. lts current value is 8, T) timesthe expectedpayof providedthat expectationsare cAlculated in a world where the expectedpri of the bohd equalsits forwardpri. 2. Considerany instrumentthat provides a. payof at time T%dependent n the interestrate observed at me T for the perid betweenT and T*. lts current value is P, T#) timesthe expectedpayof provided that expectaons are calculated in a worldwhete the expectedvalue of the underlying interestrate equals the forward interestrate. 3. Considerany instrumentthat provides a payof in the form df an annuity. Supposethat the size of the annuityis determinedat time T as a functionof the n-year swap rte at time T. Suppgsealso that annuitylasts for r, years and paymentdatesfor the annuityare the same as thosefor the swap.ne value of the instrumentis timesthe expectedpayof per year where (a) is the current value of the annuity when payments are at the rate $1 per year and (b)expectations @re taken in a world wherethe expected future swap rate equals the forwardswap rate. . results is a generalizationof the European bond ption model; the secondis a generalizationof the cap/ioor model;the third is a generalizationof the
The
srstof thse
swapon model.
28.5 HEDGINGINTERESTRATEDERIVATItES This sectiondiscusseshowthe material on Greeklettersin Chapter17can be extended to cver interestrate derivaves. ln the context of interestrate derivatives,delta risk is the risk associatedwith a shift inthe zero curve. Becausethereare manywaysin which the zero curve can shift,many
655
InteyestRate Deyivatives:The StandaydMayketModels
deltascan be calculated. some alternativesare:
1- Calculatethe impactof a l-basis-pointpamllelshift in the zero cmve. This is termtd a DVOI. sometimes 2. Calculatethe impactof small changesin the quotes f0r each of the instruments used to constructthe zero cune. 3. Divide the zero curve (orthe forwardcurve)into a number of sections (or buckets).Calculatetheimpactof slliftingthe rates in one bucketby 1'basispoint, keepingthe rest of the initial term structure unchanged.tThis is describedin Business Snapshot6.3.) 4. Carryout a principal componentsanalysisas Qutlined in Section20.9.Calculate a deltawith respectto the changesin eachofthefrst fewfactors.The flrstdelt&then parallel, slliftin the zerocurve;the measuresthe impactof a sinall,applpximately seconddeltqmeasmesthe impct of a small twis'tin the zero curve; and so on. .
l n practice? tradersten to prefer the secnd approch. They arpe that the onlyway the zerocurve can changeisif thequotef0r oneoftheinstrumentsuse to computethe zero curve changes. They therfore fel thatit Pakessense to focuson the exposures arisingfrom changes in the prices of theseinstruments. Whensveral deltameasuresare calculated,therearemany possiblegammameasures. Supposethat l instrumentsare used to compute the zero curve an tht deltasare calculatedby considering$eimpactof changesi the quotesfor each ofthese.Gamma is a secondpartial derivativef the form 211 / xi xj, wher H is the portfolio value. There are l choices for xi and l choics for xj and a total f 55 diferent gamma overload''.0ne approach ipore cross-gammas measures.This may be and focus on the 1 partial derivaves wherei = j. Anotheris to calculate a single ofthe valueof theportfoliowithrespect gammameasureas thesecondpartialderivative to a parallel slft in the zero curve.A furtherpossibilityis to calculate gammas with respectto the flrsttw factorsin a principalcomponentsanalysis. The vega of a portfolio of interestrate derivatives measures it8expoure to volatilit changes.One approach is to calculate theimpacton thi portfolio of mking the same smallchange to the Blackvolatilitiesof a1lcapsand Etlropeanswap options.However, tls assumesthat ne factordrivesall volatilitiesand may be too simplistic.A better ideais to carry out a principal componentsanalysison the volatilitiesof caps and swap optionsand calculate vegameasurescorresponding to the frst 2 or 3 factors, Ssinformation
SUMMARY Black'smodel and its extensionsprovide a popular apprach for valuipg Europeanstyleinterestrate options.The essen ofBlack'smodel isthat the value of the variable the option is assumed to be lognormalat thematmityof theoption.ln the underlying caseof a European bond option, Black'smodelassumesthat the underlyipgbopdprice is lognormalat the option s maturity. For a cap, the model assumesthat the interest ratesunperlyingeach of theconstituentcaplets are lognormallydistribuied.ln the case of a swap option, the model assumesthat the underlying swaprate is lognormally distributed.Each of thesemodels is internallyconsistent, but they are n6t consistent with each other. j
'
1.
!
CHAPTER 28 Black's model involvescalulating the expectedpayof basedon the assumptionthat the expected value f a variable equals its forwardvalue and then discountingthe expetedpayo; at the zero rate observed in the larket today. This is the correct vanilla'' instrumentswe have considered in this chapter. for the procedure However, s we shqllseein t e next chapter,it is not coMct in a11situations, Ssplain
Manel) and Problems(Anjwersin solutioqs Quekions 28.1. A company caps 3-month LIBOR at 1% per anmlm. The principal amount is $2 million.On a reset date,3-mpnthLIBORis 12% per annum.What paymentwould tllis lead t under the cap? Whenwould the paymentbe made? 28.2. Explainwhy a swap ption can be regarded as a type of bond option. 28.3.Use the Black's model to falue a l-year European put option on a l-year bnd. Assllmethat the current value of the bond is $125,the strikeprke is $11, tLe l-year interestrate is le/a per qnnup,theboqd'sforwardpricevolatilityis 8% per annum, and the presentvalue of the coupons to bepaid duringthe lifeof the option is $1, 28.4. Explaincarefully how you woulduse (a)spot volatilities and (b):at volatilitiesto value a
cap. s-year
28.5. Calculatethepriceof an option thatcaps the3-monthrate, startingin 15months' time, it 13% (quoted with quarterlycompounding) on a principalnmount of $1,. The with quarterly forwardinterestrate fortheperiodin questin is 12% perannum(quoted compounded) is 11.5% compcunding),the ls-month rk-free interestrate (continuously per annum,and the volatilityof the forwardrate is 12% per annum. 286 Abant uses Black'smodelto pri Europeanbondoptions. Supposethatan impliedprice option on a bond maturing in 1 years is used to price a g-year vclatilityfor a s-year option on the bond. Wouldyou expectthe resultant prie to be too lligh or too low? Explain. 28.7. Calculatethe value f a 4-year Europeancall option n bondthat will mature 5 years cash bondpriceis $105,the cash priceof a fromtodayusingBlack's model.The s-year 4-yearbond with the Same coupon is $102,the strikepri is $1, th 4-yearrisk-free interestrate is l% per annum withcontinuouj compounding,and the volatilityfor the bondpricein 4 years is 2% per annllm. 28.8. lf the yield volatility for a s-year put option on a bond maturingin 1 years time is specifed as 22%, how shold the option be valued? Assumethat, based on today's interestrates the mdifkd durationof 1hebond at the maturity of the option will be 4,2 years and the forwardyield on thebond is 7%. .
.
28.9. What other instrumentis the sameas a s-year zero-costcollar where the strikepriceof the cap equals the strikeprice of te :oor? Whatdoesthe common strikepriceequal? 28.1. Derivea put-callparity relationsllip for Europeanbond options.
; '
'
InteyestRate Deyivatlves:Tke
MayketModels
'fuetyl
jjy
28.11.Derivea put-call parityrelationshipf0r EuropeanSwap options. 22.12.Explainwhy thereis an arbitrageopportunhyif the impliedBlack(:at)volatility of a capis difeent from that of a ioor. Do the broker quotes in Table 28.1present an arbitrage pportunity?
28.13.Whena bond'spri islognormalcan thebond'syield be negative?Explai your answer. 28.14.Whatis the value of a Europeanswapoptionthat givesthehclderthe right to enter int where a flxedrate cf 5% is jaid and LIBORis a 3-yearan:ual-pay swap in years lnillitm. Assme that the yield curve is;at at 5% per reived?The swap lrincipalis $1 comjounding volatility cf the swapratt is 200/:.coppare and the annumwithannual with given that DerivaGem. by mur answer '4
28.15.Suppse that the yieldR on a zercoupon bond followsthe pross dR = pdt + o'dz
wherep. and c' are functicnscf R and 1, anddzis a Witnerprcess. UseIt's lemmato showthat the vclatility of the ztro-coupon bondpricedeclinesto zeo as it approaces maturity. . 28.16.Carry out a manual calculation to verify the optionpris in Example28.2. 28.17.Supposethat the l-year, z-year, 3-year,4-year,and s-year zero rates are 60/:,6.40/:, semiannualcap with a principal of $1 at a 6.7%,6.9%,and 7%.The prke of a s-year determine: cap rate of 8% is $3.UseDerivaGemto :at volatility for caps and floors (a) The s-year collar when !hecap rate is 8% (b) ne floor rate in a zero-cost s-year 28.18.Showthat Jh + J = P2, whereJ is the value of a swap optionto pay a flxedrate of sx And receive LIBORbetweentimesT1and T2, is the value of a forward swapto receive J a fued pte f sx and pay LIBORbetweentimesT1and T2,and 72 isthe value of a swap optionto r.eceivea hxedrate of sx betweentimesT1and T2.Dedu that Jh = 72 when sK tq ualsthe current fomard swap rate. 28.19.Supposethat zero rates are as in Problem 28.17.UseDerivaGemto determinethevalue of an option to pay a hxed rate of 6% and receiveLIBORon a s-year swap startingin 1 year. Assumethat the principal is $190million,payments are exchangedsemiannually, and the Swap rate volatility is 21%. '
28.20.Describehow y0u would (a)calculate cap ;at volatilitiesfrom cap spot volatilitks and (b) calculate cap spot volatilities from cap :at volatilities.
AssignmentQuestions Europeanput optionon a Treasurybondthat currently has 14.25 28.21.Consideran s-month yearsto maturity.The currept cash bondpri is $910,the exercisepri is $900,pndthe volatilityfor the bond prke is 1% per annum.A coupon of $35will be paid by the bond in 3 months.The risk-free interestyateis % for all maturities up to 1 year. Use Black'smodel to determinethe pri of the option. Considerboth the case whefethe strikepri corresponds to the cash pri of thebond and the case where it rrespcnds to the quoted price.
658
CHAPTER 28
28.22.Calculatethe pri of a cap on the go-dayLIBORrate in 9 months' time when the infornption: principalamount is $1,0. UseBlack'smodeland the following (a) The quoted g-monthEurodollarfutures pri 92 (Ipore diferens between futures and fomard rates.) (b) The interest rate volatility impliedby a g-moth Eurqdollaroption = 15% pir =
.
anllllm.
.
(c) The current lz-month interest rate '' .
with continuouscompounding
=
.
.
.
7.5% pef'
g
almllm.
(d) The cap rate 8% per annum,(Assllmean actua1/360day count) 28.23.Sapposethat theLIBORyield curveis;at at 8%withannualcompunding. A swaption givesthehilder the rkht to reeive 7.6%in a s-year swapstartingin 4 years. Paments volatility of forward The the made y. swaprate is 25% per annumand the annua11 are milvn. is $1 UseBlack'smodel to pri the swaption.Compareyour answer Principal given DerivaGem. by withthat collar that puaranteesthat the maximum 28.24.Usethe DerivaGemsoftwareto value a s-year LlBoR-basedloan andminimuminterestrates on a (withquarterlyresets) are 7% and compounded) is currently flat at 59:,respectively.The LIBORzero curve (continuously volatility of principal is $1. 2%. Assllmethat the 60:.Use a :at =
28.25.Usetht DerivaGemsoftwareto value a Europeanswapoptionthat givesyou th rkht in fed rte of 6% and reive 2 years to enter into a skyear swap in whichyou py :oating. Cashfows are exchangedsemiannually on the swap.The l-year,z-year, s-year, compounded) ar=.5%, 60:, 6.5%, and l-year zero-coupon icterestrates (continuously and 7%, respectively.Assumea principal of $100and a volatility of 15%per lntlm. Give an exampleof h9w tl swapojtion mkht be used by a corporation. What ond optin is equivalent to the swapoption?
Adjustments A popular tFo-step produre for valuing a European-stylrderivativ
:
1. Calculte the expected payof by assuming that tht txptcte,d valut of ach underlyingvariable equals its forwardvalue. 2. Dount the expectedpayof at the risk-free rate applicble for the timeperiod betweenthe valuation date and the payof date. We frst used th procedure when valuing FRAS and swaps.Chapier4 showsthat an FRA can be valued by calculating the payof on the assllmptionthat the forward interest rate will be realized and then dcounting the payof at the risk-free rate. Similarls Chapter7 showsthat swps can be valued by calculatingcash :ows on the assumptiontlt fomard rates will be realized and iscountinjtv caslkfows at riskfreerates. Chapters16and 27 show that Black'smodd providesa generalapproachto valuinga wide range of Europeanoptions-and Black's odel is an applicationof the two-stepprocedure. The models presentedin Chapter28for bondoptions,caps/qoors, and swap options are a11examplesof the two-stepprocedure. This raises the issueof whether it is always correct to value European-styleinterest rate derivativesby using the two-stepprocedure. The answer is no! For nonstandard interestrate derivatives,it is sometimesnessary to modify the two-stepprodufe so that an adjustmentis made to the forwardvalue of the variable in the frst step.Th chapter considers three types of adjustments: convexityadjustments, timingadjustments,and quanto adjustments.
29.1 CONVEXITYADJUSTMENTS Considersrstan instrumentthat providesa payof dependenton a bondyieldobserve at the time of the payof. Usuallythe forwardvalue of a variable S is calculatedwith referen to a forwar contractthat pays ofl-Sz K at time T. It is the value of K that causesthe contzactto havezero value. As discussedin Section27.4,forwardinterestrates and forwardyklds -
659
60
CHAPTER 29 are defmeddiferently. forwardinteest rate is 4heratr impliedby a forwardzeroyiei isthe yiqldimpliedbythe forward couponbond.Moregenerally,a forwardbond bond plice. prking) Supposethat B isthepri of a bondat timeT, yz isits yield,and the (bond BT relationshipbetween and yz is Bv =
o(yz)
bond pri at time zero for a contract maturingat time T Desne f' as the forward and a as the fomard bnd yield at timezero. The desnitionof a fomard bond yield meanjthat A' =
o(y)
The function G is nonlinear. nis meansthat, whenthe expected futurebond price equalsthe forwardbond.price (sothat we are in a wprld that is forwardrisk neutral withrespict to a zero-coupon bondmatming at timeT), the expectedfuturebond yield does not eqtal the forwardbond yield. This is illustratedin Figure29.1,which showsthe relationship betweepbond prices and bond yields at time T. For simplicity,supposethat there are only three possible bond prices, :1, B2, and B and that theyare equally likelyin a world that is fomard risk neutral with respect to Pt T). Assumethat the bondprkes are equally spaced,so that B2 BL = B %. The fomard bond pri is the expected bond plice %. The bond prices translateinto threeequallylikelybond yields: yl, y2, and y3. neseare not variable y2 is te fomard bond yield becauseit is the yield equally spaced. yield is the averageof correspondingto the forWard bond price. Th expected pond yI, ya, and y3 and is clearly greater than y2. Considera delivativethat providesa payof dependenton the bond yield at time T. From equation (17 20) it can be valued by (a)calculating the expected jayofin a worldthat is forwardrisk neutral with respectto a zero-couponbond matunng at time T and (b)discountingat the current risk-freerate for maturity T. %z knowthat the expectedbond price equals the forward price in the world being considered. %z -
-
'the
,
.
'
.
Figure 29.1
Relationshipbetweenbond prices and bond yields at time T.
Bond
price
l LI I l
l 1 l II I I
l
I 1 1
l
I I
y
I I I I I l l
l
.:2
.#1
.
::.
Yield
Convexity,Timing, and
661
tltmfpAdjustments
thereforeneed to knowthe value of the expectedbond yieldwhen the expected bond
. pri equals the fomard bond price. The analysis in the appendix at the end of this chaptershows that an approximateexpressionfor the required expected bond yield is Eyj
2 27,
zycy yg .l.
=
-
G''(yp
(29.1)
s,(yg)
whereG' and G'' denotethe rst and second partial derivativesof G, fz denotes in'a world that is forwardlisk neutml with respectto Pt, T), and cy is the expectations fomard ield volatility. lt followsthat the expected payof can. be discountedat the currentrisk-free rate for maturity T provide the epected bond yidd is assumed to be y -
ly2c2z 2
y
G''(y) gs)
ratherthan y. The diferencebetweenthe expectedbond yield and the forwardbond yield G''(y)
-ly2c7z 2 y gts
is knownas a cozvexityajastmezt. It correspondsto thediferen betweeny2 and the yield in Figtlre29.1.(Theconvexity>d,juqtmentis positive bcause G'ty() < () expected andt2 (y9)> 0.) '
'
/1
'
Appliction 1: lnterest Rates consider an instrumentthat provides a cash For a rst applicationof equation (29.1), flowat timeT equal to theinterestmtebetweentimesT and T* appliedto a prindpal of L. (Thisexamplewill be useful when we consider LlBomin-arrearsswaps in Chapter 32.)Noie that the interestmte applicable to the time period betweentimesT and T* is normallypaid at time T#; hereit is assume tat it is pd early, at time T. Thecash flowat timeT is LRzz, whtre z = T# T and Rz isthe zero-couponinterest with a compoundingperiod rate applicable to the period betweenT and T* (expressed of r). 1 The variable 'z can be viewed as the yield at time T on a zeio-coupon bond maturing at time T*. The relationship betweenthe price of thisbondand its yield is -=
,
G(y) = From equation (29. 1),
fz(:z)
=
R
-
1
1+ yr 1,2c2 2
ar
Or
Sr(Rr)
=
R +
a''yj
atqy)
2 Rn
2
1+
&r
r
(29.2)
whereR is the forwardrate applkable to the period betweenT and T* and c, is the volatilityof the fomard rate. l As usual, for ease of expositionwe assume actual/actualday counts in our examples.
62
CHAPTER29 The value of the instrumentis terefore Tlfazg'
8,
+
ozj
crT j
.y
txample29.1 Considera derivativethqt provides a payof in 3 years equal to the l-year zeroy compounded)at that time multiplied liy $1. Supjose 11 couponrate (annua that the zero rate for allmaturitiesis 1% per annum with annualcompounding and the volatilityof theforwaldrate applkableto thetimeperiod betweenyear 3 = T 3, z = 1, and and year 4 is 2%. In this case, R = 3 = = value of the derivave is '(, 3) 1/1.1 ()7513.The .l,
.2,
.c,
=
.
9.7513 x lo
x 1x
12 j.1+'.
(,2()2x 1 x 3 x+,..1, 1 x1
j
or $75.95.tTis compareswith a price of $75.13when no convexityadjustment is made.)
Application 2: Swap Rates Considernext a derivativeprovidinza payof at timeT equalto a swap rate observed at that time. A swap rate is a par yield. For the purposes of calculatinga convexity adjustment we can make an apprbximation and assumethat the N-year swaprate at timeT equals the yield at that me on an A-year bond with a couponequal to today's to be used. forwatdswap rate. i'ltisenables equaon (29.1) Exmple 29.2 Consideran instrumentthat yrovidesa payof in 3 years equal to te 3-yearswap tate at that timemultipliedby $1. Supposethat payments are made annuallyon the swap, ihezero rate f0r a11maturities is 12% per annum with annual compoundinz,the volatilityfpr the 3-yearforwardswap rate in 3.years(implied from swapoption prices) is 22%. Whenthe swap rate is approximatedas the yield on a 12%bond, the relevant fupctionG(y) is C
In this case the forward yield y is .12, so that O/(y) 6''(y) 8.2546.From equation (29.1),
-2.4018
=
=
2
'rtyr) 0.12+ 12x 0.122x 0.22 x 3 x =
8.2546
=
2.4018
0.1236
and
Conrexity,Timing, and
663
Adjustments Quanto
A forwardskap rate of 0.1236( 12.3,6%)rather than 0.12should thereforebe assumedwhen valuing the instrument.The instrumentis worth =
1 x 0.1236 - 8.80 1.123 a price of 8.54 obtained without any convexity or $8.80.(This comppres w1t.11 qdjistmnt.)
29.2 TIMINGADJUSTMENTS ln this section consider the situation where a market variable J? is observed at time T and its value is used to calculate a payof that occurs at & latertimeT*.Defne: Pz : Valueof P at time T E (Pr) : Expectedvalue of Pr in a world that is forwardrisk-neutzalwit,hrespect to z #(t, T)
f'r.(Pr)
:
Expectedvalue of Pr in world thatis forwardrisk-neutralwith respect to Pt T*) 1
This is the forward price of a zero-coupon bond lastingbetwee times T and T*,
Defne:
cy : Volatilityof P cw: Volatilityof W ppw: CorrelationbetweenP and F From equaon
where
the changeof numeraire increasesthe growth rate of F by ay, (27.35) tp
=
pvwcpo'w
(29.3)
Thisresult can be expressedin termsof theforwardinterestrate betweentimesT and T*. Defne:
;:
Forward interestrate forperiodbetweenT and T#, expressedwith a compoundingfrequencyof m '
tu : Volatilityof R The relationship betweenF and R is F
=
1
(1+ Rlmj'n r.-p
The relationship betweenthe volatilityof F and the volatilityof R can be calculated
6j4
CHAPTER 29 fromIt's lemmaas @w =
c k(t
=
#
.'-
r)
1+ R(m
becomes2 Henc equation (29.3)
'VRO'V%RCT'T) -
Jy
=
-
11
RIm
is the instantaneouscorrel>tionbetweenP and R. As an approxiwherepvR = mation,it can be assumed that R remainsconstantat its initialvalue, %, and that the vlatilities and correlationin th expressionare cnstant to get, at time zero, .-pyw
/uJy0k&(T*
fz(V)eXp
Sz.(V)
=
-
-
1+ Rlm
T)
T
(29k4)
Example 29.3
Considera derivativethat providesa payof in 6 years equal to the vlue of a stck index observed in 5 years. Supposethat 1,200is the fomard value of the stock indexfor a contract maturingin 5 years. Supjosethat the volatilityof theindexis 2%, the volatilityof the forwardinterestrate betweenyears 5 and 6 is 18%,and Supposefurtherthat the zero cufve is;at the correlation betweenthe twois results with annual compounding. The at 8% justprodud can be used with P desned as the value of the index,T = 5, T# 6, m = 1, Rb pvR = = and c, so that cy -.4 x 0.20x 0.18x 0.08x 1 Fz(Uz) eXP X 5 fr.(Pr) 1+ -.4.
.8,
=
-.4,
=
.18,
.2,
=
=
-
.8
1.535'r(Pr). Fromthe argumnts in Chapter27, fr(Pr) is $he or fr.(Pz) 1,2 x 1.00535= forw>rdpri of theindex,or 1,2. It followsthat Er.(Pz) again 27, theargumepts inChapter it followsfromequation (27.20) 1206.42.Using that the kalue of the derivativeis 1206.42x P, 6). In this case, P(, 6) 1/1.086 = 6302 so thatthe value of the derivativeis 760.25. =
=
=
.
,
Application 1 Revisited The analysisjustgivenprovidesa diferentway of producing the tesult in Applkation1 ofSection29.1. Usingthe notation fromthat application, Rr istheinterestrate between T and T* and Rb as the fomard rate for the periodbetweentime T and T#. From
equation (27.22),
fr.(Rr)
=
Rb
Applyingequation (29.4) with P equalto R gives fz.(Rz)
=
fzt'zlexp
2
O'RR -
z
T
1+ &z
R and F are negativly correlated.We can refkct th by setting cw -ca(T# p/(1 + R/m), 2 variable,s is a negativenumber, and setting pvw pvR. Alternativeiywe can changethe sign of cw so that it is whicll and set p:,w -pvR. In either case,we end up with the same formulafor ag. positive =
=
=
-
665
Adjustments Convexity,Timing,and Quanto
wherei
=
T#
-
T (notethat m = 1/r). It followsthat
kb sztazjexp orz j + yz =
0r
-
zz(a,) a -
oarr o,j
exp j
+
Akproximatingthe exponential functiongives
1'z('z)
=
R +
Rzzz
1+ %z
Tllisis the same resultas equation (29.2).
29.3 QUANTOS A qaanto or cross-carrency #erfycffv:is an instrument where two currencies are involved.The payof is desned in terms of a variable that is measured in one of the currepciesand the payof is made in the other currency.0ne exappleof a quantoisthe CMEfuturescontract on the Nikkeidiscussebin BusinessSnapshot5.3.The market ismeasuredin yen),but variableunderlying this contract istheNikkei225index(which the contract is settled in US dollars. Considera quanto that provides a payof in currency X at time T. Assumethat the payof dependj on the value P of a variablethat is observed in currency F at time T. Desne:
Pxt, T) : Valueat time t in currency X of a zero-coupon bond paying ofl-1 unit of currencyX at time T Pyt, T) : Valueat time ! in currency F of a zero-coupon bond paying ofl-1 unit of currencyF at time T Pz : Valueof P' at time T f'x(Pz) : Expectedvalue of Pz in a world that is forwardrisk neutral with resptct to PX(1T) '
,
'y(Pz): Expectedvalue of 7z in a world that is forwardrk to Pyt, T)
eutral with respect
The numeraire mtio when we move from ihe &(!,T) nuleraire to the Pxt, T) numeraireis F(f)
=
Pxt, T) J(t, T) s
of F per unit of X) at time!. It followsfrom whereStj is the spot exchangerate (units of F per unit of X4 th that the numeraireratio F(l) theforwardexchangeratt (units maturing Defne: T. time contract at for a
cv Vglatilityof F cy : Volatilityof 7 pyv: Instantaneouscorrelation betweenP' and F.
6f4
CHAPTER 29
the cange #romequation (27.35),
where
of numemire increasesthe growthrate of P by ay, Jy
(29.5)
P-cycw
=
If it is assumed that the volatilitiesand correlationare constant, this meansthat 3
.
XZIUT)
or as an approximation
=
XIFIVIYPWPCVT
'rtprltl
+ p-cycwT')
'z(Pr) p
(29.6)
nis equatipp will be used for the valuation of what are known as dif swaps in Chapter32. fxample 29.4 supposethat the current value of theNikkeistock indexis 15, yen, the l-year dollar risk-free rate is 5%, the l-year yen risk-free mte is 20:, and the Nikkei difidend yield is 10:. The forwardpri of the Nikkeifor q l-year contract denominatedin yen can be calculatedin the usual way from eqation (5.8)as 15,:
' (.2-.1)x1
=
15,150.75
Supposethat the volatilityof theindexis za, the volatilityof the l-yearforward In yenper dollarexchangerate is 12%,and the correlationbetweenthe twois = = and tls case 'y(Pr) 15,150.75,cz- = From cv equa) tion (29.6), the expectedvalue of theNikkeiin a world thatisforwardrisk neutral with respect to a dollarbondmaturingin 1 year is ,3.
.2,
.12
.3.
=
.3x.2x.12x1
15,150.75:
=
15,260,23
This is the forwardprke of the Nikkeifor a contract tat providesa payof in dollarsrather than yen. (Asan approximation,it is alsothefuturespri of sucha
Usinj Traditional Risk-Neutral Measures. The forwardrisk-neutml measure works well when payolh our at nly one time.In othersituations,it is often moreappropriateto use thetraditionalrisk-neutral measure. supjose the processfollowedby a variable P in thetraditionalcurrency-Frk-neutral worldis knownand we wish to estimateits pross in the tmditionalcurrency-xriskneutralworld. Defne: of F per unit of #) 5': spotexchange rate (units o.s: Volatilityof S cy: Volatilityof J?
p: lnstantaneouscorrelation betweenS and P
In tlliscase, the change of numemireisfromthe money marketqount in curpncy F to the loney market nemount in currenc X (withboth money market nemounts being in currency X). DeVe Lz as the value of the money martet nemount in denominated
'
.
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(f, but the functionsBt, T) and
=
Tk-Bt,nr
x(f,
(l,T) are diferent, le y(r-l) .j)
Bt, T) =
(/ + J)@P
(z-!)
.j; .j,
and (c+p)(T-l)/2 j)+a)j
zy 21/2
,(!, g(),+)r;),(,-,j r)
-
-
and slightly humpedyield with p = al + 2tr2. Upward-sloping,downward-sloping, m odel, possible. the longrate, Rt, T), is linearly As in the case of Vasicek's curvesare dependenton r(l). Tls means that the value of r(l) determinesthe levelof the term structureat time 1. The general shape of the term structure at time t independent of r(l), but doesdependon 1.
Two-FactorModels A number of researchers haveinvestigatedthe properties of two-factormodels. F0r Brennanand Schwartzhavedevelopeda model where the process for the example, process.dThe long shortrate reverts to a long rate, which in turn followsa stochastic rateis chosen as the yield on a perpetual bondthat pays $1per year. Becausethe yield on tll bond is the reciprocal of its price, It's lemmacan be used to calculate the processfollowedby the yield fromthe pross followedby the prke of the bond.The bondis a tradedsecurity.Thissimplies the analysisbecausethe expectedreturn on the bond in a risk-neutral world must be the risk-free interestrate. 5 of Interejt R.ates,'' seeJ.C. Cox, J. E. Inpersoll, an ' s.A. Ross, ttA Theory'of the Term structure Econometrica,53 (1985): 385-497. 6
E. s.schwar,tA Continuous TimeApproach tc Pricing Bcns,'' Joarzal of seeM.J. Brennan an(July 1979)4133-55:M.J. Brennan an E. s.schwar,An EquilibriumMoel of BankingJZW Finance, 3 Bond Pricing and a Test of Market Eciencyj'' Joarsal of Rncnckl J?l Qaantitate xncly',21, 3 (september1982):301-29.
'
CHAPTER 30
678
Another two-factormodel, proposed hy Longstaf and Schwartz,starts with a generalequilibriummodelof the economyand derivesa terl structure model where theyeis stochastic volatiliiy.?The model proves t he analytically juite tractable.
30.3 iOTARBITRAGEMODELS The disadvantageof the equilibriummodels we havepresentedis that they do nt automatically t today's term structure of interestrates. By chbosing the parameters they can be made to providean approximatefh to pany of the ttrm judicieusly, tht are encounteredin prtice. But theEtis not usully an exactone and, in structures somecases, no reasonahle Et cap he found.Mosttraders:nd this unsatisfactory.N0t theyarpe that theycan havevery lhtlecovdence in the priceof a hpnd unreasonably, optionwhenthe model doesnot pricethe underlyingbond correctly.A 1% errorin the price of the underlying hond lay to a 25% error in an optitmprie. no-arbiirage model model designedto he exactlyconsistentwith todaj's term is a A strlctureof interestrates. The essential diflbrente betreen an equilibrillm and a n0model isthereforeas follows.In an eqpilibriumlpodel, today'stermstnlcture arbitrage ofinterestrates is an output. In a no-arhitrage fnodel,today'sterm structure of intrrst ratesis an input. In an equilibrium model, the drift of the short rate (i.e.,the coecient of dt) is not usuallya functionof time. In a no-arhitrage model, the driftis, in geneml, dependent on time. This is becausethe shape of the initial zero curve governs'the averagepath takenby the shoit rate ili thefuturein a no-arbitragemodel. If the zero curve i steeply upward-slopingfor maturitits betweenil and l2, then r has a positive drift hetween for thesematurities, then r has a negative thesetimes;if it is steeply downwardusloping driftbetweenthesetimes. It ttlrns ut that some equiliblillmmodels can he converted to no-arbitrage models by includinga functionof time in the drift of the short rte. We now consider the Hiee, Hullz-Whe (one-and two-factor),and Black-Karasinskimolkls. 'lead
The Ho-Lee Modvl Ho and Lee proposed the srstno-arbitrage model of the term structurei a paper in 1986.8Theypresentedthe model in theformof a binomialtDe of hondpris with two parameters:the short-rate standard deviationand the market priceof risk of the short rate. lt has since been shown that the continuous-timelimitof the model is dr = 941)dt + o'dz
(3g.19)
whereg, theinstantaneousstandarddeviationof the short rate, is constantand 94!)is a functionof time chosen to ensure that the model 5ts the initial term structure. The p(!) desnesthe aveage directionthat r moves at time1. Thisis independentof variable the levelof r. Interestingly,Ho and Lee'sparnmeterthat conrns the market price of
7 See F. A. Longstaf and E. S. Schwartz, slntefest
Rate Volatilityand the Term Structure:A Two Factor 1992):1259-82, GeneralEquilibrium Model,''Jozrzal ofFisasce, 47, 4 (september
8 seeT. s.Y. Ho and S.-B.Lee,
tt-i'erm
and PricingInterest RateContingentClaims,'' Movements structure
Jozrnal ofFinance, 41 (December1986):11 1.49.
679
InteyestRate Dyiyatives: Models PJthe s'/lp/Ra '
.
.
Figure 30.3
ne Ho-lkeemodel. r
rate
N
Initialforwardclm
.
y
g
'
r
/
.
Time 7
rk provesto be irrelevantwhqn themodelis used to prke interestrate derivatives.Th is analogousto risk preferencesbeingirrelevantin the pridng of stock options. The variabk p(t) can be calculated analytkally (seeProbkm 30.13).It is
ot)
=
#,(, t) + oat
(3.11)
wherethe F', 1) is the instantaneousforwardrate for a maturity t as seen at timezero and the subscript t denotesa partialderivativewith respt to 1. s an approximation, p(!)equals F!(, l). This means that the avemge directionthat the short l'ate will be movingin the futureis approximatelyequl to the slope of the instantaneousforward curve.The Hiee model is illustratedin Figure30.3.The slope of the forwzd curve desnesthe average directionthat the short rate is moving at any given time, superimposedon this slope is the normally distributedrandom outcome. In the 2 Lee model zero-coupon bonds and Europeanoptions on zero-coupon bondscan be valued analytically.Theexpressionforthe prke of a zero-couponbondat time ! in termsof the short rate is -
,
Pt, Tj
where P,
ln (!,T) ln P, =
(l,. T)e -.
=
T) 1)
+ (T
-
r(!)( 2. tj
(3.12)
--
!)F(, 1) jpjtr 2 -
-
tjj
In theseequations,timezero is today.Times! and T are generaltimesin thefuturewith ! T p 1. Theequations,therefore,desnethe pri of a zero-couponbondarta futurextime in terms of the short rate at time t and the prices of bondstoday.ne latter can be calculatedfromtoday'sterm structure. #
CHXPTER 30
689 The Hull-Wbite (One-Factor)Model
ln a paper published in 199, Hulland Whiteexploredextensionsof th8Vasicekmodel that providean exact fit to the initial term structure.gOne version of the extended . Vasicekmoel thai they conzer is
dr Ep(t)Jr) dt + 4 dz =':
(3.13)
-
O
dr
=
a
0t4 -
a
r
dt + c dz
where a and c arr cdnqpts. This is known as the Hull-Whitemodel. It can be characterizedas the H-tee model with mean 'reversion at rate a. Alterntively, it can be caracterized as the Vasicekmodelwith a time-dependentreversion leyel.At time t, the short rate revertsto otjlaat rate c. The H-lee modelis a particularcase of the Hull-Whitemodelwith a = The odel hasthe snme nmount of analytictractabilityas Hiee. The 0t4 function can be calculated fromthe initialtermstructur (seeProblem 3.14): .
0t4
=
Frt, t) + JF(, t) +
y (1 e.x,)
(3.14)
-
The last termin this equation is suallyfairlysmall. lf we ipore it, theequationimplies that the drift of the process for r at timet is 1l(, t) + Jl#'(, t) r). Tllis shors that, on -
Figure 30.4 The Hull-Whitemodel.
N
Sol rate
N hal
r ----e
forwardcurve
r Time 7
.
Advkwof Jrlcacfnf 9 sCC J Hull and A. wllite pricingInterest Rate Derivative securitie,s 3, studies, 5.73-92. 4 (1990): ''
,
,
,
T'
InteyestRate Deyivatives:Models 0/ the
681
Rate
'tyl
it r followsthe slope of the initialinstaptaneousforwardmte curve..When average, from that curve,it reverts back to it at rate ai The moel is illustratedin deviates'
Figure304. . Bondpris at tlme l in the Hull-Whitemodel aa given by .
Pt, T) = where
.
(l,l'k-*'b6
(3.15)
1 e-s(r-l)
(3.16)
-
B(t, T) =
,
----.---.=
and
P,
T)
ln (t,T) ln P, t) + Bt, =
1 Q3
T)F(, t4 -
'
.
-.z
e
-.j
-
e
jyut
)
.j;
(?j.1y;
.
of a zero-couponbond at a pris futuretimet in termsof the hort rateat timet andthe ofbondstoday.Thelatter can be calculated from tobay'sterm structure. EQuations
(3.15),(3.16),and (3.17)deMe the pri
The Black-KarasinskiModel TheHiee and Hull-Whitemodels havete disadvantagethatthe solt-terni inteyest rate, r, can becomenegative.A model that allowsonlypositiveinterestl'ates is a model proposed by Blackand Karasinski:lo Jln r =
(p(t) at) -
lntr))dt + c(t) dz
(39.18)
The variable ln r followsthe sameprocess as r in the Hull-Whitemodel.Whereasthe value of r at a future time is normal in the Hly-Lee and Hull-Whitemodels, it k model. lognormalin the Black-Karasinski The Black-Karasinskimodeldoesnot haveas much analyictractabilityas H-lwee Or Hull-WLite. F0r example,it is n0t possible to produ formulasfor valuingbondsin terms of r.using the model.
The Hull-White Two-FactorModel A model that involvesa similarideato thetwo-factormodel sugyestedby Brenna and s chwartz,but is arbitrage-free,isll
JJ(r) (9(t)+ u =
=
afjdt
+ cl Jzl
(3.19)
whereu has an initialvalue of zero and followsthe process fu =
-bu
dt + pdh
As in the one-factor models justconsidered, the p'arameter btj is chosento make the modelconsistentwith the initial term structure. The stochatic variable u is a component of the reversion levelof r and itself reverts to a levelof zero at rate b. The 10SeeF. Blackand P. Karasinski,tBondand OptionPricingWhenShort Rates AreLopormal,''ktatcal AnalystsJtwrllc/,July/Augujt (1991), 52-59. 1I
ModelsII: Tw seeJ. Hull and A. White, Numrical Produres for Implementin Term structure Factor Models,'' Joarnal ofDerivatives,2, 2 (Winter1994):37-48.
682
CHAPTER 30 *
.
parametersa, b, c, nd c2 are constants and dzL and #z2 r Wieer processes with instantaneouscorrelation p. TMsmdel profidzs a richer pattern of term structult movementsand a rkher pattern of volatilitiesthanone-factormodelsof r. For tor'e informationon the model, see Technical Note 14 on the author's website. ,
' ,
30.4 OPTIONS Oj
'BPNDS ' .
some of the models justpresented allowoptionson zero-cnupon bonds to be vqlued H-LK, and Hull-Wbitepodels, the price at timezero analytically.For tht Yasicek, of a call option that matures at time T on a iero-couponhondmatulingat time s is Lp, sjNbj
KPq, T4Nb
-
-
cy)
whereL is the prlncipal of the bnnd, K is its Strike pri,
k
=-ln
LJ'(0 s) oy P, TjK
1
(39.20)
and
zo. l
,
of a put ption on the bondis The yrice Kp,
T)N-h + cy)
-
s)N(Lhj
Lp,
In 4hecase of the Vasik and Hull-Whitemodels, J
JJ,
In the cpj of the Hiee
=
-u(,-r)
(1
e
oy
c(,-
-
-
a
)
1
-
e-*T
2c
mbdel, =
r)W
Equatilm(3.2) is essentiallythe same as Black'spodel for prking bond options ip section 28.2. The forwardbond pri volatilityis cJ,/./f andthe standarddeviationof the logarithmof thebondpri at timeT is cy. Asexplned in Sedion28.3, an intere rate cap or Qoorcan be expressedas a portfolio of options on zero-cgupon bonds.lt can, therefore,be valued analyticallyusing the equations justpresenttd. There are also formulasfor valuing options on zero-coupon bonds in the Cox, Ingtrsoll,and Ross model, which we presented in stion 30.2.neseinvolveintegrals ot the noncentralchi-squam distribution.
Optionson Coupon-BearinjBonds ln a one-factor modelof r, all zercoupon bondsmove up in pri when r decreases and a11zero-coupon bonds moye downin price when r increases.As a result, a onefactormodelallowsa Europeanoption on a coupon-bearingbond to be expressedas the sum of Europeanoptionson zero-coupon bonds.The procedureis as follows:
1. Calculater#, the criticalvalue of r for which thepri of the coupon-bearingbond equalsthe strike pri of the option on the bond at option maturity.
InteyettRte Derrdw3;
683
Modtls p.f the Stvf Rate
2. Clcplate the prkes of options on the zer-coup bnds that comprise the coupon-bearingbond.,setthe stiikepri of eachoptign equalto the value the cprrespondingzero-coupon bond willhaveat time'T wn r r#. 3. Set the prke the qptionotl the coupon-bearingbondequalto the sum of te pris on the options on zero-couponbondsckulated in step 7. =
,of
allows options on coupon-bearinj bonds yo be valued for the Vasicek,Cox, Ingersoll,and Ross,Hollwee,and Hull-Wlte models s explainedin.Businessspapshot 28.2,a Europeanswap option can be viewedas an option pn a coupon-bearing using this procdure. For more details on the bond. It can, thereforeybe 15 procedure,see rechnical Note on theautor's website. Th
.valued
70.5 VOLATILjTYSTIUCTURES Thr models we havelookedat giverise to difkrentvolatilityenvkonments.Figure30.5 showsthe volgtilityof the 3-monthforwal'drate as a functionof maturity for Hiee, Hull-Whiie opt-factor and HullsWhitetwo-factr models.The term structure of rates is assumedto be :at. interest For Ho-Leethe volatilityof the 3-monthforwardmteisthe same for a11matrities. In the one-factor Hull-hite model the efkct of men reversion is to cause the of the 3-monthforwardrate to be a decliningflmctionof matmity. ln the volatility Hull-Whitetwo-factormodel when pammetersare chsen approprktely,the volatility '
'
' .
.
.
.
. .
.
-
.
'
..
Fi! ure 30.5 Volatilityof lhmonth forwardrate as a functionof matrity for (a)the Hiee model, (b)the Hull-White one-factormodel, and (c)theHull-Wlte twofactormdel (when parametersare chosenappropritely). Volatility
Volatilitz
Matuo
MatuO
(a)
(b) lrtllltilit)l
(c)
684
CHAPTER 30
look. The latter is consistent with of the 3-monthforward rate has a .impiricalevidenceand impliedcap volatilitiesdiscussedin Section28.3. thumped''
.
'
--
30.6 INTERESTRATETREES An interestrate tree is a disete-time Dpresentation of the stochasticpross for the shorttatt in muchthe sameway as a stockprice tpe is a discrete-timespresentationof theprocess followedbya stockpri. If te tlmestepon tht tpt is t te rateson the tree whena treeis arethe continuouslycompounded l-period rates. The usual assllmption stochasiic At-peripd followsthe R, rate, is process as the constructedt at the sme instantaneousrte, r, in thi correspondingcontinuous-timemodel. main diflkren betweenintertstrate treesand stockprice treesisintheway thatdiscouptingisdone.In a stockpri tree, th discountrate is suallyassumed to be the sameat eachnode (ora fundion of me).In an interestrate tree,thediscountrate varies fromnode to node. It often proves to be convenieht to use a trinomialrather than a binomialtree for interestrates.Themainadvantageof a trinomialtreeis that it provides an extradegree of freedom,making it tasier forthe treeto represent featuresof theinterestrate process such as mean reversion. As mentionedin Section 19.8, using a trinomial tree is diserencemethod. equivaltntto using the explicit snite ,
'l'he
'
lllustrationof Use of TrinomialTrees To illustratehowtrinomialinterestrate treesae used to value derivatives,consider the simpleexample shownin Figure 30.6.Thisis a two-steptreewith each time stepequal to 1 year in lengt so that A! 1 year. Assumethai the up, middle,and down =
Figure 30.6 Examjk of the use of trinomialinterestrate trs. Upperpumber at each ' node is rate; lowernllmber is valce of instrument. E. 14%
3 12%
F j2o
a
1
1.11
10% g 0.35
.
jco '
c
,
c
0
0.23
8%
o
joo
H
so
0
0 l
6%
0
.
685
InteyestRate Deyivatives:Models n./ the shoyt Rate .
.
'
.
.
,
and 0.25,respectively,at eachnode.The assumed l-period probabilitks are 0.25, rateis shown as the upper nllmberat each node.12 The tree is used to value a derivativethat providesa payof at te endofthesecond timestep of .5,
'
maxllt:
=
l
.11),
'
.
,
,
whereR is the Al-period rate. ne clculated value of this derivativeis thr lower numberat eachnode.At the snal nodes,the value of thederiyativeequalsthe payof. = 3.At earlkrnodes,the value For expmple,at odeE, the yalue is l x (.14 of the (terivative'is calculated using the rollbackprodure explainedin Chapters11 and 19.At node B,the l-yearihterestrate is 12%.Th is sedfordiscountihg t obtain vaiues the value of the derivativeat node B fromits at nodes E, F, andG as .11)
-
(0.25x 3 +
x 1+ 0.25x
.5
-.12x 1 =
k
j jj .
At nbde C, the l-yqarinterestrate is 1%. This is used for discountingto obtainthe valueof the derivativeat node C as (0.25x 1+
.5
x + 0.25x
):-'1X1
=
:
.
23
At the initialnode, A, the interestrate is also 1% and tlie value of thederivative 'is
(0.25x 1.11 +
.5
x 0.23+ 0.25x
s.lxl
):
=
g gj .
Nonstandard Branching It sometimesproves convenient to modify the standardtrinomialbranchingpattern tat is used t a11nodes in Figure 30.6..'rhree alternative branchingpossibilitiesare shownin Figure 30.7.The usual branchingis shown in Figure 3.7(a). It is one/ two/upone/straightalonf', s straightalonj/down one''. 0ne lternativeto thisid shownin Figure 3.7(b). This proves usefulfor incorporatingmean reversionwhen iterest rates are very low. A tird bracilinj pattern shown in Fipre 3.7(c) is dstraightalong/down one/down two''.Tilisis useful for incorpomting mean reversion The wheninterestrats re very high, use if diferentbranchingpAtterns isillustratedin the followingsection. Sup
$$up
' .
Fignre 30.7
(a)
Alternativebranchingmethods in a trinomialtree.
(h)
(c)
12We explain laterhow the probabilitie,sand rates on an interestrate treeare detennined.
686
CHAPTER 30
PROCEDURE 30.7 A GENERALTREE-BMILDING Hulland White haveproposed a robust tw-stage procedux for constructing trinomial t ees Sorepresent a wide range of one-factor models.l3 This section ftrst explains how the procedure can be ued for the Hull-Whitemodel in equation (30.13) and then other extended models. showshowit can be to represent
First Stage The Hull-White
for the instantaneousshort rate r i model
E(l)
dr =
-
cr) dt + o' dz .
We suppse , that. the time step on the tree is constantand equal to Assumethat the l rti, R, followsthe same process as r.
(p(!) c:) dt + o'dz
dR =
l.
j4
'
-
Clearly,this is reasonable in the lilit as Al tendsto zero. Theflrststage in buildinga treefor this model is to constructa tree for a variable R%that is initiallyzero and the process follows dR*
=
-aR%dt + o'dz
This process is symmetricalabout R%= 0. The variable Rbt + l) R'tj is normally distributed.If terms of higher order than l are ignored,the expectedvalue of # R#(!+ Al) R (l)is -J Retjt and the varian of R't + !) :#(!) is o7! The spacing betweeninterestrates on the tree, R, is set as -
-
-
.
hR r'
'
=
o' 3A!
This proves to be a good choice of : fromthe viewpointof error minimization. The objectiveof the srststage of th procedureis to builda yreesimilar to that shown in Figure30.8for R%.To do this, it is frst necessary to resolve which of the three methods shown in Figure30.7will applyat each node. This will determine branching theovergllgeometry of the tree.Oncethisisdone,thebranchingprobabilitiesmust also becalculated. Defne (i,j4 as the node where l i l and R%= j A:. @hevariable i is a positive and j is a positiveor negativeinteger.)Thebranchingmethodused at a node inust integer leadto the probabilities on all three branchesbeing positive. Most of the time, the branchingshown in Figure3.7(a) is appropriate.When a > it is necessaryto switch frm the branchingin Figure3.7(a) to thebianchingin Fipre 3.7(c) for a suciently laygej. Similarly,it is necessary to switch fromthe branchingin Fipre 3:7(a) to the branchingin Figure3.7(b) Whenj is.suciently negaiiv. Defne ymax as te value of j switch branching the fromthe Figure3.7(a) Fipre 3.7(c) branchingand to wherewe ./..,nas the value of j where we switch from the Fipre 3.7(a) branhing to the Figure3.7(b) branching.Hvll and White show that probabilities are alwys positive '
=
,
13seeJ. Hull and A. White, x'NumericalProdures for ImplementingTerm StructuredusingMcdels1: Single7-16; an J. Hull and A. White, Factor Motlelsy''lonr?ll/of Derivatives,2, l (1994): Hull-White .lnterestRate Trs,'' Joarzalof Derivatives,(Spring 1996):26-36. 14seeTechnicalNcte 16on the authcr's websitefora discussionof how nonconsunt time steps can beused.
687
Interest Raie Denvatives: Models d/ the 5'/ltelRate
!) apd ls set equal ifjmzsis set equal to the smallest integergreatir than 15Dehne h, pm,and pd as the probabilitiesof thehighest,middle,andlowest to ranches emanating fromthe node. Theprobabilitiesare chosen to match theexjted b. and varianceof the change in R*over the next timeinterval !. Theprobabilities change .184//
.jmu
-.jmax.
unity. Thisleads to threeequations in thethpe probabilities. As already mentioned, the mean chang in R* in timeA! is -aR*t andthe variance ofthe tliange is ht. At node (f,j), A#= j r. If thebranchinghastheformshownin Figure 3.7(a), the h, pm, and pd at node (f,jj must satisfy te followingthree equationsto match the mean and standrd deviation:
' must also sum t
'
-
pz: hB
pdR = .-ajR X2= c2l+JA#A
-
2+ n
T'z+ pm+ h Using hR
=
l
1
=.
c 311, the solution to theseequations is R
1/2./2!2 6 2 1+
=
ajnt)
-
aljlx? #,n= 13 -
##
1(c2/!2 1+ 6 2
=
-ajtj
15The probabilities are positive for any value of jnjzxbetween Al) and Al) and for any Al) and At). Channg thebranchingat the flrstpossiblenode between valueof Provesto be computationallymost ecient. .184/(c
ojmin
-.124/@
-.216/@
'
.8l6/(c
CHAPTER 30
688
Similgrly,if the bmnchinghas the formshownin Fipre 30.7(b),the probabilitiesare !2(J 2 2 2 pv .t+ i A! + alhtj aljlh? jaj 1Pm 3 =
=
2+ Laljlh? 2
pd
j
-
-
+ ajhtj
Finally,if the branchinghas the formshownin Figure3,7(c), th probabilitiesare pu = Pm= Pd
=
2+1@2/A 2 l3 jlhp Laljlhtl 1+ 6 2 -
-
ajhtj
-
+
xy!
ajhtj
-
T0 illustratethe rst stage'of the tree construction, supposethat c' = a= and Al = 1 year. In th case, LR = Uj-= 0173 jmax set equal to the stallest and ymin integer greater than Yhismeans that jjnzs= 2 and and the tre is as shownin Figure3.8. The probabilitieson the branches ojmin emanatingfrom ech node are shownbelowthe tree and are calculated using the equationsabovefor p., pm, and h. Note that the probabilitiesat each node in Figure30.8 depend only on j. For example,the probabilitiesat node B are the same as the probabilitiesat node F. Furthermore,the treeis symmetrkal.Theprobabilitiesat node D'are the mirror image of the probabiEtiesat node B. .l
.1,
,
.l
.
.184/0.1,
,
'-/max.
=
-2
=
Second Stage The secondstagein the tree constructionis to convert the tzeefor R* into a tree for R. Tll is accomplhed by displacingthe nodes on the P-tree so that the initial term of interestrates is exactlymatched. Desne structure
((1)
Rtj
=
=
R'tj
The c's are calculated iterativelyso that the initialtermstructureis matched exactly.lf Desne ai as i Al), te Value of R at timei Al on the '-tree minusthe corresponding valueof R%at timei Al on the r'-tree. Dene Qi,jas the presentvalue of a securitythat paysofl-$1if nde i, j4is reachedand zero otherwise.Theai and Qi,jcan be calculated usingforwafdinductiopin sucha way thattheinitialtermstructureismatchedexactly.
Illustration of Second Stage Suppse that the continpouslycompoundedzero rates ip theexamplein Figure30.8are is 1.. The value of (u is chosen to give the as shownin Table30.1.The vlue of Qg,() 16It is possible to estimate a(l) analytically.Since dR =
it follows that
19(1)nR)dt + adz -
and dR*= -aR* dt + c dz
'
da
=
((!) x(t)Jdt -
If we irore the distinctionbetweenr and R, the solution to thisis
,
a(l)
=
0, 1)+
()'2
2c (1 -
.tx
e
2
)
However, thtse are instantaneousa's and do not lead to the tree calculationsexactlymatching the term of interestrates. structure
Interest Rate Derivatives. Models n/
e
'
xsnrl
689
Rate
Table 30.1
Zerorates for examplein Fipres 30.8and 39.9. '
Matarity
Rate (%)
.5 1. l.5 2. 2.5' 3.
3.430 3.824 4.183 4.512 4.812 5.086* '
right pri for a zero-cqupon bond maturing at time.A!. That is, a is set equalto the initial Al-period interest rate. Beuse ! 1.in this exalple, a = 9.03824.Tls defmesthe position of thi iitial node on the '-tree in Rguie39.9.The next step is t calculate the values of :1,1,Q1,,and Q1,-I There is a probabilityOf 9.1667thatthe (1,1)node is reached andthediscountrate forthefirsttimestepis3.82%.The valueof = Ql, = 0k6417and Qj,-1= Q:,:is therefore0.1667: similarly, lt is chosento Once:1,1 Q1,,and :1,-1havebeencalculated,J1 tanbedetermined. givethe right pricefor a aro-coupon bondmaturingat time2Al. Bause A: 9.91732 and i :a : 1,ili jiie Of thisbondas seih at node Bis :-(1+ g gjsy gmjarjy thegriceas :-(J1 Te priceas seenat seenat node Cis :-1 andthe price as seen at node D is =
From the initial term structure, this bond prke shold be e-.4512x2= (jgjyy sdtutingfor the Q'sin equation (3.21), .
-(aI+.1:32)
1604: .
+
.
yjyy-l + g jgjz-ll
,
sub.
-0.01732)
.9
=
.
j?y
0r
1604:-631732 + e-dl( .
jjy +
.
2,1694/.01732 ) = g jjyy .
0r
a1
=
ln
g yjy + g.jg4/.1732
0.1604: -.1:32 +
.
=
-0.9137 -
-
-
--
0.05205
Th means that the central node at time t in the treefor R corresponds to an interest rate of 5.205%(seeFigure 30.9). The next step is to calculate :2,2,:2,) Q1n, Q1 and Q1 The calculationscan . be shortened by using previously determlned Q values. Conslder:2,1as an example. Tilis is the value of a securitythat pays of $1if nodr F is reached and zero otherwise. Node F can be reched only fromnodes B and C. The ihterestrates at thesenodes are and 5.205%,respectively.The probabilities associatedwith the B-F and C-F 6.937% bfanches are 0.6566and 0.1667.The value pt node B of a securitythat pays $1at node The variable p: 1 is -06937n e va1ueat node C 0.1667:-035205 F istherefore9.65.66: 0.1667:*05205 received node plus valueof -0.06937 times B $1 at tfje present 0.6566: mes node of received value is, C; that at present $1 the -1
,
-'-
-'
-2.
,
-1
.
.
'
:2,1
=
jjjyy-.515 x 0.1jgj + g
-.0693
0.6566:
.
x g jyjy :::: g j jjg .
.
Similarly,:2,2::Fz0.0182,:2,9 0.4736,:2,-1= 0.2033,and :2,-2= 0.0189. The next stepin producing the R-tree in Figure30.9is to calculate a2, Mter that, the :3,y's can then be computed. The variable a3 can then be calculated, and so on. =
Fermulas fer e's and
Q's
To expresstheapproachmore formally,supposethat the Qi,)havebeendeterminedfor i %m m k ). The next step is to detenpinean so that the tree correctlyprices a zerocouponbond maturingat m + 1)Al. The interestrate at node m, jj is am + j A:, so that the price of a zero-coupon bond maturing at time m + 1)Aj is given by nm
Jk+1
=.
E
juz-wnz
+ jhRjht pm,/expE-/?n
(30.22)
wherenm is the number of nodes on eachside of the central node at tim m A!, The solutionto this equation is ->al
J=M
nm Qn,je lnEs-nm Aj
-
jn J)m+1
691
Inteest Rate Deyivatives:Models PJthe s'prlRate
Onceam has beendetermined,the Q,j for i = m + 1 can be calculatedusing
p,,+!';E Qm,iq,.expE-ttu+l =
p
j
j
whreqk, j) is the probabilityof moving fromnode @,1) to nde
summationis taken over all values of l for which thisis nonzero.
m + 1,j) and the
Extensionto Other Models The procedurethat has jst been outlined can be extendedto moregeneralmodelsof tLeform df = (94!)JJ(r))dt + cdz (39.23) -
Thisfamilyof models has the propertythat theycan t any term structu/. 17 As before,we assume tht the l periodrate, R, followsthe same processas r: dRt
E9(f)aRdt
=
+ cdz
-
'
.
We start by setting
.z
=
J(X),so that dx
(941)cz) dt + c dz
=
-
The frst stage ij to build a tree for a variable x%that followsthe sameprocessas exceptthat 941)= and th initialvalue is zero. The produre hereis identkalto the procedurealready outlinedfor buildinga tree such as that in Fipre 30.8. As in Figure30.9,the nodes at time i ! are then dplad by an amountai to providean exact ft to the initialterm structure. The equatiopsfor determining ai and Qi,jinductivelyare slightly diferentfromthosefor the J(R) R case. Thevalueof Q that the Qi,jhavebn determined for at the frst node, :,, is set equal to 1. suppose i %m m k ). The next step is to determineam so that the tree correc prices an m + 1)l zero-coupon bond. Defne as the inversefunctionof J so 1at the !periodinterestrate at the /th node at timem ! is gvy,y j a) .z
=
The price of a zero-coupon bond maturing at time m + 1)l is givenby nm
+ p.zll) :,,+1 j=-nm X P?n,yexpE-gt(u =
(39.N)
This equation can be solved using a numerical produm such as Newton-Raphson. is J(R()). The value tz qf a when m = Oe qm has beendesrfmiped,thi Qi,jfor i m + 1 pn be calculatedusing ,
=
''Flzll! P,,+1,; 1-7 Qm.kq,JlexpE-/??l =
j
17Nct all no-arbitragemoels have thisproperty. F0rexample,theextene-clR Ingersoll,and Ross (1985) an Hull and White(199),whichhas the%rm dr
=
moel, consideredl)y&x,
E9(!)urldt + o'dz -
cannotfit Tiel curves where Re fcrwar rate tclinesshaq. when (r)is negative.
whereqk, jj is the probability of moving fromnode (m,k4to node m + 1,j4 and the pmmation is taken over a11values of k where thisis nonzero. Figllre 30 10 shows the results of applyingthe produre to the model .
d1n(r) =
whena
=
0.22,c
=
0.25,ht
=
(p(l) a ln(r))dt + c dz -
0.5,and the zero rates are as in Table3.1.
Choosing J(r) setting fj Settingf 1n(r) r leadsto the Hull-Whitemodel in equation (3.13); equation circumstances Black-Karasinksi modelin leads to the these (3.18).In most perform market models aboutthe in tting data actively traded to two same on appear inqtrumentssuch as caps and Europeanswap options. The main advantageof the f = r model ij its analytictractability.1tsmain disadvantag:isthat negativeinterest rates are possible. Ip most cirolmstans, the probability of negative interestrates occurringuhder te model is very small, but some analystsare reluctant to use a model wherethereis any chance at al1of negativeinterestrates. The f = lnr nydel has no analytictractabilhy,but has the advantage that interest rates are always positive. Another advantageis that traders naturally think in terms of c's atising from a lognormalmodel rather than c's arisingfrom a normal model. Thereis a probkm in choosing a satisfactorymodel for countries with 1owinterest rates.The normal model is unstisfactory because,when theinitialshort rate is low,the =
=
693
InteyestRate Dcyrpa/'lw; Models OJthe Styf Rate
probabilityof negativeinttrtst rates in thefutureis no longernegligible.Thelognormal the c pammeter in the model is unsatisfqctory becausethe volatility of rates lypormalmodel) is usally much gyattr when rates are.lowthan when theyare high. (Forekample,a vlatility of 1% mlghtbeappropriatewhen theshortrate is very low, While20% mightbe appropriatewhen it is 4% or more) A.model that appearsto work is chsen so that rates are lognormalfor r lessthan 1% and wellis one where normalfor r greater than 1%.18 .(i.e.,
of(rl
Uslng Ahalytic Resultsin Conjupctionwitk Trees '
.
,
.
When a tree i; constructed for the /(r)= r versionof the Hull-Whitemodel,the analyticfeplts in Sectipn39.3 can bt used to provide the complete 21711Structure and European'option prices at eachnode. It is importantto recopize that the interest rate on the tree is the l-period rate R. It is not the instantaneousshort rate r. Problem30.21)that it can beshown(see Fromequations(3.15),(3.16),and (3..17) Pt, T) = a(r,Tje
.fczy
whire
(3t).2!)
, !
.
lnj(l, T) ln =
P, t + l) Bt, T) ln P, 1) 140,t4 Bt, t + l)
P,
T)
-
-u(1 and
J(l,T)
=
-
e
-?m),(r, r)E,(r,r)
-
,(,, ,+
Bt, T) l Bt, t + aj)
r)! (3,.2. (3d2:)
gn the case of the Ho-bee model,we set J(l,T) = T t in theseequations) Bond prkes should thereforebe calculated with equation (3.25),and not with equation (3.15). -
-
'
Exampk30.1 rtes for matries betweenthose suppose zero rates are as in Table30.2. indicatedare generated using linearinterpolation. Uonsidera 3-year(=3 x 365days)Europeanput option on a zero-coupon bohd that willexpirein 9 years (=9 x 365days).Interestrates are assumedto followthe Hull-White(/(r)= r) model. The strike pri is 63, a E:z 0.1, and X 3-yeartreeis constructed and zzro-couponbond prkes are calculated c= analyticallyat thefmalnodes asjustdescribed. s shownin Table30.3,the rtsults fromthe tree art constent withthe analyticprke of the option. Th exampleprovidej a good test of theipplementationof themodelbecause atelyaftertheexpirationof the gradient of the zerocurve chanyes sharplyimmedi. construction of and use the treeare liableto havea the option.Smallerrors in the bk eflkcton the option values obtained.tWe exampleis used in SampleAjplication G of the DerivaGemApplicqtionBuilder software.) 'rhe
.01..
18SeeJ. Hull and A. White 'Taking Rates to te Limit,nik, Decemer
168-49. (1997):
694
CHAPTER 30 Table 30.2 Zero curve with a11rates continuouslycompopnded. Matarity
3 days 1month 2 monyhs 3 months 6 months l year 2 years 3 years 4 years 5 years 6 years 7 years 8 years 9 years 1 years
Tree for American BondOptions The DerivaGem software accompanyingtllis book implementsthe normal and thelognormalmodel for valuing Eurppean and mericanbond options, caps/:oors, and European swap options.Figure30.11 thows the tree produced by the software whenit is used to value a 1.5-yearAmerkancall option on a l-year bond usipg four timesteps and the lognormalmodel. The parametersused in the lopormal model are J= % and c' 2%. The underlying bopdlasts1 years, has a principalof l, and paysa coupon of 5% per annum semiannually.Theyield curve is:at at 5% per annum. The trikepriceis 15. As explninedin Section28.1the strikeprke cap be a cash strike priceor a quoted strike pri. In this case it is a quotedstrike pri. The bond prke shownon the tree is the cash bondpri. The arued interestat each node is shown belowthe tree.The cash strikeprke is cakulatedas thequotedstrikepri plvs anled The quoted bond pri is the cash bond priceminusaccnld interest.The interestk =
Table 30.3 Value of a thr-year put option on a nine-year zero-coupon bondwith a strike prke of 63: = and c = zero curve as in Table30.2. a .1
payoffrom the option isthe cash bondpri minus the cash strikepri. Equivalently it is the quotedbond price minus the quotedstrike pri. The treegivesthe price of the option as 0.668.A much largertr with 1 timesteps givesthe price of the option as 0.699.Two points shouldbe noted about Fipre 30.11: 1. The softwaremeasuresthetimeto option maturity as a wholenulhberof days.For exmple, when an option maturity of 1 yearsis input,the life of the option is assumedto be 1.5014years (or1 year and 183days).Coupondates(andtherefore aruals) dependon the valuation date,takenfromthe computer's clock. .5
t
696
CHAPTER 30
2. The price of the l-yeay bond cannot be computed analyticallywhen the lognormalmodelis ssumed. It is computed numericallyby rolling backthrough a much largertree than tat shown.
30.8 CALIBRATION
, j,tr.y 'J .
.
,
,
,h
.y.
,.
j
.
'
Up to now, we haveassumedthatthe volatilityparametersa and c are known.Wenow discusshpwthey are detenpined.Thisis knownas altbratingthe model. .T.he platility prametrs arr determinedfrommarkd dqtapn gctivelytradedoptions (i.g.,byokirtots pp js and swap options such as thostin Tablts 2t.1 and 22.2). Tese will be referred to as the calibratlng instraments.The flrst ssage is to chopsea measure. Su/pose there are rl cajbrating instnlments.A populpz zoodness-of-ftmesure is '
.
..
.
Stgoodness-of-st''
7(t1Pf)2 -
f=1
whereUiisthe market pri of the fth calibratinginstmmentand Pj isthe pri givenby the model for this instrument.The objective of calibration is to choosethe model parametersso that this goodness-of-ft measure is minimtzed. The number of volatility parameters should hot be Feater than the number of calibratinginstruments.If a and c' areconstnt, thereare only tw0volatilityparameters. Themodels can beextendedso that a or c, or both, are functionsof time.Stepfunctions can be used. Suppose,for exnmple,that a is constant and c' is a functionof time.We might choose times !1, !2, tn and assume c(!) s c for ! 11 c(l) = cj for = c(l) < ti t K lj+1 (1 i rl 1),and h for t p ls. ThereWouldthen be a total of volatility and cs. 2 + parameters:a, c, cl rl The minimization of the goodness-of-ft measme can be aomplished using the Levenberg-Marquardtprodure, 19whena or c, or both, are functionsof time, a penaltyfunctionis often addedto the goodness-of-fhmeasureso that thefunctionsare lwellbehaved''. In the example just mentioned,where c is a step function, an appropriateobjectivefunctionis .
.
.
,
,
-
,
n (U f=1
-
Pf)2+
.
.
,
n-1
n 'tpl,ftcd -
J=1
.
tn-1) 2
'tp2,i(ck-1
+
2
+ cf+l 2cf) -
i=1
The jecond term providesa penaltyfor largechangesin c betweenone step and the next.Thethird term providesa penaltyf0r highcurvature in c. Appropriatevalue! for upl,fand ?2,j are basedon experimeptationand are chosen to provide a reasonable level of smoothnessin the c function. The calibrating instrumets chosen should be as similaras possibleto theinstrument beingvalued. Suppose,forexample,that the model is to be used to valu a Bermudanstyleswapoption that lasts l years and can be exemkedon any payment datebetween year 5 and year 9 into a swap maturing 1 years from today. The most relevant calibratinginstrumentsare 5 x 5,6 x 4, 7 x 3,8 x 2, and9 x 1Europeanswap options. (An rl x m Europeanswapoption is an n-year option t9 enterinto a swaplastingfor m years beyondthe maturity of the ption.) and W.Y. 19For a gccd descripticncf this procedurq see W.H. Press,B.P. Flannery, s,A. nukolsky, CambridgeUniversitzPress,1988. Vetterling,Numerkal &cI>e, in C.. Te Art ofscientsc Computing.
Interest Rate
Mdels r ivatives:
697
OJthe s'w! Rate
Thi advantageof making a or c, or both,functionsof timeisthatthe models can be ftted moreprtcisely to the prices of instrumentsthat tradeactivelyin the market.The disadvantageis that the volatility structurelcomes nonstationary. The volatility term structumgivenbythe model in thefutureisliableto be quiyediferentfromthatexisting
in the mrket today.2: A somewhatdiferent approach to calibration is to use a11available calibrating t.worll that is .t ' risk-neatral world. In tls world refer extrs i Lte. We to th as a rolliq forward wecan discountfromtime 4+1to time tbusing the zerc ratt obstrved at time lj f0r a maturity4+1.Wedo not have to worry about what happensto interestrates between tWes tk andt k+1 t timet the rolling forwardrk-neutral world is a world thatisfomard risk neutml givesthe processfollowdby withrespect to the bond prke, Pt, !,,,(t)). Equation (31.7) Fj(!) in a world that isforwardrisk neutral with respectto Pt, lk+l). From 27.8, section rolling Fj(t) world in the forwardrk-neutral is it followsthat the processfollowedby ''
.
.
''-.
'
.
.
'
'
.
'
'''
'
'
'''
'
'
.
dlts
=
(t!%(!)(t)14+l(t)1Fl(t)dt + -
(t)Fl(t) dz
(31.8)
The relationship betweenforwardrates and bond prices is Pt, ti) + jyy-;.tj Pt, tf+1).1 Or
ln Pt,
!j)
ln C(t,tj+1)=
-
lnEl+ tTjFftlll
It's kmma can be usedto calculatethe pfoss followedby theleft-handside and the right-handsideof th equation.Equating the oecients of dz gives
vt)
--
pf+l
>-f(t)(f(t) (t) ti-F f 17f(l) =
(31.9)
l
6 In tke terminolor of Section 27.4,this world correspondsto usinz a CD'' as the numeraire.A rolling CD (rtifcateof deposit)is one were we start with $l, bu7a bondmatming at time!1 reinvestte . proceedsat timeh in a bondmaturing at timel2, relvest te proeds at time12 in a bondmaturing at time l3,and so on..(Strictlyspeaking, tiwinttrtst rate treeswe constructedin Chapter39art in a rollingfomard rk-neutral world rather thanthe traditionalrisk-neutralworld.)The numeree is a CD rolkd ovtr at the t'ndof each timt sttp. t'rollinz
,
i0g
CHAPTZR p1
the pross so that fromequation(31.8) World neu t ra l is dF (!) ' = F'ktj
k
i
f=.(,)
followedby Fktj in the rolling forwardrisk-
6.(!)(j(!)(!) dt + (!)dz 1+ h6.0) .
. (31.10)
is thelimitingcase of this as the f tend to zero (see The HJM result in equation(31.4) Problem31,7).
Forward Rate Volatilities j,
The model can be simplisedbyassumingthat (!)is a functon onlyof the number of wholenemrual periods betweenthe next Dset date and time 4. Defmehi as the value of (!)when there are i suchaccnlalperiods. Tls meansthat (ktj = Ak-m(j)is a step function. The Aj can (atleastintheory)be estimatedfromthe volatilitiesused to valuecaplets in Black' s model (ie. fromthe spotvolatilitks in Fipre 28,3).6suppose thatck istheBlack corresponds period caplet betweentimes tk and 4+I that to the volatility for the Equang variances, we must have .
,
.
t
k k
=
k A12 3f-j -
(3111)
j
j=1
This equation can be used to obtain the A's iteratively.
fxampleJ1.1 Assumethat the are a11equal and the Blackcaplet spot volatilitks for the srst three caplets are 249:, 22%, and 2%. means that Ag = 24%,Sin 'h
'fhis
A2+ A2 2x 1 =
.
222
Also, since
A1 is 19.
.8?,4.
2 is 15..23t% fxample 31.2
2+ A21 + A22 = 3 x 9.292 Ag
.
.
Considerthe data in Table31.1on caplet volatilitiescj. Theseexhibitthe hump discussedin Section28.3.The A's are shownin the second rw. Noticethat the humpin the A's is more pronunced than the humpin the c's. Table 31.1
6 In practi the A's are determinedusing a least-squarescalibration that we will discusslater.
79
InteyestRate Deyivatives..S'JM and LMM
Implementationof the Model The LI0R market model can be implementedusing Monte Caqlo simulation. is Expressedin terms of the Af's, equation(31.10)
drkt) l :-(1). f=(,) or dln Fj(l)
=
'
f 6'(l)Af.
l+
m(,)A-m(t) dt + f6(l)
(Aj-m(t))2
#
i i (l)Af-?,(l)A)-??,(t)
)7
(31.1zj
dg wj.,,(,j
1+ );15(j)
imnt)
.
z
(31.13)
dt + Aj-?,,(:)dz
-
If, as an approximation, we assllme in the calculationof the drifi of ln F'j41)that 6.41)= htj) f0r tj < t < j+l, then F(l/+1)
' =
l
htj) exp
j i Fi (jJ)Aj-y.j h-j-j
f=+l
1+ bFitj)
A2j-y-j -
.
2
tj + Ak-slf
j
(31.14)
where: is a radop
sample from a normal distributionwith meanequal to zero and
standarddeviationequal to one.
Extensionto Several Factors The LIBOR market model can be extendedto incorporateseveralindependentfactors. is the componnt of the volatilhy of Fktj Supposethat there are p factorsand becomes(seeProblem31.11) attributableto the th factor.Equation(31.10) l
dF(i)
lktt' =
j-m(,)
hbtj
;,(l)4,(l) E#=1 dt +
p
(31.15)
(kgt)Jz
l + tibtj
=1
Of as the th component the volatility when there are i accrual periods between the next reset date and the maturity of the forwardcontract. Equaiion(31.14)
Dene
j
q
then beconnes
' .
F)(.f+1) lktjt. exp =
l
j i F'i (j) j Tp =1
. i=j+L
1+
j-y-j,j-y-j, -
Etz.j
fFj(!y)
j2-y-j,
tj
2 +
p l-./-l, =1
Eq
y
(31.1@
wherethe q are random samples froma normaldistributionwithmean equal to zero and standard deviationequal to one. The approximatio that the drift of a forward rate remains constant within each accrualperiodallows us to jllmpfromone reset date to the next in the simulation.This is convenientbecauseas already mentioned the rolling forwardrisk-neutral world allowsus to discountfromone reset date to the next. Supposethat we wishto simulate theforwardrates at time a zero curve for N accrual periods. 0n each trial we start w1t11
q10
CHAPTER 31 #k-1() and are calculated fromthe initialzep curve. are F(), /-14), zero. #k-I(ll). calculate /n1411), >411), Equation(31.16) Equation(31.16) is used to is then used again to calculate #'2(), f412), , #k-1(), and so on, until /k-l(fx-l) is obtined. Note that as we move thrugh timethe zero curve gets shorterand shorter. For example, supjose each accrual period is 3 monthj and N = 4. We start with a l-year zero curve. t theGyearpoint (attime 4), the simulationgies us iilformation on 9. 4-yearzero curve. The driftapproximationcan be testedby valuing caplets usingequation (31.16) and comparin'gthepricesto thosegivenby Black'smodel. Thetalueof Fktkj isthe realized rate for thetimeperiodbetwn tk and 4+I and enables the caplet payos at time 4+l to be calculated. Tis payosis dcounted backto timezero, one accrual periodat a time. The caplet value is the average of the discountedpayofl. The results of this type of analys showthat the cap values fromMonteCarlosimulatin are not signicantly diserentfromthosegiven by Black'smodel. Th is true even when thearual periods are 1 year in lengthand a very largenumber of trialsis used.7 This suggeststhat the drift approximationis innocuovsin most situations: 'fhese
.
.
.
,
.
.
.
.
.
,
.
Ratchet Caps, Stich Caps, and Flexi Caps The LIBOR market modl can be used to value some types of ponstandardcaps. Considerratchet caps and stickycaps. Theseincoporate rules for determininghowthe cap rate for each caplet set. In a ratcbet cap it equals theLI0R rate at the previous resetdate plus a spread. In a sticky cap it equals theprevionscappe rate plus a spread. Supposethat the cap rate at time tj is Kj, the LIBORrate at tim tj Rj, and the spreadis s. In a ratchet cap, Kj+L Rj + s. In a stickycap, Kj+L ntintiy, Kjj + s. Tables31.2and 31.3provide valuations of a ratchet cap and stick ap using the LIBORmarket model with one,tw, and threefactors.Theprincipal is Tl. The term =
1 Rate Volatilities,and theImplementation of seeJ. C. Hull and A. Wite, ForwardRateVolatilities,swap 2 2): 4G-61.The only exceptionis LIBOR Jtantal ofFixed 1, lscome, the MarketModel,'' (septepber whenthe cap volatilitiesare very kh.
structureis assumed to be Qat at 5% per annum an the caplet volatilities re as in Table31.1.The interestrate is reset annually.The spread is 25basispoints.Tables31.4 ad 3'tshw h tlle voltility Ws sllitinto compnents when twoualid thierfactor MonteCarlosimqlationsincorpormodelswere used. The results are basedon l, 19.7.The standard error of antithetic variable i describtd Section technique atingthe eachprice is about A third type of nonstanzard cap is zjkxi cap. Tilis is likea replar cap expt that thereis a limiton thetotalnumberof capletsthat can be exercisedkConsideran annualpay :exi cap when the principal is $1p0,the term structure ij ;at at 5%, and the cap caplets volatilitiesare as in Tables 31.1,31.4,and 315 Supjosethat all in-the-money are exercise up to a maximum of ve. Withone, two, and threefactors,the LIBOR marketmodl givesthe price of theinstrnmentas 3.43,3.58,and 3.61,respectively(see Problem31.15for other types of iexi caps). The pricing of a plain vanilla cap dependsonly on the total volatility and is independentof the number of factors.Tlis isbecaus:theprice of a plainvanillacaplet epends on the behavior of only one forward rate. The pris of caplets in the dependon the nonstandardinstrments we havelookedat are difkrentin that t.1.11 of several probability diflrent forward distribution As rates. joint a result theydo pependon the number of factors. .
Valuing Eurapean Swap Options As shownby Hull ad White,thereis an analytk approximatinfor valuing European mode1.8Let T be thematurityof the swapoption swapoptions in tlie LIBORmarket TN. Defne and asjllme that the payment dates for the swap are T1 T2, the swaprate at time ! is given by zi= 7;.+j :r. From equation(27.23), ,
.
.
.
,
-
Pt, 7) Pt, 7k) x-j zipt, -
.41) =
Ef=
6.+1)
It is also trpe that Pt, 6.) i-1 1 Pt, T) j- 1 + z./6y(!) =
for 1 t i K N, where Gjt) is the forwardrate at time ! for the period betweenTj aqdTsl. nesetwo equations togetherdefne a relaonsbip between:(!) and the Gjt). is pplyingIt's lemma (seeProblem31.12),the variance P(l) of the swap rate s(1)
givenby
x-l Vj
, !F-71 !F-7 57(,) --
2
(,)cj(,)),j(,)
,q
.y
ajtqxq
=
=l
.
whereM is the total number of factors,hF') is the'changein the forwardrati for a forwardcontract maturing in j accrualperiods,ajq .is the factorloadingfor the /th forwardrate and the 4th factor,xq is the factorscore for the 4th factor,and
equals1 when 4j qz and wheh 41 # q2. Desne sq as the standard deviationof the th factor score.If thenumberof factorsused in $e LIBORmarket model, p, is equal to the total number of factors,M, it is correct to set =
t? = a i,tz st? @.
for 1 %j,q %M. Whenp nbii r . . (.' .. . pli, .@ . .
y j. y jt r yyjj . y... j.. jjjjjyy.gj.),.).. t , .y;gg.y r . r yj.. j . ' ; j .r .,; ). ,.jy y .y. g.yj rgj jjy,yy j. .y;,y yy.. .yyj . yy g-yyyjj,y ... .j ,j,yj g... : .. )... . .... ,)yj y y gy g ( jjjk.jyy yy. ) jj y . yy sj,.. rjyj...yjy(jj jj, yyy; .j j.g yy j . . , ;,. y. ..,. ( (r . .j jy.y,. yy., .. .rnbAl.r
expectedexcessreturn Of thes&P500Over the risk-freerate is5%.Euation estimatesthe market prke Of risk fOr the company'ssales as
(33,2)
03
' x 0.05 0.075 .2 =
Whenno historicaldata am avallablefOr the prticular variableunder censideration, Other similar variables FOr example,if plant is can sometimesbe used as proxies. ?, beingconstructed to manufcture a new product, data can be collectedOn the jales Of Other similar products. The correlation Of the new product with the market indexcan thelf be assumed to be the same as that of theseOther products. In some cases, the estimateof p in equation (33.2) If an analystis must be basedOn subjectivejudpnent. performan unrelated of convincedthat a particular variable is to the a market index, its market price of risk should be set to zero. FOr to estimatethe market pri Of rk lcause some variables, it is npt nepa the process followedby a variable in a rk-neutral world ca he stimateddirectly.For example,if the is the pri Of an investmentasset, it.stotal return in a riskneutral world is the risk-free rate. If the variableis the shrt-term interestrater, Chapter30 shws how a risk-neutral process can be estimaiedfromthe initialterm structure of interest rates. Later in this chapter we will show howthe risk-neutral fOr a commodity can be estimatedfromfuturespris. pross 'variable
33.4 APPLICATIQNTO THE VALUATION0F A BUSIjESS Traditioal methods of businessyaluation,such as applyinga pri/earnings multiplier to current earnings, do nOt work well fOr new businessvs.Tpically a cbmpany's earningsare negative duringits early years as it attempts to gain market share and establishrelationships with customers. The company must lie valued by estimating future earnings and cash iows under diferent scenarios. The real options approach can be useful in this situation. model llating the company'sfuturecash flowsto variablessuch as the salesgrowthrates,variablecostsas FOr a percentof sales, flxedcosts, and so On, is developed. keyvariables,a risk-neutral stochasticprocessis estimatedas outlined in theppvioustwosections. Monte Carlo simulationisthen carried out to generatealternativescenariosforthenet cash flowsper yearin a risk-neutral world. It islikelythat under some of thesesnarios thecompany does very well and undvr Others it becomesbankrupt and ases Operations. ('rhe simulationmust have a bulltin rule fOr determiningwhen bankruptcyhappens.)ne valueof the company is the presentvalue of the expectedcash flowin each year using the rijk-iee rate for discounting.Businesssnapshot 33.1 gives an exampleof the applicationof the approach to mazon.com.
33.5 COMMODITY PRICES Many investmentsinvolveuncertainties related to future commodity prkes. Often futuresprices can be used to estimatethe risk-neutralstochasticpross for a commodity price directly.This avoids the need to explicitlyestimatea market pli of risk for the commodity.
742
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worldcan be calculated fromthe cost data as 4., 6., re-spectively. Thevalueof the project istherefore + 4.0:-.1
-15.
xl
+
.
and 8. in years 1, 2, and 3,
(k-.lx2 + g (k.1x3= .
-,
g jj .
This analysis indicatesthat the project Should not be undertaken becauseit would reduceshareholder wealth by 0.54millin. Figure 33.3 showsthe valut of the projeit at each node of Figure33.2. This is calculatedfromFigure33.2,Consider,forexample,nodeH. There is a 0.2217probability thatthecommoditypri at tht end of tht tld year is22.85,so thatth third-ytarprofh thefeis a 0.6566probability that the comis 2 x 22.85 2 x 17 6 5.70.similarly, and thereis a modityprice at the end of thethirdyear is 16.16,so thattheprost is of 0.1217probabilitythat thecommodityprice at theend thethird year is 11.43,so that the prost is.- 17.14.The value of the project at node H in Figure33.3 istherefore -
=
-
-7.68
(0.2217 x 5 70+ .
0.6566x
+ 0.1217x (-7.68)
(-17.14))e-'1X1
-5.31
=
As another example,consider node C. Thereis a 9.1667chance of moving to node where the commodity price is 31.37. The second year cash flow is then 2 x 31.31 2 x 17 6 22.74.The value of subseqnent cash hows.at node F is 21.42. The total vglue of the project if we move to node F is therefore 21.42+ 22.74= 44.16.Similarlythe total valu of theprojectif we move to nodes G
!t
-
'
=
-
of baseproject withno embeddedoptions: p,, p., and h are r,Fijure 33.3 Valuation smiddle'' sup'' and the probabilies of movementsfrom a node. Sdown''
CHAPTER 33 resjectively.Thevalue of the project at node istherefore c andH are 10.35and (-1j.93)k-0.Ix1 lj sg :.1667x 44.14+ 0.6666z 10.35+ 0.1667x -13.93,
=
.
Figure33.3 shows that the value f the projectat the initialnode A is 14.46.Whenthe initialinvestmentis takeninto accont the value of thepiojectis therefore This is in agreement with our earlier calculations, Supposenow thgt the companyhas the Option to abandonthe projectat any time. We suppose that thereis no salvagevalue and no further pymentq are required.once theprojecthasbeenabandoned.Abandonmentis an Americanput option1t.11a strike Price of zero &nd lsvalued in Fipre 33.4.Th put olyion lhould not be exercisedat nodesE, F, and G becausethe valueOf theprojectis positiveat theseinodes.It should be exercisedat nodes H and1.Thevalue ofthe put optionis 5.31and 13.49at nodes H and 1, respectively.Rollingbackthroughthe tree, the value of the abandonment put option at node D if it is not exercisedis -,54.
(.1217 x 13.49+ 0.6566x 531 + 0.2217x
):-0.lxJ
j jj
=
.
The value of exercisingthe put option at node D is 9.65.Thisis greater than 4.64,and so the put should be exercisedat node D. The value of the put option at node C is (.1
667x +
.
6666
x ()+ 0.1667x
1)k-th1X
(5.3
l
=
.
sg
ValuationOf option to abandonthe prject: pu, pm, and pd are the probabilitiesof and movements from a node.
and the value at node A is (0.1667x ()+ 0.6666x 0.80+ 0.1667x 9.65):-.1
x1
=
j
.
94
The abandonment option is thereforeworth $1 nllion. It increasesthe valueof the projectfrom millionto +$1.4 million.A projectthat waspreviouslyunattractive positivevalue to shareholders. nowhas a Supposenext that the ompany hasno abandonmentoption. Insteadit hastheoption at any timeto increasethe scaleof theproject by 2%. Thecost of doingth is $2millin. Productionincreasesfrom2. to 2.4nlillionunits peryear.Variablecostsremain$17per unit and fixed costs increaseby 29% from $6.9million to $7.2million. Thisis an Americancall option to buy 20% of the base project in Fkure 33.3for $2million. The option valued in Figtlre33.5.At node E, the option should be exercised.The payof is x 42.24 2 = 6.45.At node F, it shoul also be exercisedfor a payoll-Of .2 x 21.42 2 = 2.28. At nodesG, H, and 1,the option shouldnot be exercised.At nodeB, exercisingis worth more than waiting andthe option isworth x 38.32 2 = 5.66.At node C, if the option is not exercised,it isworth .94
-$.54
.2
-
-
.2
-
(. 1667x 2 28 + 0.6666x .
If the optionis exercised,it i8worth fore not be exercisedat node C. At
.
.2
+ 0.1667x
.k-C'1X1 =
()34 .
The option shouldtherex 10.80 2 A, if not exerced, the option is worth .16.
-
=
rode
(. 1667x 5.66+ 9.6666x 9.34+ 9.1667x
.k-C'1Xl =
1g .
7;0
CHAPTER /3 If thi option is exercisedit is worth x 14.46 2 = 0.89.Earlyexerciseis therefore not optimal at node A. In tls case, the option increasesthe value of tht project from -9.54 to +.52. AgainFe 5nd that a projt that previouslyhad a negativevalue nw has a positive value. The expansion option in Fijure 33.5is relativelyeasy to value because,on the ptionhas beenexercised,al1subseuent cash iniowsand outiows increaseby20t%.In the case wherefued costs remain the sameor increasebylessthan 2%, it is nessary to keeptrack of moreinformationat the nodes of Fipre 33.3.Speccally we need to recordthe following: .2
-
.
1 The present value of subsequert xed costs 2. The present value of subsequentrevenuesnet of variable costs @
.
.
.
.
,
.
.
.
.
.
,
The payf fromexercisingthe option can thenbe calculated. When a project has jwo or more options, they are typicallynot independent.The valueof havingboth option A and option Bistpically not the sumof the valuesof the two options. To illustratethis,supposethat the company we havebeenconsideringhas both ahandonment and expansion options. The project cannot be expanded if it has alreadybeenabandoned. Moreover,the value of the put option to abandondependson whetherthe project 'has been expanded,z Theseinteractins betweenthe options in our example can be handledby desning fourstatesat each node:
1. Not already abandoned;not alreadyexpanded ; a1ready expanded 2. Nt already abandone81 3. Alreadyabandoned;not alreadyexpanded 4: lreabyabandoned;alreadyexpanded Whenwe roll backthroughthe treewe calculate the combined value of the options at eachnode for a11fouraltematives.Thisapproachto valuingpath-dependent opons is discussedin more detailin Section26.5. Whenthere re severalstochasticvariables, the value of the base project is usually determinedby Monte Carlo simulation.The valuation of the project's embedded options is then more dicult becausea Monte Crlo simulationworks from the beginningto the end pf a project. When we reach a rtain point, we dp not have informationon the present value of the jroject'sfuture cash iows. However,the techniquesmentioned in Section26.8for valuing Americanoptionj using MonteCarlo simulationcan sometimesbe used. As an illustrationo this point, Schwartzand Moon (2) explain how their Amazon.comanalysis outlined in BusinessSnapshot33.1could be extended to take accountof the option to abandon (i.e.the option to declarebankruptcy)when the value offuturecash fowsis negative.3At each timestep,a polynomialrelationshipbetweenthe valueof not abandoningand variables suchas the current revenue,revenuegrowth rate, volatilities,cash balans, and 1os8carry frwards is assumed. Each sinmlationtrial 2 Asit happens,the twooptions do not interqctin Figures33.4and 33.5.However,the interactionsetween theoptions would bexme an issueif a larzer treewith smallertimestepsWerebuilt. 33.4assumedthat ankruptcy occurs when the cash balance fallsbelowzero, but 3 ne analysisin section is necessarilyoptimal for mazomcom. not tis
751
Real Options
.
estimte pfthe relationshipat each providesan observationf0r obtaininga least-squares 26.8.4 time.Thisisthe Longstafand schwartz approach of section
jUMMRY Th chapter has investigatedhow the ideasdevelopedearlier in the book can be appliedto the valuation of real assetsand ptions op real assets.lt has shown how the risk-neutral valuation principlecan be used to value a project dependnt onanyset of variables. expecte growthrate of eachvariable is adjusted to refkct itFmarket price of risk.Thevalueof the asset is thenthe present value of its expectedcashiows discountedat the risk-freerate. Msk-neutralvaluation providesan internallyconstent approachto cajital investlent appraisal. lt also makesit possible to value the options that are embeddedin many of the projectsthat are encounteredin practice.Thischapter has illus%tedthe approachby applying it to the valuation of Amazon.comat the end of 1999and the valuationof a project involvingthe extractionof a commodity. 'l'he
FURTHERREADING Press, Amran M. and N. Kulatikka, eal Optiozs, Boston, MA: Harvard Businessschool
1999.
Copeland,T.? T. Kolkr, and J. Munin, Valaatioz:Measurizg azd Mazagizg te Value p.J' Comparies,3rd edn. New York: Wiley,z. '
'
'
-
'.
'
..
Practitimers tkii, New York:Texere,z). Cojlnd, T., and V. Xntikarov,Ral optiozs: Schwartz,E. S., and M. Moon,tRational Pricingof Internet Companies,''FizazcialAzalysts Jp?zr?zlz, May/lune(2): 62-75. Trigeorgis, L., Real Optioas: Mazageliql Flexibilityc??#St?ategy iz Resource zlllpccfitw, Cambridge,MA: MIT Press, 1996. .
.
.
./1
'
and Problems(Answersin SolutionsManual) Questions 33.1 Explainthe diferencebetweenthe net present value approachand the risk-neutral valuation approach for valuing a new capital invtstmentopportunity. What are te advantagesof the risk-ntutral valuation approach for valuingreal options? the volatilityof copper pricesis 2% per 33.2. The market price of risk f0r copper is and the (-monthfuturesprice is 75 nts pound, priceis the 8 cents per annum, spot per pound. Whatis the expectedpercentagegrowth rate in copper pris over the next 6 months? 33.3. Considera commodity with constant volatility c and an expectedgrowth rate that is a functionsolelyof time.Showthat, in the traditionalrisk-neutral.world, .
.5,
ln ST
'w
#((lnF(T)
4 F. A. Longstas and E. Schwrtz, 'Taluing s. of FfncacfnlStads, l4, l Approack'' uRcvfcw
-
1c2T c2rj
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2l): (spring
2
'
A simple Leastquares Optionsby simulation: l 13-47.
.
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CHAPTER 33 ..
'
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.. ..
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l
.
where is the value of thecommodityat timeT, bjtj isthe futres pric at timet for a contract maturig at time t, and @,$ is a normal distributionwith mean m and kr
variancet).
33.4. Derive a relationship betweenthe convenien yield f a compodity andits market prke of risk. The 33.5.The correlation betweena company's gross revenue and the market indexis volatilityof the market return of the market overtheriskzfreerate is 6% and the excess indexis 18$4.Whatis the market price of risk f0r the cpmpany's revenue? 33.6.A companycan buy an option for the deliveryof 1 million units of a commodityin 3 yearsat $25per unit. The 3-yearfuturespri is $24.The risk-free interestrate is 5%per annum with continuous compoundingand the volatility of the futuresprice is 2% pet annum. Howmuchis the optionworth? 33.7.A driverentering lntoa car leaseagreementcan obtainthe right to buythe car in 4 years for$1,0. The current value of the car is $3,. The value of the car, S, is expected to followthe process .2.
.
ds
=
gsdt 4. o'sdz
15,and dz is a Wienerprocess.Themarketprice of risk for the wherey, = c estimated price Whatis.thevalue of the option? Assumethat.the riskis to be car free rate fr all maturities is 6%. -.25,
=
,
-0.1.
AssignmentQuestions the spot prke, f-monthfuturespri, and lz-monthfuturespri for wheat that the price of wheat are 250,260, and 27 nts per bushel,respectively.suppose :a equation followstht process in witha and c 0.15.Construt a two-time(33.4) risk-neutralworld. prke of wheat in for the steptree a and a further A farmer has a prject that involvesan expenditure of $1, months. will wheat increase that is It expenditureof $9, harvested and soldby 6 in bushelsin 1 year. Whatis the value of the project? Supposethat the farmercan 4, abandontheproject in 6 months andavoidpaying the $90, cost at that time.Whatis the value of the abandonent option?Assllmea risk-free rate of 5% with continuous compounding. 33.9.In the exmple considere in Section33.6: (a) Whatis the value of the abandonmentoption if it costs $3millionrather than zero? (b) Whatisth value ofthe expansion opon ifit costs $5nllion rather than $2million?
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Define Risk Limits It is essential that a11companiesdefne in a clear and unnmbiguouswaylimitsto the fnancial risks that can be taken.They shtmldthen set up proceduresfor ensuring that the limits are obeyed.ldeally,overall risk limitsshouldbe set at bbard kvel. These should then be convertedto limits applicableto the individualsresponsible for managingparticular risks. Daily reports shouldindicatethe gn or lossthat will be experiencedfor particular movementsin market variables. Theseshoul be checked againstthe actual gains and loses that are experiencedto ensmethat the valuation proceduresunderlyingthe reports are aurate. lt is partkularly importantthat companiesmonitor risks carefu when derivatives aiused. Tllis is because,as we sawin Chapter 1, derivativescan be used for hedging,
756
CHAPTER 34 speculatiop,and arbitrage.Withoutclosemonitoring,it is impossibleto knowwhethet a derivativestrader haj switched frombeinga hedgerto a speculatr or skitched from beingan arbitrageurto beinga speculator.Baringsis a classicexampleof what can go wrong.Nick Leeson's mandate was to carry out lowdsk arbitrage betweenthe S1ngapote and Osaka marketj p Nikkei225futures.Unknownto his superiorsin London,Leesonswitched frombeihgan arbitrageurto takinghugehets on the future dirtion of the Nikkei225.Systemswithin Baringswere so inadequatethat nobody knewwhat he was doing. ne argumenthereis not that no tisks should be taken.A treasurerwrking for a corporation, or a traderin a fnancialinstitution,or a fund lanager shouldbe allowed to take positipns on the futuredirettionof relevant market variables. Butthe sizesof the positions that can be takenshouldbe linite and the systemsin place jlmuld aurately report the risksheingtakenk
Take the Risk Limltj Seriously Whathappensif an individualexcds risk limltsand makesa prlt? Thisis a tricky issuefor senior management. It is temptingto ignoreviolations of risk limitswhep profitsresult. However,this is shortsilted. t leadsto a culture whererisk Iimitsare not takenseriously,and it paves the way for a disaster.In manyof the situations listed in Business Snapshots34.1 and 34.2, the companies had becomecomplneant about the riskdthey were takingbecausetheyhad takehsimilarrisks in previous years and made profts. The classictxamplehereisOranzeCounty.RobertCitron'sactivitiesin 1991-93had bn very proftable for OrangeCounty,and the municipality had come to rely on his tradingfor additionalfunding.Peoplechose to ipore the risks he was takinghecause he had produd profts. Unfortunately,thelossesmaik in 1994fat exceededthe prots .
frompfevious years.
The penalties for exeding risk limitsshouldbe justas great when prots result as whn loqsesresult. Otherwise,traderswh makelossesare liable to keepincreasing theirbets in the hope that eventuallya profh will rysult and a11Fill be forgiven.
Do Nbt AssumeYouCan Outguess the Market Sometraders are quite possibly betterthan othrs. But no trader gets it right a11the titne.A trader who correctlypreicts thedirectionin which marketvariableswill move 60% of the timeis doingwell.If a traderhas an outjtanding tmckrecord (asRobert Citrondid in the early 199s), it is likelyto be a result of luck rather than superior tradingskill. j Supposethat a fnancial institutionemploys 16 traders an one of thosetraders makespro:ts in every quarter cf a year. Shouldthe tpder receive a good bonus? Shouldthe trader's risk limitsbe increased?The answerto the st questionis that the trader will reive a good bonus.The answerto the secondquestion inevitably shouldbe no; The chance of making a profit in four consutive quarters from 0.54 or 1 in 16. This means that just y' chance ne of the random trading it 16 traders will k;get it right every single quarter of tjye year. It suould not be that the trader's luck will continue and the tmder's risk limitsshould not assumed be increased. ,,
DeyiyativesMishaps and WhatF: Can Leayn #p??lThes
757
Do Not Underestimate the Benefitsof Diversificatian When a trader appears good at predicting a particular market vafiable, thereis a tendencyto incrrasethe tmder'slimits.Wehavejustarguedthat tllisis a b# idea becaus it is quite likelythat the traderhasbeenluckyrather than clever.Hwever,jte us supposethat a fund is really convinced that the trader has specialtaknts. How un diversifedshouldit allowitselfto becomein ordr to takeadvange of thetmder's specialSkills? The answeris that the benestsfromdiversiscationare hug, and it is unlikelythat any traderis so good thatit is worth foxgoingthesebeneststo speculate heavilyon justone market variable. An examplewillillustratethe point here.suppose that there al'e 20 stocks, eachof wich havean expectedreturn of l% per annumand a standarddeviationof Dturns By pf 3%. The correlation betweenthe returns from any two of the stocksis dividing n investmentequally amongthe20stoks, an investorhas an expected turn enables of 1% per annum and standarddeviationof returns of 14.7%.Diversifcation the investorSo reduce risks by over half. Another way of expressing tllis is that diversihcationenablesan investorto doubletheexpectedreturnper unit of rk ten. Theinyestorwould haveto be extremelygpod at stockpickingto get a betterrisk-return tradeof by investingin justone stock. .2.
Carry out Scenario Analysesand StressTests The calculption pf lisk measures suh as VaR should alwaysbe aompanied by snario analysts and stftss ttsting to obtain an undtrstandingof what can go wrong. These were mentioned in Chapter2. They are very important.Humanbeihjshave urifortunatetendencyto anchor o one or two scenarioswhenevaluatingdecisions.ln 1993and 1994,fpr example,Procter&Gambleand GibsonGreetingsmay havebeenso convind that interestrates would remain 1owthat thy ignoredthe possibilityof a l-basis-point incrtast in thtir dtcisionmaking. It is importantto be creativein the way scenarios alr generated.0ne approach is to look at l or 2 years of data and choose the most extremeevents as scenarios, Sometimesthereis a shortage of data on a keyvariable.It is then sensibleto choose a similar variable for whkh much more data is available and use istoricaldaily percentagechanges in that variable as a proxy for possible dailypercentagechanges in thekeyvariable. For example,if thereis littledata on the prkes of bondsissuedbya particularcountry, historicaldata on pfices of bondsissuedby other similqrcountries can be used to developpossiblesnarios.
34.2 LESSONSFOR FINANCIALINSTITUTIONS Wenow moye on to consider lessonsthat are primarilyrelevantto fnancialinstitutions.
Monitor Traders Carefully ''untouch-
In trading rooms thereis a tendencyto regard lgh-performingtradersas able'' and to not subject their ctivities to the same sutiny as other trapers. ApparentlyJoseph Jett, KidderPeabody'sstar tmder of Treasury instruments,was
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oftentttoo
busy'' to answer questionsand discussllispositionswith the company'srisk
managers.
It is importantthat a11traders-particularlythose makinghigh profhs-be fully aountable. It is importantfor the Enncial ipstitutionto knpw whether the high profhsare beingmade by takingunreasonably.highrisks. It is alsoimportantto check that theVapcial institution'scomputersystemsand prking models are correctand are in someway. nt beipgmanipulated
Separate the Front, Middle,and BackOffice The/rtl oyce in a nantial institutionconsistspf thetraderswho are executingtrabes, taking positions, and so forth. The middleo'cc consistsof risk managers who are monitoringthe rks beingtaken. The back o'cc is where the record keepingand of the worst derivatives haveocurred because nemounting takes pla. some thesefunctionswere not keptsejarate.NickLeson controlled both thefront and bak and was, as a result, able to conceal the dkastrous oce for Baringsin singapom nature of his tradesfromllissuperiorsin London for some time.Jrme Kervielhad beforebecotninza traderand took advantage workedin Socit Gnrale's backo o llisknowledgeof its systemsto hidehis positions. 'disasters
Do Not BlindlyTrust Models Someof the largelossesincurredby fnancialinstitutionsqrose becauseof the models and colputer systemsbeingused. WediscussedhowLdder Peabody was misled by its own systemson pge 13. Anotherexampleof an incorrectmodel leadingto losses is providedby NationalWestminsterBank. This bank had an incorrectmodel for valuingswap optionsthat led to sipi:cant losses. If large profts ari reported when relativelysimple tradingstrategiesare followed, ther is a good chan that the models underlying the calculationof the prtts are wrong.Similarly,if a fnancialinstitutionappearsto be pprticularlycompetitive on its quotesfor a partkular type of deal,thereis a good ch>ncethat it is using a diflkent modelfromothrr market partidpants, and it should analyzewhat is going n carefully. To the head of a tradingroom, gettinp too much busineqsof a certain type can be just as worrisome as getting too littlebusinessof that type.
Be Conservative in Recojnizinj Inception Prnfits Whena fnancialinstitutionsdls a hkhly exotic instrumtntto a nonfnancial corporaon,the valuation can be llighlydependenton the underlying model.For example, with long-datedembeddedinterestrate optbns cn behighlydependenton instruments theinterestrate mode1used. In thesedrcumstances, a phrase used to desibe the daily model. markizg markt Th isbecausethereare no market of the dealis to to marking pricesfor silnilardeal that can be used as a benchmark. Supposethat a fnancial instutin managesto sell an instrumentto a clientfor $10million more thanit is worth-or at least$1 lnillion more thanitsmodelsays it is worth.The $1 million is knownas an izceptiozprost. Whenshouldit be recognized? There appers to he quite a variation in what difkrent investmentbanks do. Some
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D,eyivatives Mishaps and What F: Can Leaynhomne??l
whereasothers are much moreconservatj ve anj . recognizethe $10million immediately, recognizeit slowlyover ihelife of the deal. tradersto is very dangerous.It enctplmges Recognizinginceptionprots immediately a nd value of m odds, the taketheirbgnuses, a nd leavebeforethemodel aggressive use scrutiny. under close much the deal come It ls bettir to recopia inption profts slowly,so that tradershavethemotivaticnto invtstigatetheimpadof sevemldiferent modelsand severaldiferentsets ofassumptionsbeforecopinitting temselvesto a deal. '
Do Not Sell Clients InappropriateProducts It is temptingto sell corporate clients iappropriate products, paiticnlafly when thty appearto hakean appetite for tht underlyingrisks. But thisis shortsighted.Themost dmmaticexample of thisistheactivitiesofBpnkersTruit(BT)intheperbd leadingup to the spring of 1994.Manyof BT'sclientswerepersuaded to buyhigh-riskand totally inappropriateproducts. A typicalproduct (e.g., the 5/30swap discussedon page 741) and a wouldgive the client a yoodchance of savinga fewbasispoints on itsborrowipgs smallchance of costiny a largeamount of poney. The produts wrked well for BT's clientsin 1992 and 1993,but blewup in 1994when interestmtesrose Sharply. Thebad publicitythat followedhvrt BT greatly.Tlleyears it had spent buildingup trustaong corporateclients and developingan enkabk reputation for innovationin derivatives salesmen.BTwas wtre largelylost as a result of theactivitiesof a fewoverlyaggressive of court. It was clients settle large o ut of its lawsuits amounts to forcedto pay moneyto takenover by DeutscheBank in 1999. t
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Do Not Ijnore LiquidityRisk and other instruments Financialengineersusually basethepricingof exotic instruments relatively of prices that trade infrequentlfon the adivelyiradedinstmments.For example:
1. A fmancialengineeroften calculatesa zero curve fromactivelytradedgvtrnment bonds (known as on-the-nm bonds) and uses it to price bondsthat tradeless frequently(of-the-run bonds). 2. A fnancial enginer often impliesthe volatilityof an asset ftom activelytraded optionsand uses it to price lessactivelytraded option. 3. A nancial engineeroftenimpliesinformationaboutthebehaviorOf intertstrates from actively traded inttrest rate caps and swap options and uses it to price productsthat are highlystructured. These practices are not unreaonable. Howtver,it is dangerousto assumethat less activelytraded instrumentscan always e traded at close to theirtheoreticalprice. Whenfmancialmarkets experiencea shock of 0ne sort or qnother thereis ofttn a to quality.''Liquidity becomtsvery importantto investors,andilliquidinstruoften sell at a big discountto theirtheoreticalvalues.nis happenedin 2*7 ments followingthe joltto credit markets cause bylackof conden in securitiesbackedby subprimemortgages. Another example of lossesaring fromliquidityrisk is provided by Long-rlkrm Capital Management (LTCM), which was discussedin BusintssSnapshot2.2.Th arbitrage.It attemptedto identify hedgefundfolloweda stratec knownas cozvergezce Stflight
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jwosecurities(orportfolios of securities)that should in theorysellfor the Same price. If themarket price of one securitywaS lessthat of the other, would buythat security and sell theother. The strategyis basedon theide that if two securitieshavethe same .it
theoreticalprice.tlkeir ketprices shouldeventuallybe the same. mar In the sllmmer of 1998LTCMmade a hugeloss.Tilis was largelybecausea default by Russia on its pbt caused a iight to qqality.LTCMtended to be long illiquid instrumentsand hort the corresponding liquidinstrllments(forexample, it was lopg of-the-run bondj Y jhort on-the-run knds). The spreads betweenthe prkes of illiquidinstrutents ad the correspondingliquidinstrpment.s idenedshaply after theRusian default.ETCMwas ilighlyleveraged.t experiencedhugelossesand there weremargin calls on jts positionsthat it was ppableto meet. ne LTCM story reinforces the importan of carryingout scenqrio analysesand stresstestingto look at Whatcan happenin the worst of all worlds. LTCMcould have trkd to examine other timesin historywhen therehavebeenextreme Kghtsto quality to quantify the liquidityrisks it was facing.
Beware When Everyope Is Followinj the Same Tradinj Strategy It sometimes happensthat many market participants are followingessentially the sametrading strategy. This cteates a dangerousenvironment wliere there are Eableto be big market moves, unstable markets,anj largelossesfor the market participnts. We gave one exampleof thisin Chapter17when discussingportfolio insuranceand the market ctash of October1987.In the months leadingup to the crash, increasing numbersof portfolio manazers were attempting to insuretheir portfolios by creating spthetic put options. They bousht stocks or stock indexfuturesafter a rise in the marketand sold them aer a fall. This created an unstable market. A relativelysmall declinein stock prices could lead to a wave of selling by portfolio insurers.The latter wouldlead to a furtherdeclinein themarket,whkh copld giverise to another wave of selling,and so on. There is littledoubt that without pqrtfolio insurancethe crash of Uctober1987would havebeenmuch lesssevere. Another exampleis provided by LTCMin 1992.lts position Wu & more dlcult by the fact that many other hedgefundswere followingsimilar convefgece arbitrage straiegies.Afterthe Russiandefaultand the flightto quality, LTCMtried to liquidate part of its portfolio to meet margin calls. Unfortunately,other hedgefundywep facing similar problems to LTCM and trying to do similar trades. This exacerbated the situation,causing liquidty spreads to be even highr than they would othemise have beenand reinforcing the iight to quality.Cnsider, for exqmple, LTCM'S positionin U.S.Treasury bonds.It was longthe illiquidof-tht-run bondsand short theliquidonthe-runbonds.When a :ight to qualitycausedspreadsbetweenyieldson thetwotypesof bondsto widen, LTCMhad to liquidateits positionj by sellingof-the-run bnds and buyingon-the-run bonds.Otherlargehedgefundswere doingthe same. As a result, the priceof on-the-nm bonds rose relativeyoohkhe-run bondsand the spread betweenthe two yieldswidened evenmore than it had donealready. A furtherexample is provided by the activitiesof Britishinsuran companks in the late 199s. These insurancecompanies had entered into many contracts promising that the rate of interestapplicableto an annuityreived by an individualon retirement wouldbe $e greater of the market rate and a paranteed rate. At abot the snmetime, a11instlran companies decidedto hedge part of their risks on these contracts.by
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Them DerimtivesMishapsand What F: Can fzg?l Jrp'??l
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' buyinglong-datedswap optionsfromfmancialinstitutions.Thelancial institutions theydealtwith hedgedtheirrisks bybuyinghugenumbersof lonpdatedsterlingbonds. As result, bond prices rose and long sttrling rates declined.Molr bondshad to be boughtto mainin the dyn>michedge,longkerlingrqtes declinedfurther,and so on, Financialinstitutionslostmorey and, becauselongrates declined,insurancecompanies fou d themslvesin a worst pdsition on the risks that theyhad chosennot to edye. The chief lessonto be ltarnedfrointhesekoriesis (hatit is imjornt to see te big pictureof what is going 0n' i' snncilmarkets nd:to understandthe risksinherentin where many marketparticipants are followingthe snmetmdingstrgtegy. situations
D0 Not Finance Lonj-'term Assetswith Short-Termtiabilities As discussedin Section4.1, it is impornt fQr a snancial institutiplto match the maturitiesof assets and liabilities.If it does not do this, it is subjting itselfto rist. Savingsand Loansin theUnitedStes ran intodiculties signihcantintertstrate in the 196s, 197s, and 19808betuse theysnanced long-termmoitgageswith shorb term deposits.ContinentalBank failedin 1984 for a similar reason (seeBusiness Snapshot4.3). Duringthe period leadingup to the credit cnmch of 2007,therewas tendencyfor subprimemortgages and other lonpterm assets to be Vand by commercialpaper wile they were in a portfolio waiting to be packaged into structuxd products (ke Business Sngpshot23.3). Conduits an special purjdse vehicles had an pngoing requirementfor this type of snancing. Thecommercialpaper would typicallyberolled overevery month. fbr example, the purchajers of commercialpaper issuedon pril 1 wouldbe redeemedwith the proceedsof a new commercialpaper issueon M>y1.This newcommercialpaper issuewould in turn be redeemedwith another new commercial paper issue on June 1, and so on. When inkestorslost cnsdence in subprime mortgagesin ugust 2007,it becnmeimpossibleto roll over commercialpaper. In manyinstans bankshad provided guamnteesand had to pfovidefmancing.Tllisled to a shortage of liquidity.As a result, the credit crunch was moreseverethanit would. havebeenif longer-term nancing had beenarranged.
Market Transparenc ls Important 0ne of the lessonsfrom the credit crunch of 2007is that market transparencyis important.Duringthe period leadingup to 2007,investorstradedlghly stnlctured productswithout any real knowledgeof the underlying assets. A11theyknewwas the credit rating of the security beingtraded,Withhindsight,we can say that investors shouldhavedemandedmore informationaboutthe underlyingassetsand shouldhave more carefullyassessed the risks they were taking-but it is easy to V wise afterthe event! The subprimemeltdown of Aupst 2997caused investorsto losecolden in a11 structuredproducts and withdraw fromthat market. This1edto a marketbreakdown wheretranchesof structured ppducts could only be sold at prices well belowtheir theoreticalvalues. There was a ght to qualityand credit spreadsincreased.If there had been market transparencyso that investorsunderstood the asset-backedsecmities theywere buying,therewould still havehen subprimelosses,but theiight to quality and disruptionsto the market would havebeenlesspronounced.
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34.3 (ESSON! FOR NONFINbNCIALCORPORATIONS We now consider lessonsprimarilyapplicableto nonfmancialcoiporations. .x
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Make Sure You FullyUnderstand the'Trades YouAre Doinj Corppiationsshpuld never undertake a trad ot a trading jttategy that they d: not '
fu11 y understand. Thisis a somewat obvioujpoint, but it is surprisinghow often a trader working for a nonhnancial corporation will, after a big.loss, admit to not knowinj what was really goinp on and ciaim to have been misled by investment bankers.Robert'Citron,the treasurerof Opnge Co'unty did this. sodid the traders workingfor Hammersmithand Flllhnm,who in spite of their huge positionswere surprisinglyuninformed about howthe swapsand other interestrate derivativesthey traded really worked. If a senior manager in a corporation ds not understand a trade proposedby a subordinate,the tradeshouldnot be approved.A simplerule of thumbisthatif a trade andthe rationale for entering intoit are so comjlicated that theycannot beunderstood by the nunagezj it is almost rtainly inappropriatefor 4hecorporatkn. The trades undertakenbyProcter& Gambleand GibsonGxetingswould havebeenvetod using this criterion. One way of ensuring that you fullyundetstand a hnancialinstrumeptis to value it. If a corporationdoes not have the inouse capability to value an instrument,it shouldnot trade it. In practice,corprations often rely on theirderivativesdealersfor valuation advice. This is dangerous,as Procter & Gambleand Gibson Greetings found out. When they wanted to unwind their deals, they found tey were facing pris produced hy BankersTrust'sproprietarymodels, which theyhad no way of checking. .
Make Sure a Hedjer Does Not Becomea Speculator
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One of the unfortunate facts of life is that hedgingis relativelydull, whereas speculaon is exciting. When a company hires a trader to manage foreignexchanse, commodityprice, or/ interestrate risk, there is a dangerthat the followingmight happen. At flrst,the tmder doesthe job diligentlyand earns the confdence of top management.He or she assessesthe company's exposums and hedgesthem.As time goesby, the traderbecomesconvinced that he or she can outpess the market. Slowly the trader hecomes speculator.At rst thingsgo well, but then a lossis made. T0 recoverthe loss,the traderdoublesup the bets.Furtherlossesare made-and so on. The result is likelyto be a disaster. As mentioned earlier,clear limitsto the risks that can be taken shotlldbe set by senior management. Controlsshould be put in pla to ensure that the limits are obeyed.The trading strategy for a corporation should start with an anqlysisof the risksfacingthe corporation in foreignexchange,interestrate, commodity markets,and so 0n. A decisionshould thenbetakenon howthe risks are to be redud to acceptable levels.It is a clear sip that somethingis wronp within a corporation if the trading strategyis not derivedin a very directway fim the company's exposures. .
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' Be Cautious about Makinj te TreasuryDepartment a Profit Center
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ln the last 20 years therehas bee a tendencyto make the treasurydepartmentwithin a coaoration a proft center. Tllis appears to have much to recommend it. The treasureris motivatedto reduce lanting osts and manage risks s proftably as The problem is that the potential fpr the treasurerto makeprofts is limittd. possible. When raihg funds anti invrstingsurplus cash, the treasureris facingan ecient markd.The trtasurer can usually improvethe bottomline only by takingaddhional risks. The company'shedgingprogmmgives the treasurersome scope for making shrewddecisionsthat increaseprofhs. But it shoul be rememberedthat the goal of a hedgingprogram is to reduci rks, not to increaseexpected prtts. s pointed out in Chapter3, the decion to hedgewill lead to a worse outcomethan the decion not to hedge roughly 50% of the time. The danger of makingthe treasurydepartmenta proft center is that the treasureris motivated to becomea syculator. Thisis liableto lead to the type of outcomeexperiencedby OrangeCounty,Procter& Gmble, or GibsonGreetings. '''
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SUMMARY The hugelossesexjerienced fromthe use of derivatives havemade many treasurersvery wary. Following some of the losses,somenonfnancial corporations haveannounced plans to rezuce or eve elilninate their llse of derivatives.This is unfortunate because derivativesprovide treasurerswith very ecient ways to managerisks. The storiesbehindthe lossesemphasizethe point, made as earlyas Chapter1,that derivativescan be ustd for tither hedgingor speculation;thatis,theycan be used either to reduce rks or to take rks. Mpst lossesoccurred becausederivativeswere used
inappropriately.Employeeswho had an implicitor explicitmandateto hedgetheir company'srks detidedinsteadto speculat. The keylessonto be learnedfromthe lossesis the importanceof izterzal coztrols. senior management within a company Shopld issuea clear and unambiguous policy statementabouthowderivativesare to be used and the extentto whkh it ispermissible for employeesto take positions on movementsin market variables.Managemptshould then institutecontrolsto ensurethat the policy is carried out. lt is a recipefor disatter without a,closemcnitoring of therisks to giveindividualsauthority to tradederivatives beingtken.
FURT'HER READING Dunbar,N. InyentingMony: Te Story p./ Lozg-TermCapital /lt7gcpe?l/ azd te Legcntf. Behind1t. Chichester,UK: Wiley,2. Jorion, P. Big Bets (zp?l:Bad: Deriyatis azd Bazkruptcy iz Orange County. New York: Academic Press,1995. 1999). Jorion,P. $'HowLong-TermLostIts Capital,''Risk (september
CHAPTNR 34
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Vale at Risk to ControlRiskTakiz: HowWrongCanYpu Ju, X., and N. Pearson. Risk, 5-36. 1 (1999): Be?'' Journal:.J c?l#ManagingFinancial Persaud,A, D. (ed.)LiquidityBlack S&e..' Ilnderstanding, Quantfying LiquidityRisk. London,Risk Books,23. Tomson, R. Apoclypse Roulette: Te ffcl Worldofberivatives. London:Macmillan,1998. ttusinz
Accrual Swap An interestrate swap where intereston one side accrues only when a certain condition is met. . r Accrued Interest The interestearned on a bnd since the last coupon paymentdate. '
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Adaptive Mesh Model A mdel develped by Figlewskiand Gao that grafts a hightree on to a lw-resolutiontreejo that thereis more detailedmodelipg of resolution theasset pricein critical regions. Ajency Costs Costsarilingffm a situationwhere the agent (e.g., manager) is nt shareholder). motivatedt gct in the bestinterestsof the principal (e.g., American Option An optionthat can be exercisedat any timeduringits life. Amortizinj 5wap A swap where the notional principal decreasesin a predetermined
wayas time passes.
Analytic Result Restllt where answer is in the frm of an equation. Arbitrage A traing strategy that takes advantage of two or more securities beig mispricedrelative to each other. Arbitrajeur
An individualengaging in arbitrage.
Asian Option An ption with a payos dependent n the averageprice of theunderlyig asset duringa specifkd period. Ask Price The pri
that a dealeris ofering t sell an asset. Asked Price SeeAsk Pri. Asset-BackedSecurity Stcuritycreated from a prtfolio of loans,bonds,credit card receivables,or other assets. Asset-or-Nothinj Call Ojtion An ption that provides a payof equal to the asset priceif the asset price is aboveth strike price and zero othrwise. Asset-or-Nothing Put Option An ptin that prvides a payos equal to the psset ' pri if the asset price is belowthe strik'e price and zero oterwise, Asset Swap Exchangesthe coupon n a bondfor LIBOR plus a spread. As-You-Like-ltOption SeeChooserOptin.
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0./ .' Glossayy Teyms
At-te-Maney Option An option it whkh the strike price equals the price of the underlyingasset. AveragePriceCallOption An option givinj a payof equal to the greater of zer and tht amount by whkh the average pri of the asset exceedsthe strike prke. j AveragePr ce Put Option An option givinga paynf equal to the greater of zero and the amount by which the strike priceexceedsthe avepge price of the asset.' Option An option that provides a payof dependenton the difkrence Averagestrike betweenthe finl asset price and ti e averageasset pr ice .
Backdating Practice(often illegal)of marking a documentwith a datethat precedes the current date. BackTesting Testinga value-at-riskor other modelusing Mstoricaldata. '
BackwardsInduction A procedure for working fromthe end of a treeto itsbeginning ' in order to value an option. Barrier Optbn An option whose payof dependson whether the path of the underlyingasset has reached a barrier (i.e.,a certain predetermined level). Base Carrelatian Correlationthat leadsto the plice of a CBXI to ##/()CDO ttanche beingconsistent with the market for a particular value of X.
Basel11 New international regulations for calculating bank capital expected to come into efect in about 2007. Basis The diferencebetweenthe spot price and the futuresprice of a commodity. Basis Point When used to describeaii intelesirate, a bas point one hundredthof i/n one percent (=t0 ()1 ) Basij Risk The risk to a hedgerarising from uncertainty aout the basis at a future time. Basis swapA swap where cash flowsdeterminedby one ioating reference rate are exchangedfor cash iows determinedby another ioating referencerate. .
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Baskt Credit Default swapCreditdefault swap where there are several reference entities. Baskd Optbn An option that providesa payof dependenton the value of a portfolio of assets. Bear jpread A short position ip a put option with strik price X'j combined with a longposition in a put option with strikeprice #2 where K2 > rl. (Abearspreadcan also e created with call options) BermudanOptian An option that can be exercisedon specifed datesduringit$life. Beta A measure of the systematicrisk of an asset. Bid-AskSpread ne amount by wlch the ask price exceedsthe bid price. Bid-oifer
spreadSeeBid-AskSpfead.
BidPrice The price that a dealeris prepared to pay for an asset. BinaryCredit Default swapInstrumentwhere thereis a sxed dollar payof in the particular liy f default event a company. a Binary Ojtian Optionwith a discontinous payof, e.g., a cash-or-nothingoption or an asset-or-nothing option.
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GlossnyyOJTeymt
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Binomial Model A model where the pri of an asset is monitored over successiye shortperiods of time. In each short peiiod it is assumed that only two price movementsare possible. BihomialTree A tree that representshowan Assetpri
can evolveunder thebinomial
model.
Bivariate Nprmal Distribution A distributionfor two correlated variables, each of whichis normal. '
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Black's Approximation An apprqximate produre developedby FischerBlack for stock. valuinga call option on a dividend-paying Black's Model An extension of the Black-scholesmodel for valuing European optionson futurescontracts. As describedin Chapter26, it is used extensivelyin practiceto value European optigns when the distributionof the asset prie at matur ity is assumed to be lognormal. Black-scholes Model A model for pridng Europeanoptions on stocks,developedby Fischef Black,Myron scholes, and Robert Merton.
Board Broker Theindividpalwho hanles limitorders in some exchanges.Theboard brokermakes informationon outstanding limitorders available to other traders.
Bond Option An option where a bondis the underlying asset. : . Bond Yield Discountrate whlch,when applied to all the cash Powsof a bond,causes thepresent value of the cash llowsto equal the bond's market price. Botstrap Method A procedurefor calculating the zero-coupon yield clrve from
niarketdata. Boston Option See DeferredPayment Option. Bpx Spread A combination of a spread created from calls and a bear spread createdfrom puts. Break Forward See Deferred Payment Option. 'bull
Brownian Motion See Wiener Pross. Bull Spread A long psition in a call with strikeprice (l combined with a short positionin a call with strike pri Kj, where Kj > #1 (A bull spread can also be createdwith put options.) Buttey Spread A position that is created by taking a lopg position in a call with strikeprice #l, a long position in a call with strike pri K, and a short position in + K). two calls with strike pri #2, where K > #2 > #1 and #2 with options.) also created be put (A butter spreadcan .
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CalendarSpread A position that is created by takinga longpojition in a all option that matures at one timeand a short position in a similr call option that matures at a diferent time.(A calendar spreadcan also be created using put options.) Calibration Method for implyinga model's parnmeters from the prices of actively traded options. CallableBonz A bondcontainingprovisionsthat allow the issuerto buyit back at a predeterminedprice at certain timesduringits life, CallOption An option to buy an asset p,ta certai pri by a rtain date. CancelablvSwap Swapthat can be canled by one sideon prespecied dates. :
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Teyms GlossayyO.J
Cap SeeInterestRate Cap. Cap Rate The rate determiningpayofs in an interestl'ate cap. Capitil Asset PricingMedel A mdel relating the expectedreturn n n asset to its heta. Caplet 0ne componentof ap interestrate cap. Cash FlowMapping A ?rodure for mpresentingan instrumentas a poryfolio of Tiro-coupon hondsfor the purpose of calclting value at risk. Cash-or-Nothing CallOption An optionthat proyidesa fxed predeterminedpayof if the snalasset prke is ahove the ttrike pri and zero othemise.
Cash-or-Nothing Put Option An option that provides a fxed predeterminedpayof if the fnal asset prke is helowthe strikeprice and zero otherwise.
Cash Settlement Procedurefor settling a futures ontract in cajh rayher than by deliveringthe underlyingasjet. CATBond Bond whefe the interestand, possibly, the principal paid ar reduced if a insuran clails xceed a certain nmount. particulgrcategory of dtatastropllic''
Coolingdegreedays.th e maximllmof zer and the amount by wllich the daily 65O averagetemperatureis greater than Fahrenheit.The average temperatureis the to midnight). averageof the higheqtand lowesttemperatures(midnight CDO SeeCollateralizedDeht Obligation. cDo Squarek An instrllmentin which the default risks ip a: prtfolio of CDO tranchesare allocatd to new securities. CDXNA IG Portfolioof 125 North merican companies.
CD;
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Cheapesbto-Deliver Bond Thebondthat is cheapestto deliverin the CllicagoBoard of Tradebond futui'escontmct.
CholeskyDecompesition A method of snmplingfrom a multivariate normal dis-
tribution.
CheoserOption An option where the holderhas the right to choose whether it is a callor a put at somepoint duringits life. Classof Options SeeOptionClass. CleanPriceof Bond The quotedpri of a bnd. ne cash prke paid for the bond price) is calculated by adding the accrued interestto the cleanprice. (or zirty
Clearinghouse.A fil'mthat guarantees theperformanceof the parties in an exchangetraded derivativestransaction(alsoreferred to as a ckring corporation). ClearingMargin A mar/n postedby a member of a clearinghouse. CMO CollateralizedMortgageObligation. Cellar SeeInterestRate Collar. CollateralizationA systemfor posting collateralby one or both paytiesin @derivatives tranjaction. Dqbt Obligation A way of packaging credit risk. Severalclasses of collateralized securities as tranches)are created from a portfolio of bonds and there are (known rulesfor detrmining howthe cost of defaultsare allocated to classes. '
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Glossary0./ Teyms
' CollateralizedMortgageOligation (CMO) A mortgage-backe surit whefeinvestorsare dividedinto c1asses and thereare ruks for determining1)0wprincipal
epaymentsare channeledto the classes. Comination A position involvingbot callsand puts on the sameuntking asset. CommissionBiokers lndividualswho executetradesfor otht: peopleandcharge a commissionfor doing so. CommodityFtures Trading CommissionA body that regulatestradinsin futufes contractsin the United states. CommoditySwap A swapwhefecash qowsdepend n the pri of a commodtty. ComlmundCorrelation Corfelationimplie frop themarketpli of a CD0tanche. CompoundOption An option on an option. Compoundinjfrequency Th dehneshowan interestl'ate is miasured. where iterest compotmdsinstead f hing gid. ComjmundiqjSwap swap ConditionalValueat Risk (C-VaR) ExpectedlossduringN dayscondltbnalon being X)% tail of thedistributionof prosts/loses. ne variale# i8thetime in the (1 #9/ horizonand n is the condence level. Confirmatiqn Contractconsrming verbal agxement betweentwopaties to a tradein -
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market.
Constnt Elasticityof Variance (CEV)Model Modelwheretht varianceof thechange in a variable in a shrt period of timeis proportional to the value of thevarkbl. Consnt MaturitySwap(CMS) A swap wherea swaplate is txchane foreithera ratt or a ioating rate on each pament date. flxe,d onstant Matufity Tasury Swap A swap where the ykld on a Trealurybondis exchangedfor either a sxed rate or a Eoatingrate on eachpaymentdate. ConsumptionAsset An asset heldfor consmption rather thaninvestment. ontango A situationwhere the ftures price abve the expectedfuttlfespotprice. ContinuousCbmpounding A way of qupting interestmtes. It is the limitas te assumedcompounding intervalis madesmallerand smaller. ControlVariate Technique A techniquethat can sometimesbe used forimpfoving the of a numerkal accuracy
procedure.
ConvenienceYield A masure of tht benets fromownershipof an assettat a:e not obtainedby the holderof a longfuturescontmct on the asset. ConversionFactor A factor ustd to determlnethe number of bondsthatmust be deliveredin the ChicagoBoard of Tradebondfuturescontmct. Convedile Bond A. corporatq bond that can be converted into a pfedetermined amount of the company's equity at certaintimesduringits life. Convexity A measureof tht curvature in the relationsip betweenbondpris and bond yields. ConvexityAdjustment An overworkedterm.For example,it can Dfer to te adjustment necessary to convert a futuresinterestrate to a fomardinterestrate, lt n also refer to the adjustment to a forwardratt that sometimesnecessarywhenBlack's modelis used.
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Gkssary p./ Terms Copula A way of desningthecorrelationbetweenvariableswith knowndistributions. j Cornis-Fiser Expanson An approximaterelationshipbetweenthe fractilesof a distributionand its moments. probability Costof Carry The storagecosts plus the cost of snancing an assetminusthe income the earnedon asset, counterpartyTe oppositesidein a ancialtransaction. 'J
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Coupn Interestpament made on a bond. Covariance Measure of the linear relationship betweentro variables (equals the variables timesthe froductof their standarddeviations). correlajonbetweenthe CoverekCall A short position in a call option on an asset combinedwith a long positionin the asset. Codit Contajion The tendencyof a defaultby onezcompanyto lead to defaultsby othercompanies. CreditDefault Swap An instrumentthat givesthe holderthe rkht to sell a bond for itsfacevalue in the eventof a defaultby the issuer. Credit Depivative A derivativewhose payof dependson the creditwortllinesso one
or more companiej or countrks.
measureof the creditworthinessof a bond issue. Cxdit Ratihjs TransitionMatrix A tableshowing theprobabilitythat a company will movefrm one creditpting to anotherduringa certainperiod of time. Cfdit Risk The risk that a loss will be experiend becauseof a default by the counterpartyin derivativestransaction. Option Optionwhosepayof dependson the spread betweenthe yields Cit spread earned n two assets. Credit Value at Risk The credit loss that will not be eieded at some specised covdence level. CreditMetrics A proceduxfor calculatingcredit value at risk. Cross Hedging Hedgingan exposure to the price ot one asset with a contract on another asset. CumulativeDistribution Fundion ne probability that a variable will be lessthan as a functionof x. Swap where interestand principal in one currency are exchanged CurrencySwap for interestand principalin anothercurrency. Day Count A convention for quotinginterestrates. Day Trade A trade that is entered into and closed oui on the snme day. Credit Ratinj
.z
Default Correlation Measuresthe tendencyof two cmpanies to defaultat about the
sametime,
Defaultlntensity SeeHazardRate. DefaultProbabilityDensity Measuresthe unconditionalprobability f defaultin a futureshort period of time. DeferredPaymentOption An option where the pri paid is deferreduntil theend of theoption's life. '
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Gssayy oj Teyms ' .
Defefr,d Swap An agreementto enter into a swap at some timein thi future(also calleda foaard swap). Delivefy Price Priceagreedto (possibly some timein the past) in a forwardcontract. Delta The rate of changeof the price of a derivave with the pri of the underlying '
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Delta Hqdginj A hedginyschemethatis designedto make thepriceof a portfoiioof derivativesinsensitiveto small changesin the pri of the asset.
rpderlying
Delta-Neutral Poftfolio A portfoiiowith a delta of zero s that thereis n sensitivity to small changes in the pri of the underlyingasset.
DerivaGem The Softwre accompanying thisbook. Derivative An instrumentwhose pri dependson, or is derivtdfrom,the pri of anotherasset. Deterministic Vafiable A variablewhose futurevalue is lown. Diajonal Spread A position in two calls where both the strike pris and timesto maturityare diferent. (A diagonalspread can alsobe createdwith put pyions.) Diffefential Swap A swap where oating rate in one currency is exchanged for a qoatipgrate in anothercurrency and both rates are appliedto the same principal. Diffusion Process Model wherevalue of assetchangescontinuoujly (nojumps). Dirty Priceof Bond Cash price of bond. Discount Bond See Zero-coupon Bond. Discount Instrument An instrument,such as a Treasurybill, that provides no COUP0ns. Discount Rate The annuaiizeddollarreturn on a Treasurybill or similar instrument expressedas a percentaze of thehnalfa value. Dividend A cash payment made to the owner of a stock. Dividend Yield The dividendas a percentage of the stock price. Dollar Duration The prduct of bond'smodiled durationand the bond jrke.
a
Down-and-ln Option An option that comes into existen when the pri underlyingasset declinesto a prespecifkdlevel,
of the
Down-and-out Option An option that teasesto exist when the price of the underlyingassetdeclinesto a prespecifkd level. Downjrade Trijjer A clausein a contract that states that the contract will be terminatedwith a cashsettlementif the credit rating of one side fallsbelowa certain
level.
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Dfi Rate ne averageincreaseper unit of timein a stochasticvariable. Duration A measure of the averagelifea hond.It isalsoan approximationto theratio of the proportionalchange in the bondpriceto the ahsolutechangein its yield. Duration Matchinj A proceduref0r matching thedurationsof assetsandliabilitiesin
a hnancialinstitution. Dynamic Hedjinj A procedure for hedging an option position by periodically changingtlie position held in the underlying asset. The objective is usually to maintain a delta-neuttalposition.
772
Glossa'yp./ Te'ms EarlyExercise Exerciseprior to the maturitydate. Efficient Market Hypothesis A hypothesisthat asset prkes refkct relevant informa'*
j
t on. Eledronic Trading Systemof tradingwhere a computeris used to match buyersand
sellers.
Embedded Option An option that is an inseprable part of anotherinstrument. Empirical Research Researchbasedon historicalmarketdata.
stockOption
A stock option issuedby companyon its own stock and employees givento its as part of their remuneration.
Employe
Ejuilibrium Mpdel A modelf0ytrlrbthaviorof interestrates derivedfrop a model of theeconomy. EquityswapA swapwhere the retmn on an equity portflio is exchahgedforeither a xed r a foating rate of interest. Eurocurmncy A currency that is outside the formalcontrolof the issuingcountrfs
monetaryauthnrities. Eurodollar A dollarheldin a bnk outside the UnitedStates.
Eurodollar FutuxgContrad A futurescontractwritten on a Eurodollardepsit. Eurodollar Intest Rate Theinterestrate on a Eurodollardelosit. European Option An ptionthat can be exercisedonly at the end of its iife. EWMA Exponentiallyweightedmoving average. Exchange Option An option to exchangeone assetfor another. Ex-dividend Date When a dividen is declared,an ex-dividend date is specifkd. Invekors Fho own shaDs of the stcck justbeforethe ex-dividenddate reive the
dividend.
ExerciseLimit Maximumnumbr of option contractsthat can be exercised.within a
fwe-dayjtriod.
ExerciseMultiple Ratio of stock priceto strikeprke at timeof exercisefor tmployee stockoption. '
ExercisePrice The prke at which the underlying asset may be bought or soldin an option contract (alsocalled the strike prke). Exotic Option A nonstandardoption.
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Expedations Theory Thetheorythat fomard interestratesequal expectedfuturespot '
interestrates.
Expeded
See ConditionalValueat Risk. shortfall
Expeded Valueof a Variable The averagevalue of thepriable obtained by weighting the alternative values by theirprobabilities. ExpirationDate The end of lifeof a contract. Explicit FiniteDifference Method A method for valuing a d.erivative by solving the diferentialtquation. The value of the derivativeat time t is related to underlying threevalues at time t + !. It is essentiallythe snme as the trinomialtree methodk
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ExponentiallyWeijhted Movinj AverajeModel A model whete exponenalweighting is gsed to provide forecastsfor a variable fromhistoricaldata.It is sometimes appliedto variances and covarians in value at risk calculations. ExponentialWeijhtinj A weightingschemewhere the weightgivento an Observation dependson howrecept it is. The weightgivento an observationi timeperiodsajO is timesthe weight given to an Observation i l timeperiods agO where t: 1. Exposure The maximuin lossfromdefaultby a counterparty, -
ExtendahleBond A bond whose lifecan be extended at the option of thehplder. ExtendahleSwap A swap whose lifecan be extended at the Option of Ohe sideto the contract.
Fador
of uncertainty. source
Fador analysis An analysis aimedat fmdinga smallnumbef of factorsthat describe to a principal most of the variation in a largenumber of correlatedvariables(similar
componentsanalysis). FAS123 Accountingstandard in UlzitedStatesrlating to employeestockoptions. FAS133 Aounting standard in United statesrelpting to instrumentsused fOr hedging.
Board. FASB Financial Aounting standards iinancialIntermediary A bank or other fmancialinstitutionthatfacilitates thellowOf fundsbetweendiferent entities in the economy. Finite Diffence Method A method for solving a diflbrential equatitm. FlatVolatility The name givento volatilityused to price a cp whenthesamevolatility is use for each cklet. Flex Option An option tradedon an exchangewith termsthat aredifkrentfromthe standard options tradedby the exchange. Flexi Cap Interestrate cap where thereis a limit on the total numberOf caplets that can be exerised. '
Floorlet Ope component of a floor. Floor Rate The rate in an interestrate flooragreement. Foreign CurrencyOption An option on a foreignexchangerate. Forward Contrad A contract that obligates iheholderto buy or sellan asset fOr a predetermineddeliveryprice at predetermined futuretime. Forward Exchanje Rate The fbrwardpri
of
unit Of a foreip currency. Ferward Interest Rate Theinterestrate for a futureperiod of timeimpliedbythe rates Ore
prevailingin the markettoday.
Ferward Price Thedeliveryprice in a folwardcontmct that causesthe contract to be worth zero. Forward Rate Rate of interestfor a period of timein the futureimpliedby today's ZCr0 f ates .
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Glossaryp./ Terms ForwardRate Agreqment(FRA) Agreementthat a certain interestrayewill apply to a certainprincipal amount for a certain time period in the future. World A world isfomard risk-neytl'alwith xspect to a certain Forward Risk-Neutral assetwhen the market prie of risk equalsthe volatility of that asset. Frward Start Option An option desiped so that it will be at-the-money at some
timein the future.
Forward Swap SeeDeferredSwap. Futures Contract A contract that obligatesthe holder to buy or sell an asset at a predetirmineddeliveryprice duringa spcied futuretimeperiod.The contract is
setileddaily.
.
FuturesOption An option on a futurescontract. Futures Price
ihe
elivev pri
currentlyapplicqble to a futurescontract.
Futurestyle Option Futurescontract on the payof from an option.
Gamma The rate of change of deltawith respect to the asset pri. Gamma-NeqtralPodolio A portfolio with a gnmmaof zero. GARCH Model A modelfor forecastingvolatilitywhere the variance rate followsa
mean-reverting process.
GaussianCopulaModel A modelfor dening a correlation stnlcture betweentwo or more varbles. In somecredit derivativesmodels, it is used t dene a correlation structurefor timesto default. Procedure for integratingo#era normal istribution. GaussianQuadrature GeneralizedWiener PYcess A stochasticprocesswhere the change in a variable in time t has a normal distributilmwith mean and varian bcth projorlional to !. GeometricAveraje The nth root of the product of n numbers.
Geometric Brbwnian Motion A stochasti process ofte assumed for asset prkes wherethe logarithmof the underlyingvariable followsa geperalizedWienerprocess. Girsanov'sTheorem Result showing that whenwe change the measure (e.g.,move fromreal worldto risk-neutral world) the expectedreturn of a variable changes but thevolatility remains the same. Greeks Hedgeparameters such as delta,gamma, vega, theta; and rho. Haircut Dcount applied to the value of an asset for collaterl purposes. Hazard Rate Measuresprobability of defaultin a shortperiodof timeconditional on no earlier default.
HD9 Htatingdegreedays.The maximumof zer and the amount by whch the daily 65O averagrtemperatureis less than Fahrenheit.The akerage temperatureis the of the highestand lowesttemperatures(midnight to midnight). average Hedje A trade designedto redu risk. Hqdjer An individualwho entersinto hedng tmdes. Hdje Ratio The ratio of the sizeof a position in a hed/ng instrllmentto the size of
thepositionbeinghedged. Historical Simulation A simulationbasedon historicaldata.
Glonayyp./ Teyms
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Historic Volatility A volatiEtyestimate fromhistorkaldata. Holiday Calendar Calendr desningwhichdays are holidaysfor the puposes of determiningpayment datesin a swap. and Dember. IMM Dates Trd Wednesdayin Maph, June, september,
lmplicitFinite DifftrenceMethod A methobfor valuing a derivativeb solvingthe diferenal equation. Thevalueof the derivativeat timet 4. Al is related underlying
threevaluesat time 1. lmpliedCorlation Correiationnumber impliedfromthe price of a credit erivative qsingthe Gaussiancopula pr similarmodel. to
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Implied Distribution A distributionfor a future asset prke nplied from opon Prkes Implied Tree A tree describingthe movementsof an assetprke thatis constructed to beconsistent with observd opon prices. .
Implied Volatility VolatilityimpEedfroman option price using the Black-scholesor a similar model.
lmpliedVolatilityFunction (lVF)Model Modeldesiped so thatit matchesthe market pricesof a11Europeanoptions. lnception Profit Profh created by selling a derivativefor more than its theoretical ' va.1ue lndexAmortizinj 5@ap seeindexedprincipal swap. .
Index Arbitrage An arbitrage involvinga position in the stocks comprising a stock indexand a position in a futres contract on the stockindex.
lndexFutures A futurescontract on a stock indexor other index. lndexOption An option contract on a stock indexor other index. Indexed Principal Swap A swapwherethe principal declinesover time.The reduction in the principal oll a payment datedependson the levelof interestrates. Initial Marjin The cash required froma futurestrader at the me of the trade. Indantaneous Forwrd Rate Forward rate for a very short period of time in the
future. lntest Rate Cap An ption that provides a payof when a speced interestrate is abovea certain level.The interestrate is a ioating rate that is reset periodically. Interest Rate Collar A combination of an interest-ratecap and an interestrate ior. Intest Rate Derivative A derivativewhose payoss are dependenton futureinterest
rates. lnterestRateFloor An option that jrovidesa payof when an interestrate is belowa certainlevel.The interestrate is a Qoatingrate that is reset periodkally.
lntecstRateOption An option whete the payof is dependent n tl levelof interest rates.
lntest Rate swapAn exchange of a flxedrate of intereston a cettaln notional principalfor a ioating rate of intereston the same notional principal.
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Glouayyp/ Teyms International Swaps and DerivativesAssociation Trade Associationfor over-thecounter deriv>tivesand developerof master agreements ustd in over-the-counter
contracts. ln-tlp-MoneyOption Either(a)a call option were the asset price is greater than the strikeprice pr (b)a ppt option where the assetprice is lessthan the jtrike pri. Intrinsic Value For a call option, tis isthe greater of the excessof the assetprice over the strike price and zero. For'a put option, it isthe greater of the excessof the strike price over the assd pri and zero. lnvertedMarket A market where futurespfices decieasewiih maturity. .
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Investment Asset An assetheldby at leastsome individual;for investmentpurposes. securitywhere the holderreives 10 InterestOnly.A mortgagezbacked cashqnws on the underlying mortgage pool.
only interest
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and DerivativesAssociation. ISDA See Iniernationalswaps It5 Process A stochastic pross where the change in a variable duling each short periodof timeof length f has a normal distribution.The mean and varianc of the distrlbutionare proportional lo ! and are not nessarily constant. lt's temma A-resultthat enables the stochasticpross for a functionof a variable to be calculzted fromthe stochastic pross for the variable itself. lTraxx Euxye Portfolioof 125investment-grade Europeancompanies. Jump-DiffusionModel Model where asset pri has jumpssuperimposed on to a difusion pross such >s geometricBrownianmotion. Kqrtosis A measure of the fatness the tails of a distribution. '
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,of
LEAPSLong-termequityanticipationsecurities.Theseare relativelylong-termoptions onindividualstocks or stock indis. LIBID Londoninterbankbid rate. The rate bid by banks on Eurocurrencydeposits (i.e.,the l'ate at whicha bank is willingto borrowfrom other banks). CIBOR Londoninterbank fered rate. osered by bankj on Eurocurrency willing which bankis t he rate at to lend to other banks). a deposits (i.e., LIBORCurve LIBORzero-coupon interestrates as a functionof maturity. rl'ht
'rate
LlBoR-in-Arrears Swap Swapwhere the interestpaid on a dateis determinedby the interestrate observed on that date (notby the interestrate observed on the previous paymentdate). Limit Move The maximum price move permitted by the exchange in a single trading session. k' timit Ofdkr An order that can be executed only at a specised pnce or oe more favorableto the ihvestor. tiquidityPoference Theory A theoryleadingto the conclusion that fomard interest rates are aboveexpected futurespot interestrates. tiquidity Pmium The amount that forwardinterestrates exceed expected future spot interestrates. tiquidity Risk Risk that it will n0t be possibleto sell a holding of a particular instrumentat its theoreticalpri.
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GlossayydJTeyms
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ocals Individualson theioor of an exchangewho tradefor theirown aount thanfor komeoneelse.
rather
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Lojnormal Distribution A variable has a logormal distzibutionwhen the logarithm of the yariable has a normaldistributin. ong Hedje A hedgeinvolvinga longfutuls position. Lonj Position A positioninvolvingte purchaseof n asset. Lookback Option Ap ption whose payof is bependent on the maximumor minachieved price rtin duringa perio. imumof the asset lasi-random ow Discrepancy sequenc:See sequence. Maintenance Marjin Whenthe balancein a trader'smargin neztount fallsbelowthe maintenan margin level,the traderreceivesa margincall mquiringthe nemount to be topped up to the initialmargin level. Marjin The cash balan (orsecurity deposit)required from a futures or options trader. '
Marjin Call A request for extra margin when the balancein the margin aunt beloWthe maintnance margin level.
falls '
Market Maker A'trade?who is willing to quoteboth bid and ofer prices for an asset. M#rket Model A model most commonlyused by traders. Market Price of Risk A measure of the trde-ofs investorsmake betweenrisk and return.
A theoty that sort inferest rates arr deterniined independentlyof longinterestrates by the market.
Market
Teor sejmentation
Marking to Market The practice of pvaluing an instrumentto yeiect the current valuesof the relevant market variables. Markov Pocess
A stochastic process where the behaviorof the variable over a short peod of time dependssolely on the value of the variable at the beginningof the period,not on its past history.
Martinjale
& zero drift stochasticprocess.
Maturity Date The end of the lifeof a contract. Maximum ikeliood Metod A method for choosing the values of parameters by maximizingthe probability of a set of observationsoccurring. Mean Reversion Thetendencyof a maket variable (such as an interestrate) to revett back to some long-runaveragelevel. also called a probability measure, it defmesthe market prke Measure sometimej of risk. Modified Duration A modihcation to the standard dutationmeasure so that it more accuratelydescribesthe relationship betweenproportional changes in a bond price and actual changes in its yield. Themodication takesaccount of the compounding frequencywith which the yield is quoted. Money Market Account An investmentthat is initiallyequal to $1 and, at tim l,,z increasesat the very short-term risk-free interestrate prevailingat that time.
778
GlossayyOJTeyms Mante Carla
A proceduref?r randomlysnmpling changes in market simulatian
in order to value a derivative. variables
Modgage-BackedsecurityA seculky that entitles the owner to a sharein the cash :ows realized from a pool of mortgages. x
Naked Pasition A short ppsitionin a call option that is not comhinedwith a long ' position i the underlying asset. Netting Theabilityto ofset contractswith positiveand negave values in the event of a defaultby a counterjlty. Newtan-RaphsanMethod An iterave procedurefor solvingnonlinear equations. '
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No-ArbitrageAssumption The assumption that there are no arbitrage opportunities inm arket pris. No-Arbitragelnterest Rate Mdql. A potkl. fpr tlw b:ehavior of interestrates that is eractlyconsistetdtwith-tllinitrialie' strucire of interestmtes. ,
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Nonstationary Madel A mdel whie the volatility pargmeterjare a functionof time. Nonsystematic Risk Rk that can be diversisedaway. Normal Backwardatin A situation whert the futures pri future spot pri.
is belowthe expected
Normal Distribution The standard bell-shapeddistributionof statistics. Normal Market A market Wherefuturesprices increasewith maturity. Notional Principal The principal used to calculate payments in an interestrate swap. The principalis becauseit is neither paid nor received. Sdotional''
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Numerai Dents the units in whkh securityprkes are measured. For example, if price the of lBM ij the numeraire,a11securitypricesare measuredrelativeto IBM.If IBM is $8 and a particular security pticeis $50,the securityprice is 0.625when 1BM'isthe numeraire. Numeric/l Pracedure A method of valuingan optionwhen no formulais available.
OCC OptionsClearingCorporation.SeeClearinghouse. Offer Price SeeAsk Price. Open Interest The total number of long positions outstandingin a futuresctmtract (equalsthe total number of short positions). of tradingwhere tradersmeet on the floorof the exchange Open Outcry system
Optian The right t buy or Sell an asset. Optian-Adjus'ted spreadThe spreadoverthe Treasurycurve that makes the theoreticalpriceof an inttrest rate delivativeequal to the market price. OptionClass A11optionsof the same tpe (callor put) on a particular stock. Option seriesA11optionsof a certainclass withthe samestrikeprke and expiration date. Order Book Ocial See Board Broker. Out-of-the-Money Optbn Either(a)a call optionwherethe assd priceis lessthan the strikt price or (b)a put option where theasset prke is greater than the strike
pri.
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GlossayyOJTeyms
Over-the-counterMarket A marketwheretradersdeal by phone. The tradets are institutions,corporations, and fund managers. usuallysnancial
Package A derivativethat is a portfolio of standardcalls and puts, possiblycombined with a positionin forwardcontracts and the asset itself. Par Value The principal amount of a bond.
coupon on a bond tat makesits price tqti'l te principal. Parallel i A movement in the yieldcurvewhereeachpoint on the curve chansesby th sameamount.
Par Yield
-m-f'he
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Path-DependentOption An optlon whosepayof depends n thewholepath followed bythe tpderlying variable-hot justits snalvalue. Payoff The cash realized bytheholderof ql option or other derivative at theend o its life. uked describep.stjndprd deaj PlAinV j+, '
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Real-worldmeasre.
PO Principal Only. A mortgage-backedsecurity where the holder reives only principalcash :Ws on the lmderlying mortgage pool. Poisson Process A pross describinga situationwhere eventshappenat random.Te probabilityof an event in time Al is Al, where is the intensityof the pross. Podolio Immunization Makinga portfolio relativelyinsensitiveto interestrates. Podolio Insurance Entering into trades to ensure that the value of a portfolio will not fall belowa certain level. Position L,imit The maximum position a trader(0rgroup of tmdersacting together)is
allowedto hold.
Premium The prke of an option. Prepyment fundion A functionestimating the prepayment of principal on a porb folio of mortgagesin terms of other vaiiables. Principal The par or fa value of a debtinstrument. Principal ComponentsAnalysis An analysis aimed at nding a small numlr of factorsthat describemost of the variation in a largenumber of correlatedvariables (similarto a factor analysis). Ptogram Tradiqg A procedure where trades are automaticallygenerated by a computer and transmittedto the tradingQoorof an exchange. Protedive Put A put option combied with a long position in the underlying asset. Pull-to-par The reversionof a ond's pri to its par value at mattrity.
Put-call Parity The relationsp betweenthe pri of a Europeancall option and the prke of a European put option when theyhavethe same strikepri and maturity date. Put Option An option to sell an assetfor a rtain price by a rtain date. Puttable Bond A bon where te holderhas the right to sellit back to the issuerat rtain predetermined timesfor a predeterminedprke. Puttable Swap A swap where one side has the right to terminateearly.
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Glossay :./' Tevms Q-Measure Rk-neutral measure. QuantoA derivativewhere the payof is desned by variables associated with one but is paid in another currency. currency (yasirrandomSequences A sequens of numbrs used in a MonteCarlosimulation that are representativeof alternave otcomes ratherthan random. RainbowOption An option whose payof is dependenton two or more undeylying variables. '
Ranje ForwafdContrad The combination of a long call and short put or the of a shoytcall and long put, combination '
Ratchet Cap lnterest rae cap where the cap rate applicable to an accrual period equalsthe rate fof the prvipus qenrual period plus a spread. RealOption Opon involvingreal (asopposed to Enqncial)assets. Rral assets include lpnd' plant, and maclnery. '
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Rehalancinj The processof adjusting a tmding position periodically. Usuallythe pufposeis to maintain del neutmlity. ' RecoveryRate Amountrecoveredin the event of a defaultas a pernt
of the face
Value.
ReferenceEntity Companyfor which defaultprotection is boughtin a credit default SWaP.
Repo Repurchaseagreement.A procedure for brrowing moneyby sellingsecurities to a counterparty and agrtting to buythtm backlattr at a slightly lgher pice. RepoRate The raie of'interestin a repo transaction. Reset Date The date in a swap or cap or :oor when the ioating fate for the next periodis set. ReversionLevel The levelthat the value of a market variable (e.g,,an interestrate) teds to revert. Rho Ratt of changt of the price of a derivativewith the interestrate. RightsIssue An issueto exting shareholdersof a stcurity giving thtm tht right to buy new shares at a certain price. Risk-EreeRate The rate of interestthat can be earned without assuming any risks. Risk-NeutralValuation The valuation of an option or other derivativeassuming the worldis risk netral. Risk-neutralvaluation givesthe crrect prke for a derivativein all worlds, not jst in a rk-neutral world. Risk-keatralWorld A world where investorsare assumed t require no extra retufn on average for bearingrisks. RollBack See Backwards Induction.
scalper A traderwhoholts positions for a very shol'tpeliod of time. scenarioAnalyjis An analysisof the efects of possible alternativefuturemovements in markd variabks on tht valut of a portfolio. 5EC Securitiesand ExchangeCommission.
Glossy
p./
Teyms
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Seqlement Price The average9f the prkes that a contract tradesfor immediately before te bell sipaling the close of tradingfor a day.It is used in mark-to-market
calculations.
ShortHedge A hedgewhere a short futurespositionis taken. SllortPosition A noition assumedwhen traderssellshares theydo not own. Shori Rate The interestrate applyingfor a very shortperid of time. Short Selling Sellingin the market shafes thayhavebeen borrowedfrom another
investor. Shorblrm Risk-FreeNte SeeShortRate. Sout Option An option whert the holderhas the right to lockin a minimum value for ihepayof at one timeduringits lifq. Simulation SeeMonteCailo Simulation. Specialist Ap individal responsible for managinglimit orders n jome exchanges, The special'i doesnot make theinformationon outstanding limitorders availableto Other traders. Speculator An individualwh is taking a position in the market. Usually the ' individualis bettingthat the pri of an asset will go up or that the price of an asset will 2o down. Spot Interest Rate See Zero-cupdn InterestRate. SpotPrice The price for immediatedelivery. SpotVolatilities Thevolatilitiesused to price a cap when a dferent volatilityis used
foreach caplet.
SpreadOption An option where the payof is dependenton the diferen between
twomarket variables.
SpreadTransadion A position in two or more options of the same type. StaticHedg A hedgethat doesnot haveto be changed n it is initiated. StaticOptionsReplication A procedure for hedginga portfolio that involvesfnding anotherportfolio of approximatelyequal value on someboundary. Step-up swapA jwap where theprindpalincreasesover timein a predeterminedway. Sticky Cap Interestrate cap where the cap mteapplicableto an arual period equals the capped rate for the previus arual period plus a spread. StochasticProcess An equation descrihingthe probabilisticbehaviorof a stochastic variable. StochasticVariahle A variable whose futurevalue is unrtain. Stock Dividend A dividendpaid in the form of additional shares. Stocklndex An indexmonitoringthe valueof a portfolio of stocks. Rock Index Futures Futureson a stockindex. Rock lndex Option An option on a stock index. StockOption Option on a stock. StockSplit The conversion of each exting share into mre than one new share. StorageCosts The csts of string a commodity.
782
Glossaryp./' Terms Straddle A long positio itl a call nil a put Withthe samessrikeprice. Stranjle A long positionin a call and a put with diferent strikeprices. Strap A long positionip tw0 call optitmsand
put option with the same strike
0ne
pri. Stress Tekinj portfolio.
Testingof the impactof extrememarket moves on the value of a
StrikePrice The priceat wlch the assetmay be boughtor soldin an optiontontract (alsocall' the exercisepri). '(1
Strip A long positionin one call option and tko put optitms with the same strike Pfice .
Strip Bonds Zero-couponbonds created by sellingthe coupons on Separtely fromthe principal. .
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Subprim Mortjaje Credit history.
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Mortgagegranted to bofroer with a poor ciedit historyor no
Swaz An agreement to exchangecash Qowsin the futurenemording to a prearranged formula. SwapRate Thefxed rate in an interestrate swap that causes the swapto havea value Swaption An option to enterinto an interestrate swapwhere a specifkdfxed rate is exchangedfor Coating. Swinj Option Energyoption in whkh the rate of consumption must be betwen a minimumand maximllm level.Thereis usually a limit on the mlmber of timesthe optionholdercan change the rate at wlch the energyis consllmed. Spthetic CDO A CDO created by sellingcredit defaultswaps. Spthetic Optian An option criated by tradingthe underlying asset. S#ematic Risk Riskthat cannot be divrsifkd away. Tailingthe Hedje A procedureforadjustingthe number of futurescontracts used for hed/ngto refkct dailysettlement. Tail Loss SeeConditionalValueat Risk. Take-and-payOption See Option. swing
Term Strudure of lltteres't Rates The relationship betweeninterestraies and thek
matmities, Terminal Valqe The value at maturity. with the passage The rate of change of the price of an optin or otherberivative of time.
Teta
Time Decy
SeeTheta.
Time Value The value of an optionarising fromthe timeleft to maturity (equals an option'sprice minus its intrinsicvalue). Timin! djustment Adjustmentmae to the for ard value of a variable to allow for the timingof a payof hom a derivative.
tt3
Glossayy0/ Teyms
Total ReturnSwap A swapwhere the return on an assetsuchas a bondisexchanged for LIBOR plus a spread.The return on the assetincludesincomesuchas cotlpons and the change in value of the asset. Tmnche 0ne of severalsecuritiesthat havediferet rk attributes.Examplesal'ethe tranchesof a CDO or CMO. Transeion Cost The cost of carrying out a trde (commsions plusthe dference betweenthe price obtained and the midpoint of the bid-oser spread). Treasury Bill A short-termnon-coupon-bearing instrument suedbythe govemment its to nance debt. Treasury Bond A long-trm coupon-bearinginstrumet suedby te gcvernmentto '
OVCC
it debt. Treasury BondFutqres A futurescontract on Treasurybonds. Treasury Note See TreasuryBod. (Treasurynotes have maturitiesof lisj than 10 years.) Tasury kte Futures ikturecontract on Treasurynotes. Tree Representation of the evolution of the value of a market variable for te Purposesof valuing an option or othe? derivative. TrinomialTree A tree whtre thereare threebranchesemanatingfromeachnode. It is usedi the same way as a binomialtr for valuing derivatives. TripleWitching Hour A term given to thetimewhen stockindexfutures,stockindex options,and opons on stock indexfuture!a11expire together. Underlying Variable A variable on hichthe price of an option or other derivative depends. Unsystematic Risk SeeNonsystematic Risk. ':
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Up-and-ln Option An option that comes into existencewhen the price of the tmderlyig asset incpases to a prespecifkd level. Up-nd-out Option An option that ceases to exist whenthe pri of the underlying assetincreasesto a prespecied level. Uptick An increasein price. Value at Risk A lossthat willnot be exceededat some specitkd coldence level. Variance-covariance Matrix A matrk showingvariancej of, and covarius 1tween,a number of diferent market variables.
Variance-GammaModel A pure jllmpmodel where smalljumpsoccur often and largejllmpsoccur infrequently. Variance Rate The squareof volality. Variance Redution Procedures Proceduresfor reducing the errorin a MonteCarlo
simtllation.
Variance Swap
where the realize swap
variance rate duringa period is exhanged Both for a flxedvariancerate. are applied to a notional principal.
npztount VariationMargin An extra margin required to bringthebalancein a margin up to the initialmarginwhen thereis a margin call.
Vega The rate of change in the pri
of an option or other derivativewithvolatility.
GkssaryOJTe?.ps
784 *
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Vega-Neutial Portfelio A portfolio with a vega of zero.
VejtingPerind Periodduringwhich an optioncannot be exercised. VIXIndex Indexof the volatility of the S&P5. Volatility A measure of the uncertainty of the return realize on an asset. VolatilitySkew A term ustd to describethe volatilitysmilewhen it ij nonsymmetrical. VolatilitySmile The variation of implkdvolatilitywith strike price. VolatilitySurface A tableshwing the variationwofimpliedtolatilitieswith strikepri andtime to maturity. VolatilitySwap Swapwhre the realizedvolatilityduringa period is exchangedfor a flxedvolatility. Botb percetage volatilitiesare appliedto a notioal principal. with time to matuity. VolatilityTermStructure The variation of impliedvolatility t An optipn issue by a company or a fmancialinstitution.callwarrants are warran frqtlently ljued by companis on teir ownstck. Weather Drivative Derivativewherethe payof dependson the weather. WienerProcess A stochasticpross wherethe change in a variable durihgeach short periodof timeof lengthAl has a normal distributionwith a mean equal to zer and a variance equal to !. WildCard Play The right to deliver0p a futurescontract at the closing price for a period of time afterthe close of tradipg. Writingan Option Sellingan option. Yield return provided by an instplment. YieldCurve See TermStructure. Zero-couponBond A bond that provides no coupons. Zero-cbupon Interest Rate The interestrate that wouldbe earned on a bond that providesno coupons. Zero-cbupon Yield Curve A plot of the zero-coupon interestrate againsttime to '
m
aturity.
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ZeroCurve See Zero-cbupn YieldCurve. ZeroRate See Zercoupon Interest Rate.
Der iva Gem joftFare The softwareaccompanyingthisbookisDerivaGemfor Excel,Version1.52.It requires Excel 2 or laier. The jftware consists of threefiles: or later and Windows2 b DG152.DLL, G152.xls,and DG152functions.xls.A set-uproutine is providedwith the software. Thiswillload DG152.DLL intoWindowsystemor WlNNrlyystem3z.l Before using the software, Oce 2007usersmust click on the Optionsbutton to Enable fcrtv. If you art not using Uce 2007,you should ensure that Securityfor Macrosis set at Mediamor Low.CheckToolsfollowedby Macrosfollowedby Secarity in Excel to change th. Whileusing the software,y0u ma be asked whetheryouwant to enable macros. You should click Ezable Mt7crpks. UPdates to the software can be dokliloadedfrom ie author's website: http://-.rotmu.utoronto.ca/-hu There are two parts to the software:the OptionsCalculator(DG152.x1s) and the Both Builder Applications (DG152functions.xls). parts requi/ DG152.DLL to be loadedinto the Widowsystem or WlNvlhsystem3zfolder. New users are advisedto start with the OptionsCalculator.
THE PTIONS CALCULATOR DG152.xlsis a user-frievdlyoptions calculator. It consts of thlreworksheets.Thefirst worktheetis used to carry out computations for stockoptions, currencyoptions, index options, and futures options; the second is used for European and merican bond options;the third is used for caps, ioors, and European swapoptions. ne softwareproduces pris, Greekletters,and impliedvolatilitiesfor a wide range of dferent instruments.It dplays charts showingthe way that option prices and the Greeklettersdependon inputs.It alsodplays binomialand trinomialtrees,showing howthe computations are carried out. 1 SometimesWindowsExplorer is set up so that *.(j11 fles are not dplaye. Tis oes not appear to hea problemin WindowsYista.To change the settingin earlierversionsof Winows so that the ;le can be by Pkw,followedby Sow Hidden followed seen,proceed as follows.ClickTools,followedby Folderoz/folu', lc.yand Abltir.y. #.d11
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DeyivaGemSoftuaye
LeneralOperation To use the options calculator,you should choose a worksheet and click on the appropriatebuttons to selettOptionTpy, UnderlyingTpe, and so on. You should hitSnteron your keyboard, then epterthe parameters fortheoption you are considering, and click on Calcalate.DerivaGemwillthendisplaythe pri or impliedvolatilityforthe optionyou are considering, togetherwithGreekletters.If the price hasbeencalcglated froma tree,and you are using the rst or secondworksheet,yopcan thenclkk on Display displaysof thi treeare shownin Chapiers11 and 19. Many Treeto seethe tree. sample diserentcharts can bedisplayedin a11threeworksheets.To displaya chart, you mustflrst choosethe variableyou require on the vertical axij, the variable you requife on the horizontal axis, and the range of values to be consideredon the horizontal axis. Followingthat you should hit Enter on your keyboardand click on Draw Grap. Note that, wheneverthe values in one or morecellsare changed, it is nessary to hit Enter on your keyboardbeforeclickingon one of the buttons. If yor versionof Excelisliterthan7., you willbe askedwhether you wat to update to the new version when you frst sve thesoftware.Youshould choose the F:J button. .
Optibns on Stocks,Currencies,Indics, and Futures The rst worksheet (Equity-Fx Index-Futures)is used for pptiops on stocks,currecies,indices,and futures.To use lt you shld srstselecttheUndeying Type (Equity, Currency,Index, or Futures).You should then select the Option Type (nalytic Europea, BinomialEurpean, BinomiatAmerkan,Asian,BarrierUp and In, Barrier Up and Out, Barrier Dwn >nd In, Banier Dwn and but BinaryCash or Notlng, Binary.X. sst py N,g-y.pg,.ct/rr, CompoundOptionon Call,Compqund.Optionon Lookbact). Put, or #oushould itn enterth dau on the underlyingasset and data on theoption. Note that a11interestrates are expressedwith continuous compounding. In the case of European ald'ieriian iuityoptions,a tablepops up allowingyou ex-dividenddate (measured of each dividends. the time in years from E nter enter to today)in the srstcolumn and the mount of the dividendin the second column. Dividends must be entered in chronological order. You must click on buttons to choose whether the option is a call or a put and whetheryou wish to calculate an impliedvolatility. If you do wish to calculate an imjliedvolatility?the option prke should be enteredin the cell labeledPrice. Onceall the data has beenentered you shouldhit Etlter on your keyboardand click ,
on Calcalate.If ImpliedVolatilitywas selected,DerivaGemdisplaysthe implied volatilityin the Volatility(%per year) cell. lf ImpliedVolatilitywas not selicted, it usesthe volatility you entered in this ll and displaysthe optionpri in thePrke cell. Oncethe calculations havebeen completed, the tree (ifused) can be inspectedand charts can be displayed. WhenAnalyticEuropean selected,DerivaGemusesthe Black-scholes equations in Chapters13, 15,and 16to calculateprices,and theequationsin Chapter17 to calculate Greekletters.When BinomialEuropean or Binmial Amerkanis selectedj'a binmial tree is constructed as describedin Chapter 19. Up to 50 timesteps can be used, The input data are largelyself-explanatory. In the case of an Asian option, the CurrentAvirageis the averageprice sin inceptitm.If the Asian ption is ntw (Timt sin Inceptionequals zero), thenthe CprrentAveragecellis in-elevantand can be left
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hlank.ln the caseof a LookbackOption,theMinimumto Date is usedwhena Callis valuedand the Maximm to Date is used when a Put is valued. For a new deal,these shotld be set equal to the currentpriceof the underlyingasset.
Bond Options Thesecondworksheet (Bod-options) isusedfo: Europea and Americanoptions on 1)0n (1s The alternative models are Black's one-fator normal, and a oni-f>ctr lognormalmodel. You should flrstselecta pricing model(Black-European,NormalAnalyticEuropean,Normal-rfreeEuropean,Normal-merican,Lopormal-Emopen, Data. or Lognormal-mericah).You should thenentertheBondData andthe Wtion The couponis the rate paid per yea: andthefrequencyof paments can be selectedas semi-Annual,or Annual. zero-coqponyieldcurveisenteredinthetable Quarterly, in yeafs)intheflrstcolumnandthe labeledTermstructure. Enter maturities(measufed corfesponding cotinuously compounded rates in the seiond colllmn.Thematurities shouldbe enteyedip chzonlogical ordef. DerivaGemassumesa piewis linea:zer curve sirnila: to that in Figufe 4.1. Note that, Fhep valuing intefestfate derivatives, DerivaGemrounds a11timesto the neafest wholentlmberof days. Whena11datahavebeenentered, hit Ester on your keyboafd.ne quoted bondprice pe: $100of Principal,calculatedfromthe zerocufve,isdisplayedwhenthe calculations are complete.You should indicatewhether the option is a call or a yutand whetherthe strike price is a quoted (clean) strike pri 01. a cash (dkty) stnke pli. (Seethe discussion and exampk in Section28.1 to understand thz lkteen th of enteled prindpal. strike price priceis thb two.) Note that the Yu pe: $100 as shouldindicatewhether you are considering a call : a pgt option and whethe: yu wishto calculate an impliedvolatility. lf you select impliedvolatilityandthe normal modelor lognormalmodel is used, DerivaGemimpliesthe sht-fate volatility,keeping the revefsion rate fxed. Once a11the inputs a:e complete, you should hit Ester on you: keyboafdand clkk Calculate. fte: that the tfee (ifused) can he inspectedandchals can he displayed. Notethat the tfee displayedlastsuntil the end of thelifeof theoption. DelivaGemuses a much lafge: tree in its computations t value the undeflyingbond. Note that, when Black's model is selectd, DerivaGemuses te equations in Section28.1. lso, theprocedurein section 28.1is used forconvertingtheinputyield volatilityinto a plice volatility. j
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Caps and Swap Options Thethird worksheet (Caps-and-swap-options)isusedfo: caps and swapoptions.The alternativemodels are Black's, one-factor nomal, and a one-factorlopoaal model. You shbuld rst select the OptionTpe (SwapOption0: Cap/Floo) and Pricing Model (Black-European,Norml-Eufopean, 0: Lopofnial-Eufopean). You should Freqnency indicates then enter data on the option you aie considering.The settkment the frequencyof paymentsand can be nnual, SemAnnual, lafterly, o: Monthly. The softwafecalculatespaymentdatesby workingbackwardhomthe end of thelifeof the cap o swap Option. Theinitialaccrualperiod maybea nonstandardlengthbetween .5 and 1 tims a normal accrual period.The softwarecan be used to implyeither a volatilityor a cap rate/swap rate fromthepfice,Whena normalmodel or a lopormal .5
788
DerivaGemsb.fffzur: is usd, berivaGemimpliesthe short rate volatility keepingthe reversion rate model The zero-couponyield curveis entered in thetablelabeledTermstructure. Enter flxrd.
in years) in the flrstcolumnand the corresponding continuously maturities(measured compoundedrates in the secondcolumn,The maturitks should be entered in chroologicalorder. DerivaGe ssumesa piewise linear zero curve similar to that in
Figure''4.l .
Onceal1theinputsare complete,you shouldclick Calcalate.Mter that, charts can be displayed.Note that, when Black'smpdelis used, DerivaGemudes the equationsin Sections28.2and 28.3.
Greek Letters In theEquity-Fx lndex-Futres worksheet,theGreeklettersari calculatedas follows. -7
Ddta: Changein option pri per dohr incrrasei underlying asstt. Gamma: Changein delta per dollarincreasein underlying asset. volatilityincreases Vega: Changtin option prict per 1%incrtasein volatility(e.g., from2% to 21%). Rho: Change in option pri per 1% increasein interest rate @.g., interest 5% increaseshom to 60:). Theta: Changein option prke per calendar day passing. In the Bond-optionsand Caps-and-swap-optionsworksheets, the Greeklettersare calculateday follows: .
DVI
:
Changein option pri
per one bgsispoint upward parallel shift in the
ZCFO CUrVC.
Gnmmal : Changein DV1 per one basis point upward parallel shift in the zero curve,multipliedby 100. Vega: Change in option pri when volatility payameterincreasesby @.g.,volatility incrtastsfrom20% to 210/n)
THEAPPLICATIONSBUILDER The ApplicationsBuilderis DG152functions.xls.It is a set of 21functionsand seven sampleapplications from which users can buildtheir own applications.
The tunctions The followingis a list of the 21 functionsincludedin the ApplicationsBuilder.Full detailsare on the hrst worksheet (Functionspecs). 1. Black-scholes.Th carrks out Black-scholescalculations for a European option on a stock,stock index,currency,or futurescontract. 2. TreeEquityopt.This carrks out binomialtree calculations for a Europeanor Atnericanoption on a stock,stock index,currincy, or futurescontract.
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3. Bjnaryoption. Thiscarries out calculationsfor a binaryoptionon a stock, stock index,currency, or futurescontmct. 4. Barrieroption. Th carries out calulations for a barrkr option on a nonstock, stockindex,currency,or futurescontract. dividend-paying sk Avtrageoption.This carries out calculations for an Asian option on a nondividend-paying stock,stockiydex,currency,or futurescontract. carries out calculationsf0r a chooser option on a non6. Chooseroption.Yhis stock,stockindex,currency,or futurescontract. dividend-paying 7. Compoundoption.This carrirs out calculationsfor compound optionson nonstocks, stock indices,currecies, and futures. dividend-paying 8. Lookbackoption.This carrks out calculations for a lookbackoption (flxed or :oating) on a non-dividrnd-paying stock, stock index, currency, or futurs contract. 9. Eportfolio.This carries olt calculations for a portfolio of options on a stock, stockindex,currency,or futurescontpct. 1. Blackcap. This carries Out calculaons for a cap or ioor using Black's model. 11. HullWhitecap.This carrks out calculations for a cap, or floor using the Hull-Whitemodel. 12. Treecap.This arries out calculationsfor a cap or Qoorusing a trinomialtree. I3. Blackswapoption. This carries out calculation! for a swap option using Black's ppdrl. J l ' 1i. HllWhitesWj. Thij carrks out calculations for a swap option using the Hull-Whitemodel, 16. Trees' apoption. This carrks out calculations for a swap option using a trinomialtree. 16. BlackBondoption. This carries ollt calculationsfor a bonddptionusing Black's model. 17. HullWiteBondoption. Th canies out calculationsfor a bond option using the Hull-Whitemodel. 18. TreeBondoption. This carries Out calcultions for a bond option using a trinomialtree. 19. Bondprice. This values a bond. 2. Swapprice.Th values a plain vanilla interestlate swap. Note that it ignores cashCowsaring from reset datesprior to start time. 21. Iportfolio.Th carries Out calculationsfor a portfolio of interestrate derivatives. .
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Sample pplications DG152functions.zsincludessevtnworksheetswith sample applications: the convergen of the binomialmodel A. Binomial Convergence. Th investigates in Chapters11 and 19. B. GreekLetters.Til profidescharts showing tht Greeklettersin Chapter 17.
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DeyvaGemSo/fgture theperformanceof deltahedgingas in Tables17.2 C. Delta Hedge.This investigates and 17.3. theperforman of deltaplus gamma D. Delta andGnmmaHedge.Thisinvestigates hedgingfor a positionin a binafyoption. and Risk.This calulates Value at Riskf0r a portfolioconsistingof three E. Value optionson a single stockusing threedifkrentapproaches. F. Barrier Replication.Th carries 0ut calculations for the staticoptions replication exainplein Section24.14. G. TrinomialConvergen. Tllisinvestigatesthe convergen of the trinolnialtree modelin Chapter3.
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Tradlng and Options
Majo: Exhages .
Futres
merkan stock Exchange ustralianStockExchanje Bolsade Mercadoriasy Futuros,Brazil Bura Malaysia Chiago Board of Trade ChkagoBqardOptionsExchage Exchange ChicagoVefcantile Eurex Euronext
HongKongFuturesExchange lntercontinental Exchange
InternatiunalPetroleumExchange, London Exchange Internationalsecurities KansasCityBoard of Trade LondonMetalExchange MEFF Renta Fija and Variable, spain MexicanDeriyativesExchange MinneapolisGrin Exchange MontrealExchanye Ntw YorkBoard of Trade NewYorkMercantileExchange NewYork stock Exchange NordicExchange OsakaSeculitiesExchange Exchange Philadelphiastock SingaporeExchangr sydneyFuturesExchange TokyoGrain Exchange TokyoFinancialExchage WinnipegCommodityExchange '
Therehas keena great dtal of internationalconsolidationof dtrivativtstxchpnges in the last fewyears. For example, knOctober2006,CBOTand CMEannound theirintentionto merge their t form the world's largestdelivativesexchange; EURONEXTand NYSE announc.e.d with July Jne merged in 2006)AsX SFEin intentionto merge 2006;ICE agreedto acquire 2006and IPE in June 2001; EUREX jointlyopemted by utsche NYBOTin september Exchange;EURONEXTowns the LondonInternational Financial Brse AG and sWx swiss Futures Exchange (LIFFE) as well as two French exchanges; NYSEacquiredthe Pacifc 2005.No doubtthe consolidationhas beenlargelydrivenby economies Exchangein septelber lowertrading of scalethat lead to costs.
791
le or
x
e x
w
This table shows values of Nx) f0r % The tableshould be used with interpolatinn.F0r example, .x
Grahap, J. R., 67 Grinblatt,M., 95, 144 Guan, W., 393 Gupta, A., 174 Hamilton,DkT., 492 Hnley, M., 58 Hanison, J. M., 63 Hasbrtmk,J., 486 Haushalter,G.D 67 Hayt, G. s.,67 Heth, D., 445,462,793,719 Hendricks,D., 463 Hston, S.L., 539,69 Hicks,J.R., 119 Hillrd, J. E., 346 tl()j. T s Y 139 678 698 flopper,G., 463 Hoskins,W., 144 Hotta, K., 174 Hua, P., 463 Hull, J. C., 329,367,384,389,413,425,436, 439,458,463,494,495, 512,526,541,542, 543,544,589,591,594j692,693, 649, 61, 689,681,686,691,693,698,71, 712,715, .,
yield curve, 78-82 zero-coupon Inteast rate caps and ioors, 644-50, 775 cap as a portfolioof bond options,645 cap as a portfolio of interestrate options, 44
ioors and collars,645 forwafdrisk-neutfalvaluatitmand Black's model,648-49 impactof day count conventions,649-50 put-call parity, 646 spot volatilities vs. ;at volatilities,647-48 valuationof cap and floors, 645-46 Interestrate collar, 645,775
Normal market, 33, 778 Noticeof intentionto deliver,22 Notionalprincipal, 149,518,778 Numeraire,621,778 annuity factoras the numeraire,624-25 impactof a changein numeraire,628-29 intemstrates when a bondpri is the numeraire,623-24 moneymarkt account as the numeraire,
29 numeraireratio, 629 bond price as te numeraire, zero-coupon 622-23
Numericalprocedure?399,778
OCC, see Options ClearingCorporation,778 October 19, 1979,29, 374, 376,327,452, 76 Ofer, 2, 188,778 Ofsetting orders, 189 Of-the-run bond, 759 On-the-nmbond, 759 Openintefest,33, 189,778 Open order, 36
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Package,547-42,779 Par value, 779 Par yield,79-80, 779 Parallelshift,451-52, 459-61, 779 Partialsimulationapproach,Monte Carlo 457 simulation, Pass-throughs,71
SubjectIndex
810 594-8, 779 Payh-dependentderivative, Payos, 779 Perfecthedge,45 Philadelphia stockExchange(PHLX), 184,
Treasurybond and note futures, 132-36 Treasurybonds, 131 Us dollar swaps, 152-53 USD-GBP exchangerate, 4 pri, bond and Treasurybill, 131, Quoted 641,694-. See also Cleanprice '
Tail ioss, 445,782 Tailingthe hedge,58,782 Take-and-payoption, enerr and natural gas market,577,782 Tax, 38-39 planningstrategs 193-94 TaxpayerReliefAct of 1997,193 Tenor, 644 Term repo, 75 Termstructuremodelof interestrates, 673,782 Term structuretheories,shapeof zero curve,
91-92
Terminalvalue, 782 Theta, 359-61,782 usingbinomkl tree,46 estimating, with delta and gamma, 365 relationship Timedecay,359,782 Time value, 186,782 Time-dependentparameters,417-18 Time-of-dayorder, 36
814 impjkd,296-97. See also Volatilitysmile matchingvolatilitywith z and d, 248-51 portfolioinsuranceand ssockmarket volatility,374-75 swap,173,559-70, 784 termstructure,volatilityof stock return, 388-89,481,784 .715,