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3.1 A Filst Glance at she Arbitrage Theorem 3.2 Reevance of the Arbicrnge Theorcm of Synrhecic Probabilities
21
and Submartingales
3.5 Normalization 3.6
24
3.7 The No-Arbitrage Contlition 4 A Numerlcal Example PossibiliLies
4.l Cttse 1) Arbitrage 4.2 Case 2) Arbitmge-Frcc
Prices
IncleLerminacy
with
26
27 27 28
29
Lattice Models
29 32
;
CHAPTER
Dividends
6.2 The Case with Foreign Currencies 36 7 Some Genemlizations
34
8 Conclusions: A Methodology for Pricing 37 Assets
Theorem Exercises 11
4
72
Pricing Detivatives 77
2 Pricing Functions 78 78 2.1 Forwards 80 2.2 Options 3 Applicatlon: Another Pricing Metlzod 85 3.1 Example 86 4 The Problem
37
38 9 References 10 Appendix: GeneMization of the Arbitrage
.
1 Inttoduction
of the World
7.3 Discounting
47
Modib und Notatien
i
')
'
7.1 Time Intlex 7.2 States
66 5 Partial Derlvatives 67 5.1 Example Differentials 67 Total 5.2 68 5.3 Taylor Series Expansion 5.4 ordinaryDifferential Equatiorus 73 6 Conclusions 74 z References 8 Exetcises 74
25
6 Payouts and Forei> Currencies 6.1 The Case
46
57 4.2 The Chain Rule 59 4 3 The lntegral 65 Intcgration by Parts 4.4
' '
24
Rates of Retum
4..3 An 5 An Application:
45
46 2 Some Tools of Standard Calculus 47 3 Ftmctions 48 3.1 Random Futwtions of Functions Examples 49 3.2 and Limit 4 Convergence 53 4.1 Tle Dcrivative
E
3 A Basic Example of Asset Pricing
()f
Flows
1.2 Modcling Random Behavifar
17
Equalization
Calculus in Deterministic and Stochastic Environments
1 lntroduction
World
2.4 Portfolio
3.4 Martingales
3
13
and Payoffs
3.3 The Use
*
4,1 A Fint Look at Ito's I-emma 88 4.2 Conclusons 88 5 References 89 6 Exercises
E
38
40
'(
..
84
Contents
X
CI4APTER
.
5
Contents
Tools in Probability Theory
1 lntroducdon 2 Probability
4 Relevance of Martingales in Stochastic MMeling
91 91
126 4.1 An Example of Martlngale Trajectories 5 Propeuies of Examples 130 Martingales 6
92 2.l Example 93 2.2 Random Variable 3 Moments 94
3.l First Two Moment.s .3,2 Higher-ordcr Moments
130 6.1 Example l : Brownian Motion 132 6.2 Example 2: A Squared Ptocess 6.3 Exampe 3: An Exponential Proccss 6.4 Example 4: Right Continuous Mattingales l34 7 The Simplest Martingale
94
4 Conditional Expectations
97
97 4.1 Conditiona Probability of (n-ontlitional Expectarions 4.2 Propertics Some Important 100 Models 5
.5.1 Binomial Distributitm in Financial Markcts 101 5.2 Limiting Propertics 102 5.3 Moments 103 5.4 The Normal Distribution 'Fhe l06 Poisson Disttibutitm .5..5 6 Markov Prxesses and Theit Relevance 109 6.l The Relevance I 10 6.2 The Vcctor Case 7 Convergence of Random Variables 7.1 Types of Cxnvergence and Their Uses 7.2 Weak Convergence t 13 116 8 Conclusions References 116 9 Exercises 1 17 10
135 7.1 An Application 136 7.2 An Examplc Martingale 8 Representations
100
137 8.l An Examplc 8.2 Doob-Meyer Decomposition 9 The First Stochastic lntegral
9.1
PTER
*
6
108
2.2 Continuous--fime
112
1l
.
140
to Finance: Tmding Gains
A Summary
.1
12 Concluslons 13 References 14 Exetcises
E
'
143 144
145
C
P
1.52
152 153 154
R 7 Differentiation in Stochastic Environments *
156 1 Introductlon Metivation 157 2 3 A Fmmework for Discussing
119 120 120 Martingalcs
137
.
Representations
2.l Notation
134
147 1l l A Hedgc 147 11.2 Time Dynamics and Normalization Prohabilior Risk-Neutral 11.3
Martingales and Martingale
1 IntrMuction 2 DeVitions
Application
127
133
10 Martingale Methods and Pricing 146 11 A Pdcing Methodology
'
C
124
l11
3 The Use of Martingales in Asset Pricing
122
4 5
Differentiation 161 of The Incremental Errots 167 One lmplication fsize''
164
Conrent 6 Putting the Results Together l 70 6.1 Stochastic Diffcrentials 17 Concluslons 1 7 l 71 8 References 17 1 9 Exerclses
CI'IAPHR
*
8
The Ito Integral
l69
,
2 Two Generic Models
4.l Nonnal Events 4.2 Rare Evenlns
The 1to Integr'al ls
209
a
213
220
Martingale
220
224
Other Properties of the lto lntep-al
4
226
5 lnterals with Respect to JumpProcesses 228 6 Conclvsions References 228 7
184
226
226
4.1 Existence 4.2 Correlation Ptoperries 227 4..3 Adtlition
.
21 1
Rtemann Sums
8 Exercises
227
228
187 189
in Stochastic
lntegration Envitonments The 1to lnfegrul
1 lntroduction
Integral and
3.2 Pathwise lntehyals
175
C
2O4
PTER
.
10
lto's Lemma
1 Introduction E
196
3.2 '
:
First-order
231
232
3 lto's Lemma .3.l The Nttion
199
230
2 Types of Derivatives
2.1 Example
E
9
208
214 2.4 An Expositoty Example lntegral lto he Properties of t 3 E
190 5 A Model for Rare Events 193 6 Moments That Matter 195 Conclusions 7 8 Rate and Normal Events in Practice l96 8.1 Thc Binomial Model 197 8,2 Nonnal Events 198 8.3 Rare Events of The Behavior Accurrtulated Changes 8.4 202 9 Refetences Exercises 203 10
CHAPTER
of the Ito lntegral
2.3 Def nirion: The Ito Integml
174
l76 2.1 The Wierker Process 178 2.2 The Poisson Process 180 2.3 Examples Back Rare Events 182 to 2.4 3 SDE in Discrete lntervals, Again 183 and Normal 4 Characterlzing ltare Events
.
2.1 The Riemann-stieltjes 2.2 Stxhastic Integration
'
173
1.1 Relevance of the Discussion
and SDES
1.2 The Practical Relevance 2 The 1to lntegral
The Wiener Process and Rare Events in Financial Markets
l lntroduction
xiii
Contents
tf
232 ''Size'' in Stochascic Calctllus
Terms
237
3.3 Second-order Terms
238
3.4 Terms lnvolving Cross Product.s 210 3.5 Terms in the Remainder 240 4 The 1to Formula of 241 5 Uses lto's laemma 5.1 Ite's Formula as
a
Chain Rule
241
242 5,2 6 lntegol Form of lto's Lemma 244 Fomuuln Complex Settings 7 lto's in More Ito's Formula as an Integration
Tool
245
K.
Contents
xiv
MuLtivariate Case
7.2 Ito's Formula 8 Conclusions 9 References 10 Exercises
CHAPTER
*
11
and
245
1.l
4 Partial Differential Equations
248
250 251 251
252
on at and
253
tz'r
2 A Geometric Descrlption SDES
of Paths
SDES
255
3.1 Wha: Does a Solutton Mean? 256 3,2 Typcs of Soluriorvs 3.3 Whic Solution Is to Be Prefetzetl? 3.4 A Discussion of Strong Solutions 3.5 Veritication t)f Solutions o SDES l6? 3.6 An Important Example
4 Major Models Lincar Constant
of
SDES
Coefficient
258 2,61
*
12
of
290 292
292 PDES
292
10 Exercises '
13
280
294
The Black-scholes PDE An Applfcltion
2.1 A Geomerric Look
3
PDES
296 PDE
4 Exotic Options Lookback (Jptions
296
Black-scholes Formula
at the
in Asset Pricing
.3.l Constant Dividends
276
293
293 294
8 Cotclusions 9 References
1 lntrMuction 2 The Black-scholes
Eqwafo,xs
1 lntrvuction 275 Riskree Portfollos 2 Forming 3 Accumcy of the Metbod
Example 1: Lincar, First-cder PDE 5.2 Example 2: Linear, Second-order PDE
7 Types
255
4.1 267 4.2 Geometric SDL 269 4.3 Square Root Process 270 4.4 Mean Reverting Process 27l 4.5 Omstein-uhlenbeck Process 271 5 Stochastic Volatility 272 6 Concluslons 272 7 References 273 8 Exercises
CHAPTER
lmplied by
254
3 Solution of
282
?83 4.1 Why ls tle PDE art 'sEquarton''? 283 the Btaundary Conditicm? Wha Is t 4.2 PDES 284 5 ClassiEcation of
The Dynamics of Derivative Prices stocetic Dffferenfzl Iqutfffond
Conditions
282
3.1 An Interpretation
Jumps
1 lntrodttction
XV
Ctnntenrs
299 300
301 30l
4.1 301 4.2 Ladder Opions Trigger Knock-in Options or 4.3 302 4.4 Knock-out Options 302 Exocks 4.5 Other 303 4.6 The Relevant PDF-S PDES Practice in 5 Solving 304 5.l Closed-Fonn Solutians
302
304
298
xvi
Contents 5.2
Numerical Solutions
6 Concluslons 7 References 8 Execises
CHAPTER
*
14
Contents
5.1Detennining ?
306
309
3.2 The lmplied
.
310 310
4.1 Cakulation
as
'sMeasure''
5.1 Equivalence of rhe Two Approaches 5.2 Crtitral Sreps of the Derivation 5.3 Inregml Form of the ito Formula
312
2 Changing Means
517 321
5.1 A Normally Distributcd Random Variable 325 3.2 A Normally Distributed Vcctor 327 3.3 'I'hc Radon-Nikolym Derivative Measures Equivalent 328 3,4 4 Statement of the Girsanov Theorem 5 A Discussion of the Glrsanov Tlzeorem
5.1Appliation
to
Probabilides
C
PTER
.
15
CHAPTER
353
*
16
331
5 Conclusions 6 References 7 Exercises
337 340
364
New Results and Tools for lnterest-sensitive Secutities
1 Introduction 467 2 From Conditional Expectations to
' E
2.1 Case l : Constan.t Discount Factors
434
2.2 Case 2: Bond Pricing
435
472
2.3 Case 3: A Generalizacion
:'
2.4 Some Clatifkations
437 :
460
461
rt Complex 3: More Forms Case 5.3
;
2.1 Example 1 428 429 2.2 Exampe 2 2.3 The General Case 429 the Modcl Race Spfat 2.4 Using 432 2.5 Comparison wkth the Black-scholes World 3 The HJM Approach to Temn Structure
455
zto
5+2 Casc 2: A Mean-Revcrcing
Classical and HJM Approaches to Fixed lncome
444
Classical PDE Analysis for lnterest Rate Derivatives
5 Closed-Form Solutions of the PDE
: .
20
451 1 lntroduction The Fmmework 2 454 3 Market Price of lnterest Rate Risk 457 4 Derlutlon of the PDE 459 4.1 A Comparison
420
5 Conclusions) Relevance
*
'
117
4 Forward Rates and Bond Prices
441
4 How to Fit rt to lnitial Term Structure 445 4.1 Monte Carlo 446 4.2 Tree Models 447 4.3 Closed-Fonn Solutions 447 5 Conclusions Refetences 447 6 448 7 Exetcises
Modeling Term Structure and Related Concepts
1 lntroduction 407 2 Main Concepts 408 nree Curves 409 2.2 Movements on the Yield Cunre 3 A Bond Pricing Equation 4l4 (zonsrantSpot Rate 416 3.2 Scochastic Spot Rates
443
475
475
PDES
469
Contents
XX
W'hich Drifc?
476
2.6 Atlorher Kmd Price Formula 479 2.7 Wlch Forrrtula? PDES 479 to Conditional Expectations 3 From and Other 4 Generators, Feynman-lfac Formula, Tools 482 482 4.l Ito Diffusions 483 4,2 Markov Property 483 4.3 Generattlr of an Ito Diffusion A 484 4.4 A Reprcsentation for 485 4.5 Kolmogorov's Backward Equation 487 5 Feynman-lac Formula
492 3 Stopping Times of Times 493 Stopping Uses 4 494 5 A Simpliied Setting The Model 494 5.l 499 6 A Slmple Example and Martingales Stopping Times 7 7.I Mmingales
7.7 Dynkin's Fonuula
8 Colulusions 9 References 10 Exercises
BIBLiOGRAPHY 513 JNDEX
504 504
491
492
5O4
nis edition is divided into hvo parts, The lirst part is essentially the revised and expauded vrsiotl of the rst edition and consists of 15 chapters. The sccond part is entirely new and is made of 7 chapters on more recent and more complex material. Overall, the additions amount to nearly doubling thc content of the first edition, The srst 15 chapters are revised for typos and othcr errors and are supplemented by several new sections. The major novelty, howevcr, is in thc 7 chapters contained in the second part (f the book. These chapters tzse a similar approach adopted in the Erst part and deal with mathematical tools for fixed-income sector and interest rate products. The last chapter is a brief introduction to stopping timcs and Arnerican-style instruments. The other major addition to this edition arc thc Exercises added at the ends of thc chapters. Solutions will appear in a separatc solutions manual. Several pcople provided comments and helped during thc proccss of revising the first part and with writing the seven new chapters. I thank Don Chance, Xiangrtmg Jin, Christina Yulzzal, and the four anonymous referees Who provided very useful comments. The comments that I received from numerous readers during the past threc years are also greatly appreciated.
505 505 505 509
Kx
This book is intended as background reading for modern asset pricing thcory as outlined by Jarrow (1996),Huli (1999),Dufie (1996),Ingersoll and other excellent sources. (1987), Musiela and Rutkowski (1997), require, by their very nature, models hnancial derivatives Pricing for utilization of continuous-time stochastic processes. A good understanding of the tools of stochastic calculus and of some deep theorems in the theory of stochastic processes is necessary for practical asset valuation, ncre are several excellent technical sources dealing with this mathematical thcory, Karatzas and Shreve (1991),Karatzas and Shreve (1999), and Revuz and Yor (1994) are the first that come to mind. Others are discussed in the references. Yct cvcn to a mathematically well-trained reader, these sources are not easy to follow. Sometimes, the material discussed has no direct applications in Iinance. At otlner times, the practical relevance of the assumptions is difficult to understand. The purpose of this text is to prtwide an introduction to the mathematics utilizcd in the pricing models of derivative instnzments. ne text approacbes the mathematies behind continuous-time finance informally. Examples are given and relevance to hnancial markets is provided. Such arl approach may be found imprecisc by a tcchnical reader. We Simply hope that the informal treatment provides enough intuition about Some of these dicult concepts to compensate for this shortcoming. Untflrtunately, by providing a descriptive treatment of these concepts, it is diflicult to emphasize technicalitics. This would defeat thc purpose of the book. Further, there ac excellent sources at a tcchnical level. What sccms to be missing is a text that explains the assumptions and concepts behind XX11l
lntroduction
xxiv these mathematicai tbeory.
Introduction
Trading became cheaper. ne deregulation of the hnancial scrvices that 1980s was also an important factor here. Bathcred steam during the nree major steps in the theoretical revolution led to the use of advanced mathematicalmethods that we discuss in this book:
tools and then relates thcm to dynamic asset plicing
1
Audience
xKarbi-
'inew''
*
Black-scholes pzee/ (Black and Scholes, 1973) used the method of arbitrage-free pricing. But the paper was alm inquential because of the techriic.al steps introduced in obtaining a closed-form formlzla for options prices. For an approach that used abstract notions such ms Ito calculus, the formula was accurate enough to win tlze attention of market participants.
@
ne methodology of using equivalent martingale measures was developed later. nis method dramatically simplised and generalized tbe original approach of Black and Scholes. With these tools, a generat method could be used to price any derivative produd. Hence. arbitrage-free prices under more realistic conditions could be obtained,
'
fluring the past two decades, some major developmcnts have occurred in the theoretical understanding of how derivative asset prices are determined and how these priceg move over tne. There were also some rccent institutional changes that indirectly made the methods discussed in the following pages popular. ne past two decades saw the freeing of exchange and capital controls. This made the exchange rates significantly more variable. In the meantime, made the elimination of currency risk world trade p'ew sigafcantly, a much bigher priority. During this time, interest rate controls were eliminated. This coincided with increases in the government budget descits, which in turn 1ed to large new issues of government debt in all industrialized nations. For this reason (among others), the need to eliminate the interest-rate risk became more urgent. lnterest-rate derivatives became very popular. It is mainly the need to hedge interest-rate and currency risks that is at the origin of the reecent prolc increase in markets for derivative products. This need was partially met by :nancial markets. New prducts were developed and offered, but the conceptuai understanding of tbe strucure, functioning, and pricing of these derivative products also played an imporapplicable to tant role, Because theoretical valuation modeis were directly able price to these new products, nancial intermediaries were understanding the of clear and successfully market them. Without such a developsimilar what extent a conceptual framework, it is not evident to ment nlight have occurred. As a resuit of these needs, new exchanges and marketplaces came into of new products became easicr and iess costly. elstence- lntroduction
f/lctprczzzl
gives the formal conditions under which trage'' prohts can or cannot exist. lt is shown that if asset prices satisfy a smple condition, then arbitrage cannot exist. This was a major development that eventually permitted the calctklaticm of the arbitrage-free derivative product. Arbitrage pricing must bc conpricc of any trasted with equillbrium pricing, which takes into consideration conditions other than arbitrage that are imposed by general equilibrium.
. The arbitrage
ne text is directed toward a reader
K, the option will have some value. One One can buy the underlying security at Sv. Since there are no commissions or K. Market participants, being be K usz on thc option, and wc have uy
-
should clcarly exercise the option. price K and sell it at a higher price bid-ask spreads, te net prct will aware of this, will place a valuc of
We can use a shorthand notation wrjtisg Cv
Value ef
Jd
v
K
>
==>
f,'F
=
uy
-
/
J
a.
40 39
/
Oplion'q value expiration
brorc
6
20 10 / i
0
20
-
;
40
/
/ ./
,
optitm's
FlG UR1
IX trike Price
120
IK
.
(6)
swaps
Swaps and swoptions are among some of the most common types of deriva-
value at expiration
tives.But this s not why we are interested in them. It turns out that one method for pricing swaps and swoptions is to decompose them into forWards
Sk
K
60
K, 01
zsz
C.w1Itlplion 50
-
This means that the Cz will equal the greater of the tw'o values inside the brackets. In later chapters, thig notation will be used frequently. Equation (6), which gives the relation between Sr and Cz, can be graphed easily. Figure 3 shows this rclationship. Note that for Sp :GK, the C1. is zero. For values of such that K < &, the CT increases at te SV. Hence, for this range of values, the graph of Eq. (6) is a same rate as straight line with unitary slope. Options are nottlirtear instruments. Figure 4 displap the value of a call option at various times before exPiration. Note that for t < F the value of tlle function can be represented by a smooth continuous cuwe, Only at expiration does the option value become a piewise Iinear function with a UI'I.k at the strike price.
(5)
K.
max (&
.'
-
.
=
to express both of these possibilities by
and options,
'ritis
illustrates the special role played by forwards and basic building blocks and justifies the special emphasis put on as titemin following chapters.
'
options
3
'L
; ..
C H A P T ER
10
*
1
Financial Derivatives
9 Exercises
DEFINITION: A swap is the simultaneous selling and purchasing of cltsh qows involving various currencies, interest rates, and a number assets. of oter snancial
Tls very basic intcrest rate swap consists of exchanges of interest paycounterpatics borrow in sectors where they have an advantage m cnts.Thc then exchange the interest payments, At the end both counterparties and W gl secure lower rates and the nvap dealer will earn a fee. It is always possible to decompose simple swap deals into a basket of simplerforward contracts, The baskct will replicate the swap. The forwardscan then be priced separately, and the corresptnding value of the from these numbers. This decomposition into swap cam be determled of buildingblocks fonvards will signilicantly facilitate the valuation of the
Even a brief summary of swap instruments is outside the scope of tMs book. As mentioned earliez, our intention is to prtwide a heuzistic inlroduction of the mathematics behind derivative asset pricing, and not to discuss the derivative products themselves. We limit our discussion to a typical example that illustrates the main points.
swapcontract.
6.1 A
smplelnfcresf
Rlfe Jtt'up
'g
a swap into its constituent components is a potent example engineering and derivative asset pricing. lt also illustrates the
Decomposing
of snancial special role played by simple fomards and options, We discuss an interest rate swap in detail, Das (1994)can be consulted for more advanced swap structuresJ ln its simplest form, an interest rate swap between two counterpardes and B is created as a result of the following steps:
.4
needs a $1 million tloating-rateloan. Counterparty 1, Counterparty flxed-rate loan, But because of market conditions and B needs a $1 million with relationships various banks, B has a comparative advantage in their rate/ :oating bon-owing at a and B decide to exploit this comparative advantage. Each counter2. party borrows at the market where he had a comparative advantage, and then decides to exchange the interest payments. 3. Counterparty borrows $1 million at a flxed rate. ne interest payments will be received 9om counterparty B and paid back to the lending bank. 4. Counterparty B borrows $1 nlillion at the lloating rate. Interest payments will be received from counterparty W and will be repaid to the lending bank. 5. Note that the initial sums, each being $1 million, are identical. Hence, are called notmal pzntn/f-. they do not have to be exchanged. the also in 'rhe interest payments are same currency Hence, the counterconcludes the interest dlfferentiala. only the interest parties exchange rate swap.
conclusions
'
we have rcvicwed some basic derivative instruments. Our first, to give a brief treatment of the basic derivative twofold: purposewas securitiesso we can use them in examples; and second, to discuss some notation in derivative asset pricing, where one first develops pricing formulasfor smplc btlding blocks, such as options and forwards, and then decomposesmore complicated structures into baskets of forwards and opway, pricing formulas for simpler structurcs can be used to value tions. complicated stnzctured products. more
!
Hull (21400) is an excellent source on derivatives that is uniquc in many ways. Practitioncrs use it as a manual; begirming graduatc students utilize it as a textbook. It has a practical approach and is meticulously written. Jarrow and Turnbull (1996)is a welcome addition to books on derivatives. Duffie (1996) excellent is an sourcc on dmamic asset pricing theory. Howeverpit is not a source on the details of actual instruments traded in the markets.Yet, practitioners with a very strong math background may fmd it useful Das (1994)is a uscful reference on the practical aspects of derivative instruments.
In this chapter,
.
rrhis
..4
I
.4
..4
rrhey
rrhis
sother recent sources on practical appiiraations of s'waps are Dattatreya et aI. Kapnef and Mnmilall (1992). 6nis means that has a comparalive advantage i.n borrowing at a EXGIrae, .z4
(1994)and
,
9 Exercises 1. Consider the following investments:
;
'.
* An investor short sells a stock at a price and wites an at-the-money call option on the same stock with a strike price oj K ,,
Financial Derivatives
CHAPTER'I
.i>.
j'A
* An investor buys onc put with a strike price of K3 and one call option at a strike pricc of Kz with KL S Kz. price A'1, * Am investor buys one put and writes one call with strike and buys one call and writcs onc put with strike price KZLKL :s #2).
,
(a) Plot the cxpiration payoff diagrams in each case. (b) How would these diagrams look some time before expiration?
*
rl *
2. Consider a fixed-payer, plain vanilla, intercst rate swap paid in arrears with the followlg characteristics'.
r ltra
* ne start date is in 12 months, the maturity is 24 months, * Floating rate is 6 month USD Libor. * The swap rate is x 5%, (a) Represent the cash flows generatcd by this swap on a graph. (b) Creatc a synthetic equivalent of this swap using two Fomard Rate Avcements (FlkA) contracts. Describe the parameters of the selected FRAS in detail, (c) Coeld you generate a synthetic swap using appropriate interest rate taptions?
a,
.
r
&>
.
.w.. .
:*
er on t e e eore
=
3. Let the arbitrage-frec 3-month futures price for wheat be denoted by Ft. Suppose it costs c$ to store 1 ton of wheat for 12 months and per year intcrest rate applicable to traders to insure the srtme quantity. The (simple) of spot wheat is r%. Finally assume that the wheat has no convcnience yield. .$
(a) Obtain a formula for Ft. 1tX)$,c 150$ and the spot price 1500, r 5%, J (b) Let the Ft of wheat be St 1470. ls this Ft arbitrage-free? How would you form an arbitrage portfolio? (c) Assuming that all the parameters of tlze problem remain the same, what would be the profit or loss of an arbitrage portfolio at expiration? =
=
=
=
=
1* 4. An at-the-money call written on a stock wit.h current price St trades at 3, ne corresponding at-tbe-money put trades at 3.5. There are no transaction costs and thc stock does not pay any dividends. Traders can brrow and lend at a rate of 5% per year and all markets are liquid. (a) A trader writes a forward contract on the dclivery of this stock. ne delivery will be within 12 months and the price is Ft. What is the value of F/? 101 for this con(b) Suppose the market starts quoting a pri Ft arbitrage portfolios. Iw/ tract. Form =
=
1
Introductlon
A11 current methods of pricing derivativc assets utilize the notion of arbifrtzpc.ln arbitrage prcz;g methods this utilization is direct. Asset prices are obtitled from conditions that preclude arbitrage oppornmities. ln equiIibrium pricing methods, lack of arbitrage opportunities is part of gcneral equilibrium conditions. ln its simplest form, arbitrage means taking simultaneous positions in different assets so that one guarantees a riskless profit Mgher than the risklcss return given by U.S. Treasury bills- lf such prosts exist, we say that there is an arbitrage opportunity. Arbitrage opportunities can arise in two different fashions. ln the firqt way. one can make a series of investments with no current net comntment. yet expect to make a positive proht. For example, one can short-sell a stock and use the proceeds to buy call options writtcn on the same securi. ln this portfolio, one snancesa long position in call options with short Pitions in the underlying stock. lf this is done properly, unpredictable mtwements in the short kmd long positioms wili cancel out, and the portfolio Will be riskless. Once commissions and fees are deductcd, such investment Opportunities should not yield any exccss prolits. Othenvise, we say that tkere are arbitrage opportunities of the fil'st kind, In arbitrage opportunities of the second kind, a portfolio can ensure a negative net commitment today. while yielding nonnegative profits in tlie
future.
13
14
C H A P T ER
+
2
A Primer on
the
Arbitrage Theorem
Wc use these concepts to obtain a practical dehnition of a price'' for a linancial asset. We say that the price of a security is at a level, or that the security is correctlkpriced, if there are no arbitage opportunities of the Erst or second kind at tlaose prices. Such arbitrage-free asset prices will be utilized as benchmarks. Deviations from thesc indicate opportunities for excess prohts. In practice. arbitrage opportunities may exist. nis, however, would not prices. In fact, determining arbitragereduce our interest in aarbitrage-free'' valuing derivative assets. We can imagine at free prices is at the center of least four possible utilizations of arbitrage-frec prices, One case may be when a derivatives housc decides to engineer a new fmancial product. Because the product is new, thc price at which it should be sold cannot be obtained by obsening actual trading in Iinanci::tl markets. Under these conditions. calculating the arbitragc-free price will be very helpful in deterrnining a market price for this product, A sccond example is from tisk management. Often, risk managers would like to mcasure the risks associated with their portfolios by running some case'' sccnarios. These snulations are repeated periodically. Each time some benchmark price needs to be utilized, given that what is in qucstion is a hypothetical cvcnt that has not been observed.l A third example is marking lTp market of assets held in portfolios. A treasurer may want to know the current market value of a nonliquid asset for which no trades have been obselved lately. Calcuiatirig the corresponding arbitrage-free price may provide a solution, Finally, arbitrage-free benchmark prices can be compared with prices obsen'ed in actual trading. Signcant differences between obscwcd and arbitrage-free valucs might indicate excess profit opportunities, r'Iqhfsway arbitrage-free prices can be used to detect mispricings that may occur during short intenrals. If the arbitragc-free price is above the obsenred price, the derivative is cheap. A long position may be called for. When the opposite occurs, the derivative instrument is ovcrvalued. The mathematical environment providcd by the no-arbitrage theorem is the major tool used to calculate such benchmark priccs.
15
7 Nota tion jng of tbe notation is sometimes as important as an understanding mathematical Iogic. underlying
'ifair
d
2
the
A Primer on
In general, fnancial assets will have different values and give diferent payouts at different states of the world wi. It is assumed that there are a finite number K of such possible states. It is not very dimcult to Wsualizc this concept, Suppose that from a instant. Clearly, trader's point czfview, the only time of interest is the securities prices may change, and we do not necessarily know how. Yet, iri a adowntick'' small time interval, securities priceg may have an or a or may not show any movement at all. Hence, we may act as if there are a total of three possible states of the world. dxnext''
5
i'uptick''
2.3 Rehms
ne states of the world wi matter because in different states of the world returns to securities would be different. We let #fj denote the number of units of account paid by one unit of security i in state j. These payoffs will have two components. ne first component is capital gains or losses. Asset valucs appreciate or in the asset, an appreciation leads depreciate. For an investor who is capital and leads depreciation gain to a capital loss. For somebody to a a who is in the asset, capital gains and losses pill be reversed.z The second component of the dq is payouts, such as dividends or coupon Some assets, though, do not have such payouts, call and interest payments.S fptions and bonds among these. discount put The existence of severai assets, along with the assumption of many states of the world, means that for each asset there are several possible kj. Mafrce. are used to represent such arrays. the payoffs dij can be Thus, for the N assets under consideration, grouped in a matrix D:
columnwise. Each column of D represents payoffs to different assets in a given S tate of the world. If current prices of all assets are norlzero, then one can divide the /th the corresponding Sitj and obtain the gross rc/lgrn,&in different row of D by will have a I subscript in thc general case when state s of te world. The D time. depend on Payoffs
2.4 Portfolo :
uzzd Payogs
17
Example of Asset Pricing 3 A Basic
Arbirrage Theorern
A portfolio is a particular combination of assets in question. To form portfolio, one needs to know the positions taken in each asset under a considcration.The s'ymbol 0i represcnts the commitment with respect to the fth asset. Idcntifying all .(pf,i 1 N) specilies the portfolio. A positive % implies a long position in that asset, while a negative #;. impliesa short position. lf an asset is not included in the portfolio, its correspondingei is zero. If a potfolit delivers thc same payoff in all states of the world, then its value is known exactly and the portfolio is tiskless. =
Itlong''
Rshort''
' ' :
.
.
3 A Basic Example of Asset Pricing
' '
,
We use a simplc modcl to explain most of the important results in pricing derivative assets. Witb this example, we hrst intend to illustrate the logic used in derivative asset pricing. Second, we hope to introduce the mathematical tools needed to carry out this logic in practical applications. The model s kept simple on purpose. A more general case is discussed at the end of the chapter. We assume that time consists of and a period'' and that tbis tlzesetwo periods are separated by an inten'al of length book will reprcsent a but noninfinitesimal inten'al. We consider a casc where the market participant is intcrcsted only in tnow''
Wnext
'rhroughout
#ll
#1x.
.
(5)
D=
ds
'
three assets'
dNK
One can look There are hvo different ways one call visualize such a mat. of a given unit payoffs matrix D as if each row represepts to one at the world. of Conversely, the security in different states one can look at D
dismall''
.
1. A risk-free asset such as a Treasury bill, whose gross return until next Period is (1+r).4 in that it is constant rcgardless nis return is of the realized state of the world. 2. A.n underlying asset, for example, a stock St). We assume that during the small intewal Stj can assume one of only fw't:) possible values. This miltimum Iw'o states of the world. of a is risky because its payoff means is different in each of the two states. IGrisk-free,''
'
5
o
1)
+
=
=
c(/)
-
(1 +
r)
(43)
(1+
r).
(44)
that ratios suck as
yt
+ j)
- 5'tf )
szt
-
'
,5'(1)
..j.
j
)
(45)
are the gross rates of ret'urn of in states 1 and 2, respectively. The (43) and (u) imply that if one uscs , /2 in calculating the expected values. a1I assets would have the same expected return, According to this new result, utmder all expected returns equal the Hsk-frcc , #a,7* rturn r.13 nis is anothcr widely uscd result in pricing linancial assets. st)
tqlmlities
'
Y
I-L
=
'znertz arc other ccmditions that a martingale must satisfy. In later chaplers, we discmqq hem in demil. Ir1 tlne meantime, we assume impdtly that thse condilional expectations t e xist-that is, they are flnite.
+
ytfl
,
is important to realize that, in linance, the notion of martingale is always associated with two concepts, First, a martinga le is always defned with respect to a certain probabiliT.Hence, in Scction 3.4 the discounted stock price. gj X St.vs., (41) t+s (j +. r)x
,
c /+d
with respect to some other probability, say #*? The answer question is positive and is quite useful in pzicing interest sensitivc this to derivative instruments, Essentially, it gives us the flexibility to work with with an asset of our choice. a more convenient probability by normalling But these issues have to wait until Chapter 17.
xt-vsis denned by Xt-vs
I+S
t)c a martingale
Wfair
-Y,
'l'he
=
EQ gXf+.vlfllAE Xt
equalit.y martingale
25
, '
lln
a'Ild
.
'h
.
probability theory, the phrase
#z.''
Ttunder
#I,
6''
means
d 0 that can be plugged , such of j s. sets are many I.n order to determine the arbitrage-free value of the call premium C, ucorrect'' j. ln principlc, this can be done onc would need to select the usjng the underlying economic eqtlibrium.
,
E
4.2 Csc 2: A&'bitrtlge-lk-rcc Prices Consider the same system as before
1 100
1.1
5 An Applicatlon: Lattice Models
1.1
/1
100 150
=
C
0
(60)
.
/2
50
Simple as it is, the eumple just discussed gives the logic behind one of the ?'rl)&l.14 Common asset pricing metbods, namelyj the so-called lattice The binornial modcl is the simplest example. We brictly show how this pricing methodology uses the results of the arbitrage theorem. call considera call option Ct written on the underlying asset St. option has strike price and time < f cxpires T; F. It is known that at at G expiration, the value of the option is given by most
E, E
But now, instead of starting with an observed valuc of C, solve thc Erst two equatitms for 1)1 #a. ncsc form a systcm of tw'o equations in two unknowns. Rnheuniquc solution gives
.
'rhe
,
.7273,
l Now use the
SO Iution:
(Jz
=
third equation to calculate a value of
c
consistent
cz
mx
=
gkz c(),oj
(64)
-
.
We rst divide tlae time interval (r f) into n smaller intewals, each of size a. choose in the sense that the variations of s, during a we A Can be approximated reasonably well by an up or down movement orlly. According to tis we hope that for small enough the underlying asset Pnce St cannot wander too far from the currently observed price Thus we assume that during k the only possible changes in S J are an up moveme'nt by =o or a down movement by -o-X&
with this
-
''small''
'
j
C
=
9.09.
this price, arbitrage prolit.s do not exist.
At
(61)
.1818.
=
(62)
Note that, using the constants #j, #z, we derived the arbitrage-free prico C 9.09. ln this sense, we used the arbitrage theorem as an asset-p ri cing tool. lt turns 0ut that in this paticular case, t-he reprcgentat ion given by the with positive and unique #i. nis may not arbitrage theorcm is gatished be always true.
=
:
,
.
%%.
'
''
=
Sl+. ''Also called tree zyltiel.
'
i.
=
st +
g.y
St -
G'U'X
.
(65)
r
C H A P T ER
A Primer
2
.
on the
5 An Application: Lattice Models
Arbitrage Theorem '
su4
values of the call option at any time l + A to 1he aties's two (arbitrage-free) rbitrage-free) the option as of time /. The lhp is known at this value of (a order equation make the usable, we need the two values Ctlh In to PO int. Cdovm Given these, calculate the value of the call option Ct at we can an d /+
Su3 Su2
su2
'
timc f
Xu
su S
yigure 2 shows thc multiplicative lattice for the option price Ct. ne arbitrage-fzee values of C3 are at this point indeterminate, except fo the ln fact, Sventhe lattice for St, we can determine the expiration using the boundary condition expiration the of C3 at values
s
a
''nodes.''
Stl
Sl
E
Sd2
-
Cp
sd2 sd3
=
max (&
01 - G,
Once this is done, one can go backward using sd*
G
F 1G U R E
Clearly, the size of the parameter (r determines how far St-vhcan wander during a time inten'al of length k&. For that reason it is called the volatilit. parameter. The (m is known. Note that regardless of tr, in smaller intervals, St Will change less. The dynamics described by Equation (65)represent a lattice or a binomial frcc. Figure 1 displays these dynamics in the case of multiplicative up and down movements. risk-free rate r for the Suppose now that we are givcn tbe (ct-mstant) probabilitiesyls risk-adjusted the detezwne pcriod Can we We know from the arbitrage theorem that the risk-adjusted probabilities and l'bovmmust satisty'
1 (1+ r)
=
Uasfahownwhj up
'
.
.:r4''il
+
hownst-
c'vr-all .
f cuS
Cu
=
1
(j
.j.
rj
downj
g/urfri+a up
+
dovln.u
l5ln the second half of the bOOk. we will relax the assumpton now we maintain this assumption 'ERemember hat Pgov.. 1 #up. =
that r is constant.
K)
i'.
(Su2
- K)
(s -
K)
cu
rrhtls,
Ct
(su4-
cu2
(66)
ln this equation, r, St, fJ-, and k are known, ne til'stthree are observed in the markets, while A is seleded by us. the only unknown is tlle /.s, which can be determined easily. 1 t$ Once this is donc, the lhp can be used to calculate the current arbitrage-free value of the call option. In fact, the equation
(69)
.
'
#up
+ st 1 +1r gzurt-.
doum
+
Repeating this several times, one eventually reaches the initii node that gives the cun-ent value of the option. Hence, the proccdure is to u:e the dynamics of St to go forwardand determine the expiration date values of the call option. Then, using the risk-adjusted probabilities and the boundary condition, one works backward with the lattice for the call option to dete=ine the current value Ct. It is the arbitrage theorem and the implied martingale equalities that make it possible to calculate the risk-adjusted probabitities Pupand gaown.
::.
.
-
(68)
.
C Cd
cd
E
cd2
f6y)
(*d2-
K)
cd'
But for E
(Sd*
F 1G U R E
. :
.
.
2
- K)
'
'
.
32
C H A PT E R
*
A Primer on
7
the
.
:
Arbitrage Theorem
'
payouts
6
cln this procedure Figure 1 gives an approximation of all the possible paths that may take during the period F - 2. The tree in Figure 2 gives approximation of all possible paths that can be taken by the price of an option written is small, then the lattices will be close the call on St. lf approximations to the tnle paths that can be foilowed by St and Ct.
Foreign Curencies
and
1
=
ut
33
(1+ r)
gsv:nu
,
-U'
'
LW
r'd ga-l U *-
gj)
1 '
where # is the risk-neutral probability. and where we ignored the time SUbscripts. Note t jaat the qrst equation is now different from the case with no-dividends, but that the second equation is the same. According to this, an asset has some known percentage payout d during the period e ach time risk-neutral discounting of the dividend Jyyjn: asset has to be done the a using thc factor (1+ J)/(1 + r) instead of multlplying by 1/(1 + r) only. It is also worth emphasing that the discounting of the derivative itself did not changc. Now consider tbe following transformation!
:
*-'
6 Payouts
and
.
Foreign Curtencles
In this section we modify thc simple two-state mo del introduced in this chapter to introduce hvo complications that are more often the casc in practicalsituations. The 6rst is the payment of interim payouts such as dividendsand coupons. Many sccurities make such payments before the cxpiration date of the derivative under consideration. These payouts do change tlle pricing formulas in a simple, yet at ftrst sight, counterintuitive fashion. The second comp licationis the case of foreign currency denominated assets.Here also thc prking fonnulas changes slightly.
6.1 The C-e u'itlv Dztrleads ne setup of section3 is first
k
(1+ r) (1+. d)
s
,
wliich means that the expectcd return under the risk-free
.
given by;
Stu st
E
r
(1 +
rL) (1-f- dtl
=
Qearly, as a first-order approximation,
'
ycar,and are small:
modilied by addzg a dividend equal to Note tw'o points. First, the dividends are not lump-sum, J, percent of the dividend : are paid as a percentage of the price at time t+,&. second, subscrpt r instead of f + a, According to this, the d is rate f known as of time f Hence, it is not a random variable given the informatlon : set at 1t. ne simple model in (10)now becomes: ,
ut-vh.
sdl'sd
Supa +
=
-
.
1+
rh
2 1+
1+ d
but paymenthas
Using this in the previous equation; S.
iq
.
E'-
'*''
.
s
or
measure is now
,
if d, r are defned over, say, a
(r -
dlh.
.
1 +. (r
l
-
djh,
.sp
u u
#J
Bt..h
St
'hn
cl
=
d
Bt-vh +
.$d
d t su ,+a
+ dt
/+a
cd/4-?k
cu/->k
.d
r+a
p r+a j a s
1/32 d y,
6
,
equations:
'
p
where B, 5', C denote the savings account, the stock, and a call option, as usual, Note that the notation has now changed slightly to reNect the discussion of Section 5. Can we proceed the same way as in Secticm 3? The answer is positive. With minor modilications, we can apply the same steps and obtain *0
According to lf we were term for Jxi, Price, will be written as; :7
!
S
=
tl (y
+ d'' ..4.
yj
j
-.,-
+
s
,+a
o st +. (,
uncse .
dlsth +
a,yrapus.
-
ds l
(7t))
-
tz.-s/api.f-a:
this last equation, we can state the following, to let go to zero and switch to continuous time, the dnT wllich represents expected change in the underlying asset's given by (r djstdt and the corresponding dynamics can be
.'
Wbere us,/'sj
f +. r - ajsth, unpredictable component,
or again, after adding a random,
dt represents stooassc
.js
.dlstt
+
(ryapy,
an innnitesimal time period.
disozential equations
wsl,c
stufsetIwi,,l more detail irl later chaptem.
34
C H A P T ER
2
*
the
A Primer on
Arbitrage Theorem
nere is a second interesting point to be made with the introduction of payouts. supposenow we try to go over similar steps using, this time, the equation for Ct shown in (7i); c
=
1
(j
End
paycuts
E
opportunities
gc upu
cldj
+
1
c
E
/
Ct
C
',
ut
C'+A
)
z
(,
-
dlh (;
c;
4 ru #/,,
,
cs,q. 2k
w-
=
=
(1+ rTl (j .j. pj 1 (1+. r)
(e upu
d
z
d
s
+ c
(CN/?
+ C
x 1 + rh.
(1+ r,j < 1 + (r (1+ rTA) we again obtained a different ne exqected rate of retum
.
.
probability P:
E p ea et
Forefgn Currencies
by adding an investment opportunity The standard setup is now moded savings foreign account. in a currency In particular, suppose we spend et units of domestic currency to buy one the et is the exchange rate at time t. Assumc unit of foreigncun-ency. U.S. dollars (USD) is the domestic currency. supposealso that the foreign savings interest rate is known and s given by rT. raus
(1+ r y )
f+.
/
j
1 .
,
1+
modiscations in the formulas, but in practice they may These are make a signihcant difference in pricing calculations. ne case of foreign currencies below yields similar results.
6.2 TRe Ctzse
ed
(1+ r y )
r)
Again, note that the ftrst equation is different but the second equation is the same. Thus, each time we deal with a foreign currency denominated the risk-neutral discounting of the forcign asset that has payout rl during asset has to be done using the factor ( 1 + r)/(1 + rT). Notc the first-order approximation if rf is small:
i
glight
lvifl
(1 +
w-i4.zs
'
and Ct during a period A are . The expected rate of returns of the risk-free /: dkprent under probability the now rA) dLj
r)
wllere the Ct denotes a call option on price et of one unit of foreign c'ur#.1S renc'y. The strike pricc is we proceed in a sinlar fashion to the case of dividends and obtain the following pricing equations:lg
Thus, we see that even though there is a divided payout made by the underlying stock, the risk-neutral expected retllrn and the r isk-free discounting remains the same for the call option written on this stock. Hence, in a risk-neutral world fumre returns to Ct have to be discounted exadly by the same factor as in the case of no-dividends. In other words:
(1 + (1 +
o, cu
ct
Ct-vh = 1 + r. ct
=
C/+. =
.
We would obtain
Stn St
u
.
.
E>
(1 +
1 '
+ r)
35
in investment and the yields of these investments ovcr using the following setup; summarized be a can now
ne
'
Foreign Currencies
'
!:
y
)a.
result.
of the et and C are dkyerent under the 2 1+
s> G+a. z t
:
r
-
(g
-
r
y
)
1 + za.
Here the K is a strike price on tilc exchange rate ';. If the cxchange ratc exceeds the X' at time t + the tmyerof thc caE will reccive te difference e,+ - K times a notional amount N. t9As usual we omt the time subscrips for convenience. ,
,
k
k
cHh
36
PT E R
-
2
A Primer on
t
heArbitrage Theorem
8
A Melodology conclusions:
for Pricing Assets
2.3 Didcotmting
switch According to the iastremark, if we were to iet go to zero an d given by rctdt. But the drifl to SDE's, the drift tenns for dCt w iil be denominated asset, det, will ow bave to term for the foreign currenc.y rflcfttf. be (r .'
models usingcontinuous-time fact, j.f is continuous,
Ieads to a change in the way discounting then the discount factor for an intenral t In js done, w'ill be given by the exponentialjnctm of lcngth
-
e 7
general, such simple U P to this point, the setup has been very simple. In rcal-life financial assets. Ixt us brielly examp les cannot be used to Price consider some genera lizations that are necded to do st.
8 Conclusions: A Methodology for Pricing Assets
ln 1, 2, 3, with f Up to this point we cons idercd discrete time continuous-t ime asset pricing models, this will change. We have to assume that f is continuous:
l e (0,x) ter, we
.
The arbitrage theorem provides a powerful methodolor for determining values of major jr market fmancial in practice. assets steps of this fa ne mcthodology as applied to inancial dcrivatives can be summarized as follows:
.
Consider
,
to track the dynamics of the underlying obtaina model (approximatc) price, asset's 2. calculate how the derivative asset price relates to the price of the 1
(72)
.
'
.'
with this chapto t-he asmall'' time interval A dcalt Symbolin Jl. the injnitesima l intcrvals denoted by
This way, in addition Ca:II
.
underlying asset at opiration or at other boundaries. 3 obtainrisk-adjusted probabilities. 4 calculateexpected payoffs of derivatives at cvrtlfoa risk-adjusted probabilities. 5. Discount this expectation using the risk-free return. .
,
.
.
7 2 Stutes of fh,e World
.
limited In continuous timc, the valucs that an asset can assume are not of continuum and uncountably many POSS ibilities a to two. There may be states of the world. stochastic zs./.erTo capture such generalizations, we nee d to introduce example, as ment ioned above increments ill security ential equations. For using Prices S: may be modeled dSt
=
p'tvh
dt +
o-t
ul
,
change in the price of where t he symbol dSt represents an insnites imal inlinitesimal dt is thc predicted movemen t during an t lae security, the 1 uuuvztn cyaaxavintcrval #/, and gtst JIFI is an unpre dicable,. inlmneslmal stochastic differdefming It is obvious that most of t he concepts used in Step. ential cquations nee d to be developed step by ut
.
-
,.
=.
-,
.--aw.m
'
'jau.d
7
ojaeaov
E .
,
using
these
In order to be able to apply this pricing methodology, one needs familiarity with the following types of mathematical tools. First, the notion of time needs to be defined carefully. Tools for handlingchanges in asset prices during time periods must be developed. requires continuoua-time analysia. nis Second, we need to handle the notion of randomness'' during such jusaktesjmal periods. conceptssuch as probability, expectation, average value, and volatility during inlinitesimal peliods need to be carefully deEned Tllis requires the study of the so-called stochastic calculus. we try to discuss the intuition bchind tlae assumptions that lead to major results in sthastic calculus. we laeed to understand how to obtain risk-adjusted probabilities and how to determine the correct discounting factor. ne Girsanov theorem states the conditions under which such risk-adjusted probabilities can bc used. ne theorem also gives the form of these probability distributions. ''inlinitesimal''
(73)
#W$
(74)
,
.
7.1 Tlme l
.
-rA
ne r becomes the continuously compounded interest rate. lf tere exist dividends or foreign currencies, the r needs to be modied as explained in Section 6.
Some Genetalizatlons
=
37
r
C H A P T ER
38
.
2
A Primer
on the
ArbitrageTheorem
lppendix: Generalization
jc
Further, the notion of martingales is esscntial to Girsanov theorem, and, world. consequcntly,to te understanding of the of various movements Rnally, there is the question of how to relate the this is standard done using calculus, quantitiesto one another over time. In equivalent concept is differentialequations, In a random environment, the equatkm (SDE). a stochastic lz//rezc/ Needless to say, in order to attack these topics in t'urn, one must have calculus. concepts amd results of somenotion of the wcll-known thc notion of derivative, of notion the (2) There are basically three: (1) integrala and (3) thc Taylor series cxpansion.
xowdcfine
.
''risk-neutral''
of the
a portfolio, :,
39
s the vector of commitments
to each asset:
0L
'
#
=
(,20
.
oxv ln dealer's termfnology, 0 gives the positions takcn at a certain time. Multiplyill# the 0 by Stt we obtain the value of portfolio 0:
d'standard''
'
./# t
x =
j=l
'
9
ArbitrageTheorem
References
sit )tj.
qy
This is total investment in portfolio : at timc /. TN 2t) In matr . payoff to portfolio 0 in state j is izzz1 d ij f3i expressed as
,
*
lngcrsoll ln tis chapter, arbitrage theorem wa,s treated in a simple way. accessible, quite is that detailed treatrncnt (1987) provides a much more with a strong quantitative background may Readers beginncr. to a even The original article by Harrison and Kregs ( 1979) prcfer Dufse (1996). material can be found in Harrison may a lso be consulted. Other related RutkowRki Musiela and in chapter (1997)is , and Pliska (1981).The first read afler this chapter, cxcellent and very easy to
jqj
.
o'j?
.
DEFINITION:
.
either
:
Accordirig to the arbitrage theorem, if there are no arbitrage possibilities, therl there are state prices, jsij., such that each assct's price today etjuals a linear combination of possible future values. Thc theorem state prices then there js also true in reverse. lf there are such (supporting) are no arbitrage opportunitics. ln tis section, wc state the gencral form of thc arbitzage teorem. First we brielly dehne thtt underlying symbols. * Desne
a matlix
o, N' js thc total number thc.
Wol1d.
Of
=
securities
: is an arbitrage portfolio, or simply an arbitrage,
one of the following conditions is satisqed:
if
.z
According to this, the portfolio /? guarantees some positive retttrn irl all states, yet it costs nothing to purchase. Or it guarantees a nonncgative teturn whil having a negative cost today.
q ,
'l'he
'
dcussed
following theorem is the generalization of the arbitragc conditions earlier.
THEOkRM: dbK
'
.
dx
deline an arbitrage porolio..
'
payoffs, D1
fll
()x
s'v ...s () and D'o > () a s,o (j ajalj p,o u (j. '
'
,
xvr
1.
.
'Isupporting''
faf
we can now
(78)
'
dl K .
:.,
oj
.
:
,
10 Appendix: Geneulizatlon of the Abltrage Theorem
#xj
form tls is
(75)
Jow and K is the total number of statcs
j Skch
'
@
:
opportunities,
s
! E
jj tjwre are no arbjtrage that
wNotc the differenc.c Ixtwzen summaton
to j.
=
thcn there exists a
D*. with respect
>
0
(,79) o
and summation
with respect
.:
C H A PT ER
40
*
A Primer
2
on the
2 If the condition in (77)is true, then there are tunities. *
Arbitrage Theorem no azbitrage
world there cxist
This means that in an arbitrage-free #11 5) =
dNK
dsj
oppor-
:suc ja that .
(b)
*K
D
#2r .
(d)
and
1)1
d:
-1-
.
.
.
+
.
#r,
=
f+
.
1
=
y
..4.
r;
E zaxgct.v aj.
use the normalization by St and Iind a new measure # under wlajch the normalized variablc is a martingale. whatis the martingale equality that corresponds to nonmalization
,
2, In an economy there are two states of the world and four asscts. You are givcn the following prices for three of these securities in different states Of the world:
:
: ,
'
(82) : Price
-
g
)-q
j
=
(83)
(Jv.
A security secmrity s
2=1
Security (
The 40 is the dixount in riakless mrrtlwnp t:
1 1 Exerclses nondividend
1' You are given the price of a world where t here ropean call option Ct in a
'
state1 jx
8() q)
Dividend State 2
State 1
State 2
yj
1
($:
j
ls()
1()
rp
current prices for A, B, C are 100, 70, and 180, respectively. (a) Are the prices of the three securities arbitrage-free? ''current''
pay ing stock St and a Euare only two possible states:
320
if u ocmzrs
260
if d occurs.
=
=
risk-neutral martingale measure P* using the normalization by risk-free borrowing and lending. Calculate the value of the option under the risk-neutral martin#itle measure using
define
Sr
.5,
=
hy s:? Calculate the option's fair market value using the #. (e) that the option's fair market value is independent state Can we (t) of the choice of martingale measure? (g) How can it be that we obtain the same arbitrage-free price although we are using two different probability measures? (h) Finally, what is the risk premium incorporated in the option s p rjce? can we calculate this value in the real world? Why not?
, , 7.
,
=
(#'T
(c) Now
'
(81)
Jn this matr ix D the hrst row is constant and equals 1. This implies that State of the world the rcturn for the flrst asset is the same no matter which security riskless. is is realized. So, the tirst Of D with the Using the ar bitrage thcorem, and multiplying the hrst r0w Obtain I), we state pricc Vector S1
at time
=
c,
' Jxi
ne
'
if eac h statc under considcration has a nonzero probability of occurrence. Now supptse we c onsider a special type of return matrix where 1 #21
280.
with A
E
=
#d .5J,The annual interest rate is constant at r 5%. 3 months. ne option has a strike price of
of the two states are given by
(a) Fin d the
(80)
all i
i/f > 0 for
=
=:
have
Note that according to the theorem we must
probabiiities is St current timc is discrete, ne K 280 and expires
vjwtrue prife
.
*1
dLK
*
SN
/
41
11 yxercises '
;
:
(b) lf not, what type of arbitrage portfolio should one form? (c) Determine a set of arbitrage-free prices for securities A, C.
(d) supposewe
B, and
introduce a fourth security, which is a one-period ftltures contract written on B. What is its price?
7
c HA PT E R
.
2
the
A Primer on
Arbitmge Theorem
43
1j Eercises
5. You arc given the folloeng information concerrting a stock denoted
(e) Suppose a put option with strike price K 125 is written on C. ne option expires i.lzperiod 2. What is its arbitragc-frce price? =
by st.
value 102. . currentvolatility 30%. =
3. Considcr a stock and a plain vanilla, at-the-money. put option written on this stock. ne option expires at time /+, where denotes a small intenal. At time f, there are only two possible ways the St can move. lt can either go up to 5 lu+ or go Jtlwa to Sd/+.a. Also available to traders is borrowing and lending at annual rate r. risk-free sb
. '
xrmual
=
5%, which is known to be you arc also given the spot rate r months. during the 3 next stant con lt is hoped that the dynamic behavior of Sf can be approximated reasonably observation intervals of Iength 1 wellby a binomial process if one assumes =
@
'
,
(a) Using the arbitrage theorem. write down a three-equation system . withlw'p states that givs the arbitrage-free values of s, and E (b) Now plot a hvo-step binomial tree for St. supposeat every node the markets are arbitrage-free. How many thrceof the treesptems similar the preceding could thcn be to case equation entire for the tree? written (c) can you find a three-equation sptcm with 4 states that corredsto the same trcc? spon How do wc know that all the implied state prices are internally (d) consistent?
month.
.
a European call option written on ne call has a 120 and an expiration of 3 months. Using the price K and thc risk-free borrowing and lending. Bt, construct a portfolio thatreplicates the option. the replicating portfolio price this call. Using (b) 100 such calls to your cusSuppose you sell, over-the-counter, (c) tomers. How would you hedge this position? Be precise. (d) Suppose the market price of tMs call is 5. How would you form arbitrage portfolio? Considcr (a) Strike
.
.
,
4, h four-step binomial trce for the price of a stock usingthe up and down ticks given as followsz u
=
#
1.15
=
nese up and down movements apply to one-month = 1. We have the following dynamics for St, k up St-a
=
uSt
down
s'aa
=
-
.%
is to be calculated
6.
supposeyou are
at
-k-free
5%.
QJrest
=
6%,
*
u periods denoted by
ds t,
u$j
.
wsy..j
;
,
+1
.
Jwn
zzz
ySt +
whcre the e is a serially uncorrclatcd following values! 6
up and dtaw'rldescribe the two states of the world at cach node. Assume that time is measured in months antl that t - 4 is the expiration for a suropeancall option c, written on st. nc stock does not pay '-markct oarticioants''j to erow at dividends and its orice is expec-ted (bv r-is c-onstant of ris kn 15 Tlie to rate rate -%.
given tlle following data:
* Risk-free yearly interest rate is r The stock price follows:
.
where
date anv an-annual
./
an
'
1
vt.
=
=
cglt s ( y
binomial process assuming the
with probability p sjth probability 1 p. -
-
-be
-
ne 0 < p < 1 is a paramcter. Volatiiity is 12% a year. The stock pays no dividends and the current stock price is 100.
Now consider the following questions.
E
(a) According to the data given above, what is the (approximate) annual volatilty of St if this process is known to have a log-normal distribution ? ('b) calculate the four-step binomial trees for the St and the Ct. (c) Calculatc the arbitrage-free price C, of the option at time t 0.
(a) supposeJt is equal to thc
risk-free interest rate:
:
,
an d oat the st is arbitrage-free, What is the value of p? (b) Would a p 1/3 bc consistent with arbitrage-free St3
=
=
..L
'(.
7 cHh (c) Now suppose
J,t
P T ER
-
2
A Prinwr on the Arbitcage Tlaeorem
''
!
k7.
is givcn by: Jt
=
r + risk
'N
oywwygpwztyyfyswzw.
xwpmww..m -%
,m
-
4--x
premium '
(d)
What do the p and 6 repesent un der these conditions? ls it possible to determine the value of p?
alcxllxlsln .
7 Using the data in the previous question, you aTe now asked to approximate the current va lue of a European call option on the stock ,St. ne option has a strike price of 100, and a maturity of 200 dap. an appropriate time interval /, such that the binomial has 5 steps. tree What would be the implied u amd t? probability? What is the implied the tree for the stock price S:. Deternne Determine the tree for the call premium Cf.
RR
eter *
lnlstlc *
a:tjc
.
nV1rOn
.
ents
'
(a) Determine (b) (c) (d) (e)
tOC
.
: '
Wup''
,
1.
1 Introduction
:
The mathematics of dcrivative assets assumes that time passes continuously. As a reslzlt, new information is revealed continuously, and decision-makers may face instantaneous changes in random news. Hence. technical tools for pricing derivative products require ways of handling random variables over ienitesimal time intcrvals. The mathematics of such rarldom variables is known s stochastic calculus. Stochastic calculus is an intcrnally consistent set of tperational rules that are different from the tools tf calculus in some fundamental
.
Istandard''
,
wap.
2
At the outset ? stochastic calculus may appear too abstract to be of any use to a practitioner. first impression is not correct. continuous time nis is both simpler and ticher. market fmance participant gets some praca once ticeit is easer to work with continuous-time tools than their discrete-time equivalents. In fact. sometimes tbere are no equivalent results in discrete time. In thissense stochastic calculus offers a wider variety of tools to the fmancial
.
E
,
For example, continuous time aaalyst. mrtfolioweights. This way, reyicating
permits inlinitesimal adjustments in assets with Prtfolios becomes possible. In order to replicate the underlying option, an
q . i
! 4
:
' .. '
.
5
''nonlinear''
''simple''
C
: .c'
46
C H A PT ER
*
3
Deterministic and Stochastic Calculus
asset and risk-free borrowing may be used. Such an eact be impossible in discrete time.l
1
1 Infov'nwtion
replication
j yunctons
VII
@
.
.
s
the response of one variable to a (random) is, we would likc to be able to differentiate
'rhat
cbange in
of interest. variotlsj'unctions wtAu'd
like to calculate sums of random increments that arc of We jjyterest to uS. This leads to the notitan of (stochastic)tegral. arbitrary function by using simplcr wc would likc to approximate an Taylor scries approximations, ftmctions. vjjjsIeads us to (stochastic) We model wOu ld like thc Finally, dynamic behavior of to * z-nntinuolls-tj me rasdom variables. This leads to stochastic differential eqlt *
It may be argued that the manner in which information llows in linancial markets is more consistent with stochastic calculus than with caiculus.'' interval'' may be diffcrent on diffcrcnt For example, the relevam trading days, During some days an analyst may face morc volatile markets, . the basic obscrvain othel's less. Changing volatility may require chanpng tion period,'' i,e., the of the previous chapter. Also, numcrical methods used in pricing secqlrities are costly in terms of computer time. Hence, thc pace of adivity may makc the analyst choose coarser or finer time intervals depending on the level of volatility. Such approximationg can best be accomplished using random variables defned will be needed to over continuous time. The tools of stochastic calculus defme these models.
;
( dstandard
Wtime
*'
'-----sns.
if
1.2 M.odeing Rl
like to calculate wcwould anothcr variablc,
':
z someTools of
:'
:
E
.
Bfltwitn'
A more teclmical advantage of stochastic calculus is t.hat a complicated E randomvariable can have a very simple structure in contitmous time, once the attention is focused on infinitesimal intervals. For example, if the time Perjod under consideration is denoted by dt, and if Jl is then asset prices may safcly be assumed to have two likely movements: ': ! tmtick or downtick. strudure may be a good apUnder some conditions, such a interval #l, but not necessarily inflnitesimal dering an proximationto re ality l.2 denoted by time'' inten'al large in a thc 1to integralFinally, the main tool of stochastic calculus-namely, the Riemann linancial markets than appropriate in be to use may more caiculus. used standard in integral nese are somc reasons behind developing a new calcu1us Before doing this, however, a review of standard calculus will be he 1p11 After all, althoughthe rulcs of stochastic calculus itre diferent, the reasons ftr devcloping such rules are the same as in standard calculus:
Standard Calculus
calJn tus section wc review the maior concepts of standard (deterntinistic) xlus. Even if the reader is familiar with elementary conccpts of standard cajculus discussed here, it may still be worthwhile to go ovcr the examples in this section. The examples are dcvised to highlight exactly those points at which standard calculus will fail to be a good approximation when underlying variablcs are stochastic. .
,
i'infinitesirnaly''
: '
ibinomial''
''discrete
:
'
3 Functions
suppose and B are two sets, and let be a rule which associates to every elemeut x of ,4, exactly one elemcnt y in B.3 Such a rule is called kfunction or a mappiag. In mathematical analysis, functions are denoted by .d
f
.
:
-->
B
or by
,'
:
.,4 '.
y
=
f (A),
(2)
If the sct B is made of real numbers, functionand write
then we say that
f
is a real-valued
.
Would be thlz case Will the undcrlyillg State Space s itself discrete. nis values in the fut-ure. number possibie of ' ther underlying asset pricc can assumc only a Iinite values, and it may be ' ptlksible c)f the variable two random assume binomial can one 2A easicr te wor k with than, say, a rimdom variable that may assumc any onc of an signiqcantly uncountablenumber of Nssible values.
IVIIICSS, Of Course.
.'
f
..4
R, If the sets and B are themselves collcctions os functions, then f transforms a function into another function, and is called an operator. Most readers will be familiar with the standard notion of f'unctions. Pewer readers may have had exposure to random functions. ',
.->
.,4
'.
aThe
set
ad
is called the domain, and the set B is called the ranne of
)'.
C H A PT E R
48
3
Deterministic
Stoclastic Calculus
and
Flznctm.s
3.1 Ru
.
In the function y
E
(4)
,d,
flxt,
=
x.
Often y is assumed to
y. once the value of x is given, we get the element signi:cant following number. consider the Now be a rea f nere is a set p: wherc w c Wz-denotes a state of the Jzi(s depends on w e JF; f on x c R /
or
y
=
fxb
:
R x l#r
u?),
->
alteration. world. The function
(5)
R
x s R, w
q
''plug
wherc the notation R x JF implies that one has to variables,one from the1,) set H,', and the other from R.
in'' to
f.j
'
''
two
has the following property: Given a w QE I,F-.the 1J?) of x only. nus, for different values of w G HJ function becomesa f', different functions of x. Two such cases are show'n in Figure 1. we get tql and ftxa fx, tzlzl are two functions of x that differ because the second elementw is different. Wlien x represents time, we can interprct f @,t&ll and f (-r.u7z) as two differcnttrajectories tha t depend on differcnt states of the world. vJ) Hence, if u? rcpresents the underlying randomness, the function flx, Anotber name for random functions is be called a random hmction. . The function
stochastic processes. With stochastic proccsses. x will represent tne, and limit our attention to the set x k 0. we otken fun damen taI pout. Randomness of a stochastic process is in this Note trajector.y of the as a whole, rather than a particular vftlue at a specsc tejxns words. other the random drawing is done from a collection ln time. Pointin Of trajcdtries. Choosing the state of thc wozld, 1:), determines the complete trajcdoly
.;
(6)
I'I'C
49
3 ytmctions
2 Emmple: of Fuxcfimzs
There are some important mctions that play special roles in our discussjon. We will brie:y review them.
. ExponentiGl 3.2,1 Z71& The insnite sum
f (x,
Function
7
(7)
.,
) ''
.
,
.
.:
can
Converges to an irrational number between 2 and 3 as n --> x. nis number is denoted by the letter e. ne exponential functionis obtained by raising c to a powcr of .r:
y
..
ffkx.w)
E
;
.ew
y
!'
s p
(yj
.
'I'liis function is generally used in discounting asset pricts in continuous
ime. ne exponential function has a number of important properties. It is infmitelydlfferentiable. is, begirming with y elx) the followng opy' CF36On rnjat
;
'
=
Carl
be repcated
intinitely by recunively letting d
)' =
Jx f
tx-p'Ll
e%ez =
:
x 1
dfxt dx
(g)
.
We exponentjal function also has the interestng multiplicative
:
.'
F l Gu R E
(?'f(x)
be the right-hand
property:
eA+Z
(10)
,
.
Bhally if js random variable x a ,
.
7.
then
.p
=
ex will
be random as well.
50
C H A PT ER
3
*
Detenninistic
Stochastic Calculus
and
51
t, jutwtions
3.2.2 F/zcLogarithmic Function The logarithmic function is delincd as the inverse of the exponential function. Given
Thus, fundions of bounded variation are not excessively function will be of bounded variation/ fact any
'
. R.
wjjen () < (
'
,
(13)
:J:
f
u,jyagujaro,,
,
.
N
5 It can be shown that /(f) is not of bounded variation, That this is the case is shown in Figure 2. Note that as I --> 0, f becomes excessively ne concept of bounded variation will play an important role in our dismlssions jater, one reason is thc following: asset prices in continuous
3.2.3 Funcffon' of Bounded Pzo/ft?rl The following ccmstruction will be used several times in later chapters. Supposc a time interval is given by (0,F1. We pardtion this inten'al into n, as n subintcrvals by selecting the ti, i 1, =
of thc intewal , herethc maximum is taken over all possible partitionsvariations (0, F1, ln this in f (.), sense, L$ is the maximum of al1 possible Speaking, tf On and it i9 finite V is the tottll . g0,F1. Rougitly V measures the ltmgth t'f the trajectory followe d by (') 11S / 80e9 from 0 to F, W
yjjeu
oo
vauatjoa
I-)yytjj)
sAlhlf#lrl
.
-o
:
Ve
'
.
:: '
:. .
right-hand
- yt, ) j
side of this equality
I
=
4
0, there exists a N < x guch that (x) < arbitrary if for x* c to
for all
Lct
.
Thc notion of conveqence of a sequence has to do with the tx). In the case where xn represents rcal numbers, we V aluc of x n as n -can state this mtre formally;
e
chain r//d.
,
ne
Ixu- x*I s get smaller. the base of the rectangles will get smaller. More rectangles will be available, and the area can be better approximated. Obviously, the condition that ftj should bc smooth plays an important path followed by /41)r role durlg this process, ln fact, a very method. Using the ter- ' may be much more diflicult to approximate by this k minology discussed before, in order for this method to wor the fundion (f ) must be Riemann-intcgrble. A counterexample is shown in Figurc 9. Here, the function /(f ) sho< j gteep variations. If such variations do not smooth out as the base of the L. rectanglcs ge ts smaller, the approximation by redangles may fail. with the in dcaling will that be important EE We have one more comment Ito integral later in the text, The rectangles used to approximate tlie area
: : :
: : : : : I 1
'
:
,
under the curve were constructed in a particular way. To do this, we used thc value of fltj evaluated at the midpoint of the inteaals (. 1/-: Would the same approximation bc valid if the rectanglcs werc dclined in a different fashion? For example, if one detined the rectangles either by
E
T'l G U R b
I
: : : : 1
:
:
: :
:
: ; :.
2
: : : l : :
'
ytillti
-
,
Or
.
(38)
/f-l),
(39)
by
fti-jjti
-
would the intcgral be diffcrcnt? To answer this question, consider Figure 10. Note that as the partitions gct liner and finer, rectangles defmed either way would cventually approximate thc s'ame area. Hence, at the limit, the approxnation by rectangles would not givc a different integral evcn when One uses different heigbts for deining the rectangles. lt tums out that a similar conclusion cannot be reachcd in stochastic environments. Suppose f p)) is a function of a random variable Wz)and that we are interegted in calculating
.
'irrcgular''
:
'
,
'
v
'
ln
ytpl,llt
(4p)
.
Dnlike the deterministic case, the choice of rectangles
.
/(p;,)(p;,
.
,
)
j..j
-
)...
-
p;,-,
)
delined by
(41)
('
:
' u
:
62
C H A PT E R
ftt) ca
Deterministic
3
.
and
..
Stochastic Calculus
4
upwr reaangle
0.25
0
.k,)
i
i
(
i
i
0.15
Roxaugleusing
i
j
'
j
I-ower Rectangle
()
O.4
0.6
FIGIJRE
1o
0.2
the where
:
0.95
hxj
E
0.8
1
'
t
Aq
.
T
n
.
0
l'Ntate that
(14$ i -
Wzi and f 1)
#K are i
=
A(x)dx.
f=1
g
fi + / i -
a
1
fti) -
.f(J2-1)).
(49)
time. Hence, it may appcar that the Riemann-stieltjes integral is a more appropriate tool for dealing with derivative asset prices. However, before coming to such a conclusion, note that alI the discussion thus far involved deterministic functions of time. Would the same dcfinitionsbe valid in a stochstic environment? Can we use the same rectangles to approxnate illtegrals in random environments? Would the choice of the rectangle make a difference? e answer to these questions is, in general, no. lt turns out that in Stochastic environments the functions to be integrated may vary too much for a straightforward extension of the Riemann integral to the stochastic A new definition of integral will be nceded.
: '
':
.
(43)
.
E
rjwjj
(44)'
case
correlated.
-
.
i.( E
#(-) #.f(.)
Because of these similarities, the iirnit as maxj ti I - ff-j I --> 0 of tbe right-hand side is known as the Riemann-stieltjes integral. ne Riemann-stieltjes interal is uschzl whcn the integration is with respect to incements in /(x) rather than the x itself. Clearly, in dealing with Iinancial derivatives. this is often the case. ne price of the derivative asset depends on the underlying asset's price, which in turn depends on
We have alrcady discussed the equality
dfx)
(48)
.
.
f (47).
yzlx)dx.
=
This definition is not very different from that of the Riemann integral. In fact, similar apprtmmating sums are uscd in both cases, lf x represents tlrrle r, the Stieltjcs integral over a partitioned interval, y, wj,is given by
.
+
(47)
Jy(x)
(42)',
4 3 2 The Stieltjes Integral integral is a differcnt definition of the intcgral. Define the 'I'he stieltjes diyerentialdf as a small variation in the function (x) duc to an infmitesimal variation in a':
g(x) df (.t),
With
;
To see the reason behind this fundamental point, consider the case where . 1#) is a martingale. nen the expectation of thc term in (42),conditional on information at time fj.-l, will vanish. nis will be the case becausc, by dchnition, future increments of a martingale will be unrelated to the current : information set. on the other hand, the same conditional expedation of the term in (41) will, in general, be nonzcro. 11 Clearly, in stochstic calculus, express ions that utilize different deEnitions of approximating rectangles may lead to . different results. result. Note that when Finally, wc would like to emphasize an important /(.) depcnds on a random variable, thc resulting integral itself will be a 5 random variable. In this sense, wc will be dealing wit.h random integrals.
fx
(46)
xa
E
H6,-1). f ('r'1k1)(11$ -
dfxj
gxjfxx).
'Ihc.n the Sticltjes integral is defined as
(
-
=
:'
will in gcneral result in a different cxpressicm from the rectangles:
dA)
(45)
function /?(.v) is given by
'
=
hxtdxxg
Inipoilt
i
.
.1
Limit
(Note that according to thc notation used hcre, the derivative fxxj is a of x as well,) Now supposc wc want to integrate a ftmction /l(.r) orjctjon rCSPCIX to 3:: W ith
.
o.2
and
ccnvergence
'
.E
.
64
C H A P T ER
.
'
:
Detenninistic and Stochastic Calcolus
3
*
:
4
cxmvergenceLimit Elnd
65
J.J, J Evnmple In this scction, wc would like to discuss an example of a RiemannStieltjes integral. Wc do this by using a simplc function. Wc 1ct g(8)
;
.lit)
avt
=
(50) '
,
where a is a constant, This makes g(.) a lincar function of value of- the integral
.5/.
'
12
w'hatis the
t1
s /
A
(51)
J.s(/)
gus.
-
,
V-
'!
.,
2*
.
0
a
=
s
-'r-
,
kv.r
IY-Y
if the Riemann-stieltles' dehnition is used? Directly zttaking''the integral gives
auhd'St )
W
1 2 -St
.a:'''
*
'tYr=
.
(52)
a
ko.v.F2-Il>rzs-zsuxasprovs: .,
!
()
--
VS
,
2
'
So T
t
0
dvbjt)
=
1 1 a -S2 2 z'- -&2
z
(53)
xjujxnt (sooss/a
:
F l G U R lJ
Because g(.) is linear, in this particular case the approximation by redangles works well. This is especially true if we evaluate the height of tlle redangle at the midpoint of thc basc. Figurc 11 shows this sctup, with 4, a Due to thc lincarity of. g(.), a single rcctangle whose heigbt is measured the midpoint of the intewal & 5'wis sufficient to replicate the sladed at area.In fact, the area of the rectangle 5'().,dS,$z is
4 4 Iwtcgvution !r-yPuvjs ln standard calculus there is a useful rcsult known as integration by parts. lt can bc used to transform some integrals into form more convenient to a deal wit. A smilar restgt is also very useful in stochastic calculus, even tlmugh the resultng formuia is different. Consider two differentiable functions (f) and (/), where / (E (0,F1 represents time. Then it can be shown that
:
=
,
-
1
2
1
l5'z- s'lpj a -sv 2 - 2.-s =
!
T ./',(1)(1)
g,
ne Riemann-stieltj 'es approximating sums measure tangle exactly, with no need to augment the number rec rectangles.
,
(54)
thtz area tlnder the of approximating
.
()
dt
=
E./'(F)(F)
-
/'(())(0)j
I (f) and ftt) where With
r -
(j
/7/(f)y(r) dt,
(55)
ftmctions are the derivatives of the corresponding respect to time. 'They are themselves functions of time /. ln the notation of the Stieltjes inlegral, this transformation means that 2.11expression that involves an integral
'
1
.
I 25'
11
by
rectangles.
.5l
Sv
'
.
Now, let us see if we can get the same result using approximation
+ vb'v a 2
E
z-vcwx,
ji--kka.'
.
r
r
! i
r
1'
G
i
Equal eiangLes
is a f'unction of time.
f.l .
;
l'
'
'
htj
dft)
U'
66 Can
c HA PT ER
Deterministic and Stochastic Calcultls
3
.
j parrial Derivatives
flnancial markets. However, partial derivativcs are very useful as inPrjce in termediary too ls They arc useful in taking a total changc and then splitting from different sources, and they are useful jt into components that comc in tos/ diycyentiations. on seforedealing with total diffetentiation, we have one last comment changes, ytjal derivatives. Because the latter do not represent pa stochastic or dctcrministic envitjwj.e is no difference between thcir use in yosments. We do not have to devclop a new theory of partial differentiation cnvironments. stochastic ja j'sake this clearer, considcr the following example.
now be transformed so that it ends up containing the integral
z n
.
.
'
f f) dh($).
(57) .
The stochastic version of this transformation is very useful in evaluating , 2 Ito integrals. In fact, imagine that is random while (.) is (conditionusing integration by parts. we ally)a detelnninistic function of timc, stochastic intevals as a function of integrals with respect to can exprcss variable. ln stochastic calculus, this important role will be deterministic a playedby Ito's formula. .f(.)
Gobscrved''
'lnhen,
'
.
5 Partlal
vo
5 j sxampje -
Derivatives
Consider a call option, Timc to expiration affects the price (premium) of call in dfferent First, time tlw two expiration date will as ways. passes, the h and the remaining lifc of the option gets shorter. This lowers the approac But at the same time, as time passes, the pricc of the underlying premium. will change. This will also affcct the premium. Hence, the price of a asset callis a function of tw'o variables. It is more appropriate to write
Considcr a ftmction of tw'o variables ys / f)
'
,
.
=
FCS:
,
f
),
+. t2
(61)
,
St is the (random)price of a tinancial asset and / is time. involves simply differentiating F(.) Taking the partial with respect to With respect to OFCSf,t) (62) = PS? Here cut is an abstract incrcment in St and does not imply a similar acttzal change in rcality, In fact, the partial derivative Fs is simply how much thc function F(.) would have changed if we changed the ; by one unit. nc Fs is just a multiplier. Whcre
,
G
,yyy
=
.t
.
.%:
.3.
(58) '
where Ct is the call premium, St is the price of the underlying asset, and f is time :E ' Now suppose we the time variable f and differentiate Fxt /) with rcspect to St. The resulting partial derivative,
.
iiftx''
,
r
oFst,
f)
0'% -
(59)
F,
s.2 Total Digeventias we obsewe a small suppose
,
change in tlic price of a call option at time Let this total change be denoted by thc differential dh. How much of this valiation is due to a changc in thc underlying asset's price? How much of the variation is the result of the expiration date getting nearer as time Passes? Totai differentiation is used to answer such questions. Let f (x;,r) be a function of the two variables. Then the total differential is detined as
'
effect of a change in the price of the unwould represent the (theoretical) derlying asset when time is kept ftxed. nis effect is an abstraction, because 1 in pradice one needs some time to pass before can change. ) The partial derivative with respect to time variable can be dehned simi: larly as , fFst, tj Ft (60) = pf
f
ct
,
.
N O te that even though St is a hmction of time, we are acting as if it does not change.Again, this shows the abstract charader of the partial derivative. As t changes, St will change as well. But in taking partial derivatives, we behaveas if it is a constant. Because of this abstract nature of partial derivatives, this type of differentiationcannot be used diredly in rcpresenting actual changes of asset
dj-
yst
,
/)
ofst,
tj
dt. (63) 0St ln other words, we take the total chfmge in St and multiply it by the partial derivative Js. we take the total change in tne Jf and multiply it by tbe partial derivative j;. The total change in f(.) is thc sum of these two Poducts. According to this, total diffcrcntation is calculated by splitting an obsewed chan ge into different abstract components
'
' ,
=
dSt +.
t?f
.
.
y...
'.. (
' .
g'
68
C H A PT ER
Deterministc
3
*
Stochastic Calcolus
and
69
j partial Derivatives
Tlae convcntion in calculus is that, in gcneral, tcrms t:f order ((;r)2or negligible if is a determiniaic variable,l4 Thus, if hi#her are assumed to be t jaat x is detenninistic, and lct (x xa) be small, then we could we assume Taylor scrics approximation:fjrst-order use the
5.3 Taylfr Sevies Ea'panxsitm
.x
bc an infinitelydifferentiable function of arbitraryvaluc of x; call this x(). Let
fxj
The Taylor scries expansion of
DEHNITION:
deuedas flxt
fxj
=
.'z(.r)(.K
+
1
-i'Tx=(Ab)(,Y
=
f=0
-(f
1
-
fxj
R, and pick an
:
around
irtl
:
R is
fxt
'
1 2
-/.(.b)(,Y
+
a+
.t())
+ 3. x
Ab)
-
(E
.x
*b) 2
-
nis
'
'
(.x
-
i.
Under thcse conditions.
:.
i
I0)
)jx)
becomes an equa lity if the
(64)E
'
E
i (.ra)(Jr
-
,
Jx g)g(x
respect
to x
one js
and the
'
Jf(xl
:
d.f(A)
Ixre,
1lZ
wc surely have
fxx
=
>
I-ri-
>
.
'
'
5.3.1 The equation kcctpritf-tlrer
(x) flxoj +
.Jf(-),the
latter become constaats,
xtxoltx
-
x()) +
1
jxxtxeltx -
,
..
j,l
'
'.
lf so the terms (J.x)5 '
'
.
: '
(J2)
x(j)
w())2.
(73)
point is quitc relevant for the later discussion of stochastic calculus. ln fact in order to prepare the groundwork for Itffs Lemma, we would like to consider a speczc example.
,
oncc xa is plugge,d in
-
'rhis
:7
lhat
LLnj,
j cajud a frst-order Taylor series approximation. often,a better approxis mato,i can be obtained by including tlae second-order term;
;
the result even smaller,) Under these conditions, we may want to drop some of the terms tm tlle ' right-hand side of (64)(we c,an argue that they are negligible. To do this, E We mllst adopt a dConven tion'' for smallness and then eliminate al1 terms Small When that are
5 paxial
vjoure 13 plots the exponential cun'e with the second-order
pP
This fundion begins at t 0 with a value of Bij 100c-rT. Then it increases T, the value of Bt approaches 100. at a constant percentage rate r. As t visuaiized value. Hence, Bt could be as the as of time 1, of 100 to be paid value of at time F. lt is the present a default-free zero-coupon bond that matures at time F, and r is the corresponding continuously compounding =
sl u
i
=
:
We are interested in the Taylor series approximation of Bt with respect to f assuming that r, F remairl constant. A hrst-order Taylor series expansion around / t will be given by
,
.
'
=
(r)l00e-r(F-:)(f
-
j()),
t
(u
'
(75)
((),F),
where the fil'st term
:
/o. The on the right-hand side is #/ cvaluatcd at f ' second term on the right-hand side is the hrst derivative of BI with respect to f, evaluated at /a, times the increment t /(). : Figure 12 displays this approx'lmation. The equation is represented by a F T. The Erst-order Taylor series approxconvcx curve that increases as I a4. imation is shown as a straight line tangent to the cun'e at point Note that as we go away from fll il either directitm, the line becomes a worse approximation of the exponential cuvc. At t near l0, on the other hand, the approximation is qlzite close. =
'
.p < /
-
dBt Bf
j g
j(;, sj.
=
1
-(
1. -( r-/)2(r-o)2 J,
,)(r-rj))+
r-
,
f
s p, wj,r
,
-(T N
-
tjtr '
'
1
- aj + -4F 2
-
*'
/) c
(r
150 .
-
r(,) a ,
90
'
8c
;.
t
e
((),T1, r
'
A
Parabola
s0
..
:
50 40 20
5
1(1
15
''-''
FlGU RE
25
:
30
j
30
:.
2:
A
ztn
. L(
i: O
Time )
l2
5
10
$
IS
FlGURE (
j
70
q,1.
60
s
.
t ''
().
(10()e-rn(F-?))
Rr
BI
c
>
>
0,
This expression provides a second-order Taylor series expansion for the Percentage rate of change in the value of a zero coupon bond as r changcs
'
1O0
(76)
.jyjz,
(r )1.f)()g
-l)j
jj()(je-ro(T
by oividing
--y
c
fa)
0,
5
+
.,jw./*)(j
-
-
.
100c-dF-la)
1. z
(r)1(J()g-r(r-/o)jf
..j.
The r;ghsjaand side of this equation is a parabola that touches the exponent jaj curve at point A. Because of the curvature of the parabola near 10, we expect this cun'e to be nearer the exponcntial function. Taylor Note that the difference between the first-order and second-order hinges on the size of the term (.1 /(J)2. As / nears Sef jes approximations Smaller. More importantly, it becomes smaller faster f this terms becomes f()). t han the term (f - approximations show how the valuation of a discount Taylor series xesc bond changes as thc maturity dase approaches, A second set of Taylor scties approximations can be obtained by expan djyjg s, wjth respect to r, keeping /. T flxed. Consider a second-order approimationaround the ratc rt);
.
yieldto p'tzfzrf/A'.
t
1(j(u-r(r-/o.) +.
--/
B
Taylor series
-rmoximation'.
0,
(74)
-
Derivatives
'
:.
20
25
30
Trilth=
s.'
E
C H A P T ER
*
Deterministic
3
imd
Stochastic Calcultts
6 Ccn
side measures the percentage fatkt of change infinitesimally. The Iight-hand in the bond price as r changcs by r - r(T, Where rlj can be interpreted as the current rate. We see two terms containing r - ra on the right-hand side. In hnancial markets the coefficicnt of thc s1'St term is called the modilied duration. The second tem is positivc and has a coefficient of 1/2(F - f )2. It represents the so-called convexity of the bond. (verall, the second-order Taylor series expansion of Bt with respect to r shows that, as interest rates the value of the bond decreases (increases). increase(decrease), nc of the that the bigger these changes, the smallcr their implics bond vexity'' relative effects,
clusiom
ju an ordinary dterential
equation,
dxt .-dt
.
wherc tjw unksown
s xt, a
axt + b,
=
function.More
'.
.x/
In tbe case of the ODE,
:
CCViCW
thC
dB3
Sf
Of
Concept
dB t
=
dt
-r,s,
Bv, rt
with known
>
changes ' This exprcssion states that Bt is a quantity that varics with /-i.e., in Bt arc a function of t and of Bt. The equation is called an ordinary dfferential equation. Here, the percentage vaiation in Bt is proportional to r: some factor rt times dt UV (78)E t -% dt. ,
'
'
s
Now, we say that the fundion #,, dchned by
Bt
e-
=
..(J'
ro du
(3
.
6
Here
t he solution
,
b
=
,
,
,4
.4
.
xt'
(85)
.,
E
E .
k. .
l ,
conclusions
equivalents. 'Ihe other jmportant concept of the chapter asmallwas the notion of ResS.'' ln Particular we need a ctmvention to decide when an increment is Small enough to be ignored.
(S1)
0,
thc unknown element is a vector. Under appropriate cond it ions thc solunlultiplied by the vector b. -1 y--.i e the inverse of tion would bc a' =
=
chapter reviewed basic notions in calculus. Most of these concepts elementary. While the notions of derivative, integral, and Tayior series may all be well known, it is important to review them for later purposes. Stochastic calculus is an attempt to perform similar operations when the utjderlyjug pjjenomena are continuous-time random processes. It turns out that in such an environment, the usual dehnitions of derivativc, integral, and Taylor scries approximations do not apply. In order to understand stochastic Vtrsions of such concepts, one first has to understand their deterministic
is '''
=
-
an + btds
Wee
(80)
x.
to be determined.
,1,Y
(64)
.
'l'his
'
the unknown is x, a number .x -1/2, ln a matr equation,
sta
(79)..
,
rrhat
=
FD
where t h e un known xt is again a function of r.
,
.:
solves the ODE in (77)in that plugging it into (79)satisfies the equality (77). is, Thus, an ordinary diferential cquation is hrst of all an equation. it is an equality where there cxist one or more unknowns that nced to be ' determined. A very suple analogy may be useful. ln a simple equation, 3.):+ 1
jy
''E
(
t 1
/
C-
1, was
Readers wjjj recognize this as the valuation function for a zero-coupon bond. This example sbows that the pricing functions for hxed income scctzritics can be characterized as solutions of some appropriate differential equations. ln stochastic settings, we will obtain more complex versions of tjlis result. Finally, wc need to defme the integml equation
,!
=
=
(83)
'
(77)
0.
=
E
Calculus
dt,
-r/f(
=
tjw solution, with the condition Sz
,
Standafd
third major notioll
is
.
''
that we would like to from equation differential (ODE). For exrdinay an ample, consider the expression WC
preciscly, it is a ftmction of t !
(f).
=
'icon-
5.4 Ordinur.y Differertial Equation.s
(8a)
:
.
.
..
cHAP
4
'r
ER
Deterministic
3
.
and
calculus stochastic
7 References
4.
(a) Xn
(b) (c) Xn Xn
=
an,
=
(1+ l/njn
=
(-1)''-'1/n!.
n
=
5.
1, 2, 3, where
with s'j t
(-LJ, given abovc,
'
convergent?
(c) xs
(d) xu
=
sin
/lW')
.
converge as N
( g$
series h'
.x?
-->.
7. Suppose x
=
1, 2, 3
.
xu.v
g.
considerthe
functon; /(X)
.
%
=
.j
'
,
()
:
@) Now consider
. ,
Is this sequence bounded?
(; < .x:
V-n )/ V-n
1im -->
(x)
.u
. F
c(r-,)
uu
(5 '
1
.
One question here is the presence of rt. In reality, this and S$ are rando variables, and one may ask if the use of a standard limit concept is vali lgnoring this and applying the limit to the left-hand side of the expressio in (3),sve obtain Sv
Fl.V, F).
=
:
(6
According to this, at cxpiration, the cash price of the underlying asset an the price of the forward contract will be equal. nis is an example of a boundary condition. At the expiration date-i-es pricing functien Fst f) assllm at the boundary for time variable f-the a special value, Sp. The boundary condition is known at time t, althou the value that St will assume at F is unknown.
'
r
,
2.2
tzns
i ':
E .
Determining the pricing fundion F(St f) for nonlincar assets is not easy as iri the case of fomard contracts, This will bc done in later chapte At this point, we only introduce an important property that the F(St, t should satisfy in the case of nonlinear products. This will prepare th gmundwork for further mathematical tools. Suppose Ct is a call option written tm the stock Ss. Let r be the czonst Iisk-free rate. K is thc strike price, and F, t < F, is the expiraticm dat nen thc pricc of the call option can be expressed as s ,
Ct
=
Fvh
,
f
(
),
L'
'
'
.1 ..
The pricing function F(St /) for options will bave a fundamental property Under simplifying conditions, the St will be the only sourcc of randomne ,
5ne
intercst
rate r is constanl and, hence, is droppcd as an argument
of +-.).
:
.
..
.
:
pI ou R .E
i... E
. y
.$.
.
C H A P T ER
*
E
Pricing Derivatives
!
.
83
? pjcing Functions
The hrst panel of Figure 1 displays the price F(St /) of a call option written on St. At this point, we leave aside how the formula for F(St /) is obtained and graphed/ Suppose, originally, the underlying asset's price is 5. nat is, initially we . on the Fst /) cunre. lf the stock price increases by dvb ac at point thc short position will lose cxactly tlle amount dS, But the option position gains However, we see a critical point. According to Figure 1, when St incrcases by ds the price of the call option will increase only by dct; this latter change ls smaller bccause the slope of the curve is less than one, i.e.,
DssjxlrjajoNi Offsetting changes in C( by taking the opposite posiof the underlying asset ls called delta hedging. Such a tjou jn yv unjts . is dclta neutral and the parameter F is called the delta. joljo port
,
,
.,zl
,
'
.
'
.
.
dCt
a.+v-D==zra====foav=*n+M2=2mv4r732x+.
.
mAowxwwu -
v. .
..
:
:
*
:
oo s
E
):
ln
*
*
1 lt
ro a
eOr
: ::
f ;
: :.
:
1 lntroductlon In this chapter, we review some basic notions in probability theoly. Our ftrst purpose here is to prepare the groundwork for a discussion of martingales and martingalc-rclatcd tools, ln doing this, we discuss properties of random variabies and stochastic processes. A reader with a good background in probability theory may want to skip these sections. The second purpose of this chapter is to introduce the binomial process, which plays an important role in derivative asset valuation. Pricing models for derivative assets are formulated in continuous time but will be applied in discrete time intenrals. Pradical methods of asset pricing usin difference methods'' or lattice methods fall within this category. Prices of underlying assets are assumcd to be obsen'ed at time periods separatedby smttl.l Iinite intewals of length In such small intcrvals, it is further assumed,prices can have only a limited numbcr of possible movcments.i Th ese methods a1l rely on the idea that a continuous-time stochastic proCPSS representing the pricc of the underlying asset can be approximated arbitrarily well by a binomial process. This chapter introduces the mechanicsof Justifying such approximations.
. '
p' '
5
' :'
(<small',
)
.
''finite
:
.
,
. !
.
i
.
2 larobabillty
'
o erivative products are contracts written on underlying assets whose prices fluctuate randomly. A mathematical model of randomness is thus needed.
'lotexample, prices can move
$.
'
.
up and down by some preset
.
p ..
f'
.
j
amounts.
:
c H A PT
ER
.
E.
Tools in Probabilit'yTheory
5
:
'
E
Somc clementary models of probability theory are especially well suited to pricingdcrivative assets. This can be a bit surprising, given that many investorj appcar to be notions of probabilities rather than by an axiomatic drivenby and formal probabilistic modcl. However, the discussion in Chapter 2 1n- ; probabilities area if therc are no dicated that no matter what the the fair market value of linancial represent opportunities, one can arbitrage asynthetically.'' Hence, rcgardassetsusing probability measures constnlcted mathematmarkct participants, chances perceived by of subjective any less products. pricing derivative modeis natural in probability have use a ical ) In working with random variables, one first deines a probability space. the notion of chan E That is, onc explicitly lays out the framework where witlnout falling into be dehned rcsulting probability the some incan and
pepending on what is in the report, we can call it either favorable or unfavorable, nis constitutes an example of an evertt. Note that there may rs's that may lead us to call the hawest report It is ye sevcral ts's. collections of that events are sense thjs jyj report.'' ylence, we may want to know the probability of a given by is nis Jyharvest report favorable), (3)
;
''favorable.''
''intuitive''
Ifavorable
''true''
,
'
consistcncies.
=
Finally, notc that in this particular example the f' is the set of al1 possible reports that the USDA may make public.
In general, there is no reason for a probability to be representable by a simple mathematical fonuula. However, some convenient and snple mathematical models are found to be acceptable approximations for representing probabilities associated with Enancial data,z A random variable X is a function, a mapping, defined on the set G. G 3, a random variable will assume a particular numcrical Given an event value, Thus, we have X 2 -->. B, (4)
2
'
.
.
./,1
,
Where
B is the set made of all possible subsets of the real numbers R. In tel'ms of the eumple just discussed, note that a harvest report'' may contain several judgmentalstatements besides some accompanying numbers. Lct X be the value of the numerical estimate provided by the USDA ald let 100 bc some nnimum desirable harvest, Then mappings
,
J7(.,4) k: (),
any
Wfavorable
(1).
CE3,
E
dPA)
=
(2/
1.
=4G
vuruuc
2. : Ra
To deline probability models formally, onc needs a set of basic states of r tlle world, A particular state of the world is denoted by the symbol 'The symbolI represents all possible states of the world. The outcome tf an, cxpcrimentis determincd by the choice of an fs. ne intuitive notion of an event corrcsponds to a set of elementary %.: ne set of a1l possible events is represcnted by the symbol 3. To each event' .,4e one assigns a probability J'(.,4). These probabilities must be consistently defmed. 'Bvo conditions of consistencyarc the followinp -4
93
2 pobability
,
such as
The firgt of these conditions implies that probabilities of events are either, should sum to onc. zeroor positive. The second says that the probabilities L Here, note the notation dPA). This is a measure theoretic notation an dl be read as the incremental probability associated with an event a E may triplet According to this, aE P3 is called a pro bability ;. space. ne pointoa of l is chosen randomly. rtz1lawhere z4. (E t1, represents fhs probability that the chosen point beltmgs to the sct
favorablc rcport
.(fl,
,
azl
.
G@)
tS
Note tat
:
commodity future during a suppose the price of an exchange-traded he report t U S Department of Agrigiven day depends only on a harvcst that public day. will make during Cu lture (USDA) The specihcs of the report written by the USDA are equivalent to a,n .
2ne
:
Pttant
'
.
3l-lere,
u
.
.
'
'
.6
-4
Moments i'moments.''
r
t '.
3.1 First T't,tloMtmkcnt.
'
Thc expected value E (-Y1of a random variable X, with density called the hrst moment. lt is defned by x
A('Y)
fxj,
isr '
d.Y,
z.ly .sjxjj:
-c.c
ne vazonco t y(x) is uaecorresponuingprobabilidensiwfunction,. momear first tiw second arouna moment is the mean. vhc - sl-jl while theE
here a w of
Gnsider the nonsymmekic
J'x
d.r by dF(x$.
density shown in Figure 2. lf the mean is the of gravity and standard deviation is a rough measure of the width of the distribution, then one would necd another parameter to charaderize the skewness of the distribution. Third moments are indeed informadve about such asymmctries. Ceter
,
;If the densty does not exi%t, we replace
.
3.2 I-jzgjeg-tlrtlzx Momeats
2
-
()
Cal
i'center
-
=:
In fact, an hiyaer-oraer moments of mormally distrouted ranaom variables be expressed as functions of J,t and fw2. In other words, given the lirst two moments , hkher-ordcr moments of normally distributed variables do nOt provide any additional information.
Fg.' of gravity'' of the distribution, the a random variab le is'nformation about the way the distribution is sprea ds secoodmoment gives . out. ne squarc root of the second moment is the standard deviation. It is a.: t#' obsenmtions X/'n the mean. ln hnancial.' measure o f thc average deviation of markets, the standard deviation a price change is called the volatility. distributed random variable X, of normally the For example, in a case formula ll-known thc density function is given by thc we ' -:54(.v-Jz)2 1 (8) fxj e 2*t7-2 =
8'
G2 is the second moment around the mean where the variance parameter and the parameter Jz, is the first moment. Figure 1 shows exa mples of normal distributions. Integrals of this formula determine the probabilities associated with various values that the random variable x can assume. Note that fx) depends g on only two parmeters, c- and m.Hcnce, the probabilities auociated with a nonnally distributcd random variable can be inferred if one has the sample estimates of these two moments. A rjojaually distributed random variable X would also have highcrorder moments, For example, the centered third moment of any normally distrjhuted random variable X will be given by
There are different ways one can classify models of distribution funcSome random .,bL. tions. One classification uses the notitm of Others neeo. characterized momentsvariables theirhrst by can be fully m'fz characterization. higher-order moments for a full
=
6
,'
.j
ZEYI
4
17j G u R y) l
r
3
2
-2
,
'E;'
96
c H A f, T E R
.
5
Tools in lrobability Theory
4
Ija other words, a casual obsen'er is more likely to be observations in the case of heavptailed random variables. extremc
tsurprised''
. (
'
dcnsity A Tlonsyrrzmetric (An F(5.3) distribution)
.
4 Conditional Expectations of taking expectations of random variables is the formal hcuristic notion of (Torecastinps' To forecast a random variablc, one utilizcs some information denoted by thc symbol 1t. Expectations calculated using such information are called conditional expectations. expectation ne corresponding mathematical operation is thc utilized be, and in general is, difoperator.''s Since the infonnation could another, conditional expectation the operator is fcrent from one time tt
ne operation equivalcnt of the
. Itormal
by
'
A Neavyaailou l.tlistlibution of fmedom with 5 dchzrrecs
A Ktndfird
97
Expectations conditional
E
densy
KGcondititmal
0 y?I o v Rz
E
z
itsclf indexed by the time index. In general, thc information used by dccision makers will increase as timc passes. If we also assume that the decision maker never forgets past data, the information sets must be incrcasing over time!
S :.
ln financial markets, a morc important notion is the phenomenon of hcav. taib'. Figure 2 displays a symmetric densit which has another chazu acteristictbat differentiates it from normal distributions. The tails of this iL' distributionare heaver relatwe ' to the middle part of the tails. Such dcnsities and called heavptaikd are fairly common witb financial data. Again, are would need a parameter other than variance and the mcan to char- i one acterizethe heavy-tailed distributions. Fourth moments arc used for that ) end. i
/.J
.
.
,
gt,
y.,
g
.
.
.
,
tjij
i = 0, 1, are times when the information set becomes available. ln the mathematical analysis, such information sets arc called an increasof slkma hel. When such infonuation sets become available 111:Seqlence continuously, a different term is used, and thc family It satisfying (9) s Where
.
!,
.
calleda hltration.
' i
The conditional stcps.
J.2. l Nef;py Tails What is the mcaning of heavy tails? A distribution that has heavier tails than the normal cuwe means a point should be carehigherprobability of exlreme obscrvations. alsoBut thistails the normal density that extend to plus madc, has that Note fully and minus inhnities. Thus, a normally distlibuted random variablc could also assume extreme values from time to time. However, in the case of a E heavy-tailed didribut-ion, thcsc extreme observations bave, relatively speak- E ing, a bigllcr fretFenc. nGrmal But thcfc is more to hcau-tailed distributifms than that, In a around distributim, tbsenrations Of would naturally be occtirring the most t he the occurrence t7f extremcs is gradual, Z ccnter. More irnprtantly, Obstrrvations that to large, and then to extreme from frdinaly the PaGage in OCCUI'S a gradtlal fashion. In case of a heavy-tailed distribution. on the hCr Ot to extrcme observations is mofe hand, the passage from Suddcn. The middlc tail region of the distribution contains relatively less i . weight than in the normal density. Ckmpared to the normai density, one s likely to get many extreme obselwations.'' ,
:i
.
,
expedation
can then be defined in scveral
opcrator
#-1 CtmditzontzlProbeility
,
'
srst, the probability density functions need to be discussed furthcr, Is is a random variable with density fttnction /(x). and if a;tl is one possible value of this random variable, then for small dx, we have x
p
'
jx -
wjjj s
dx 2
2
jjxoj dy.
jjt)l
'I'IIiS
is the probability that the x will fall in a small neighborhood of ;r1). neighborhood is charactered by the #x. These quantities are shown in Figure 3. Note that although fxj is a Rtmlinear curve in tlzis ligure, for small dx it can be approximated rcaSonably well by a straight line. Then, the rectangle in Figurc 3 would be
'
'l'he
Gtdistancel'
ttordinary''
.
5./%n
'
:
operator is a function that maps funuions into funclions. That is, it akes as irlput a as output another function.
anu produces fuucion
. 0, F(/) will be given by
:'
is either an uptick, and Plices increase according to ntj
r :
.,
=
+IJ'v''XI #,
Pit)
=
-aX-j
ne time index f starts from k
f
=
fo, fp +
k.
=
(1
and increases by multiples of -
.
.
,
/() +
nL,
.
-
,
.
(20)
stochastic
process s a sequence
of random variables indexed by time.
t
F(f
' .'.'
('small''
: J ;
:
..q .
psee,
.' '.
: j
)
=
F(fo)
dFsj.
+ Io
nat is, beginning from an initial Jprce F(fn), we obtain the price at time f by simply adding a1l subsequent inhnitesimal changes. Clearly, in continuOuS time, therc are an uncountable numbcr of such inhnitesimal changeshence the use of integral notation. Also, at the lintit. the notation for iucremental changes F(f) is replaccd by #F(l), which represents iafiltitesimal changes. Finally, consider thc following question. At the limit, the inlinitesimal Changes dFt) are still vcry erratic. Would the trajectoriey of Ftj be of btwnded variation'?l The qucstion is important, becausc otherwise, the Riemann-stieltjes way of constructing integrals calmot be exploited and a TW dehnition of integral would be needed.
L'. ,.
E
,:
At each time pojnt a new F(f) is observed, Each increment F(f) will equal A-avIf the LFt) are independent of eacb other, the or ,-awCS. either tf incremcnts rFlfl will be called a binomia I stochastic pmcel-, sequence or simply a binomial rmccsat' that a eltemember
E
(19)
p).
-
.
l
!:
(18)
=
.
If/x-I
..
ZF(f) represents the change in the observed price during the time interval A11other outcomes that may very well occur in reality arc assumed for the time being to have negligible probability. Then for fixed f, /, the AF(/) becomcs a binomial random variable. 1n particular, Ll't ) can assume only two possible values with the probabilities
,
,
l
(17) Wsmall''
#(AF(f)
markets.
5.2 Limtfng Propc'rties
'
Considcr a trader who follows the price of an exchange-traded derivative asset F(f) in real time, using a service such as Reuters, Telerate, oT Bloomberg. 'rhe price F(f) changes continuously over time, but the trader is assumed to have limited scope of attention and checks the market price every A seconds. We assume that A is a small time interval. More importantly. we assume that at any time l there a<e two possibitities:
adual
turnover, therc even very oa a giventime periods where #(/) change. Or, in does not some special arc many change by However, it than downtick. may mtre an up or c jrcumstagces, will with complications be dealt later. the being, For time we considcr sucja sjmpier case of binomial processes. oe
'
Disfrirlztio'a in Finfmcnl Murkets
Models
these assumptions are somewhat articial for xotctbat trading day, in markets with high
:
Tilis section disctzsses some important models ior random mriables.These models are useful not only in thcory, but also in practical applications of asset pricing. tn this section we also extend the notion of a random variable to a random process. 5.1 Binemll
someImportant
:. '..
claapterg.
!
C H A P T ER
102
.
5
Tools in Probahility Theory
.
:f
5
Another important point is the following; the integral in (21)is taken with respect to a random process, and not with respect to a deterministic variable as is the case in standard calculus. Clearly. tMs intdgral is itse a random variable. Can such an integral be successfully dehned? Can we lzse the Riemann-stieltjes procedure of approximations by appropriate redangles to construct am integral with respect to a random process? questions Iead to the 1to integral and will be an-ered in Chapter 9,
Important somlt
Models
1
.
o'ie'' a 0'8
'.
0.7
'rhese
o.6
:
Nuauce
0.5
.
o.4
5.3 Merrmts
var(A#(l))
=
pa
proponional toA2 variance
o.2 01 .
c
'
(1
s
0.3
One question that we answer here concerns the moments of a binomial : . proccss. variance of fXF(1) value the and are si Lct t be flxed. nen the expected defmed as follows: plavsj pj-axj, FIAF(r)! + (22) =
9.2
-
XS)l+ (1
-
pl-axjl
-
gFtlFtfjjjz
(z3j'
y?l(; g
'y:)
(24)
e jourth-order
moment
'
wsa
itgelf.
(+(z,Zi)4 asl
(%)
=
=
>
obseaations imply that for small intervals higher-order moments of a binomial random variable that assumes values proportional to vq' Ol'l be ignored.
-t)
d
,',
,
.
,&
5.# The No
j
Distdbafzon
Now consider the following experiment with the random variable F(I) djscussed in the pyevious section. We ask the computer to calculate many reajjzauons of Then, begnrjirig from the same initial point F(0), we yt ). pjot these trajectories. Reginnillg from /: (), in the immediate futurc, F(/) has only two possibje vajues:
,
,
r
=
.
.5
=
E =
4
nese
,r
had ipstead Euctuaed behvecn, sayy +z; and -4A, 2 For sma 117, the value of A2', wouldbe much smaller. When -->. 0, the variance would go to zero sig-. nihcantly faster. Under such conditions, it can be maintained without at itself is not. contradiction that the variance cf x F(/) is negligible, whilc Heuristically speaking, a random variable with a variance proportional. gj , to 12 wgl be approximately constant Jn infmitesimal time xntervazs, i between the difference ance 4 illustrates propo rticmal to Fkurc a var 1. negligible foz' latter becomes Thc proportional and lne) (the 45@ to onc small uj. This last point is also relevant for higher-order moments of the binom valu for simplicity. 'Fhen the expectcd process. Again assume that p be will third moment the is zero and Svenby Wit.h #
jk:
1
() tjje jouujsoykjer moment will become negligible. It is prothat goes to zero faster than the timc inten'al to a power of
...s
portional
.,
.5.
o.8
is obtained as
7(F(/)14
expccted value will equal 0 while the variance is given by alt. t. It is important to rcalize that the variance of the binomial process iigi As A approaches zero, a variance that is proportional: proportional to think';;C will to ,'X go toward zero with the same speed. nis means that if we quantity, then the variance will also V. small but ttonnegligible of
and the
would be proportional to
0.6
a
.:j
=
the variance
0.4
F
.
lf wre have a 50-50 chance of an uptick at any time t, then 1 #
as a nonnegligible. Jn contrast, if
preponiosal toa
p-avzl' = fk.F'(z)J3
the third moment
+
(1 -
pl-at-s.
; (2.5),
F(0 +
''
blence pt)
',y
equals zcro.
'
,
j:. L
jfxey
)
F(0) + ax:-
with probability p
n
with probability 1
=
F(0)
-
a
s unomialat t
-
() +
a.
- p.
(gy)
T r'
.
)04
c H A P T ER
.
5
Tools in Probability
:
'l-heory
5
But if wc Ict some more time pass, and then look at F(r) at, say, f 2, F(f) will assumc one of three possible values. More precisely, we have the , following possibilitics'.
,&
:
F(2A)
a'Vs+ axS
F(0) - aF(0) - av-
=
with probability pl
+ a-
.->
with probability 2/7(1 - p) with probability (1 #)2.
av-
(28) E
.
)
-
.
That is, F(2A) may equal F(0) + 2aXS,F(0) - za-, or F(0), Of these, tlielast outcome is most likely if there is a 50-50 chance of an uptick. E Now consider possible values of F(/) oncc some more time elapses. Sew eralmore combinations of upticks and downticks become possible. For ex-' ample,by the time t 5, one possible but extreme outcome may be
IEconvergence
convelyence.
=
F(5)
-
Another
F(0) + avF(()) +
+ a-
+ a-
+ awq + a'-i
sa-.
(29) . (3t))(.
=
F(0) - a4-
More likely arc combinations F5h)
=
80)
-
-
a-
-
av-
-
a-x
-
av-.
of upticks and downticks. For eumple, a-
+ a
- a,
+ axs
According to the central limit theorema the distribution of Fnhj proaches the normal distribution as n, --> x. and that Assume that p
ap-
F(0)
(34)
.5
=
)
in a row: extreme may be to get five aownticks F(5a)
105
---> x, the time period under consideration in wsch was constant and n indehnitely, sed and looked at a limiting F(f) projected toward a we incrca sdjstant'' future. one question is what happens to the distribution of the random variable remains flxed'? A somewhat different question is #(a) as gy x and of F(aA) as --.> 0 while ni is fixed.g the disttibution to what bappens remembcr Ftt was binomial, but a little tarther otigin tlzat the at Now, number of possible the outcomes grew and it became multinomial. away probability distribution also changed accordingly. How does the fonn ne of tbe distribution change as n --> x? W'hat would it look like at the limit? of random Questions such as these fall in the domain of variables.'' Therc arc two different ways one can investigate this issue. ne jirst approach is that of thc central limit theorem. The second is called uzdt?/
=
F(0) +
somelmporranr Models
+ as/-
E
and Then, for fixed arge n, t hc distribution of FnL) can be approximatcd by a normal distribution with mean 0 and variance alnh. The apProximating density function will be givcn by
(31)
d(l
It
2:
c
?$
(32).
g(F(n,)
''
=
x)
1
=
l'rralni
::E
or F(5a)
=
F4()) - axi- + a-
+ av-h
-
a-
+ a-
are two different scqucnces of price changes, each resulting in the mqme. E Price at timc t 5Z. rlnhere are sevcral other possibilities. In fact, we can considcr the generalt fmkez case and tl'.yto sfldthe total number of possible values Fnt) can nume! Obviously, as n may take any of a possibly inEnite x, Fnt) of values. A similar conclusion can be reached if 0 and n --> t:xz while the product tn remains constant. ln this case, we are considering a time interval and subdividing it into Ener and finer part itions.8 For the case =
?
.-->
:.
->
gucre too
(
fact, this latter type tlf convcrgcnce is ef inlerest lo us. nese types of experimenl wfll in the dumain of weai convergence and give us an apl3rox imatt distribution for a '. Of random variablcs obsen'ed during an ntewul. sequence . 81n
fall
:
E
e-
za
xa
(x)c .
(35)
ne corresponding distribution hanction does not have a closed-form formula. It can only bc representcd as an integral. The convergence in distribution is illustrated in Figure 5. It is important for practical asset pricing to realize the meaning of this convergence in distribution. We observe a sequence of random variables indexed by n.10 As n increases, thc distribution function of the nth random variable starts to resemble thc normal distribution.ll It is the notion of wc;k convelgence that describes the way distributions of whole sequenccs of random variables converge.
!
(33)
' '
0.
-
5
lar.r.jj
.
:
n
-..
x.
at is, we have a stochastic process.
'Again we emphasize tbe whole tFt()j,
.'.
,
'y. .:
svquence
. .: :'.
. .
.. .
that we arc dcaling with the distribution of F(z1), Ffnhj Fqhj, c(:zl), .j.
.
.
.
.
.
and not with
'
r
h S--i-F,R
cu
lz .s
Toou
5
.
lvobabiliw
in
'riwory.
j
inaplies that the Claussian rn odcl is useful Vzhen nexv infornnatio n tvjving during inhnitesimal periods is itself infinitesmal. As illustrated for a .-->. 0, the values assumed by Ft) tje binomial apprtximation, with jj bccom e smaller and sma jj er an tjje variance of new information givcn by
;
at n= 1
Distributiop
!
:
.
,a>'h-
vartastfl)
aa/-
0
zs
...
a
=
..-
-
-
'
-
--
-
-
..
..-
-
-
-
-
-
--
-
.-
-
..-
-
--
-
-
-
-
-
-
-
.-
-
-
-
-
-
--
-
-
-
-
a1h
-
(36)
to Zefo. That is, in inhnitcsimal intenrals, the F(/) cannot Jump. cjyayjgos are incremcntal, and at the limit they converge to zert. Continuous-time vcnions of the normal distribution arc vcrjr useftzl in assct pricing. Under some conditions, however, they may not be sufficient to approvimate trajedoics Of asset Prices obsen'ed in some financial markets, we may need a modcl for prices that show as well, Thcrc were during the October 1987 crash of stock many cxamples of such markets around thc world. How can we represent such phcnomena? The Poisson distributitm is thc second building block. A Poissondistributed random process consists of jumps at unpredictable occurtence The jump times arc assumed to be independent of 1, 2, tnes (, one anothcr, and each jump is assumed to be of the same size.12 Further, durhlg a small time inten'al the probability of obscrking more than one jump ls negllglble. Ll'he total number ol Jt1mP9 Observed up to time f is called a Poiason counting process and is denoted by N,. Fb a Poisson proceF, thc probability of a jump during a small inteal wll be approximated by g oes
is a positive constant called the intensity. Notc the contrast with normal distribution. For a normally distributed variable, the probability of obtaining a valuc cxactly equal to zero is nil Yet with the Poisson distribution, if is this probability is
7
%
j----..-.
-
1)
where
.,
----
=
.
s,
,-
Phhj
7,
k
f builde '
0)
=
1
-
A.
(38)
Yence during a small interval there is a p robability that nt jump Will occur. Thus, the trajectoly of a Poisson proccss will consist of a cont.u uous path broken by occasional Jumps, Ihigh's
,
: In dcaling with continuous-time stocbast ic pr ocesses, we need 1wta ing blocks. One is the continuous- timc equivalcnt of the normal distribtltio flm. known as Brown ian motion or, cquivalently. as the Wiencr process. M Of these processes trajectorie: indicates. vious disctlsgion in the pre section continuous. be likely to are
,
.
u BO1h lu :
C: E
..
of tlwse assumptions can be altered. need to l)e independent.
jump times
However, to keep the Poisson characteristc,
:
108
c HA PT ER
probabilty tlmt during a finite intewal by
ne givcn
PiNt
n)
=
v
(39)
,
Fl!
+
=
,
,
w hich
is tbe correspondirig distlibution.
6 Markov Processes and Thelr Relevance
'
-
'''-'
.
The discussion thus far has dealt basically with random variables. Yeta this although, it does constitute a too simple to be useful in nnance. block for more complex models. ln finance, what we really need' are': a model of a sequence o random variables, and often those that
6 1
E
tic proccsses. distribution function, F(x1
,
,
.
'rhc
.
,
xr)
'.
=
@
Probxj
.S
.Yj,
.
.
Xt S x/),
,
.
as / --+ x. In case the indcx f is continuous, ontt is dealing with uncountab many random val-iables and clearly the joint distribution fundion of such procesg should be carefully as will be illustratcd lor Wiene
r J.+
! :,
Process.
'
'? .
gectiony
We diSCu55
:
.l,
.
.
.
.
,
.
.
i
.
,
I J;j
=
o-(1t, t ) = ajrt,
,
(41)
,
grt
,
flk
(42)
-->
dr /
.
14
S
uws.x.-
'
:
.
...$.
' .
=
jxtrt,
j)#/
+ ato,
t l#jj?;.
(44)
We are not talking about the dependence of means or varances of zwg only. ne morc t past shoultl no! innltence statemznls concerttirlg thc whole probabilislic behavior or a -*%'''Y Pttss.
djstan .
:'
E
(43)
steps will bc discussed in more dctail when we develtp the notion Of stochastic differential equations in Chapter 11. There, letting A 0, we u Outain a standard stochastic differential equation for r/ and write it as
=
13Itis quite important that the prtxetks one is modcling in Enance is a Markov process. ) valid oa# for such procG Feyuman-lac theorem tllat we will see in latcr chapters wll be -' not Markmr,Yet it can be shown that shorl-lerm interest tat processcs are, in seneral, fOr short ratc procelsesimposes limitatons orl tbe numerical methods that can be applie d
t).
'rhese
-r:)
-
fljj;
,
'
-
,
a'lid
'
,
+ slst
.
E ((G+s - rt)
with joint X:, DEFINITION: A discrete time process, (XI, j'. xtj, said be F(x: is distribution hmction, to a Markov probability process if the implied conditional probabilities satisfy ;r/5), P-vs PX:+s :!! xt-vs Ix:, S xt-vs l (40 :. '' irdbTwhere 0 < s and P' I 1t) is the probab ility conditional on the mation 9et It i .
j Itj
tjaqyuyj; js somc unpredictable random variable with variance Tlwn, tlie clt , t) ;& wgl be the standard deviation of interest rate incremcnts. ne lirst term on thc right-hand side will represent expected chnnge in interest rate mtwements, and the second term will rcpresent the part tlat is unprcdictablc given It. It turns out that if rt is a Markov process, and if It contains only the current and past values of rt, then the conditional mean and variance will be ftmctions of rt only and we can write:
Z dctail a c1aSS Of stochastic prtcesses thaf lays an important role in derivative asset Pricing; namely, the Marko P uL, :jj : Whicb Vll be in discrete time, wlll try to mtzvaw Processes, Otlf dinlssionj will also clarify some some important aspects of stochastic processes and with continuous time models fo#; notions that will be used later in dealing Catc defivatives.l3 Ztercst In this
rtj jtry-ju -
..s
-
W jwa
j,
'dconstructed''
'or
-re
::
.
Rejevance
SUPPOSe
.
process
.
..
How do these notions help a market practitioner? the Xt represents a variable such as instantancous spot rate r,. assuming that rt is Markov means that the (expected) Then, futu behavi of rt-s depends orlly on tlae Iatest obsewation nnd tilat a condition such as (40)will be valid. We can then proceed as follows. we split changes in interest ratcs into expectcd and unexpected compoyjej.j ts:
is concept building f is continuous time. ovcr observed of random variables (.-,J indexed by an index t, where t ik sequences 0, 1, or continuous, t iE y, x), are called stoch as:': eitherdiscrete,A f stochastic is assumed to have a well-defned Joint ,
.
-,
@
.
.
:
E .
=
109
The assumption of Markovness has more than Just theoretical rele1, 2 vancc in asset pricing. In heuristic terms, and in discretc time t (A-r), is a sequence of random variables such that Markov process, a knowlcdge of its past is totally irreievant for any statement concerning the 3L 0 < % giVell the last obsen'ed value, xt. ln other words, any probabiljty sutement about some future Xt-s, 0 < s, will depend only on the latest obsenratifm xt and on nothing obsen'ed earlier.l4
,
'
'
-
and Their Relevance tj Markov Processes
:
, is there will be n Jumps
.&
a( aln
c=
'
Tools in Probability Theory
5
*
'
4.
C HA PT E R
1 J0
5
.
'(
Tools in Probability Theo
y
5
x
But, if interest rates were not Markov, thcse steps cannot be followe since the conditional mean and variance of the spot rate could potential dcpend on observations other than (he immcdiate past. ls Hence, thc assumption of Markovness appears to be quite rckvant pricing derivatives, at lcast in case of interest ate derivatives.
.
-
'.
rp.j.
qL
TRe Vector Ce
-
for the Rt substitute
; rrhere is another relcvant issue coneerning multivariate Markov pr alld u eesses. We prcfer tta discuss it again in discretc time, f, t + variables. motivating intcrest rates as t'mr Below we will show that, although fw'o processes can be jointl. Mark . when we model one of these processes in a univariate setting, it will, t gcneral, cease to be a Marktw process. ftxed-income. t)f discussed in this can best be ( rrhe relcvance elsewherc) a central concept is the yicld clfzw. The so-called classical a proach, attempts to model yicld cun'e using a single interest rate pr : such as the rt discussed above, On the other hand, the morc recent Hea (HJM) approach, consistent with Blackcholes philosop , Jarrow-Merton models it using k separatc fonvard rates. wltich are assumed to be Mar ' . jointy. But as we will sce below, the univariate dynamics of one elem of a k-duensional Markov process will, in gcneral, not be Markov. Hen Markovness can be maintained ill HJM methtadology, but may fail Z; short-rate based approach. Suppose we have a bivariate process, Irl,S21, whcre the r; represents rate. Suppose also that jointlyt.h rate and the Rt is the are Markov: '
,
.
.
Reevance
'
rt-n
rtn
+ pzpt-x + pb gtnrp-
alrt +
=
=
pj azrt-h
ajr; +
m
R r4-,:
a 2r / +
pzR(
lfl-h
g.c
/2 :- a
.y
,
R tn
,
,
V+A
-
rt
such owjously,
ISAIs o if interest ratcs are not Markov, a very imporlant correslxmdence l'etwee,ll a equations (PDE) and a class of conditionai expecmtions cannot te f yoao p artial differenial PDE'S j tls common.y tablishe d uootccarlomcthods cease to bccome equvalcnt to te tile field of interest rate derivatives.
.
JLE .
l6u
.: ..
..
.j
jjsx
.-ha-utstj
.!
r
:
.
.
..
0,
7 Convergence of Random Variables
etr-ltnz-vaa) >
:.
,
.:
.
aad Tlxeir Uses
7 1.2 Eample Let St be an asset price obsezved at equidistant timc points:
*
.
of three diffcrent convergen In pricing Iinancial securities, a milmum used. criteria are is a criterioo utilized to desn The lirst is mean square conveence. stochastic differenti utilized cbaracterizing in the lto integral. The lattcr is equations (SDEs). As a result, mean square convergcmce plays a fundrlmo tal role in numerical calculations involving SDES.
ftl 0, the appro complicated random variable with a simpler model. imation improved. ln this section, we provide a more systcmatic treatmenh of these issues, Again. the discussion here should be considered a bzief an heuristic introduction. j7 .2.!: itf 7.1 Types Cwcrgence
Xn converges
(5
St #,2.
(57)
/(,
But there is fkmdamental a
will have a smaller and smaller variance as n goes to infinity. Note that for fnite n, the variance of eu may be small, but not necess In doing numcrical calculatio . zero. Tltis has an important implication. imation errors into account explid one may have to take such approx Onc way of doing this is to use the standard deviation of e,, as an estimat '
difference. The
A
sum now involves ranOm processes, Hcnce, in taking limit of a a new type of convergence (56), zriterion
.
should be used.
not applicable.
,1
.
.'
Which (random)convergence
'!l (. ;k. '
..
: '
.
.
ne
L
standard dennition of limit from calculus is criterion
should be used?
.
.
;'
.'
..
'
(2 H A P T E R
114
*
5
Tools in Probabilit'y Theotj
)
It turns out tbat if Sf is a Wiener process, thcn Xn will not converg almost surelyjl? but a mean square limit will exist. Hence, the type of a proximation one uses will make a difference. This important point is take
up during the discussion of the Ito integral in later chapters.
''
r
? '
.(
'rhe notion of m.s. convergence is used to hnd approximations to vl assumedby random variables. As some parameter n goes to infmity, valu assumedby some random variable Xn can be approximated by values variable X. somelimiting random In t he case of weak conveqence (the third kind of convergence), what beingapproximated is not the value of a random variable Xn, but the pr p abilityassociated with a sequence A%, , Xn. Weak convergence is used 1 'matingthe diatribution znc/fon of families of random variables.
q3f
11 5
ylcnce, in dening an lto integral, values assumed by a random variable of fulldamental interest, and mcan square convergence needs to be aj.e uscd. At other times, such specisc values may not be rclevant. Instead, onc conccrned only with opectationa-i.e., some sort of average--of may be
'
7.2 Weuk Cwergence
convergenceRandom Variables
Vantlorl
HOCCD
.v)
(, j;
'
For exarlzple, y( y1 w) may deuote the random price of a derivativc product at expiration time F. ne derivative is written on the underlying asset SI.. We know that if there are no arbitrage oppolunities, then therc exists a d'r isk-neutral'' probability # such that under some simplifying assumptions, the value of the derivative at time t is given by x
,
,
,
'
,
-
,
.
apprtm
=
E p (.(Ak)) n
--->
E
J'
uy
$
(5
lim Pn fl u--,cxk where P is the probability distribution of X if
r
E
.
:.
(5
,
cw
.
.
'
(/'()J
Wus, ingtead of bcing concerned with the exact future value of Sv, wc need to calculate thc emectation of some function F(.) of Sv. Using the concept of uzetz/cconveqence, an approxmation S. of can be utilized. nis may be dcsirable if S). is more convenient to work with than tlle actual random variable Si For example, may be a continuous-time random process, random wbccas Sv may be a sequence deqned over small intervals that depend on some pararneter a. If thc work is done on computers, Sn will z be easier to work with than Sv. nis idea was utilized earlier in obtaming a binomial approximation to a continuous normally distributed process.
'
DEFINITION: Let Xn be a random variable indexed by n with probability distribution Pn. We say that Xg converges to X Breakly and
r,
t
,
: 7
.:;
where f') is any bounded, continuous, real-valued hmdioni EPn t.'l..Yklj is the expectation of a function of Xn under proba@ P is the expectation of a ftmction of X bility distribution #s; E under probability distribution P.
7.J,2 An Example
coasjdera time interval g0,11 and let t c g0,11 represent supposewe are givcn n observations ei, i 1, 2, jyom the uniform distribution 1./(0, 1 ).19 udepeutjently
M(AR)1
According to this dehnition, a random variable Xn converges to X >4 if functions of the tw'o random variables have cpectations that are cl values that are v enoug h 'Thus X n and X do not necessarily assume probabilities cx). arbitrarily close governed by as n close, yet they are .
,
Iar time.q'
=
Next define the random variables Xiltl, i
lpfzk Conveence often interested in values assumcd by a random variable as so We are Paramctcr n gOCS to infinity, F0r example, to defme an ltf) integral, a
YKs FZZUOIX Utjm VZSZFIC Wlt 0. Si&PIC SCIICOFC is fst COIRXXXCU. able wili depend on some parameter zl. In the sccond step, one shows the Ito integrai in the as n - x this simple variable converges to
x (,)
,
z'
.,
YWe
displays occas
ionaljumm.
Rmesents :
hxs
'! '. :. :
to this, Xit)
valueassumed by ei.
:.
SCnSC.
vt
accOr du g
j
,
17The same rvsult applies if, n addilion,
1,
.
.
,
,
.
.
a particun drawn ,
n by
,
...+
72.1 Relevance of
=
.
;. ....
for
. N
(
. '. .' '*'
';z
mayj for examplc, the present. meaus
I
if e; :G f
()
otjjejwjse
.
'
is either 0 or 1, depending on the t and on the
!e( the exyiratitm
2 S t :E 1.
time of some dcrivative ccmtract br 1, while 0
hat probtej
*'.8
(6z)
s
1)
=
j,
jfijl
f2 H A P T E R
116
0
: l
:
:
: 1 l
l
: '
: : ;
,
07
C5' S..q
W
.('
..) ' ?
j
.,'.
l ylou Usillg #/
(f), i
=
1y
.
.
.
p
&, WC
Snt)
=
dcllne the random
1 y-
*
n
n f-l
(-V(l)-
Variablc
.$)
il
(1) 7
..'
..
'
t (63.
f).
example
In the remainder of this book, we do not require lmy further results on pyohability than what is reviewed here. However, a fmancial market paticstudent will always benelit from a good understanding of yant or a Nnance Stocha8tic Of WOCCSSeS. zn excellent introduction ig Ross (1993). the theoy IJIVSCF all tj sjusayev(1977)js an cxccllcnt advanced ltroductitm. cnlar another (19Jg) is source for the intermediate level. 'T'he book by Brzezis a good source for introductory stochastic nuk and Zastawniak (1999) also See the new book by Ross (1999), Processes.
! '
RE
proccss. Tls
9 Ref erences
;
2
I
discussed an impoztant binonal
used to introduce the important notion of convergence of stochaswas exeetmple discussed here also happens to have tic PEOCC sses, The binomial since it is very similar to the binomial tree-models ractical implications, P used in pricing fmancial assets. rotltksejy
'
:
83
second,we
:
s l
117
jtl Exercises
.: :
:
k :
: l0.2
P1 L
Tcols in Probabilicy 'Tleory/
:
:
'n
5
.
.!.
7. Notc that Snt) is a rfec r Figure 6 displays this construdion for n wfaecontinuoua functionwith jumps at %. and the As n x, the jump points become more frequcnt morc pronounced. Thc sizes of the jumps,however, w 111diminK of normally distribut At the limit n x, Sntj will be very close to a will be continuous variable for each t. I'nterestingly, the process rall 7.0 equal to zero. ! the limit, the initial and thc endpoints being identically Clearly, what happens here is that as n --> x, the Snt ) starts to beha normally distributed process. For large n. we m more and more like a with than find a limiting Gaussian process more convenient to work example, as n increases, lt should also be emphasized that in this applications whe ln num ber of points at which Snlt) changes will incrcase. analyss, t tinuous-time we go from small discrete intervals towar d con would often be the case, =
,'
10 Exerclses
''oscillatio
-->
vbtt
1. You are givcn two discrete random variablcs X, F that assume thc Possible values (), 1 according to thc followinglbnf distribution:
'
--y
dom
;
#(Y
1)
=
P F
=
0)
'
astz
plxc
1)
.c
P(=0)
.4
.15
.25
'
(a) What are the marginal distributions of X and F? (b) Are X and J'v independent? (C) Cakulate SIA'Iand FEFJ. Calculatc 11. (d) the conditional distribution 'I-YIF the F (e) obtain conditional expectation FLA-I 11 and the conditional variance FlrlAAlF 11.
;
',
8 Conclusions
=
This chapter brieqy revicwed some basic concepts of probability theo. of pro We spent a minimal amount of time ori t he standard definitions i pojnts, important rnade number of However, a we ability, variables an First, we characterized normally distributed random Nocksbuilding Poisson processes as tw'o basic
=
=
.
Ksuch
bridge. a process is called a Brownian
2. We let the random variable Xn be binomial a process,
.
xn
. . ;
m
i: . '.
: .
J..
!
.
,
a
Bi
=
2=l
,
.
C H A PT E R
118
5
*
TYIS
j'
Probability Theo
in
%*47-
:
W herc each B; is indepcndent and is distributed according to 1 with probability p Vi 0 witll probability 1 p. 0, 1, 2, 4 and pl Calculate the probabilities #(-Y4 > kj for k the distr ouuonsncuom b) Calculate the expected value and the variance of x s or n =
#(Z
.-
-.
2
.a.
3. We say that Z is exponentially distribute d w ith paramcter the distribution function of Z is given by:
1
.
.
=
t
'V
MM
.
5
-
ta'
Lv.
)
*M
i
Pr(Z).
5',1 .E'Eq%+ -
('
where
:
a js a
<usmalln
=
0,
zterval, is well dehned, Yet, writing
ELQS;4 g ::::u
:
'
'
'
:
is irtformal since ds is only a symbolic expression, as we will see in tl'le delirtition o'f the Ito
.
E
Jntcgral.
.. :
t
',1(.
E.
1l 9 ';
:
' ..
:
.
C H A PTE R
12O
@
.. '
Martingaltts and Martingale Representatio
6
2 Defnltions
J.2 C'orztxxtklu-rllme Murfugtlcs
; E.
gsing diferent inftrmation sets, one can conceivably genezate different Of a Process (&).nese forecasts are expressed using conditiomal expcctdtit-ms. ln particular,
Martingale theory classilies observcd time series according to the way th . mariingale if its trajccteries tren d A stochastic process behaves like a A periodicities. process that, on the average Play no discernible trends or supcrmartingale repesents pro submartingale. The term increases is calied a desnitions o formal gives section This decline. cesscsthat, on thc average, notation. ; these concepts. First some
Ttforecasts''
'
''
.
E J jysj =E
!
Is
lt ; Iv
.
-
.
f
'
j y js known, given 1t. 2 Unconditional
?
(fforecasts''
(1:
jltn t
(E
(0,F))
( 0,
'
m
-
0 is zzero ju otjjer words, the directiolls of the future movemcrlts in martin' al es are mpossible to forecast, This is the fundmnental characteristic of -
'
7 .; .! ( '.I
.
k
.r.
Et
j.s ,ua j'v'p Luu
-
Z-jg.j are assumcd to be taken with respect
-adaptedj.
:
'
infonnati problejx a! hand, thc L will represent different types of obtain from infonnatson onc, can the will lye to represent ne most natural use of Jr markets time up to reaiized prices in fnanciai
,
Bllt f J. : j s a forecast of a zandom variable whose value is alrcady vealed', gsace Hence it equals If slt) is by dennition 1t a martingale .Et g,jv j would also cqual This gives t+u
,
lpeylcnljjj)g on
'k.j
x
p
,
(5)
rding to this defmition, martingales are random variables whosc Yttkre vahtion. are completely unpredidable given the current information Set. For examplc, supposc St is a martingale and consider the forecast of the change in t over an interval of length u > 0)
.
Wnew's
w,
nat is, the best forecast of unobsenred futurc values is the last obsewation on St.
.
I-CWCSCIX Vzfiotls
.c:
vith probability 1.
(
T
jor aIj t
,
'!
.
('
. '
. '.
. .
a'
C H A P T ER
Martingales and Martingale Representation
6
.
F'
'
j
.
'f'hu
t.jse
Martingales
of
in
Asset Pricing
123
L
thc trajectories of a process displ processes that bebave as martingales. lf atrcndsx'' thcn the process is not ak clearly recognizable long- or short-run ' martingale. 2 i Before closing this section, we reemphasize a very important property o .i' the dehnition of martingales. A martingale is always dehned with respect t probability If with measure, respect to some w some information set, and . change the information content and/or the probabilities associated with tlz l martingale. Process, the process under consideration may cease to be a rrhe opposite is also true. Oiven a process Xt that does not behave probability measure a martingale, we may be able to modify the relevant i martingale. and convert Xt into a :
Ij asset prices are more likely to be sub- or supermartingales, then why martingales? an interest in lt mrns out that although most linancial assets arc not martingales, one them into martingalcs. For example, one can Iind a probability yy cz convert 73 such that bond or stock prices discountcd by thc risk-free sstrjyutjoxmartingales. If this is done, cqualitics such as ra te become Sl7Cjj
'.
f tjl
'
y
Et
(
,.
According to the dehnition above, a process St is a martingale if s ftl movementsare completely unpredictable given a family of information sct Now z we know that stock prices or bond prices are not completely unpre The price of a discount bond is expeded to increase over time. dictable, the are expected to increase . generai, same is trteBt for stock prices. They of ice represents if Hence, t e a discount bond matu li pr theaverage. < time T t F, '
':
(8
(#sJ
Et
'
(14
0.
Because of this, martingales havc a f'undnmental rolc to play in practic assct pricing. But this is not the only reason why martingales are useful tools. Mai is very rich and provides a fertile environment for discuss' tingaletheory variables im continuous time. In this section, we discuss tbe stochastic : usefulteclmical aspccts of martingale theory. Let -Y, represent an asset price that has tlae martingale property wi fzf).and with resped to the probability >, respedto the rtltration
'T-m'e
17,ouaE
i
Figure 2 digplays an example of a zight continuous martingale. Here. the trajectozy is interrupted with occasional jtzmps/ wlaatmakes the trajectory rkht ctntinuous is the way jumps are modeled. At jump times fa. tz, the martngale is continuous rightwards (butnot Ieftwazds.)
.
,
,
E > x ,+a
(1 .
1zt 1 x r,
((
,
=
,
wherc A > 0 represents a small time interval. What type of trajectori would such an X3 lmve in continuotls time? To answer thig question, tirst dctine the martingale /ercncc xt, .-Y/
=
Xtwh
X,
-
'
Xt
:
2
. .
i, 1 tii
b-.
.
(16.J
,
and then note that since Xt is a martingale,
1
i.
.
.
E
p
(1
(Aa'V1610. =
0.S .
k1 '
earlier, this cqtiality implies that increments of a martv should be totally unpredictable, no matter how small the time fnterval consid ' is. But, since we are working with continuous time, we can indced irregular trajectories. vety small A's. Martingales sbould then display very trends lspection, even duri by discenble display should fact, Xt not any If it did, it would become predictabl infinitesimally small time intervals Such irregular trajectores can occur in two different ways. former leads to condnlm J be contittuous, or they can display jumpz. martingales, whereas thc latter are called rkht continuous rzicr/rlpls/e.. . Figare 1 displays an example of a continuous martingale. Note that o. 0. continuous, in the sense that for As mentioned
o
.
s
-n
.
-1
.
-1.5
rrhey
.
'rhe
'
'
-2
c
.E..
trajectoriesare
PLLX:
>
ej
.-->.
0.
for al1 e
>
0.
(18. '
:.1
o.c
a.a
o.4
c.s
F I G U RE
'',
.-..>
SNote that tjw process still does not have a tzend.
t: .
j ':
().*
2
0.7
a,a
().q
1
rIJ
'.
cHAPT ER
126
Martingales and Martingale Representati
6
.
,.
. .1
')
5 poperties
:
This irregular behavior and thc possibility of incorporating jumps in tll trajectoriesis certainly desirable as a theoretical tool for representing assd : prices,especially givcn the arbitragc theorem. But martingales have significance beyond this. ln fact, suppose one is dealing with a continuous martingale Y, that also has a linite secon
Martingale Tmjectories
vosee this, note
AMl
j (.42
(38) 1
This, according
?
to Eq. (37.),means that unless V gets very large, P- will toward in zero some probabilistic sense, But this is not allowcd bccausc go Xt is a stochastic proccss with a nonzero variancc, and consequently 1,,-2> 0 even for very 5ne partititms of g0,F1. nis implies that we must have P'' x. NOw consider the same property ftr higher-order variations. For examPle, consider 1,,-4and apply the same as in (37)-.
,
: ''
.'
q
-->
.
's
'
9.
(3j)j
(40)
means that 1./-4will tend to zco. The same argument to aIj varjatjons of order meatezthan two.
:
=
j2 4,.,v2
YYS
'
V
:
As long as 72 converges to a well-defined random variable 7 the right-hand sidc of (39)will go to zero. The reason is the same as above. ne x / s a continuous martingale and its increments get smaller as the partition of the inten'al I0,Fj becomes hner. Hence, as fj --> lf-j for all i'.
.'
g
P
As Xt approaches
max IXtj - Xf
3
.
p.x
..
. .
. j :
.
. .E..
: . .
nd doe,s not converge
o infinity.
jx
jxad
ay
cjIOOVIlg
thC
can be applied
SW.YCEECRt lllfgcst OUMCI'VCU
E.
.i
: :'
.J
130
C:H A P T E R
*
6
Martingales
and
:
:
Martingale Representati
Examples
h
:
131
cyartge jndepcndent acr()sg time. lw
if is a small interval, the increments xt 0, E .fd+x1
=
,
,
It turns out t hat the sequence of forecasts, for 0 < x'. EP
(
I61
some probability #,
isy,
E
G 1T.
IT-L
Wforecasts,''
Next consider SUCCZSSWC ma de at diferent tnes,
with respect to
.
of the logic used in the previous
jz' r a e (70) nis is a sum to be received at time F and may be random if c is stochastic, ueresz is assumed to be known. l'ina'ly, consider the ratio cr/sz, wluch is a relauve price. In this ratio, we have a random variable that will be revcaled at flxed time F. As a we get more information on thc underlying asset, St, successive conditional eoedations of this ratio can be calculated until the Gv/B,r is known exactly at time F. Let the successive conditional expectations of this ratiox Calculated using different information sets, be denoted by M t gT EP Mt (JI) j It Bv Where 1? denotes, as usual, the infonuation set available at time /, and P is an apmopoate mobabgity. Accordiug to the previous result, thesc successivc conditional expectations shouz form martingale: a Bv
i
646+1
Applkafien
Next, consider the investment of $1 that grows at the constant, continuously compounded rate rs until timc F;
'
1t
martingale.
gz
There is a Simp le mmingale that one can gcnerate that is used in pricing complicated interest rate derivatives, We work with discrete intervals, distribution #. yz bability considera random variable l'w with pro Suppose we keep getting n be revealed to us at some future date F. denoted by z, concernirs v. as umepasses, ,, t + 1, w ? such tuat. w 7
(66), (68)*
payoffs at rmiteexpiration dates r. Many Most derivatives have do not make any interim pamuts until cxpiration either. Suppose this is the case and let the expiration payoff be depcndent on some underlying asset price % and denoted by
. frcquen
inormation 1
=
nere are manywithhnancial applications one common case. we deal random
. ,.L
..
=
side of
section.
'
.
.
(6,)
,
(65)on the right-hand EP lrr Iz,J Mt. I
gyyy
ntus,Mt js a
gsz'gyyj gtysj j y'jj
J
that are unm x,* a lso has increments clearly,themartingale. willbe a martingale. variance finite, and it is lts right-contkiuous It is a d ictable. . square integra ue.
7 The Slmplest Martingale
=
whjch is tr jviajjy true. But Mt-vsis itself a forecast, Using EP gF# It-vsj J,j
(6..
l,
-
best forecast of a future forecast is what we gMisxl we have =p?p
ik the Poisson counting process Nt discusscd in t
We consider again chapter. Clearly, Nt will increasc over time, since it is a counting pro will grow as time passes. Hence. Nt cannot b? and the number of jumpsupward trend. has clear l t a martingale, Yet the compensated Poinon Fzrt?ce.f denoted by N:,
135
whicj, says, Applying the Z this to forecast now. r E E u t+s j gl j
/'
1:
6.4 F-rramp!c 4: Rig ltt Centinueu.s Muuingales
Martingale vhesimplest
( .
..: ,
C
Mt
::t
'Bm.
'' ?
? k
J :.
(67)deleted
=
,
p E g.$J/+, I fij
in proofs, all other equaton
,
numbers
s
>
(),
remain
(72) unclaanged.
r'
S'1
i'
C H A P T ER
.
t
Martingales and Martinga le Rep xesentati
6
S'
! :'
7.2 A Rerrzfrk
'2:J
.i
.,
Gy
J!
'
,
, (,
value of the bond. Then, the M3 is the conditional expectatitm tf the discounted payoff maturi undcr the prtabability #. It is also a martingale with respect to
the par
,
. according to the discussion in the prcvious section. g whethcr azbitragetake Mt the The intersecting question is as we can Price of thc discount bond at time /? ln otbel- words, letting the F-maturi default-free discount bond price be denoted by Bt, T), and assuming th #(f, F) is arbitrage-free, ca,n say tbat ; '
Bt, F)
8.1
book wc will see that, if the expectaon In the secmd half of t.1:1,s under probability #, and if this probability is thc real Af'/ calculated a will gcneral, Mt equal the fair pricc B(t, Tj. robability, then not, in # used calculating in Mt is selccted judiciouslyas if probability But, the arbitrage-free =equ ivalent'' probability # then
l
FJ3 =
100 1 It p
.
UT
previous examples showed that it is possible to transform a wide varie appropria ,of con t inuous-timeprocesses into martingales by sttbtracting
ne
meang ln t itis
sectim, We formalize thesc decomposition.
D 00 b-Meyer
gpecial
cases and discuss the so-ca-f
.
,
x,
this
:
::
k
As k
14
':'
' .
j. ..'
;
'..
..
'.
.
Addition or probabilities is permitted if the underlying cvents are mulually exclusive. In p artic-ular case different trajectories satisfy his condition by lefinition. exam' Ii-aks is aa Ple of weak convergcnce.
vua
-..,0
t
7
!:
. '''
:
140
C H A PT ER
*
Vartingales and Martingale Representati
6
. ingaltt Representations 8 Mart
r
!
.
shown earlier, we can write w as ylk 1 - 2p)(k + 1) +. z(, (94) aje. jyj Hence, decomposed mart we g a submartingale into where Ztk a o components, ne srst term on the right-hand side is an increasing deterministic variablc, The second term is a martingale that hs a value of st, + (1 2 ) at time /u. Th.e expression in (94) is a simplc case of Doob-Meyer decompositioml;
$ 8.1.l Is Stk a Mttrlrlgtsd? martingale with in.f respect thc delined in Eq. to Is the Lutk (82)a J price changes LSL k ? mation set consisting of the increments in Consider the expcctations under tbe probabilities given in (86) E
NO
-(
,
,
,-tk-jl
.
.
=
,
.
+ (-1)(1 ((+1)17 -
+
-fk-l
#)1
(
,
side is the expectation of hstk, here the second term on the information 1/ at time ltk-, Clearly, if p increment given the un known and have this term is zero, we Iight-hand
W
/'Es
E
tk
I.s ?c'
hst .
,
.
.
.
,
a.sk-ll sz -
1
.;)
-
,
=
,
nte General Case decomposition of an epward-trending submartingale into a deterne ne istic trcnd and a martingale component was aonefor a process obsewed at a jinite uumber of points during a continuous interval. Can a similar dewhen we wok with continuously observed O mposjtion be accomplished processes? ne Doob-Mcycr theorem provides the answer to this question. We state tlw thcorem without proof. Let VrJbe the family of information sets discussed above. THEOREM: If Xt, 0 s l s x is a right-continuous Numartingale with respect to the family j.1fj.,and if E z't-jq < x for all t. then Xt admits the decomposition
r
x7/
u
r
(
1,
which means that ) will be a martingale with respect to the info tion set generated by past price changes and with respect to this parti
t5l
probability distribution. will cease to be a martingale with respect to 1/2, the lf # k) However, the ccntered process Ztks defincd by
t&
.S.r,) + (1 2,)1 -;i!:',r ,: 1) -
k
J7 gksl+ (1 -
+
::::::
.
2#))
:
jh' '
' k,
(
f=l
; ;'.
it Zt,
willagain be a martingale 8.2 Dot-Meyer
=
z,,.16
. .3
(
1/2.
>
p
:
( !
:tE
which means, EPLSL,
Stb Ixsla, .
since lp Inlt
1Z,k-
>
1 according to
can be checked I
1.
lhat
ktk..ll -
(91).This
-
-
,
>
implies that
the expectat jo j of
.(z/ J, k
(
Sv-, AlJ
I'.
.'.
:
j'
cw
,
rtingale-
'::
S
will
on past .(Zrk J7
''
at expjration date F
4..
jy
'. '..'
.
(95) .p,
=
qstkJ is a su jy
conditional
..
-d2,
8.2.2 The Use of Doob Decomposition We fact tbat we can take a proccss tbat is not a martingale and convert t into oue may be quite useful in pricing finucial assets. In this sedion we consider a simple example. m Re assume again that time / c (0,F1 is continuous. The value of a call option c written on the underlying asset St will be given by the function
:
Thcn, as shown earlier,
Mt +
M: is a right-continuous martingale with respect to probability and vtj is an increasing process measurable with respect to z,. This theorem shows that even if continuously obsewed asset prices contairj occasional jumpsand trend upwards at the same time, then we can convert them into martingales by subtracting a process observed as of time 1. lf the original continuous-time does not display any jumps,but process i: continuous, tlAc.n tlne resulting martingale will also be continuous.
vb
so that we expect a genera 1 upward trend in observed trajectories:
=
Where
,?
Consider the case where tbe probability of an uptick at any time t somew hat greater than the probability of a downtick for a particular as >
XJ
:
Dectmvpositimz
1
'*''f'
(
Stk + ( 1 - 2#)(k + 1),
wit1arespect to
,
-
'
k5';:,sk Ufp
,
=
tdpast''
EP
141
E
.
'.'.
(..! ..
.(j. t).
kqq'.rK, (jj -
(96)
.
Tbi: term is often used for martingalcg in continuous p artjtjoo of a continuous-tme intewal.
(liKrete
r.
max
time. Here we are working with a
'
: :!
:
.
C H A PT ER
142
*
6
11
=
lslmaxlkr
K, 01 1T,1,
-
'
(
9
h..
.,4/
;:
Acall
.
.
C
' .
t ?
(
E
where thc expectation is taken with respect to the distribution ftmction governs the price movements. Given this forecast, one may be tempted to Pask if the fair market v Ct will cqual a properly discotmted valuc of E (max (.V- K, 01 i121. risk-free intcrest rate r. the cxample, For (constant) suppose we ese ' write to A', 01Ifr1, discountS'Imaxgs'r
=
-
give the fair market value Ct of the call option? is a martingale with depends on whether or not e-rt The an-er is, have the It, pair P. lf it we spect to
wouldthis equation
Fplc-rrcwltq
or,after
e-rct,
=
/
results thus far to define a new martingale Mt,. be any random variable adapted to lt Let z, bc any mar-tingale wxtlt rcspect to It and to some probability measure P. the processdelined by
Let Ht j
EP
j-r(T-8
G'rj
cwj
Cf
=
(E; .'
'fhen
&
Mtk
a
(1
(1 I.
-rts r
will bc a submartingale. But, according to Doob-Meyer decomposition, wc can decompose c -r/
k
j
/
.f
(1,. .
e .-rts t
aa
t +
zt
is a,n increasing lt measura ble random variablc, where tingale with respect to tbe infonnation 1t. .,zl,
(1
,
and Zt is a
y. y) u
Fyf
1
j
i-,
f=1
jgy .gy ,
f-1
yj
.
t
ulnside
,
..
.
j
s
gz z j -
/j
.
Eto
((
i
j
jj
..
j
=
gatpj .$J,., =
'i
lawe
uq..,
..
r.
t
s
.
;
:
(1.:7)
.
I@'E)ti..
..6 :
1
remirjd ttje reader tlaa tus means, gs'en the information in wgj 4x, kaowa exac-tly.
qe member ttjat
s js v-. y.j)
.s
J(,
za
j.j.
jjgy
(),
5). thus has tbe martugale property. ,
:
jjtjtg
ju zj are unpredictable as of time lj-2..1..9 A1so,,,)Hcj is pted Tltis means we can move the E 2f-, I-j operator
-ada
yjjjs jm jus P
.j.
4: ;
sa
)....
.
..
k
.Htj
(.
''
'.
' .
x o j&j 4
sut jjwyemosts
I
;
.
to obtain
rjahen,
,
k
:;
tl
(105)
'rl,e
dfconstantf'
Fj
'5
That is,
)
,
i-'
(1
sf.
Zq
-adapted
b
>
-
j
.
jyj
(zt
with resped to 1t. The idea behind tlus representation is not difticult to describe, z, is martingale and lms unpredictable increments, fact that Ht is incremenf-tl in means Ih given h.L are h-j Zq will be uncorrelate with Ht as well. Using thesc observations, we caa calculate
;
gc-rlT-dlyz,
Hli.L
wul also be a martingale
'g
Then e -& C/ will be a mmingale. Bet can we expect e-r'st to be a martingale under the true probability As discussed in Chapter 2, under the assumption that irwestlzhrs risk-averse, for a typical risky scctlri'ty we have '1 EP
A/lrl+
=
f=1
'
j
,
;
,
(
:r)
,18
,
E
by e-rt,
mult p ly ing both sides of the equation
i
0 feprcscnt a mall, linite We proceed in disclvte time by letting ZtelYals teWal and WC Sllbdivide as in the Period (l,F1 into n such tf a deriva pricc revious Section. current represent and St the The Cf P sec urity and the Price of the underlying asset, rcspectively. The Ct s unknown of the problem be low. Thc F is the expiration date. At expira the derivative will havc a market value equal to its payoff,
ahedge''
.
.
A Priclng Methodology
is to constnzct a synthetic
for the securify Ct. We do this by using the standard approach utilized in Chapter 2. Let Bt be the risk-free borrowing and lending at the short-rate r, assumed to be constarit Let the Sti be the price of the undcrlying security observed at tille ti. Thus, the pair fBt Sti is known at time ti. wl now, suppose we select the at. pti as in the prcvious section, to fonn a t'eplicating portfolio!
'
W here
Mb
-
'x
v..-.
of the normalized Ct, i.e., of the ratio Ct/Bt. ' This suggests a way of obtaining the pricing Eq. (112),Given a deriva security Ct, if we can write a martingale representation for it, we can try to lind a normalization that can satisfy tbe conditions in (113)and (1 under the risk-neutral mcasure # We can use this procedure as a gen ! way of pricing derivative securities. ' . In the next section we do exactly that. First we show how a ma
11
Mt,-'
=
be of any use in detennining How could this representation arbitcage-free Pfi Of the derivative sec-urity Ct'l
..
,
T
(116)
f,
f=j
that
and n is stlcb
wherc tbe b is the trend
repegentat
#tcij lNj )--q
+.
'
W
l
t
ot
j=)
Xv'rf
.
Et/
FI
means
;'
It turns out that Shis equation can be obtained from (111) Note 'that Eq. (112),it is as if we arc applying the conditional expectation omra EP of Eq. (111)after normalizing the Ct by Bt, and l (.1to bot.h sidcs letting ;J3
(111)is then
'
> -.1 Et Bv #, =
in
gwen
(p
'
C
representation
G ; during the period (:, 7-j. We can
write
..
i
of .
n
cv
: 'F
' :
rr .
,. 't
.
=
c
/
+
)'')hct J=o
i
,
(j19)
y'
:.
; :.
:
. :
148
IaT E R
cHh
becauscLCt,
=
n
c7,+
=1
Pro duct
1-1lle'''
Bq 1+ gt,,.
f=1
'
E
'rlms
n
.%1
f=1
+ j=(
.?
cz
.;.
(1
and third terms on thc right-hand
n
(trjlj)
q
i==l
(S)
1'1
(At:'&)
k& Emfkh-/fl f=tJ =
Y
f,xu.tl
+
.:
('%)
t'/f
.
f=1
(-4
kr.s,.1 gt,,+.2$..
a Atz,i
=
fe2s+, -
./,j
ati
=u
way of obtaining
/.?I.+,
-
Bt
,;
= 5 1,.- 1.
-
Sti
.
', 9
'$ .
..:
.
equations beltw is b )? simple algebra. Given
j,;#q1 -
and subtract a,(s,(+,
a dd
a:.Bz;.t
- a.i
.
t
%
.
-
cr,,)
=
=
(cr;... -
) (.::a/f
B%+ +
srj., +
ati
(B,, -,
-
S6) :
.
..:
'
j ,
,j
'
.
.j
ic
' .
..
L 1. .
l
yjye jxacketed terms i.n (126)will not, in eneral, vanis.h under such an operation But at this point there are t'Wo tools available to us.
E
,
on the rip-hand
q%;,
p'ven the information set it involves the price changes hz,, as,, that occur aper ,j. and may contain new information not containcd in h However, although own, thcse pricc changes are, in general, predictable. Thus we cannot expect he second term to play the role of dMt in the martingalc representation leorem. The second bracketed term will, in general, have a nonzero dri and will fail to be martingale. a Accordingly, at this point we cannot expect to apply an expectation operator E /p (.j where P is real-life probability, to Eq. (126)and hope to end up with something like
..
1he
jj
(Aa,f)
tlnknown
&,because hetlcc
i ;,
-
and k'a/, '
The second bracketed term will be
i.
#p.! #&
=
gatj
hnttj + at,
(111).
j
'
.
and obtain:
,
: t'
' ..
(#,.1
-
gtari)
)+j+
'
.
-
(lksjj ) .
k%;
' fs
used the notation, .1
(12j)
cvq. i-
)
'rhe
-
'
r
LBtj
'rhc
,
n
ati /-t)
Now consider the terms on the right-hand side of this expression. Ct is tbe un known oj tjw problem, We are. in fact, looking for a method to dctermine an arbitragc-free value for this term that satisses the pricing Eq. (112). two othcr terms in the brackets need to be discussed i:l tltlttil. Consider the first bracketed term. Given thc information set at time Ij-yja every clement of this bracket will be known, The Bt are prices obkpri is the rebalancing of thc rcplicating scwed in thc markets, and the iatt, PO rtfolio as described by the financial analyst. Hence, the first bracketed term has some similarities to the Dt term t-le martingale representation
(1
and
Can
i=
rc:ct,
k.'
We
Ct + +.
-
.4;
r,
(tr,) .Bfo:+ f==tl
=
j==(I
lhat
=
(..
)
of (121)22
n
r;
n
N
.
'
+ (aaj)sb-vj
n
-
:r
du.v + u-pv.
tzcq
j Bt,.j +
Regrouping,
.
,
:';
l
Applying tbis to the second
note
f-tl
r;
..
(1
.
G+
...
.
#(u,.t))
z'Another
=
.
lchange''
=
wherewe
bc rewlitten as;
(121)CM
Cz
(1
Z/3
149
t .
the operation of taking first differences. in a product, &.v, can be calctzlated u
where the A represents Now, rccall that the L j-j
:
(tnd
151
. l
..;
))
a,;+,a,f., +
.
.LLt . '..
pt,.wsbw, a,;s,,+, -
+
#,f.s,..
,
(14n)
7
E
152
C H A PT ER
.
6
:
.
Martingales and Martingale Representati
..'
.
Let Sf be the price of an assct obsen'ed by a trader at time f, During jfjjijaitcsimal periods, the tradcr rcceives new unpredictable information on These are denoted by l
is, the time ti4 1 value of the portfolio choscn at time tt for all f. suficient to readjust the weights of the portfolio, Note that this exactly writtcn lbr the nonnormalized prices. This can be done beca is equation :. normalization the we used, it will cancel out from both sides. whatever 'rhat
153
yj gcfrences
E .,..
'
'
k..z
*'
cjst
.
tyst
a(o
,
.?
11 4
A
szzmmury
whcre h is volatility and JH( is an increment of Brownian motion. Note tjot volatility has a time subscript, and consequentiy changes over time. Also no te that dst has no predictablc drift component. longer period, such unpredictable inforrnation will accumulate. overaintenral F, the asset price becomcs Aftt:r ajl
'
E
we can now summarize the calculations from the point of view of pricing. First thc tools. ne calculations in the previous section dcpend basi martingale reprcsentation tll on three important tools. The first is the it into decomposc kn that, given This a a proccss, we can says rem. trend and a mart ingale. This rcsult, although teclmical in appearanma'' in fact quite intuitive. Given any time series, one can in principle sep . it into a trend and deviations around this trend. Markct participants , work with real world data and who estimate such trend components tincly are, in fact, using a crude form of martingalc representation theo The second tool that we used was the normalization. Martingale resentationtheorem is applied to the normalized price, instead of the served price. 'This conveniently eliminates some unwanted terms in the tingale represcntation theorem. The third tool was the measure change. By ca 1cu la ting cxpectations J ing the risk-neutral probability, wc made sure that the rcmaining unwan terms in the martingale reprcscntation vanished. ln fact, ut ilizatitm of Of risk-neutral measure had thc effect of changing thc opected IrcaJ St process, and the normalization made sure that this new trend was e inated by the growth in Bt. As a result t'f a1l this, the normalked Ct en up having no trend at al1 and became a martingale. This Sves tlze PH .1 . Eq. (126),if onc uses self-hnancing replicating portfolios. t, '
d+.z'
St-vv
#Bi.
o
(108).lf evely incremental ncws is unthen thc of incremental news should also be unprcdictable sum Pre (aS ( j tjme ,). But this mcans that St should be a martingale, and wc must '1h
js equation has the same form as (jjctable,
jave
.
/ iT
s
fr
t
u
dw u
=
().
This is an important propcrty of stochastic integrals. But it is also restrica tion imposed on linancial market participants by the way information Ilows in mazkets. Mattingale methods arc ccntral in discussing such equalities. They aAe also essential for practitioners.
,
.
'
St + ;
.
.:
v
=
,
.
.'
13
?
qq
A reader willing to learn more about maningale arithmetic should conthe introductory book by Williams (1991),The book is very readable aad Provides details the mechanics of al1 major martingale results using on Simple modcls. Revuz and Yor ( l 994) is an exccllcnt advanced text on mar t). A-E'-rl
Suppose f (.x)is a function of a random proccss x. z N()w suppose want to expand fx) around a known value of x, say xz. A Taylor seri
';r)
term f x x, Consider the second term lafxxtA.Ylz. If the variable x were deterministic, One could have said that the (Ax)2 is small. This could have been term normegligible, yet small enough that its Xsttied by keeping the size of Square (.r)2 is negligible. In fact, if x was small, the squae of it would be even smaller and at Jopze point would become negligiblc. However, in the present case x is a random variable so changes in x will also be random. Suppose tbese changcs ave zero mean. nen a random variable is random, becauseit has a positive variance:
'
represcnts the change in thc function as x chang time, then the derivative is the rate at whic if represents Hence, x by interval.l ln this case, time is inhnitesimal changing during is /x) an ' tandard'' calculus. variable and s deterministic one can use
0,
(3
F. ' .
K
L
to highcr-order
derivatives.
=
l,
.
.
.
,
a,
jnjependent
... '.. .! (
:.
aj.
(35)
of the asset price during thc most volatile
w zlg
j
(j
.
=1
-42
,, (P;;)
>
( . .
(
''A3lz'rrw'
2
h'max
>
F
.
a'lla'1s h. F
u
>
(49) (5p)
.
r
clearlythc variance
term lzk has upper and lower bounds that are proportional to h, regardless of what n is, This means that we should be able to find a constant n depending on k, such that Jz'vis proportional and ignoring tlle (Smaller) to higher-order terms in h, writc'. j
(4
(;
.
h
xalgva!l
>
-v -!k >
.;
1,1
;
FglHzkj
=
2 rkh.
=
(51)
t..
i. Then,
'
;
1 > n X h -7. > -
r
4:
v'fs
-43
Jzrnlx
>
(
p'k
I'-k .
.
n
Fi
W
z4
nOW
Obt
,
k=1
,
.
1.
,
k=z 1
lz'k >
(4
-4
)
.'.
.'
prk.
r
'
.
--1u
.:'
.
: :
-
,
#-j
U
..
'lhis
=
a)p-now has variance k
-
,
''
sk
yg .yg.) /z
:
tnle Wc11 >
Where g
p
;
..
nl.z p1 zN
Titis proposition has several implications. An immcdiate one is the following. First remember that if the corresponding expectations exista one can always write
:
(
j
5 One lmplication
:
(4
We rfhis gives an upper bound on lzi that depends only on also depends only on h. We lcnow that that bokmd lower a
.
.
zla .,4
f
azlz
iS
h
1
zls
3 '-' u 1
Rllerefores
.1
ar4
n
p-p>
)'. ,
Now divide both sides by n,,4:2 j
A
>
means that
'fus
)
k= 1
-
a pux
.,
(3.
(48)
.
-''''
.4
Jzjr>
:
>
z'lauslux
.
Sum both sides over al1 intenrals: n Fi > l-dnzr'dx' k Msumption 2 says that the left-hand side of this is bounded from a
Note that n
.:'
'
Pk
=
K. >
..
3:
(47)
Use Msumption 3:
E,
of the protf.
>
MIJX
.
.
smaller.
Sinc,c this is a central result, wc prtwide a
'.
'
According to this proposition, assct prices become Iess v olatile as Sketch
X1 lt T
'$
-
.
::a
Ek.3
guk y-j j +
.C0
After dividing
yg
.j
.
-
jju g
.yv
.y
j +.
(u.ijq
(5g)
,
both sides by
2
cygy ju
jy
.
this equation, the parameter rk is explicitly made into coemcicnt of the H?ktcrm, a is a trvial transformation, because tlx term rzklppk w-ill now have a vaziance equal to h.
q
(' y'
:'
, .
'
f2 H A P T E R
*
Differentiation
7
in
Stochastic Environm
..3'
E
(
=
tlse
attirg t lae
6
.'
.
Results Together
169
'
.E'(Al,''f,$ g,1 h. this to
:
.
,
But, according to the proposition,
Suppose we
'
C
'
.
j
justifythe approximation; apz2 k x
justructivc
qujtc unpregjctablc
,
: ...
2
'.
ddnews
because it shows that the fundamental characteristic of in infinitesimal intervals, namely, that ,,
k k jz slo-auz
.
:
('
.
ajh
=
'
insurmountable difhculties in dehning a stochastic equivalent
Iead to may of tbe time derivative,
.
(In Chapter 9 we show that this approximation is valid in the sense of m , square convergence.) ln Chapter 3, w hen we dehncd the standard notion of derivative, we h go to zero. Suppose wc do the same here and pretend we can take of the random variable:
6 putting th e It esu jts Together
''
Gllimit''
W%-1)s+/: Bk-ll lim h ->() Then this could be interpreted as a time derivative of M. The appro tifm in (55)indicates that this derivative may not be well dehned:
Up to this point we have accomplished two things. First, we saw that one stochastic process St and write its variation during somc fmite can take any interval h as
-
.
(
.
t '
'
1
hJ'J,(.-1)&+/, - '1,#r(k-:)/l1 - h --o
Um
.+
az..p;-;
'lsmall
fhl'.
, ? .'
,
h;
l
-
VrtHil
E
t
.
h
t :
t
s
E
Ek..j
.,
Where
:.
hm
,
J. .
. h ()
.14
().:1
0 6
FlGURE
.
0 0 .
1
..
....
' .. r :(
'
2
. :
zk-jj.
(5p)
-
ps-jj
=
.,4
(&.j, ).
z1tft-l,
..
...:
hb
=
-4(f,.-l,
t)) + alk-klh
'
:
t1
Agstlmirlg hat the corresptmding
.
:
.
.:
.
(60)
-d(.)
,
L.
.
:
(k%
!1
..;
'.
.
(kS-
represents some hmction. ucwedthis way, it is clear that 'f is a smooth flmdion of it will have a Taylor series expansitm around (j
a4(')
=l&h
f(b)
2
-1
j
i
E) .
f (h) explcxles
(58)
.
Ths term is a conditional expectation or a forecast of a changc in asset prices. 'The magnitudc of this change depends on the latest infonnation set andon the length of t'he time intewal one is considering. Hence, E'v-:g.% k-.1)can be written as
.'
f 0
=
In order to obtain a stochastic difference cquation delined ovcr hnite intewals we need a third and final step. We necd to approximate thc first term on the right-hand side of (57),
,
4
5yj
''
,
:.
8
sg..jj +. aawk.
-
whcre the term JP''k is unpredictable given the information at the beginning tjw time ilatervttl,ll the unpredictable innovation second we showed that if is term has a variance that is proportional to the lcngth of the time inten'al,
Clearly, as h gets smaller f () goes to inhnity. A well-dehned llmit d :;' anotexist. of course, the argument prcsented here is heuristic. The iirniting o ation was applied to random variables rather than deterministic f undio and it is not clear htw one can formalize this. But the argument is s
,
j,u
Oj
hll =
-
K.
Figure 2 shows this graphically. We plot the function /()
s# sk - 1
':'
cxpectations
cxist.
-I-.A(z:-,,
).
(61)
7
?.
.
:
. 7..
C H A PT ER
170
Differentiation
7
.
in
Stochastic Environ
.
'
:E
t
h) with rcspect to h evalua is the Iirst derivative of Hcre, J(&-j) the remainder of the Taylor series expansi ht is Rlk-j h 0, ne at and predicted chang in asset p thc will not if h 0, time Now, pass words, other In will bc zero. X(Q-1,0) 0. (
.'
.
7 Conclusions
J
..4(f1-:
,
=
'
:
,
piffercntiation in standard calculus cannot be extended in a straightforStfchastic derivatives, becausc in infinitesimal intetvals the ward fashim to random processes does not equal zero. Further, when thc tlow variance of Of neW irdbrma tjorj oyeys some fairly mild assurnptions. continuous-time become very erratic and time derivatives may not exist. random proccsses As thc latter becomes smaller, thc raju small intewals, AH$ dominates tio of llfrp to is likely to get Iarger in absolute value. A well-dehned limit czmnot be found. On the other hand, the difliculty tf delining the differentials notwithstanding,wc needed few assumptitans to construct a SDE. In this sensc, a that can stoc jostic differential equation is a fairly gcneral representation be written down for a large class of stochastic processes. lt is basically constructed by decomposing the change in a stochastic process into both a predictable pa,l't and an unpredictable part, and thcn maklng some assumptions about thc smoothncss of the predictable part.
*
F
=
'
; .
=
' Also, the convention in the litcrature dealing with ordinary stb having that equations is differential any deterministic terms powers o 8reater than onc are small enough to be ignored,l3 Thus, as in standard calculus, we can let '
)
A(-l,
and obtain the first-order Taylor series approximation: Ek-Lf-k
-
17
kk-j
?(/k.-1
,
.
'.
(
0,
.
.
khlh.
Utilizing thesc results together, we can rewrite (57)as a stoc j.ja suc ence equation: j4 X 445//r-) khlh + tzylllz Sk jh 1. - rr'(k-1 u$)-1),
,
(j.
171
9 gxercises
(E
.
g :
;,
(
0 and obtain the inlinitesimal versio'' In later chaptcrs, we let : (57), which is t'he stochastic diffcrcntial equation (SDEI: -->
8 References
'.
dstj
=
alt.
This stochastic diffcrential equation ion (J.; component. di
t ) dt + h dkrt
The proof thata under the three assumptions, unprcdictable errors will have Thc chapter in Merton a variance proportional to h is from Merton (1990), (1990)on lhe mathematics of continuous-time finance could at this point jye use s! to tjae xader.
(
).
is said to have a J8# alt, tj aa
,
,: !(.
6.1 Stochzssffc Differentzls
.,
. At several points in this chapter we had to discuss limits of ran ch incrcments. The need to obtain formal dc*nitions for incremental such as d.t, #1Ft is twident. E.k How can these tcrms be made more explicit? It turns out that to do this we nccd to desne the fundamental conce the 1to integral. Only with the lto integral can we formalize the notiol ' vtlchastic ff//regzltfl: gtlch as dSI JW?),and hence glve a so 1id interpreta of the tools of stochastic differtmtial equations. This, howevera has to .2 until Chapter 9.
q Exexjses 1. We consider thc random process S(, whch plays fundamental role a in suck-scholes analysis'.
.'t.
,
st
.
wure p,t js a
*
wjenerprocess
.
,
,
1
(lzlj
.
*
.
'
..
'4l-lere, denccof these
tsrnls on
explicitly.
..
'
v
:
-
'.
,7
,
'
': $
.: .
,
.
'..
.
so
cE&J.l-twr/;;1 ,
with Ht Jfr,)
..w.
=
N(0,
0,
g,
(/
is a
dtrend''
factor, and
.)),
wlcll says that the increments in have zero mean and a variance equal I,IZI l to s Thus at t the variance is equal to the time that elapsed since 1#' observed we also know that thesc wienerincrements are independen't over time Accordin to this s/ can be regarded as a random variable with lg-normal P distribution. SCN we would Iike to work with the possible trajectofojjowed tjs t)y process.
.
-
lzcliverl fjr-j, wr arz dealing with norjrandom quanlites, and the dcrivativesin the ;' seriesexpansion call be taken in a standard fashion. ,:. l3sirlce ' is a deterministic funclion, this is consistent witll the standard calculus, m '' igneres all second-order terms in differeniatioll. for X and hK Tls shpws tlle the l ill tht! ntAtatit:n we are reintroducing
-
=
.. '.
, ,
7
'.
:
L
h
172
C H A P T ER
*
Differentiarion
7
in
Stochastic Environm
fr and f Let Js 1. Subdivide the intenral subintervals and select 4 numbers randomly from: .01,
.15
=
=
=
.z7 N (0, x
(a)
Constnzd the JF; and St over
bers
t.
.J
.
(0,11 int
'
-
'
'> .
.2.5
). the (0, 11using
'
;:!';
)'
.i
.E
L
these random a
'
4
lener
e
'
Plot the H( and St. (You will obtain piecewise linear trajecto b that will approximate the true trajectolies.) (b) Repeat the sme exercise with a subdivision of (0,1j into 8 ia i vals. (c) What is the distribution of l.. '.
,
#
St
log
for
LGsmall''
(d) Let
0 .25.
=
.
'
1
change
(g)
..
1 Intrvuction
.
log- - logs'/ . -, as time passes? 0, what happens to the trajectories of the
.''
.'I
; (;.
:
.
..
log
.
'
'
t
) : well-de in tbe previous question is Do you think the term a
randomvariable?
.
i'ex-treme''
'
.
.$,-a
?
j
',
.q;. For example, z&I may represent an uptick, u?a may be a downtick, certainly change'' in asset prices. ln real time, tbese are may represent routine developments in linancial markets. 'I''he rcmaining possibilities, z%, w5, are reserved for various types of special evcnts that may occur rarely. For cxample, if the underlpng semlrity t is a Jerivativc written on grain futures, 1174 may be the effect of a major drought,thc 1.% may be the effect of an unusually positive crop forecask # and so on. Clearly, if such possibilities rcfer to extreme price changes, and % if they ar rarc, then they must lead to price changes greater tha,n one tick- f) Othemise, price changes are caused by normal events Iz;l, ztla, uy. ' t h e pro babilistii This sctup will be used in the next scction to determlne
Using the important proposition of the previous chapter, this means m
dtnormal''
nno
.
(17)
'rhe
Rweights''
,
!
..
nlh,
=
woerethe parameter m is the number of possiblc statcs. The left-hand side oj Eq. (17)is simply the weighted average of squarcd dcviations from the arc probabilities associated mea n which in this case is zero. outcomes.ls with possible side of (17) is a sum of m Iinitc, nomncgative numxow,the left-hand of such numbers is proportional to h, and if cach element yers. If the sum zero), then each term in the sum should also be proportional js posive (or should equal to zero. ln other words, each #/w2j will be given by or
,
.
piw;
j=1
:
.
p i wli
=
cih,
(18)
',
structure
Of
wjwre () < ci js some factor of proportionaliyl 2 are linear functions of h. s quatjon (18)says that all terms such as rjlslpr Tjwnj one can visualize the pi and the wi as two functiotls of h, whosc
'j
rare events.
j . . .
'
)
: .
product is proportional to h.
'
4 Characterizing Rare
r. :':'. .
and Normal Events
.,i'
assumptions
''.
/Jf
.
Fgty'k H'r k 12
o-lh k
piht
(19)
u?ytl,
(20)
and
.
wi
.t
(15)J ?
,
=
'
:
=
is,
J
,. '. ( '.
1-3 of the previous chaptcr, an important result wa: . $ proven. lt was shown that the variance of tzkH'k, Under
rrhat
such that
.
:!
.'
=
pihtuhhtz
was proportional to the observation interval h where o'k was a known Pa'd rameter givn the information set 1k-L. Thjs rcsult can be exploited furthcr if we use assumption 4. In fact, , tjjjj c . very explicit characterization of rarc and ntrmal events can be glven Howevetk) unpleasant. the reader hnd the notat ion a bit may ' way, although tMs is a small price to pay if a uscful characterization of rare and norm a1'k j events is eventually obtained, ;), valr number of fnite assumption lAP'v According to 4, can assume only a explid f ues, ln terms of tz;f and the corrcsponding probabilities, #j, wc can 'j' write the variance as 'r (16 varlfqlfrkl piwi2
cih.
=
(21)
)L
We follow Merton ( l 990) and assume specilic exponential tjyese junctjons pjhj and u?j(J;)..
.
?.tjl
.
lj)
j
hr'
(2g)
pihj
=
pihq
(;g)
,
Where
r f and qi are nonnegative constants. zi and jt are constants that may depend on i or k but are independcnt of the size of the observation interval. ,
,
,?
f 1
J
=
'
t
-
..(I
l4Both u?i and /7. caa very wcll be made tt'ydcmnd t)n he inforrnation sGt 1k. H()w cumber*mgz this would add a k subscript to thesc variables and make the notation more of and independent k. make this, wc n, p, avoid
=
and
...
=
)
forms for
.
' .
.,.:
'blrlgencral,
subscript.
:. :
( :
: .
;
lswe
show the potentia! dcpendence of 4: tbne passes, L)yadding the subscript to
':
.
,,,..,
p on the irtformation hat becomes available
vk.
c,. wiil depend on k as well. To kecp notation
simple, we eliminate
the k
E
186
C H A PT ER
The Wiener Prkxess and Rare Events
8
*
1.4
z-.'''''
1.2 G
'',
Rare and chaxacterizing
Btlt this implies that
.
>
qi + 2rj and
:
-..-
,
.,''
.,''
x.
cj
. '
.,...---7 h
.
''
0.4
0.6
1 .2
1
0.8
1.4
,
j
2/
j
(2:)
.
(29)
-
:
h
=
Thus- the parameters q;, ri must satisfy the reseictions 1 () r/ s s 2
i
..-v h. Aording $o Eqs. (22)and (23),both the size and the probability of the As h gets larger, then the : event may depend on the intcwal length, change and its probability will k magnitude obsewed price of the (absolute) get larger, except when rj or qi are zero. y To charaderize rare and normal events we usc the parameters rf and qi. Both of these parameters are nonnegative, ri governs how fast the size of the event gocs to zero as the obsezvation intewal gets smaller, qi governs r .t how fast the probability goes to zero as the observation iriten'al decreases. It is, of course, possible that ri or qi vanish, although they cannot do so at ; t he same time.17 We now show explicitly how restrictions on thc parameters ri, qi e-rm ' distinguishbetween rarc and normal events, ', made such variance is of of ll'rpin tcrms as ne (18)
wefind that
=
=
ri
,
:
(30)
1,
:s
there are, in fact, only two cases of interest-namely,
.5,
=
(27)
1
=
..
-.--
.2
182
Normal Events
.,---M
9.6
0
.,..----h''2
f(*) h 1/3 .M
4
4
rn
0.8
0.2
.
1/2,
=
qi
(3j )
0,
=
and
'k
ri
',
.
vjahc
:.
of
(),
=
qi
tirst case leads to events that we call
t
1/2. To interpret this, consider what happens when wc select rj First, we know that the q; rnuuf equal zero. jtr As a result, the fu uctjons tbat govern the size and thc probabilify of the outcome u become, respectively,
(24)
.2
=
A:rj
b
( :!.
=
hh
1f z
.tg
=
i
y-y
(gj)
.
:L rrhat
(45)
.
'l'he
t
yxm
V-'-'
lh g
=
':
7.
uf
::
).
=
ti
'
.
that is they wgl not depend on thc Iengtll of thc inten'al h. We make the following observations conccrning rare events.
'
,
,
.5 .'
E L,b
. .g.
:'!: ' .. ,
L
.:
.
(46)
;'E'
c H A PTE
190
R
.
8
Tbe Wiener Process and Rare Events
5 A Model for R.are Events
Paths 4.2.1 sample of an innovation term that contains rare paths snmple In discontinuous. fact, tbe sizes of those wi with q 1 do
events will be = not depend on As h goes to zero, kp-p will from time to time assume values that do not . get any smaller. The sizc of unexpected price changes will be independent a jump. of whensuch rare outcomes occur, H'; will have of probability these 1, the other hand, if the jumps will depend qi on obsering a jump VII on h, and as the latter gets smaller, the probability of also go down. Hence, although the trajectory contains jumps, these jumps
that is predictable given thc information at that time. and another unprcdictable. In small intewals of lcngth h, we write
:'
Sk -
:
As
qi
. 0, wi will go to zero, In terms of size, they are .: not rare evcnts. E Note that for such outcomes the corrcsptmding probabiiities a 1so go to S zero. Thus, these outcomcs are not obsen'ed frequent ly. But given tat E their size will get smaller, they are not qualified as rare events. l:, .
.
.5
,
k
-
1, one has
,
5 A Model for Rare Events
Nk - N&-1
Where
we jrt
with probability Ah
1 =
with probability 1 - /z A does not dcpcnd on the information set available at
hxk
j'
(51)
,
()
=
Nk
-
time k - 1.
(52)
Ng.k.
-such
What type of models can one use to rcpresent
asset prices if there are rare
tNk
(
events?
Consider what is needed. Our approac 1,tries to represent asset prices by E'?' observed changes into two components: one an equation t ha t decomposes ' .
.y :
represent
jumps of
size 1 that occur with a constant
.19
rate
m-rlw rate of xcurrcnce of tlw jump during an intewal h can be calculated the corresponding probability kh by
E
.
r.
.
K
'' ..
by dividing
:'
192
H A PT ER
The Wiener Process and Rare Events
8
.
193
Moments That Matter
..
'g
It is clear that Nk can be modeled using a Poisson counting Poisson process has the following properties:
process. A
6 Moments That Matter
, ,
1. During a small interval h, at most one event can occur with probability 21J vcry close to 1. 2. The information up to timc f does not hclp to predict thc occurren (or thc nonoccurrencc) of the event in the next instant 3 cvcnts occur at a constant rate
and rare'' cvents ia important for one Tlw dfstinction betwccn other reason. Fractical work with obsen'ed data procecds either directly t)r indirectly of the undcrlying processes. ln Chapter 5, lv using appropriate representing various expectations of the the term as defined we uaderlying process. For example, the simplc expected value F(-Y/1 is the rst moment. The variance fntrmar
:
'.
;
:
.
JKmoments''
,
K'moment''
,
.
'l'hc
L
.
:
ln fad, the Poisson process is the only process that satishes all these con- : ditions simultaneously. It seems to be a good candidate for modeling jump discontinuities. We may, however. need two modihcations. First, the rate of occurrence of jumpsin a certain asset price may change and cannot i over time. ne Poisson process has a constant rate of ourrence L . accommodate such behavior, Some adlustment is needed. Sccond, th incrcments in N/ have nortzero mean. The SDE approach dcals with innovation tcrms with zcro mean only, Anothcr modillcation is needcd to eliminate thc mean ol dNt. Consider the modised variable
.sg-v,112
varlxr)
:
.;
is the second obtained by
'
t
'
b
N, -
=
f
).
(53) ('' ;
'
The increments h will have zero mean and will be unpredidable. Furtber, if we multiply the Jt by a (time-dependent) constant, say, r.rzt%-j kj, the sizc of thc jumps will bc timc-dpcndcnt, Hence, e(&-1 kjuk is an apPropriate candidate to rcprescnt unexpected jumps in assct prices. nis means that if the market for a tinancial instrument is affected by sporadic rare events, the stochastic differential equations can be written as
';,
,
,
,
Sk - Sk
1
=
aLuk1. +
As h gets small,
k)
trzt-v-:
+ ojuk ,
ktij-lk
la ,
klH''k k
=
1 2. ,
s
:
(54) -
.
.
.
,
.
n.
'$..
E 1rn.'H'k Vargfn
*
1kl
=
ast
,
f)
dt +
tn
St
,
/)
#H?;+
(55)
tj #./;.
and This stochastic differential equation will be ablc to b an dle (drare'' events simuitaneously. Finally, note that the jump componcnt dh and the Wiencr component dv t have to be statistically independkmt at every instant /. As h gets smaller, the size of events has to get smaller, whilc the size of rare evonts remains the same, Under these conditions the tw't types of events cannot be to each other, Thcir instantaneous corrclation must be zero.
.
6h-lj
IzP-z. + h-lk
.
1
=
(,I
+
?Jp.j
2+ ) gpl'tt'j =
.
.
.
-
+ pm zt7ull
=
.
.
(58)
0
2,
(59)
+ pn, 137,,,1,
independence of Hzrkand uk is implicitly used. Now consider the magnitude of these moments when all events are of the anonual'' typc having a size proportional to hll That is consider the case when tll qi (). 'Fhe hrst moment is a weighted sum of m such values. Unlcss it is zero, it will be proportional to /11/2:
'
,!
'
:!
'
11
=
.; '
' .
? .' .
SlGl
j'.
drelated''
0, this probability will become .1.
(57)
11',
wherethe
:.
idnormal''
-->
+
't
Qnormal''
zilAs
Ekx,
are
-
this becomes Jzvb't,
-
(centered)moments
where k > 2. As mentioned earlicr, moments give information about the procegs under consideration. For example, varilmce is a measure of how volatile the pices are. The third moment is a mcasure of the skewness of the distribution of pricc changes. The fourth moment is a measurc of heavy tails. In this section, we show that when dealing with cbanges over inlirtitesimal intervals, in the case of normal events only the first rwwmomcnts matter. Higher-order momcnts are of marginal significance. Howcvcr, for rare cvents, ali momcnts need to be taken into consideration, Considcr again thc case where the unpredictable suzprisc components are made of m possible events denoted by wi. The first two moments of such an unpredictable error tcrm will be given byzi
'
Jt
Higher-ordcr
Tr.v',
,'
.
-
(centercd)moment.
Et
(56)
FE.A-,
=
pj
kll'rk j
=
1/2(yp
1 t'?j
+
.
.
.
+ pmlmj.
(60)
.t'
2'In the remaining
.J
.E .j' :E,
.
>
:
(.
.
part of this section,
fnlur,
t, i
=
1, 2 wll be abbreviated
zhi%..
.:
194
C H A P T ER
*
Ctmclusions
Te Wiener Process antl Rare Events
8
As we divide titis by we obtain the average rate of unexpectcd changes in prices. Clearly, for small the X-his lalyer than and the expression ,
representing the lirst moment and the second representing Parameters, tne will variance, be sufflcient to capture al1thc rclevant information in price thc The Wiener process is then a natural choice if there are small for data no rare cvcnts. lf there are, the situation is diferent, Supposc a1I events are rare. By defmition, rarc events assume values wi that do not dcpcnd on h. For the second rnomcnt, wc obtain
,
,
E l'H$l
tnkL/ztn.
=
zi. .
'r
.(.I
(64) .;..
2, for small h we have
'
'
j
(65)
A-IJLAM'S.J (s-2)y2 = h '
h
Titis rate will depcnd on h positively. .As
tv''
by h, we obtain t he
( .E
..'.,
.
f=l
(66)
'
g
;
gets smaller. hn-1)/2 will co'n-
to Zero. :2 higher-ordcr moments of unpredictable Price Consequently, for small will useful information if the underlying events aro changes not carry any model that depends only on fA probabilistic of the type. a1l
?
J
/E
,
zzWherl n is grcater than 2, tlze exponent
of
will be positivc.
(68)
.
'.
Verge
''normal''
)'-'!zqpi
tznormal''
'
Consequently, as we divide higher-order moments corresponding rate:
h
Tis is thc casc because with rare events, te probabilities are proportional to /7, and thc latter can be factored out. With n > 2, higher-order moments /I. As we divide higher-order moments of Llk by h, thcy are aL() of order smaller will not get any as h ---> 0. Unlikc Wiener processes, higher-order of moments h cannot be ignorcd over inhnitesimal time intcrvals. nis that if prices are affccted by rare events. higher-order moments may means prtwide uscful information to market participants. This discussion illustrates when it is appropriate to limit the innovation tcrms of SDF,: to Wiener processes. If one has cnougb conviction that the events at the roots of the volatility in linancial markcts are of the typc, thcn a distribution function that depends only on the Iirst two moments will be a reasonable approximation. The assumption of normality of #P; will be acccptablc in the sensc of making little difference for the cnd results, because in small intcrvals the data will depend on the hrst two momcnts anpvay. However, if rarc cvents are a systematic part of the data, the use of a Wiener process may not be appropriate.
i
E (C'L Hi j/
=
3=1
',
7 Conclusions
.
'
',' '
..
.
111the next two chapters, we formalize the notion of stochastic differential equations. nis chapter and the previous one laid out the gmundwork
;' (.
!
c H A PT
196
ER
.
8
The
wienerProcess ana
8 Rare and Nonnal Events
Rare Eveno
in
Practice
197
'.I
for sDEs. We showed that the dynamics of an captured by a stochastic differential equation, dSt
=
a-t,
tj dt +
f) #'F; + !6.1-1((.5'1,
asset price can always be htt
,
/)
t' ,,
8
.
.
,
. smaller consider As smaller time intervals. 0, the Si will , as we follow a stable path during a tinite interval. Yet, even with vcry small there is a small probability that a event will occur becausc according to (80): eas.s i ) 1 A., Pro b (<j+j (81) which, depending on is perhaps very closc to one. nis is the case because in small intenrals'.
C
:'
for al1 i.
,
',
(75)
,
(sx/-j /f.
?j
i
for alI i, e and the probability pi can be chosen as: =
and the probability, pi, of an ;
for a11f
r
-cvZ
events, the growth
ddnormal''
(:4)
1, which, in a sense, rcpresents thc accumulated n successive periods of length / has passed.
('. First, some comments about why we nced to investigate the behavior of such a random variable. Clearly, the modeling of as a two-state process may be a reasonable approximation for the immediate future especially if the is small, but may still ieave the market practitioner in the daz'k if the trading or investment L horizon is in a more diatant that occurs after n steps of length For example, the interest of the market profcssional may be in the value of ! %., t < 7) at expiration, rather than the immediatc Sf, and tbe modeling of I immediate one-step probabilities may not say much about this. Hence, a market professional may bc intcrested in the probabilistic be- lt havior of the expiratitm point value Sv as wcll as in its immediate behavior. And the probabilistic behavior of thc accumulated changes may be quite different than the #;. that govcrns the immediate changes in i. l n1s 19 tlle because in n Peiodsr thc Si may assume many Values different from .Li case just u 2.Si or dui. . Thus, we considcr the probabilistic behavior of the ratio: .
With a linear equation we can calculate the mean and variance of the Si random variable log J. eas jjy:
.
a
t
k
ui
.
''
E log
,
.flzpglrc
.
Ftzr log
F'(7r Herc, rcmcmber that
.z'
jr
'::' u
::
)
=
.
&
= Z log d + -
Z) log (/
zs ltg
d.
(%) t .
E
:
(87) h
As discussed in thc previous paragraph, this last equation is rlow a linear t function of thc random variable Z. 1og converts a produc! into a O'I'he ut, 4 are multiplicative parameter's. Taking whichis easier to analyze in asympotic theon'. Central limit theorems ure formulatei general, in terms of sums.
m
'
:
nptl
- p).
(92)
F
=
-
.
g.r
(94)
(rl'l
(95)
,%..
-
value is easy to calculate. If wc have rl independcnt trials, each with a tlacn he total number of apected movements will be np. The Vrf Z) is slghtly variance of z ftyr a singlc trial is: morc cornplicatcd. 'tup''
'ne
pl 1
.
.
..
j
:
'
(91)
'tup,''
F n trials the rlpt1. p). -
'E'
in
log #
b-l-heexmcted 7rtbabilityp of
.;:
:
+
-f
'(
= I
2
=
:'
'
lhc
=
p).26
One can also get the approxmate (asymptotic) probability distribution of 1ogStwh/si. First, note that the log Si-vn/ui is in fact logarithmic changc in the underlying process: S-vn 1og log Si-n 1og (96)
,
(n -
jog
u pj
xf-q.r sj
-
This is equivalent to a process that takes steps of expected size JJ.,' over rtl, F1, and whose volatility is equal to (Z'UX at each step. Hence, the mean and variance of the rate of change of Si modeled this way will bc proportional to ,'. suchstochastic processes are called geometric processes.
'.
Z log u +
(89)
g grjy;j n
2
:.
=
l
Iog
=
N
$
'Si-n
lerlzj.
logj(,lJ?l + n log d pu1 n#(1 - p4 A glog
=
log S;
-jj
(88)
2
t.f
=
+ n log d
(93) k Replacing this and the values of u, d, p in both (91)and (92)we can get the asymptotic equivalents of the mean and the variance. In other words. with u, d, p, given in Eqs. (74)-(76),the fkst order approximation gives:
' r.
which depends on the way the main paramctcrs of the binomial-tree are modeled. The discussion will procccd in tcrms of an integer-valued random ). variable Z, which represents the number of '' up movements obsen'ed beE twccn points and + n. According to this, if beginning at point i, Si expericnces only movements, then Z r!. If only half of the movements n12 !' and so on. (.: are up, then Z We investigate the probabilistic behavior of the logarithm of Si-vnlsi, j L instead of ratio itself. because tMs will Iinearize the random cffects of uj, yrjri in terms of Z.M Before we proceed further, we eliminatc the i subscript from ui, #i. rf ( given that at least in this scction, thcy are assumed to be constant. We can now write; Gtup''
Si-v''
n
..
''
sj
S-vn - Si
E
'r::
(85)
glog gj glog
u pj Flzl
=
But we lnow that the f'gZ) is simply np and the rtzrgzl is npz Rcplacing these!
''
S.s-l-zl b.lq:
Si-vn
?:
.
.
varianc.e
-
.p)2 +
(1 -
p)40 - #)2
of the a indcpendent
=
p41
lmovenlents
-
.p).
(90)
will be n times p(1 - p),
r
E' ;
.
'j'
(--1.IA p'r E R
2c2
-
Tle
8
Rare Events
Process and
wiener
It can bc shown that if we adopt the parametrization in (74)and that corresponds to normal cvents, then the distribution of (log 0 by is approximated, as fr2), N(>(nA), glogSi-vn log-/j ks/.j-u-logl
10 Exercises
;
:
(75)
(c)
.
.; '
-+
-
-
2o3 ,
(1 -) -
n
1
-.s
Ixt the random variable X,t have a binomial distribution:
(97)
n
y
That is, (logs-h. logs'/l is approximately normally distributed. ' that lf, on the other hand, the parameterization in (78)and (79) correlogM) spondsto rare events is adopted, tben the distributionof (log given by; j approximately be 0 will as a .
-
. a
Whefe
'
vi-vn
-
Cach
Bi is independent and is distributed according
-.
'
11
,
glog
1ogi/j
-
vi-vn
'w
Bi
(98) :
Poisson.
These are *0 examp Ies of Central Limit Theorem, where the sum of a large number of random variables starts having a recognizable distribution. this divergence between thc applications of central hmit what
yt1 s i' izzz
E
!
with probability p
t)
wth probability 1
=
to
p.
-
we can look at Xn as thc cumulated sum of a serics of events that occur over time. The events arc the individual Bi. Note that there are two paramcters of interest here. Namely, the p and the n. 'T'he Erst governs the Probability of each B i, whereas the second govcrns the number of events. The question isa what happcns to the distribution of X,t as the number of events go to infinity? nere arc two interesting cases, and the questions below relate to these.
C ,
t ?
caeses tbeorems? Of independent .! It turns fmt thata in order fOr a Plrperly scaled sum random variables tt converge to a normal distribution, each element of negligibk. The condition for asymptotic the sum must be asymptotically l negligibilityis exactly the one that distinguishes normal cvents from rare events. Thus, with thc choi of para meters for ui di pi for rarc events, ;; the events are likely to be asymptotically nonncgligjble, and, convergence i, will be toward a Poisson distribution.
Gtevent''
,
.
'
,
(a) Suppose
,
9
'
)
--+.
x,
-.+
=
=
(19+:,no rare events 'rhediscussionthatcharacterizing innovation terms have a snitenumber of possible values ; tion assump the discussion signincantly. x reader interested in the formal 3. simplined justiying t laestatements made in tus chapter can consider tl,e arguments approach martingale dkscuss to Bremaud (1979). sremaudadopts a of poim processes- which can be labelcd as generalizations of . dynamics ,, Poisson processes, is covered iri Merton
now, n
=
.
References
wlle
0 such that A np remains p constant. nat is, the probability of getting a Bi 1 goes to asequencar, of getting a zero as n increases. But, the expected one remains the same. nis clearly imposes a certain spccd of ConvrBence on the probability. k) Write the implied formula What is the probability PrlXn as a function of #, n, and k. k) as a functon of the three (b) substitute np to write prxn 1. tcrms shown in ouestion (c) t-ct n -+ x and obtain tlae poisson distzibution..
.
.,
-
-
,
,.
prxn
,E
(d )
!,
:
10 1, Show that as n 1 ( a) 1 -u
...+
tx
(1 ) (1 (b)(y -,,) -
-
.
A
-
n
-->
.
'k j
k
-
.
c-
sxercises
:
p
1
)
-->
1
,
? ,,
'
.
uemem yej. tjaat jjurjjs
-
k)
=
c-
k!
, .
tjljs jjmjting process, the p --> 0 at a certain speed. How do you interpret this Iimiting probability? Where do rare events ,t in?
r'
1
1 Introtluction Wuw.
G-ww
.w.
''':*v.
x .
w
.
:
,m4X.
expansions. After taking into consideration any rcstrictions imposed by the theory under consideration. one gets the differential equation. At the end of thc agenda, the hmdamental theorem of calculua is proved to show that there is a close correspondence between the notions of integral and derivative. In fact, intcgral denotes a sum of incrcments, while dcrivative denotes a rate of change. lt seems naturat to expect that if one adds changes dXt in a variablc Xt, with initial value A% 0, one would obtain the latest value of the variable:
E *
*
te ratlo
tOc
ln
:
*
!
astlc
,E
e ts
V1rO
=
1.
'
@
.
l
dXa Xt. (2) () This suggests that for every differential equaticm, we can dcvise a corresponding integral equation. In stochastic calculus, application of the same agenda is not possible. unpredictable arriveg continuously, and if equations representing lf the dynamics of the phenomena undcr consideration are a function of such noise, a meaningful notion of derivative cannot be deMed. Yet, under some conditions, an integral can be obtained successfully. This pelnnits replacing ordinary differential cquations by stochastic differential equations =
('
Tlte Ito lntcgral
(' 1
.'
.'t :;
..(.
LKnewf'
; ' .'! i
:: .
;.
'I. .
1 lntroductlon
? '#!..
and integration operae'. One source of practical intcrest in diferentation nelil tions is thc need to obtai' n differential equations. Differential' equations used to describe the dynamics of phpical' phcnomena. A simple linear dif-'t ' ferential equation will bc of the form dXt AXt + Byt (1)
dXt
.L+h- xt
::.i
''
,4
,k'
'''
::
204
r
j
fsatu
=
(4)
l
s;
dSt
ast,
1)
dt + rst, t4 JWt, t Afler we take integrals on both sides, this equation =
l
I0,x). implies that
/
g'tuu, zls'u, u) du + dSu u) Jlzl'u. (6) t3 p Wbere the last term on tbe right-hand side is an integral with respect to increments in the Wiener process 1. =
=
)
(3)
doing
l
'
(0,x),
Now, consider the SDE which rcpresents dynnrnic behavior of some asset price
.
.
f G
St.
E
1If B 0, thG cquation is said to btl homogenous. Wlhen yt s indepcndent of /, the spte,m ; becolnes autonomous. Otizemise, it is nonautonomous. Xt. future path mind ftar 2For examplc, the engincer may have in somc desired nen :,. issle is to f nd thc proper (y,) which will cnsure thtt X( follows this path.
t/@(,
Can be used to give meaning to dXf. In fact, at various earlier points, we made use of differeatials such as dSt or #11$but never really discussed thcm in any precise fashion, The definition of thc Ito integral will permit
'
-'
n
r+
where dzjldt is the derivative of Xt with resnect to f and where y, is all.. and B arc parameters. 11 exogenous variable. Ordinary differental equations are necessarv tools fol' orac tical mod- .? . eling. For example, an enpnccr may think that there is some variable h' of Xt, determines future changes in Xk values that, togethcr with the past approximated by the differential equation, wlch r'-qft This relatitmship is ' c applications. be utilized in various equatbm differential obtain used the ordinary The following agenda is to First, a notion of derivativc is defmed. It is shown that for most ftlndioilj! of interest denoted by Xt, this derivative exists. Once existence is es F'CI'i lished. thc agenda proceeds with approximating dxt/dt using Taylor ''''
at dt +
.
,
''''
=
wherc futume movements are expressed in terms of differentials dxt, dt, and 4/14,; instead of derivatives such as dxtldt These differentials are desned using a new concept of integral. Fbr example, as h gets smaller, the increments
,
dt =
205
1
+YVVFW'VXYW
:.
: ''
( r.
*'
.:
206
C HA PT ER
.
9
Integration
in
Stochastic Environments
Introduction
.'
.'
The intcmretation of the integrals on the right-hand side of (6) is not immediate,As discussed in Chapters 5 through 7, increments in p) are too erratic during small intervals h. ne rate of change of thc H'; was, on the smaller.3 If average,equal to h -1/2 and this became larger as h became inlinite? would their be not erratic, incremcnts sum these are too nis chaptcr intends to show bow this scemingly dilcult problem can be solved.
rcasons- First, the Etfst-vk 5)Jwas set equal to a jrt-order Taylor scries approximation with respect to : -
;
,
Etst-k (
,
second,the
'
asu,
Ito lnterul . 1 The
and
:'
EL
dst
i'
'it
g.t,s,
;y)
l ''l-
esu, ;
,
tJL,
.
'
,.
:
ysu, ;
uj lrzj', ,
(1g)
-
Taking the intcgrals in a straightftarward way, we would obtain the difference approximation: /)J? + ojz,
r)(p)+, -
.p;j.
EEfIiv
E .
:
. ..(
(1(9'' t
aLut,t4h
+
G(k,,
fl.f#;.
.
(11)
This is the SDE rcpresentation fn finite intervals that we often used i# previouschapters. The representation is an approximation for at least twe
:L(7
',yA; mde ocbangc we mean tue staozard dewaton o .;., - ;,; divded oc avera,e standard deviatons In Chapter 6 it was shown that under fairly/IL/'general assumptions, thc ). Unpfedictzble shtcls Werc Proportiona1 to
ssy
ne
.
>
...
''
'
.
f
: t. :.
.(
zl/rp;.
(14)
u .Practiee :1 the 1to integral is used less frequently than stochastic differemial equations. Practitioners almost nevcr use the Ito integral directly to calculate derivative asset pris. z.ts will bc dismzssed later, arbitrage-free prces are calculated either by using partial differential equation methods or by using marungale transformations. In neither of these cascs is there a jxkz . caleujate any Ito integrals drecuy. lt may thus bc difficult at this point to see thc practical relevance of tbis concept from the point of view of, say, a trader. It may appcar that denning the Ito ntegral is essentially theoretical excrcise, with no practical a implications.A practltioner may be willing to accept that the 1to intcgral existsaad prefer to proceed directly into using sDEs. reader is cautioned against this. Understanding thc delinition of .tbe Ito integral is important (at least) for two reasons. Fjrst. as mentioned eartier a stochastic diferential equation can t,e desned only in terms of the lto ititegral. To understand the real meaning behind the SDES, one has
..
?
6
u) J')p;zz v(-$,,
1.2 The Pxucf-ictzl Rclewtmce OJthe It() Infegrul
.
:$
side is deEned in the lto sense and
.measurable
'
&
;+/,
,
.
Rcwriting.
tjgj
That is, the diffusion terms of thc SDES are in fact 1to integrals approximatcd during inlinitesimal time intervals. For these approximations to make scnse, an integral with respect to J#; jy jhrst jd j)e Jejined formally, Second, we must impose conditions on the s ou way ast t) and ojS(, t4 move over time. In particular, wc cannot allow ty cse y.t parameters to be too errac.
(8) g(
j :, intemal. linite time h is where some t//-crczicl7 apprimation that wo From here, one can obtain the hnite . k smali, and h is lndced, if 8. several Chaptcrs times in 7 used auu, uj and , 1.f) (rt-u, may not change vcry much during u E (/, t +. J,cspecially if they 1 are smooth ftmctions of Su and &. Then, we could rewritc tbis equation a? t+k /+n JFx du + abt, /) (9) l r) au l 1'+ /1 k E :
s-vs st < az,
Jjjt,
2+
7.1
0'6Su, u)
alSo, u) #a +.
-
(
,',
/'-.1-/1
=
crqvhg t)
'
f.ho
:.'
.t
ayy tj dt +
tjw second jntegral on the rfght-hand that as h --> 0,
:
(p)
tju
precise way, then onc could integrate both sides of
St-vh-
=
..
0
is defined in gome soE in (5)..
(
j
I
(St, tjh.
wjl jn fact mean that in the intqral equation, t+h a/j ay, u) du + du 'z =
,
Obtaining a formal defrtititn of tlle Ito tcgral will make the notion of a stochastic diffcrential equatitm more precise. Once the integral
=
u), tz.t,5k, ula u + ) were approximated by their 1. Both of these approximations require some smoothness value at u conditions on asu, u) and u). AIl these imply that when we write su,
;
SDES
)1 e g/,f
=
'
1
-
.
E'
c H h P z ER
208
-
lntegration
9
in
Stochastic Environmens
of the Ito integral. Otherwisc, errors can be made in applying SD& to practical problems. This brings us to the second reason why the lto integral is rclevant. Given that SDES are defmed for inhnitesimal intewals, their use in linite intervals require some approximationa. In fact, the approximation in (14)may may Thcn a new approximatitm will have to be not be valid if is n0t delined using the lto integral. This point ig important from the point of view of pricing fmancial deriva(IOCS Calclzlations using tinite inteNals. tivess gince in Pradicc O11e aiWal's dayo is Ciearly nOt an inhrlitcsimal inten'al, and the utiFor examplep lization of SDES for Such periods may require approximations. The preciK approximations will be obtained by taking into consideration form of these of lto integral. the delinition To summarize, the ability to go from a stochastic difference equatioa defined over the linite intezvals,
to havc some
a-spakk
k
u,-
+ h
k,
1, 2,
=
.
.
.
,
(2&
ast,
#f
1)
ojsu
4.
,
JW';,
t
e
'
E ,
y, 7
2.1 Th e Riemunn-:feltjes
.
L' .
and vice versa, is the ability to interpret (/H?;by dehning Jj in a meaningful manner. 'rhis can only be done by constructing integral.
In this partic-ular casc where the derivative sticltjesintertl can be written in two ways:
, u) #1Fk'97 a stochastik :
z .
.? ! '
'
..1 f
(;
/ (x;)dxt
t
rnw 1to integral is one way of desning sums of uncotmtable and unpret dictable random increments over time, Such an in tep-al cannot be obtain by utilizing the method used in the Riemann-stieltjes integral. lt is use ! ' to see why this is so. rando As seen earlier, increments in a Wiener process, lpl rcpresent variables that are unpredictable, even in the immediate future. The value the Wiener process at time f written as W(.is then a sum tf an uncfmnt.a ) number of independent increments: r 0
a
.
0
(1 4
.
/
'
f) (Remember that at time zero, the Wiener process has a value f zn d writc Hence, Hz;) 0.) This is t he simplest stochastic integral one can
i
.'.e
l
).
(20)
gxt) dyfxtj.
(gj)
Rere, we have an interal of a ftmdion glxt) taken with respect to F(.). Kmilar notation occurs When we deal with expectations of random variables. l7or example, F(.) may rcpresent the distribution function of a random variable xt. and we may want to calcuLate the expected value of some
=
.:
dFh
F
.
tj)!/
=
jj
'l
,
,
16
=
exists, the Riemann-
ittegal, ln the notation on the right-hand side, the integral is taken with remect to F(.). lncrements in F(,) are used to obtain the integral. Wc can complicate tlle latter notation further, For example. calculating we may be interestcd the integral
.
l
v
.
/(.)
'rhen,
'j '
,
(19)
The integral on the left-hand sidc is taken with respect to xt, where t varies from () to value each the multipleu of at is by the x, y(.) w, ieuitesimal increment dxt. nese (uncountably many) values are used to Obtain the integral, nis notation is in general presea'cd for the Ricmann
:;
'j
s
) jjxt). =
dx l
,i
2 T e Ito lntegral
Integ'rul
Ljyjyt
'
l+h o'v
(18j
.
we have a nonrandom fundion F(x/) where xt is a deterministic suppose variable of time #(.) is continuous and differentiable, with the derivative
(16)
(0,
X),
u) Jp'x
7
..: :
',
f)
by integrating the inno-
ne jntegrals n (17)and (18)are summations of very erratic random variables, since two shocks that are e > 0 apart from each other, JW/Iand (yjjj nl.f are still uncorrelated. The question that ariscs is whethcr the sum of such erratic tcrms can be meaningfully defmed. After all, the sum of so m aa:y erratic elcments can vely well be unbounded. (uncountable) again the way standard calculus desnes the integral. consider
)
+ rrst
,
()
L'
stochastic differcntial equations, =
integral is obtaincd
.
(15)7
n,
Integral
?
tone
-
ICo
A more relevant stochastic vation term in the SDE:
understanding
Rsmall.''
to
2 The
.
:
1.
k
jv
..(,
)
side of this equation is a sum of clements such as
-
+,
.
)
-
F(x,..)1,
EIJ(sk-:
'll
-f-
,
k=l
J&
=
rl sl.-yilpxa :)7.j(tg(5't-1,
,
,
n.
(27):
t,'(-$,,-1, J'''I k=1
klEafK,l,
(28)
:1
kll
+
:
J'''ltrtus-j
k=1
(25
whereas
.
2
and (F(x,,., ) F(xj)1. The srstterm re which is the product of The second term rcsembles the point ted at g(.) l xt a rescnts eva ua g(.rto,)gF(a:2.,) F(xlf)1 can be visu elemcnt dFlxij. Eacb crcments g(x2j.I ). as a rectangle with base (F(a';,.( ) - Fxti )1and height
.
n-1
: (1
.
.
Sk on the left-hand side of
.'..
I (..r, ')IF(.x,,.
1, 2,
=
a limit
(24
f
kllJFk,
,
Can wc use a methodology similar to the Riemann-stieltjes approach and defzne an integral with respcct to the random variable St as (sometype otl
k
.
(rsk-j
,-1
l-lf-sk =1 -
$!
Thcn the finite Riemann sum Ji is deined:
k4h +
.l,
n-1
'
.
au
Suppose we sum the incremcnts
';.
(O)
=
usual, it is assumed that F
,
kll/.&B$1
,
(29)
nh'?
=
'
.g(x;f-,)
-
'
s'That is, if this limit convergcs. are Inauy different wap rcctangles can approxirnate tllc arca under a tmn'e. one t'an piek the zectangle lhr same way, but change the height of the rectangle to tlase of tlle +.x'i - eithet 1l ). ., (.vti ) or to #( a ?By considering intenrals of equa: Iength, the partition or ) Tl can be matle fmer with & x. Ot/lerwise, the conditio.n sup.. 1I/ - t.-,I 0 has to be used.
,
.
+,
.
G'riaerc
-
4Whcn the function #( second or third moment.
.) is the
squarc
or the
(n.11x,
.
:
.k '
of xt this intcgral will simply be ..? ,
:.
(.: ;.
(
iz'i
.
' 't... :
->
-...
.
.L .:
:
L
l
!
..
2: ';
.
C H A P T ER
212
*
lntgration
9
in
Stochasric Environments
2 The 1to Integral
'
@ 'J
21J
J:
'
Erst term on the right-hand side of (29)does not contain any ram dom ttrms once information in time k becomes availabk. More impon increments in time h. By deftnitantly,thc integral is taken with respect to r zt ' is a smooth function and has finite variation. nis means that tion, bme the same procedure used for the Riemann-stieltjes case can be applied to de fine an integral such as8 nc
55.
0
'
The use of mean square conveqence thc sum
11 ''
j
n
1: '
p
i)
a(Sa, u) #1,F.
'''
aS
1: 1
ujdu
lim
=
n ...s x
(31) '
lftxs'k-l1)1. ,
k= 1
; .E.
tj
.:.
jjm
,
'.'
term
Ep'k
-
prk-1
.G (32):
1
:)1pk
H,',-1J
-
:::::1
(33t
:.
to a random variable.
is an integral with resped We can ask several questions:
;
Sk-Sk.j
.
,
* Which notion of limit should be used? nc question is relevant beca the sum in (33)is random and, in the limit, should converge to a r dom variable. The deterministic notion of limit utilized by Rjem Stieltjes methodology cannot be used here. the s . Under what conditions would such a limit converge (i.e.,do in (33)really have a meaningful limit)? What are the propcrties of the limiting random variable? .
: E (.. '
.
.j'
#li-l
,
k=1
1.
(
side ca'a bc written
-'it,-
-..
T(z(-(,-I),
'= 1
kllltl
,
kljp'k
t) be nonandcipative,
aru
.(j.
g-yj
ltt Ztegral within the context of
- pk
..jj,
k
r:,.
1 , g,
of the
.
.
.
,
n,
(:!y)
f
in the sense that they are inde-
) be
Rnon-explosive'':
T
E
gjuj
aen
,
0
.
the Ito integral
t)2 dt
pa-i
(4D. '
r;-1
limE
x/,x/..
N'-Km)
.,l
,' -
jjm
t
l''
-
=
s-sx
..j
(48)y
0,
::ct
=
x,.
*1+1
x. .-
v. . *.
Below we calmzlate tMs limit explicitly. ''f
'Fhis
:
cquare
Ward
be a good candidate for Z. Takng expectations
Will
:
x2
;,.,
ab From
(44),and
lctting a Je;
=
-
1
2
=
1 2
-g(J
=
i=
and b
x
n- 1
+
ll.rtf+
b)c =
xl,+1 .
xri.., )z
-
..-:: :
gives
.'
c,
A.Y/j., :)-
(5zg.
But
xt. 1
-%'Ii+,
,
'
(5
,. .
:
:2
.2 .
t *'.
't 1
(t +. j
jj
.jy
(,jjy;
..
=
...
E j=()
?. 'h .:
'4sy
.:.. ':
.
constructon
fj)
=
z, we can
T evaluate
(59) the expectation
yv
,,-., ...
;, E
A
MA
f=0
,
k
a:,f+, =
E
(j+j -
this as a candidate for
/1 ' -- 1
tty
zs
+
x ow using
:' 4 -
f=0
to
j=o
-
xtiz,
=
Fl ..j
gj
a z - b z 1,
-
(+, j
2=0
f=tl
which simplgies
,,-1
E'j.,$xz
=
>.
.
OC
,:-1
u-1
E
(5
in a straighttbr-
Way,
:.
a + b) 2 = a z + yl + ga pp,
(57)
.
jx()
Limit Explicit Calculation of Mean random variable #' stcp by step t4': intcnd calculate the limiting to we clarify the meaning of the Ito intcgral as a mean square limit of a random'' ' sum. 'T'he Erst step is to manipulate the terms insidc P's. We begin by noting that for any a and b we havc ;J
2 4.1
2 A.rrk..!
j .
.
(56)
(j.
j
D-
E
:
.'
=
.
(49)yi!'
.
*z
a.x2 t+z ..z
f+l
.-b.L?
.x/ih1
i'=()
2
''squares''
: i.
whcrc for simplicity we let
s
1
Ij.j tjafs exxresgon, there arc two on te left-hand side, onc s due to the random variable itsclf, and the other to the type of limit we are using. Hence, the limit will involve fourth powers of x, First, we calculate the expectation:
:.
J=-fl
(55)
.
kxa
:
2
'Vl
In other words, we now have to find the Z in
.q
Or, equivalently,
a.v2 w,
-
-r'
Note that xv is indepcndcnt of n, and consequently the mean square limit of I'?k will be detcrmined by the mean square limit of thc term
.
0,
=
,,''1
1 xz 2
.-
4 x t .j. 1
t.
=
a. .j.
c z'.
,,.:
j pj
i=
j'-xg
(6n)
,,..j
jkwjz
jjax;a
+ g.z .gz y..f-,j j
z aayy i.v y
jrxt)
,
r
!
j
.,
i!
.
i
,
;'
E
'j
c HA rT ER
218
Integration
9
.
Stochastic Environments
in
-x'z/
:!
.
f+
j
l
The Ito Integcal
-
.
webtain
l)2,
3(fi+: -
-
I.. 1
Pk
.
t
Ef-('
.
l-1
2
r;-1
J.*2
E
li+l - F
f=()
=
f =p
n-1
+
i
+F Now we use the fact that (.+: ti size. We bave the following!
same
2
(f+l rl - 1
- 27-
f=0
- 3(f +l
fj)2
-
=
/./)
-
Tjle tenn on the right-hand
(63)'..
n-1 a-1
2
J=0
/j)(/y+:
j
)
-
-
'
we see
'
...'
'.? '
''f
r ()
(64)
'
,
nn - 1)/12,
J
''
(65) .
variable 2llxz F
.1
=
-
-F2
=
.-nnhD
(66)
.
-
zl-1 .';,
jjm E ''*X
2
.(
1
T
-
i+ !
1)/12 3nhl + nn -
=
hl
-
which means that
(67) r .
f=0
kt '
2
Fl-1
E y=()
This implies that a,s n
.-.+
Ax2 F (-1 -
tx),
=
(O)
2T.
2i,
-
I
F
=
1im lhT h -.+ n
=
0.
'
.
F
(73)
,
.' '
- F
=
().
(74)
w
a (tsxt)
(75)
,
()
which can be interpreted as the sum of squared increments in xt. lf this integral exists in the Ito sense. then by dcfinition,
'
2
kx, i, ()
lnhl
thc size of the intewals will go to zero, and
n-l
E lim l-.+0
=
-
2
hx2I
j=tl
t)
.'.
1 c xz 2
lt is interesting to convert this into integral notation, Assume that xt is a wiencrprocess and consider the integral
4
,
n- 1
.
r'
,
-
r).
.E.':
Put a 11these together E
2- 27- j=0 (fj.j-j /j)
=
2.4.2 An Jakywrllyu/ Remark I F1 thc previous section it was shown that
l
'
rI-1
xtdxt
In the case of the Riemann integral, there was no additional term F, This is one example where the Ito integral can be calculated explicitly using mean square limits. We 'Iind that the Ito integral is the limiting random
E':
,
F
that the ,to integral rcsults in a diffcrent expression from that in ne Ito integral is given by
standard calculus.
'
an (j
(72)
xt dxt.
()
;
(/j+,-
side is the Ito integral,
. Ei
lf).
(7j)
.
T
=.()
f
=
. 'E
(l+1-
(70)
,
1 -L x2z - T 2
F(p;Ij2
/.
?nhl
2 A-r/.a. =u0
:-+CM
Et
1
N
ff)(+1
h. for all i, since al1 intervals are the
=
Ijm
'
;,
j< /
xv
-
.
;.
z-1
=
rl-1
2
''
g, 3(/j+j - t;j
2
2
.
..xr.+1
.''
o
-
is F
jq.j
wen-1jind the mean square limit of p; by using thc mean square Irnit of c just ojxaiaed.
. .
.x2
/.()
.t
=
.
(62)
-1
Ed.
Thus, the mean square limit of Going back to )'.,
!.'
and
.E7E.;r4 1
219
(61) '
(l+.t- n)(/+1 bt,
-
F
.
'
We consider the components of the right-han d s ide of (60)individually. Reltlizing that Wiener process increments are independent, Fl.-rz,
.
.
( :
,E
..
.
.
'
(. .
(69)
N-
.
E ljm -.+
.
..j ..
:
r :
. j
n
:
. ';
-
: '
x
1
fx()
1.x2,f.I
r -
0
2
Ldxtjl
=
4).
(76)
'
?t J .:
C i4 A P T E R
220
lntegration
9
.
But wc know that
z'
in
Stochastic Environments
=
(7X
T1
;
we obtain a result used to working
0. This is the case because increments in the Wiener process have zero mean and are uncorrelated', Fl(z(B$+a
Pxoperties of
:,
l+A
tyl'l'ulH''u
.
.
t
('
.
To approximate such an integral with a Riemann sum, a rcctangle with basc B$+a W6an d hcight frgR-''lflj may be used218 z
;
-
Fxumple J. 1.3 Suppose asset prices follow the SDE xdrl
dSt
=
avh
,
f
) dt +
3
a'bbj , 1) #Hz),
0S
(93)
f,
=
J,
j .
::'
:@
(94)
sg('':+ac-''':)(w;+a
!
E
an d
.'!
dsu
lha
B$
#Su du +
=
t
J'
=
t)- l
tA-u
0.
l
'
evsu dll/u
p;2)1w;1
-sgJtw$ta- (99) (100)
and + 0. This means that tlze approximating sum has a conditional exPectation that is not cqual to zero, It ispredictable. Clearly, this contradicts
:'.
.
t-blk
H$).
1 = -,2
,
That is, both parameters are proportional to the last observcd asset price St. and integrate this SDE'. Consider again a small interval of length
lGlteznernhcr
,4$)11,;1 -
EE
(95)
=
( /p)ma-
operator F/(.1 to the right-hand
.
t) ast. zz.(-&,,
p;
g
side,
'(
IJ'S:
w;+ a +
fw
But, applying thc conditional expectation
:
where the drift and diffusion parameters are givcn as
asuh,f)
f+ac.pvadll'u
t'
:
.
(96) .'
R7llere
''I%T
.
i
:, ; :
!C . -.
;'
we abuse the notation
in writing
simplicity, we tsse one rGctangle. canbu used.
.s'(1&;)
=
5't. But it simplic,s
thc exmsition.
In fac, much fmer parlitions of the intewal
(t, z+A1
.? !'
224
c H A P T ER
.
lntegracion
9
in
stochasticEnvironments
'
3 Przperties
an innovation
i
.
the
tcrl'n.
claim that the integral on the left-hand side represents
q
q drf.:t
# ()-
()
(101)
'lhis
=
,
valuc of
'/'l.
- . 0, (114) E /'(p), flJH'; () , '
fvh,
f) +
#(r, -
)j dst
s
:'
'-xloougs
not
suarastoed.
w
.
.;.
:
.)-
=
i uuc ()
dt discussed earlier,
=
()
yt,y;,t) trs;
r
.j.
gqyt,
j;
#.;.
()
tjj'y;
JumpProcesses
f(Mt, )(A/rf+, - Mt,j,
(118)
pathwise, 'Fhis P;, will convergep and the variation of the proccss Mt will be tinite With probability one. Under thesc conditions, we say that P'u converges
:
it may exist, determiniss sucil a Iimit expliciusis
4/+/2
x-aN-1
P$
.'
=
.
9
What complicated the dcfinition of a stochastic integral was the extreme irregularity bot of contineous-time mmingales and of the Wicner process. This made a pathwise delinition of the integral impossible. Do we havc the same problem if wc have a stochastic integral with reSpect to some jump process? Could one use the Riemann-stieltjes integral when dealing with, say, Poisson processes? the answer to this question is aflirmative under some surprisingly, conditions, Supptue a process Mt is a martingale that exhibits hnite jumps only and llas no Wiener component, Trajectories of such an Mt will exhibit occasional jumps, but othenvise will be very smooth. 'Thcn, one could deline a J(,,
2
E asome''
u)2 du
.jtm,
E
5 lntegrals with Respect to
(113) 7.*..
S/f1
t =
'
k
,
convergein integral.lg
v
. '
n-1
i zzz()
a
-
Ito integral also has some propeztics similar to thosc of the Riemann-stieltjes intcgral, In particular, the intcgral of the sum of two (random)functions of St in (6) is equal to the sum of t'heir integrals:
.;
(112)
wherc (,%Jis given by (6),exist? It turns out that if thc function f (.) is contjnuous, and if it is nonanticipating, this integrai exists. In other words, the finite sums
'
rrjae
.
.
fsu,
jtp-v, u) JJ,p'.
azuiflmz
#.a
:
l
u) dsu,
-
a Note the recurling use of the equivalence
E. ,'
One can ask the question: when does the lto integral of a general random function fluts l),
ulj Ju
(j
.'
4.1 Existence
0
aE'g/'tr#-, utgl p,
=
E
The Ito integral has some other properties.
,
dW?ld #(W'u, t.) #Bi
&)
t
'
4 Other Properties of the 1to lntegral
'
/(Bi
a
';.
are given by the vari-
second moments
'rhe
where 1#)is a Wiener process. ance and covariances
.
''see
227
JumpProcesses
Respect to
with
pathwue. . .
'
!
228
C H A PT ER
Integration
9
.
in
''.
Stochastic Environments
8 Exercises
:i '1.
6 Conc lus ions
in the following-.
,
!.
nis
chapter dealt with the dehnition of the Ito integral, From the poit of view of a practitioner, there are tw'oimportant points to keep in mind. First, the error terms in stochastic differential equations are delined in the sense of the Ito integral. Numerical calculations must obey the conditions set by this delinition. Second, the stochastic differential equations routinely used in asset pricing are also dehned in the sense of the lto integral. Above all, we saw that the lto integral is the mean square limit of some random sums. These random sums are carefully put togethcr so that the resulting integral is a martingale. We also discussed several examples and showed tlaat the rulcs of integration are in general very different in stochastic environments, whcn com. pared with the deterministic case. This was the result of using mean square
(a) Write the Riemann (b) Write the (c) Calculate (d) Calculate
E. j F '
E
/0 , /'j ,
FI
i
:'
./:::u1
) ;)
i
.
.
.
,
tn-
:)
). :'
2 !
'
? :.'.
'1
.L .
... (
=
fM -
6. Can we say that this is an application
y
tn
dujv + a(Jr)
=
J'JSJS. ()
4. In the above equation there are hvo integrals. Which integral is dclined in the sense of lto only? 5. Can we say that this is a change of variables?
.:
.
1,
.
l
s#H(
''
-
-
j=1
l
''
.z.
,
j= l
rl
rj-llpq g(/./ L/.,(u$-. p;,-.lj +
3. Now use this information to show that:
L'
(0.fI:
gf/p;, -
-
'
:'i.
consider the
:1
f/-lwzt--j
#(ur)
. '2k) ;'.
8 Exercises
,
f'u--.t, tn
How is this different from the standard formuia for the differentiation of products:
.:
nere are several excellent sources on the derivation of the Ito integral. Karatzas and Shreve (1991)and Rcvuz and Yor (1994) were already mentioned. Two additional sources that the readcr may find a bit easier to read The former source could be a very are Oksendal (1992)and Protter (1990). good manual for quantitatively oriented practitioners and for begming graduate students. lt is well written and easy tt understand. Technicalities are avoided as much as possible.
tbt fi
,
wc can always write:
.( .
7 References
0
.
:,
!
: 9
Usc thc subdivision of
-
y''
examples of evaluating thc lto intcgral,
W62:M
.
and
'.
.
r
intep-al in discrete time using an lto sum. the expectation of the three Riemann sums, the expectation of the Jto sum,
2. Show that given
;
Fortunately, in evaluating Ito integrals, the direct route of obtaining the mean square limit will rarely be used. Instead. Ito integrals can be evaluated in a morc straightfozward fashion using a result called Ito's Lemma. 'Ihis will be discussed in the next chapter, where we will also discuss further
(0,FJ and
of the above integral as threc different
approximation sums.
h,
Convegence.
1. Let J1( be a Wiener process defined over integral:
229
(
.
;
.
of integration by parts?
E'
E
2 Types of Derivatives
.):
'j
In (2),dFt is uscd as a shorthand notation for dF(St, J). This should not be confused with Ft, the partial of F(.) with respect to t The third is the chain rl/tr:
'E
.
gyqst tj dt
. :.
lto's Lemma
231
.
tusy
,
=
F, + #/
(3)
FI.
A fnancial market participant may be interested in thcse derivatives for various reasons. The partial derivative has no direct real-life countemart, but gives tipliers'' that can be used in evaluating responses of asset prices to obsenred changes in risk factors. For example, L measures the response of Fut tj to a small change in St only. As such, Fs is a hmothetical concept, since the only way a continuous random variable St can change is if somc time passes, Hence, in realitjr, f has to change as well. Partial derivatives abstract from such questions- Because they are simple multipliers, there is no difference between the way stochagtic and deterministic environments define partial derivatives. A classical example of the use of partial derivatives occurs in delta hedging. Suppose a market participant knows the functional form of FS le f). Then, this mathematical formula can be differentiated only with respec to St, in order to End the partial derivative Fs. nis Fx is a measurc of how much the derivative asset price will change per unit change in St. In this sense, one does not have any of the dficulties encountered in dehning a time derivative for Wiener processes. What is under investigation is not how Fst, t) moves over time, but how F(, ) responds to a hypothetical change in st,with time tixed. notion. It is assumcd that both The total derivative is a more time r and the underlying security price St change, and tben the total rcdifferential sponsc of Fgt, t ) is calculated. The result is the (stochastic) dFt. This is clearly a vezy useful quantity to the market participant. It repTesents the obsen'ed change in the price of the derivative asset during an itjten'al d(. The chain rule is quite similar to the total derivative. ln classical calculus, the chain rule expresses thc rate of change of a variable as a chain effect of some initial variation. In stochastic calculus, we know that operations such as dlldt, dstjat cannot be detined for continuous-time square j ntegrablemartingalesa or Brownian motion. But a stochastic equivalent of the claain rule can be formulated in terms of absolute changes such as dl, udSt, Jf, and the Ito integral can be used to justify these terms. nus, in stochastic calculus, the term rule'' will refer to the way stochastic diftrentiala relate to one another. In other words, a stochastic version t)f total differentiation is devcloped.
:
.
dfmul-
' : l '? t ('. 5
1 lntroduction
:.
As discussed earlier, in stochastic environments a formal notion of derivative does not exist. Shocks to asset prices are assumed to be unpredictable, Wtoo erratic,n Thc resulting asset pris and in continuous timc they become smooth. Stochastic differentials need they not be continuous, but are may to be used in place of derivatives. lto's rule provides an analytical formula that simplihes handling stochastic differentials and leads to explicit computations. lt is the main topic of
'
i'
'..),
, '. 'r
this chapter. We begin by discussing various types of derivatives.
2 Types
of
Derivatives ut
,
pF(.%, nS
/)
Ft
,
=
c?F(,%,/) pf
.
The second is the total derivative dealing with difcrentials:
dFt
=
r '. '
i
,
=
idsmall''
Rrealistics'
and Suppose we have a function FlSt !) dcpending on fwtp variables t, whcrc St itself varies with time /, Further, assume that Sf is a random process. ln standard calculus, where aIl variables are deterministica there are tllree sorts of derivatives that one can talk about. ne lirst are the partial derivativesof Fst l), denoted by
F,
:.
k :' ): ?p '
F, dSt + Ft dt.
' ' i', :
' G.
(1) '
ichain
E
(2) : :k
.
'
)
: 5':
230 .
)
:.
f)
H A PT ER
.
Ito's Lemma
10
E
2.1 Exumplc
Ftrj, tj Let us calculate
c
-r,(
r-/) jtm
1 .
.
ut
.',
(4)
.
..
the partial derivatives Fr, Ft: Fr
al'lu
=
JF
=
=
trt
-(F
/)Ie-G(
-
obtains new information about JF; and obsen'es a new as time passes, one jjwrement, d.%t.nis will also make F(.) change. The sum of these two effects is represented by the stochastic differential dFvh, t ) and is given by the stochastic equivalent of thc chain rule. be obsewed in continuous time. We again Let the random process the time intewal F1 artition I0, into n equal pieces, each with length h. P and use the finite difference approximation. However, we write this as an equaiity
''
We discuss a simple example before gtaing into lto's formuly. The examle P will help clarify the mechanics of taking various derivatives. Let Frt, tj be the price of a T-bill that matures at time F, and let r; be a hxed, continuously compounding risk-free rate. nen
t
w-t) jjsj
(5j
tYk
.
:
.fz l
IF Jf
'
=
rl
(e-r,(T-J)1(mj
/)
-4r
=
-
/)Ee--r'(T''01()(.)j
dr l + r,(c'
r,(T..-l)1()()j
at.
E'
f (a) ftat) =
':'
whereR denotes the
(p
'(
nis example suggests that when rt is random, we may be able to define the counterpart of total derivative, using the lto integyral, which gives a meaning to stochastic differentials such as drt. This intuititm is correct, and .. ; the result is Ito's formula. However, with stochastic rt not only does the E).. of drt change,: but the formula will also be different. interpretation '
,
.
3 lto's Lemma
'
k
'
The stochastic version of the chain nlle is known as Ito's Lemma. Let Si E be a continuous time process which depends on thc Wiener process E . Suppose we are given a function of St, denoted by Flut, f). and suppose w? J.E tyf passe:i change when time #/ amount in F(.) wouldlike to calculate the Clearly, passing time would inquence thc F(&, /) in two different wayK First, there is a direct inquence thrtugh the / variable in FS t /). SeconG '
.
'
,
,
. . ' Recall that such qllantitie,s are dcfined i'fl terms of mcan square ctmveqence and w Etlti. ' .. S
i.
.
..
+
f'(.b)@ -
.Yo)
+
1
f
,.?
,
(8)
n,
(Ao)(;r
-
x(,) .z + R,
(9)
remainder.
lncidentally, some readers may wonder if h-erneirregularty of st. z''(.) can be a smoothetis function Tsmxthncss''
'
(L
,
We apply this formula to F(,;, I). At the outset, F(.) has to be a smooth functionof st.l But there are two additional complications. First, the Taylor scries formula in (9) is valid for a f (.x)which i.s a function of a single variablex, while Fvt, f) depcnds on 4w variables, St arld t. Second, the formulain (9) is valid for deterministic variablcs, while St is a random process.Before usg Taylor series, these complications must bc addressed. ne extension of a univariate Taylor series formula to two variables is Onc adtls the partials with respect to the second variable, straightfonvard. With two variables, cross partials should be included as well. The applicability of the Taylor series formula to a random envimonment ig a deepcr issue. Firsta it should be remembered that some of the terms in Taylor scries are partial derivativcs. With respect to these, one does not haveany difliculty with differentiation in stochastic environrnents. Second, We have differentials such as dst. Here, wc do need an adjustment. which is in tenus of thc interpretation of the equality and not in the Taylor series txpansionitself. ne formula for Taylor series expansion will remain the
:.
..'
.
.p,
,
E
,
dljr 1:
.
.
.:
.)
1, 2,
=
,
(6) g.
Note that these partials will be the same regardless of whether rg is deterministic or random. By taking these partial derivatives, we are simply calculating the rate of change of F( with respect to small hypothetical changes in rt or in 1. O n the other hand the total derivative relates to the actual occurrenc,e of random events. In standard calculus, with nonrandom r/, thc total derivative of this particular F(.) will be given by
k
akh + JklWzrp,
=
using thc mean square equivalence between the left- and right-hand side as --z 0, nis notation will be presen'ed throughottt this chaptcr, Also note k4 k) to ak and for trtxk-j that wc shortened the notation for tzt,v-j, to n We calculate Ito's formula in this setting, using the Taylor series. Recall the Taylor series expansion of a smooth (i.e.,inhnitely differentiable) ftmction /'(x) around some arbitranr point
r
''
..I
=
2J3
3 lto's Lemma
L
.
)..
Sxhastc
u.. ' .
Statemerlt :
.E .,j
does not contradict the exor sf and still be a ver.y irrepllar process. lrrcgtzlariy hure is in the sensc of' how F(.) claangcs over time- It is not a abou! laow relates to F(.). s
!
(-2H A
PT ER
.
Ito's Lemma
10
3 fto's Lemma
.
.'
snmc, but the meaning of the equality sign would change. ne equality would have to be interpretcd in the context of mean square convergence. where the 1, 2, We apply the Taylor series fonnula to Fuk k), k Sk is assumed to obey sk ak h + (10)
Wllat does this equation mcan? On the left-hand side, F'(k) indicates the total change in F(Sk #) due to changing k and Sk Hence. if F%, k4 is the price of a derivative security, on the left-hand side we have the change is the derivative asset's price during a sho'rt intewal. This change is exPlained by the terms on thc right-hand side. The jrst-order effects arc the effects of time, represented by FI g/zl,and thc effects of change in the underlying asset's price, Fstakh + tqHij. In thc latter we again see that changes in security prices have predictable and unprcdictable components. Second-order effects are those changes that are reprcsented, ftar the timc being, by squared tcrms and by cross products. Higher-order terms are grouped in the remaindcr R. In order to obtain a chain rule in stochastic environments, the tcrms on the right-hand side will be classed as negligible and nonnegligible, It will then be shown that in time inten'als, negligible terms can bc dropped from the right-hand side and a chain rttlc formula obtained. ln addition, as h --> t)aa limiting argument can be used and a precise formula obtained in the mean square sensc, formula is known as lto's Lemma. 'I'hc Iirst step of this derivation is to separate thc terms on the right-hand side. This requires an cxplicit criterion for dcciding which terms are negligible, Aerward, one can consider the size of thc terms on the right-hand sidc of (11)individually and decide which ones are to be dropped.
'
=
,
.
,
.
.,
'
AJP;I
n
=
.
.
:'
First, tix k. Given the information set 1k-L, Sk-L is a known numbcr. Next, apply Taylor ,s formula to expand Fuk k) around Sk-j and k 1:
.
: '
-
,
Fsk
,
k)
Fsk-j
=
,
k - 1) + Fagkk -
!L
+ Fathsk -f-i'F,,E/z12
X-11+ Ft (1 +
;
'g
1
.'
.jF,,(kV-
kk-ll
c $ t; k,,
.
- 5'-1)1 + R,
,
(11) $
where the partials Fx, F,,, Ft, Ftt Fst are a1l evaluated at Sk-j k - 1. R represents the remaining terms of the Taylor series expansion. Here we are keeping the F/, Fst, Ftt notation for convenience, although these pnrtials are with respect to k. Transpose Fuk-j k - 1) and relabel the increments in (11) as follows: ,
,
'j?
E:
rrjs
)
',
,
Fsk, kj -
k - 1)
FSk-L,
sv Notice that Eq.
vsk-j
-
.
=
FsLsk + Ft gl +
hsk
Wc can substitute the right-hand expansion of (11): F(k)
=
Fsfakh + .j
+i
qa
'
F.,IA&1
'
+ R.
ak h + o
.'
(10),and
+ .H'kl + R.
f
=
fl,ol
'
.
+
.
d
Wbere
'
R is the remainder.
=
This is equivalent
j
q
< .'%
;
(total)differendation formula (19).
.,
q. t
't;
i j.,(&)(q$)
1 y/ut,%ltA5'l
.
.2
(18)
a + R,
total dcrivatives is just
fs dS.
to assuming that whilc in the Taylor (18), is small and nonncgligible, the terms involving ae smaller and carl be ignorcd as A. -.+ 0. Consequently, termfsds is preserved, whilc all othcr terms are dropped.
c .q
+.
But the formula for
JJ
;
..
(1
.,
:
in the Taylor soriYf' tuWil
=
;'.E
.'
,k
.f(&)
-
that ...) .(
1
F.vfllllA
f (.)
(10 t
.
side of this for
t' ,
p-k
+ F?(1 + glslakh + tzkHz'l;I
F,,IA21+
:
(15J
j
=
.. t
'
stochstic Culculxs
This section discusses the convention used in determining which variables can be classilied as anegligibk'' in stochastic calculus, In standard calculus, the Taylor scrics expansion of some function fv arouud gives
j1
'
. .
1 a jyFx.vr.%l
that the dynamics of St are governed by Eq.
we have
3.1 TW Nofimz of fGze'' in
.
;
knoW
):
r;
'j
'
+ jF,?I/?1+ We
.;
(14)
.
1
ZUt
'
. '
>
kh - (k 1) Now substitute these into (11): =
!;
.I
(11)already uses the increment for the time variable:
iFjkj
..
(13)L
isk.
=
'E7 r!
(12)
F(k)
=
,
KGsmall''
'P
.
135
(19) serics expansion (A,5')2, (x)3, .
.
.
in the limit, the is the
Thc rcsult
E
!
(2 H A P T F,R
*
10
lto's Lemlua
j Ito's Lemma ?
To see why such a convention makes scnse, note that as tY gets smalier, get small faster.This is shown in Figure 1, terms such as (.u$)2 (15,)3 where the functions .
,
.
.
#1(Ak)
=
hst is rando Jm its variance will be positivc. But variance is the of tlpical (z%). Hcnce, on avcrage, assunting that negligible will is a )2 be equivalent to assuming that its variance is approximately zero-that St s, approximatcly, not random, This is a contradiction, and it defeats thc purpose of usjng soEs in markets for derivative products. After all, the objcctive is to pzice rhk, and risk is gcnerated by unexpected news. Hencc in contrast to deterministic environments, terms such as Luk jl carmot bc irored in stochastic diffcrentiation. Oiven that the terms of size h are of lirst order, and that these are by convention not small, the following rule will bc used to distinguish ncgligible terms from nonnegligible ones. IEsize''
,
t,%
47
.
'
(20)
LS
'
.:
and
t
gzut
2
(A5'1
=
'
(21)
,
ks-
'
'
approaches zero much faster and smaller.
are graphed. Note that thc function gzt/j gets smaller than the hznction :1(5') as
2,37
:,
Thus, in standard calculus, a1l terms involving powers of du highe t.11%L 1 are assumed to bc negligible and are dropped from total dcrivatives. ne ; questionis whether we can do the same in stochastic calculus. The answer to this important question is no. In stochastic settings, te : timevariablc t is still deterministic. So, with respect to the time variable, te . ) samecritcrion of smallness as in deterministic calculus can be applied. termsinvolving powers of dt higller than one may be considered negligl-ble. .' on the other hand, the same rationale cannot be used for a stochastid .J differentialsuch as dSt2 chapter9 already sbowed that, in the mean squaTe -. sense, we have #%2 dt. (22)
coxvsxrlox: Gjven a function #(.T#;;, ) dcpendent on the increwjenerprocess ;#), and on the time increment, consider
'
ments o j tjw t jw ratjo
,
:'
rtBi, h
.
h4
(23)
.
Ij tus ratio vanishes (in thc m,s. sense) as 0, then we consider , hj as negljgjbje in small intervals. Otherwise, gtNlFk, h4 is nong negjigoje. .->
'
tajjj
.
,
:' '.
=
Tus convention amounts to comparing various terms with h. In particular, if the mean square limit of the function j'(H'k, h4 is proportional to r with r > 1, it will go toward faster than h thc zcro (i.e., square of a small number is smaller than the number itselt). the other hand, if r < 1, then on the mean square limit of , ) will be proportional to a larger power Of h than h itself.S The following discussion uses this convention in deciding which terms of stochastic Taylor series expansion can be considered small, a
j'
Hence, terms involving ds; are likely to have sizes of order dt, which wal' as nonnegligible. lf terms wolving dt are preserved in Taylotk considered j.Iq.,:E the same must apply to squares of stocbastic diferent approximations. point. If is a random incremen t!;E we further emphasize this important variance of W this increment. Sin w ill be then E the (15,12 ithmean zero,
.j,(.f
j
dt.
(g2)
(),
(gg)
j ttuhpkla + ztzjytv/iauzk 2 ,-Fssoka
jg
h
z
(34)
.
Again, this approximation should be interpreted in thc m,s, sense. 1, in small intervals, the dl ference between the two sides of equality (34)hms 0. a variancc that will tend to zero as Before one can write the Ito formula. the remaining terms of the Taylor serics expansion in (24)must also be discussed. 'rhat
..
?
-..>
side of (24)by h, s ,:
E
( ..
3t
Ftth :y (>) ...jE.E 2 sincc in the numerator we liave an This term remains proportional to that dcpends on h1, a power of h higher than one, and tbe i is not random. Hcnce, this term is negligible: '
.
F h 11:, -->0
=
(24)involving cross products are also negligible in small under the assumption that the unpredictable components do not contain any The argument rests on the continuity of the sample patlas fbr st. Consider the following cross-product term in (24)and divide it by h
intenuls,
''jumps.''
(29):
().
..
,
',:
Next, consider the second-order term that dcpends on l,klz, 1 z -2 Fwgkupl expanding thc squarc, an d dividing by h, Substituting for
crossProdxcjs
The terms in
.:
tt
4 Terma Intlolttiag
,
,
incremcnt increment
.,9
Fxil/zjgtzk +
fzymj
-
.
'l''le
gaknhWk ttzq,ljr/rlz + h
yz
.
v
Rsmall-''
Gsmall,''
Qh
(x)
The numerator contains In tbis equation. the fil'st term is and not is the term ran dom. ne third te p owcr of h greater than onc, scction). The Seco next prodklct It involves a cross (see is also This is term. on the otlwr hand, contains the random va riable tAH'-k)2.
; ,
.
t
.
r
E.
..;
'
g
y'z/ ju: +
aauzpj,
(:35)
right-hand
side of (35)depends on l#k. As --> (), pk. goes to zero. ln particular, Hz-p becomes negligible, becausc as --> 0 its variance goes to zero. That is, does not change at thc limit h 0. anotjxr way of sang that the Wiener process has continuous sample Pq1jys, AS long as the processes under consideration are continuous and do not (upjay any jumps, terms involving cross products of AI'IIand h would be lcsizible, accoding to the convention adopted earlier.
usk,
az /;2 1 k + -aF x;
=
. .
'. .
Wz;I
'lnhis
=
r
1
i .:
the criterion of negligibility, we write for small h'.
-
,
terms on thc right-hand
=
;)
c!
Now divide the second-order and consider the ratio
=
/q is a term that calmt:t be considered negligible, since by dehni(jon, we are dealing with stochastic Sk, and the nonzero variance of xS implies:
!:,
Terrrts sectmd-or'der
'f
2
rj-hus,
..,
k gets larger (in a pro babilistic sense) as h becomcs smaller, since the ten'n ; /2 taf l'li is the order h 1 r A1I srst-ordcr terms in (24)are thus nonnegligible. )
3.3
H,k)
dkkz
7:
g6)
''
given thc
jt was also shown that in the mean square sense discussed earlier,
y
=
l@
lj
.
(25) r
Fh t s; /).-.s(J lg vanish as h gets smaller. are c learly independent of h, and do n0t On the other hand, we already know that the ratio
'-
-
t
and lim
''-
: :
;
..q
Fsak
=
239
square of a random variablc with mean zero that is unpredictable nast. Its variance was shown to be
E
'
ratios
3 fto s Lemma
'
lto's Lemma
,$,
that
j
.
'
,
(-aH A P T E R
240
.
1
Ito's Lemm
l0
'
5 Uses of lro's Lemma
7
3.5 Terrmxin tlkc Rcmufnder
5 llses of lto's
.
?
All the terms in the remainder R contain powers of h and of H'kgreat than 2. According to the convention adopted earlier, tf the unprcdictabl events''-powers ot type-i.e., therc are no shocks are of r,f'i greater than two will be negligible. ln fact, it was ghown in Clta/ J . ter 8 that continuous martingales and Wiener proccsses have higher-order, E 0. moments that are negligible as
'hc first use of lto's Lemma was just mentioned. ne formula provides a fool for obtaining stochastic differentials for functions of random proccsses. For example, we may want to know what happens to the price of an option if the underlying asset's price changcs, Letting Fvh, t) be the option the underlying sset's price, we can write Price, and
'drare
(Tnormal''
:
st
-->
dl-ut
J'r'
4 The lto Formula
'
.i
.
l p2F tr,2 dt. dt + z?r 2 slf ' ss / or, after substituting for dSf using the rclevant SDE, @F 1 p J.F z JF JF (rt dt + + dFt h #/P) at + pkq; 2 ch%c Jf 6st =
where the equality holds
l
,/ '
(%j' ..
:.
J :
.
t' '
,
(37 .
the mean square sense.
Fss z dt.
o
(39)
rnis
')
HF
1
j.
,
r.
=
ds ' +
Fs dSt + Ft dt +
=
,
,
PF
t)
If one has an exact formula f0r Fst f ), one can then take the partial derivatives explicitly and replace them in the foregoing formula to get the f). Later itt this section, stochastic deerential, dbut we give some cxamof of this lto's 1cs Lemma. use P The second use of Ito's Lernma is quite different. Itcfs Lemma is uscful in evaluating lto integrals. may be unexpected, because lto's formula was introduccd as a tool to deal with stochastic differentials. Under nonnal circumstances, one would not expect such a formula to be of much use in evaluating lto integrals. Yet stochastic calculus is different. lt is not like ordinary calculus, where integral and derivative are separately dehned and then related by the fundamental theorem of calculus. As we pointed out cariier, the differential notation of stochastic calculus is a shorthand for stochastic integrals. Thus, it is not surprising that Ito's Lemma is useful for evaluating stochastic intcgrals. We give some simple exnmples of thesc uses of lto's Lemma. More substaatial examplcs will be seen in later chapters when derivativc asset pricing is discusscd.
';
()
-->
ITO'S LEMMA: Let Fst f) be a twice-differentiable function of t and 5 of the random process St: .i dSt at dt + o JM, 477.4 Then we havei' with well-behaved drift and diffusion parametez's, at, =
,
,
As h We can now summarize the discussion involving the terms in (24). and we drop a11negligible terms, we obtain the following result:
dF
LenznAa
: ',
ln situations that cail for the 1to formula, one will in general be giv an SDE that drives thc process S): f ) dt + o'lut f ) #Wz; (38: dSt a Vehicle YIIUS, tl1C ltO ftrmlzla tat takes thc SDE for x can be Seen J.S a corresponds t). In fact, Eq. (37)is (. that Fbsl the dctermines SDE to l). '; F(&, for stochastic differential equation with fman dcaling useful in have tool to lto's formula is ckarly a very derivatives. The lattcr are contracts written on underlying assets. Using tb ( lto formula. we can delermine the SDE for tinancial derivatives once are given the SDE for the undcrlying asset, For a market participut of t wants to price a derivative assct but is willing to take the behavior Itt's formula is a necessary tool. u nderlying asset's price as exogenous, 'ty''
=
,
,
5.1 lfo's Fonnua as a Clutn Rule
.
'
A discussion of some simple cxamples may bc useful in getting familiar
withthe terms introduced by Ito's formula.
and
,
'
5.1.1 Ekumple l Consider a function of the standard Wiener process JP) given by
/-(1, f)
:..
'
,
.
H'2.
Remember that F; has a drift paramcter Applying the Ito formula to this function,
:
irregular. Sqll 4with this we mean that ttle drift and diffusion parametezs are not too c(.J, write t) as at notational smpiicity, we condion. For satis woud this integrability Glx Q.S G( r j t ( .
=
')( .
#Ff
=
1
g (2
dt
+ 21
(40)
0 and a diffusion parameter 1. (/p)
;'
:
:
C H A PT E R
242
.
10
.
5 Uses
'
Ito's Lemma
'
or dt + 2H/; tl4(.
=
:
(42)
,
;
Note that Itt's formula results, in this particular case, in an SDE that has
ah,
1
1
.
FLJ,#;,
s
and
r)
E: E
(44)
2W')-
=
.
dcpends on tlw inormation
the dria is constant and the difssion
uence, setz
and apply the Ito formula to F(p;, d,',
-
-
F(l#I
,
i'
f)
3 + t + chpi
=
.
F(p;,
(45)
1
0 + p; dw,i + - dt.
-
,
1 ;f, dt + ehls#H') + il e #f. '
..
(46) )
1 z j l4
' :
=
-
2
cz
eB6
dt +
dl'l').
(47)
i
t
,
a SDE for F(St , f ) with T/-dependent drift and dif-
y
()
('
al', f)
1+
=
1
2
(53) side, and using the
'
t48)
.k
.)
t
and
t
1 t 2
H?;d 1#k+
=
()
-
.
(54)
XJW$
j =
-
2
1 1C/z - h 2 -
(55)
in Chapter 9 using mcan square convergence. It is important to summarize how Itozs formula was exploited to evaluate lto integrals,
!., ,,$
ds,
wuch is the same result that was obtained
.,
'-e
o
Rearranging terms, we obtain the desired result
y'
j
1+
+. 2
:
Grouping, KIFt
t
1
u, Jp;
or, after taking the second intep'al on the right-hand delinition of F(E, f):
.
=
t =
(5z)
r#).Writing the corrcsptmding
:
! dFt
f)
;,
We obtain
thig hasion terms:
/):
nis is an SDE with drift 1/2 and diffusion integral equation,
E' .,Li'
5.l.2 Fwvzrrlr/e2 Next, wc apply lto's formula to the function
(j.j)
,
,,
/*
CaSC, We Obtail
.
z
2 rp)
=
'
rltb r)
ln
243
oejue
.,
(43)
t) = 1
:
Ito's Lemma
In Chapter 9, this integral was evaluated dircctly by taking the mean square limit of some approximating sums. nat cvaluation used straightforward but lengthy calculations, We now exploit lto's Lemma in evaluating the same integral in a few stcps.
.,
dFt
of
''
Jj .
1
/,
t)
=
eHs .
(49)
.j
.
t
dH;, 0
rrltis
(:.
suppose one needs to cvaluate the following lto integral, which was discussed in Chapter 9: ra< (JuJ
1. we guessed a form for thc ftmdion 12(14:),f).
2 Itors j-emma was used to obtain the SDE for F(H'), I). 3. wc applicd the integral operator to both sides of tMs new SDE, and Obtained equation contained integrals that an integral equation,s were simpler to evaluatc than the original integral, 4. Rearranging the integral eqtzation gave us the desired result. ,
5.2 Ito 's F tymnu ltz a.s un lntcgrfztion T'xl
' : :
. ..
g bjn
;.
amounts .
.t .
'J
1 .
.
..
factj SDE notation is Rimply a shorthand to writing thc sos i.u f'ull detail.
jbr
integra.l equations. Hence, this stc,p
7
f'
.
I .
'
E;
. :j
cHh
13T E R
is indirect but strakhtforward. ne ne technique exact form of the functicm F(I#;, f). t he of using Ito's guessing tec hnique t-emmain evaluating nis ploitcd in the nex't chapter.
.
10
:'.
J;'
.
(56)
F(W'7, f)
,
1
f dls
B/)dt +
4 j (59)
.V W;kds + JWI.. f) o Rearranging, we obtain the desired intcpal:
Jlul/l'l
=
0
t t)
=
JI'FI
.
(*);
=
-
0
't
:'
;,
...x
,.
t
r
Lz )
Again, the use of lto 's Lemrrla yields the desired integral in an indiredt but straightforward series of operations.
.
?
:
'j
L$ . ..7 'S
6
lntegral Form
of
'
lto's Lemma
q
f*' : As repeatedly mentioncd, stochastic differentials are simply shorland j Ito integra ls over small time intervals. One can thus write tbe lto form ! in integral form. ; E.'( Integrat ing both sides of (37'),we obtain
7 j Mujjsuvute
.
. .
o
.
Fst,
tj
=
Ft5a, 0) +
t 0
Fu +
F ss c2u
du +
1)
we
F
Fs dsu,
yx
ku
,..
.: .
::r
; . '
.
.
j64l
gxcsg
now extend the Ito formula to a multivariate framework and give an example. For simplicity, we pick thc bivariate case and hope that the reader can readily extcnd thc formula to higher-order systems.
r:
t
,
,
''' 't'
Ol/-
dg
Our discussion has cstablished Ito s srmuu in a univariate cage, and using that unanticipated news can be charactered Wiellcf PTOCCSS iIICFCYCIItK We can visualize two circumstanccs wherc this model may not apply, Undcr somc conditions, the Rlnction F(.) may depcnd on more than a sing le stochastic variable St. nen a multivariate version of thc lto formula needs to be used. The extension is straightfolward, but it is bcst to discuss t brieqy. Tjae second generalization is more complex. One may argue that tinancial markets are affected by rare events, and that it is inappropriate to Consider error terms made of Wiener processes only. One may want to add jllmP Processes to the SDES that drive asset prices. The corresponding Ito formula would clearly change. This is the second generalizatitan that we discuss in this section,
L
Herc the first term on the right-hand side is obtained from l //6 0. J/ll'rtl
7
under the assumption
'''
0
y uru
(2.
nancial markcts.
!
:j
H$ ds.
-
()
j -
ut
''
Jt
f
s #F;
j.
,
Fu +
Ito's fonnula is seen as a way of obtairting the SDE for a function F(.;, /), given the SDE for the underlying process St. Such a tofl is very useful when F(St, r) is the price of a snancial derivative and is the underlying asset. But the lto formula introduced thus far may end up not bcing suGciently gemeral undcr some plausible circumstances that a practitioner may face in
:(:
''' i. '
/
l
l
0) - F(&,
order to
7 lto's Formula in More Complex Settings
(58) '
integral equation,
Using the dchnition of dFt in the corresponding
Fluh, tj
(62)in
,.j
E. . :. . .'F
+ 0.
(63)
.
.,
,),
App ying Ito's Lemma to F(.),
F(&, 0).
is equality provides atl expression wllere integrals with respcct to wiener processesor o ther continuous-firne stochastic processes are cxprcssed as a srjctionof intcgrals with respect to time. lt should be kept in mind that in (j2) antj (64),Fs and F,, depend on u as we jj
:..j#: ..;':
(5X
fW';.
=
=
t)
''
'
,
-
j'
Fs dSu
';i
F( 16 f):
F(.%, f)
t
.. .
.:#1'P),
-
Wc can use t jje version of the Ito formula shown in obtain another characterization. Rearranging (62),
@ ''f E ' .,
We use 1to5s Lemma. First we de line a function
=
dFu ()
:';
.
t'
245
l
:
.. .
where J#; is again a Wiener process,
More CompLexSettings
integrals will be ex- ' .S
.E
/
in
where use has been made of the cquality
'.'
only difliculty is in
5.2.1 Another Faample Suppose we need to evaluate
dFt
ito's Formula
lto's Lemma 2.
.
'
r
246
(a-H A P T E R
.
10
Ito's Lemma
Suppose is a 2 x 1 vector of stochastic processes obeying the following equation:6 differential stochastic vh
dS, (f)
f.'qltfl
?(!) =
d-bbt)
tu +
r?z(f)
t,-21(/)
J1(r)
=
dt
+
,'22(r)
JM(f)
J'r4itfl
-
gg'jltf)tf#J(f)
+
tnc(l)
#W,$,(/)1
tjute jntewal
,
(65)
dszt) ojjL't ),
(tW21(/)
) d/ +
dlfh (f) +
tr22(l) #B$( f)1,
Hence, a limiting argument can be constructed so that in the mean square
..j ,
(/ 0. The solution determines thc exact form of tMs dependence. When H( on the right-hand side of (8) is givcn cxoacnouslv and ,S. is then xo/ufftfzy of tfe sDE.' This is we obtain the so-called strong dctcrmined, similarto solutions of ordinary differential equations. ne second solution concept is specinc to stochastic differential cquat ions,It is called the weak solution. ln the H?dtki solution. one detcrmirles thc process &
,jl
d%
'.
(11) ..
fp-u,
257
3 Solution of SDL
::E
!.
E
L 2./u
'
;
.
.'
OVCS
E'
:
(
we see in On
Can
J
,.L
k
chapter14
,
tize two
wicner procesws
As mentioned earlier, mathematiciarls
4necatzse '
d't.
of tls martingalt prowrty, still be defmed the same way.
may imply different prebability mea-
call such inlormation ,
sets c-lleld
G-algebra.
or tbe Ito integrals that are in the background of
SDES
j' r t
C H A P T ER
*
11
.
3 Solution
'Fhe lxnamics of Derivarive Prices :1
Hence, the weak solution will satisfy
dk
=
ak.
r) J,rr,
(14)
where the drift and the diffusion components are the same as in r'I''J where is adapted to some family of information sets Ht.
3.3 Gch
equation
,
EE
(8),and
dxt dt -
:
.
Notc that the strong and the weak solutions have the same drift and - will have similar statistical propen diffusioncomponents. Hence, St and S: ties. Given some means and variances, we will not be able to distingktish betweenthe tw'o solutions. Yet the two solutions may also be diferent.s 'T'he use of a strtmg solution implies knowledge of the error process F'1. If this is the case, the snancial analyst may work with strong solutions. Often when the price of a derivative is calculated using a solution to an SDE, one does not know the exact process H'r/.One may use only the volatility and (sometimes) the drift component. Hence, in pricing derivative products under such conditions, one works with weak solutions.
can
.$
tion
E
:
'
xt
,
.!
'.
:
?
.( ').
' '
o/ Stl'ong Solteions
E
'
=
&+
asu,
u) du +
.:
rqt-u,
u4 Jl'u.
t
6
p,
x),
(15)
ao
..: '
'
'ng
E(I
.
''
'
,2
;
:.,
rjwo
sof course, the rcader may wonder how he candidate soluion was obtained to begn w'ith. This topic belongs to texts orl diffcrcnlial equations. Here, we just deal with models routinely
.'
1 tion i
llS0
2 We'ak
Solutin.
Bu!
lhC.
,,
,
:'
SO B
=
.
Say
..
Anjr Strtng
first condit j oyj js satisfed.
Hencc, the candidate solution satis 'Iies the initial condition as wcll. We thus that Xt solves tbe ODE in (16),This method verilied the solution using the concept of derivative. 6 If there is no differentiation theory of continuous stochastic processes, a similar approach cannot be utilized in verlfyl' stlutions of SDES. ln fact, if that onc uses the same differentiation methodology, assuming (mistakenly) j tjons it holds in stochastic cnvironments, and trics to averlfy soEs to so u by takin g derivatives one would get the wrong answer, As seen earlier, the rules of differentiation that hold for deterministic ftmctions are not valid for gunctions of random variables.
71*
is valid for a1l t. ln other words, the evolution of St, starting from an initi al !.' point &, is determined by t-he two integrals on the right-hand side. The s0- h !. lution process st must be such that when these integrals are added together, y( . they should yield the increment St - &. This would verij. the solution. ; This approach veries the solution using the corresponding inte equation rather thm using the SDE directly. Why is this so? Note tbat accordingto the discussion up to this point, we do not have a thcog .( candiof differentiation in stochastic environments. Hence, if we have a E'. date for a solution of an SDE, we cannot take dcrivatives and see if tlm alternative routes exst. corresponding derivatives satisfy the SDE, 5
ne
,
(..-()c) x (j.
':
0
ugy atyadj
=
=
l ''l
)
which is indeed a times the function itself. Letting f 0, we get
.
jr
0
j
':r
'j
at
-t.xg
E,
Lfunknown's
st
d dt
y. ,y
Thc stochastic diffcrcntial cquation is, as mcntioned earlier, an equation. This means that it contains an unknown that has to be solved for. ln the under consideration is a stochastic process. case of SDES, the By solving an SDE, we mean detennining a process St such that the interal equation
ut
x()c
=
js a solution of (16).Then, the solution must satisfy 2$$5 conditions. First, jf we takc the derivative of Xt with respect to t, this derivative must equal = 0, the function a t jmes the function itself, Second, when cvaluated at t which is assumed to bc equal initial point, value -Ya, the jloultj b give a to known. weproceed to verify the solution to Eq. (16).Taking the straightlbrward derivative of Xt,
!
.
.
jj
(16)
axt,
=
where a is a constant and A-()is given. Therc is no random innovation term; this is not a stochastic diffcrential equation, A candidate for thc solution be veriticd directly. For cxample, supposc it is suspected that the func-
:
Solution Is to Be Pveferved?
3.4 A Dtctuuion
259
SDL
The process of verifyiog solutions to SDES can best bc undel-stood if we start with a deterministic example. Considcr the simple ordinary differential
;'
t) dt + c.r(.,
of
reverse
is n0t true.
.'.' :i
Dged in
.'.'
' ' (-
. ..1: .
f. :
.? .; .
(
j
L.
. :
'
.'.
'
snance.
';
.
.1
260
C HA PT ER
11
*
The
(
Wnamicsof Derivative Prices
f
dt
!
(20)
ax l !'
dSt
2.
x
0 ea'
;'
=
a dt +
tr
JH(,
f
:
0
)
tr,
k0,
f,
r ds ?
E
t)
.:11
(24)
mdt+ GJHI.
=
(25)
$'
&
'
'
mduy
=
0
u
0
vJ 0. Herc the drift depends on St negatively through the parameter /j, and tlze diffusion term is of the constant para' meter type. Obviously, this ls a special case of reverting SDR'' This model can be used to represcnt asset prices that Euctuate around zero. ne :uctuations can be in the form of excursions, which eventually revert to the long-run mean of zero. 'T'he parameter Jt controls how long excursions away from this mean will take. larger t'he p., the faster the St will go back toward the mean. imean
'l'he
,00
=
.
5 Stochastic Volatlllty
,001
.5(.05
=
=
-
wherc the initial point was ut Ilerhis
is oftell used
10 model
=
100.
interest ralo dynamics.
.
.
.
,
Al1 previous exnmples of SDES consistcd of modeling the drift and diffuparameters of SDF-S in some convenient fashion. ne simpiest case lhowedconstant drift and diffusion. complicated most case was the ne mean reverting prxess. A much more general SDE can be obtained by making the dri and the diffusion parnmeters random. In the case of financial derivatives, this may Sion
272
'zH A P T E R
.
The Dynamics of Derivative Prices
1l
Exercises
have some interesting applications, because it irnplies that the volatility may be considered not only time-vaying, but also random given the St. For example, consider the SDE for an asset price St, dSt
p. dt +
=
th
8 Exetclses Consider tle following SDE: d(J#;3)
(79)
JW(j,
=
ttyb -
=
3gF)Jf +
/
(80)
rt ) dt + art J1'l''i,,
where the Wiencr processes 1.f'k/4r'i! may very well bc dependent. Note what Eq, (80)says about the volatility. The volatility of the asset has a lonprun mean of tyb. But at any time 1, the actual vtalatility may deviate from this long-run mean, the adjustment parameter being ne increments #HGare unpredidable shocks to volatility that are independent The tr > 0 is a parameter. of the shocks to asset prices The market participant has to calculate predictions for asset prices and for volatilty. Using such laycrs of SDES, one can ohtain more complicated models for representing real life, Iinancial phenomena. On the othcr hand, stochastic volatiiity adds additional diffusion components and possibly new risks to be hedgedmay lead to modcls that are not ,
,
I#;2JJI .
(a) Write the above SDE in the integral form. (b) What is the value of the integral
wherc the drift parameter is constant while the diffusion parameter is assumed to change over time. More specilically, tr1 is assumed to change according to another SDE, do't
273
0
l#'. 1dw.$
.
2. Consider the geometric SDE: dSt
gstdt
=
whcre St is assumed to reprcsent index is
+ fr.S'/tfH/;,
an equity index. The current value of the
ut.
idcomplete.''
rrhis
.t
6 Conclusions
z-dayintervals
denoted by &.
(a) Use coin tossing to generate random errors that will approxmate the term Jl4z),with
SDES. We distinguished rrhis chapter introduced the notion of solutions for sliila.r solution is solutions. The to the case of strong bctween two types of wcl/ novcl. stlution is equations. ordinary differential ne We did not discuss the uzw/c solution in detail here. An important example will be discussed in later chaptcrs. nis chapter also discussed major types of stochastic differential equations used to model asset prices.
ln this chapter, we followed the treatment of Oksendal (1992),which has sevcral other exa'mplcs of SDES. An applicaticms-minded readcr will also benest from having access to the literature on the numerical solution o SDES. ne book by Kloeden, Platen, and Schurz ( l 994) is both very acccssible and comprehensive. lt may very well be said that the best way to undentand SDES is to work with their numerical solutions.
940.
It is known that the annual percentage volatility is 0, 15. The risk-free interest rate is constant at 5%. Also, as is the case in practice. the effect of dividends is eliminated in calculating this index. Yottr interest is conhncd to an 8-day period. You do not see any hnrm in didding this horizon into
four otnsecutive
7 References
=
H F
=
+ 1,
=
-1.
(b) How can
you make sure that the limiting mean and variance of the random process generatcd by coin tossing matches that of #H( as 0:/ (c) Generate three approximate random paths for ovcr tMs 8-day period. ..-.
z
vh
3. Considcr the linear pdce:
SDE that represents
dSt
.01
=
the dynnmics of a security
Stdt + .05,$,#J4$
with 1 given. Suppose a European call option with expiration F 1 and strike K 1,5 is written on tllis security. Assume tlmt tlle risk-free interest rat is 3%. xlj
=
=
=
274
The
CHAPTER.H
Wnamicsof
Derivative Prices
(a) Using your computer, generate fivc normally distributed random variableswith mean zero and variance ,.
(b) Obtain one simulated trajectory for the q Choose r L (c) Determine the value of the call at expiratlon. (d) Now repeat the snme experiment with Iivc tmiformly distributed random numbers with appropriate mean and variances. 1000 times would the (c) lf we conducted thc same expernent calculated price differ signihcantly in tw'o cascs? Why? (9 Can we combine the two Monte Carlo samples and calculate the option price using 2000 paths? ,2.
4. Consider the SDE: dSt Suppose #H') is appr/ximatcd
+ .1#F).
.05#f
=
by the following pocess + A with probability .5
ls
1 lntroduction
=
- l
with probabilitjr
.5.
1. Calculate the values of St begin(a) C-onsider intenrals of size k 1, ning from t 0 to I 3. Note that you need S and repeat the same calculations, (b) Let f (c) Plot these hvo realizations. (d) How would thesc graphs look if 1, and obtain a new (e) Now multiply the variance of St by 3, let #/ realization for =
=
=
=
.5
=
.01?
.
=
=
ut.
(To generate any needed random variables you can togs a coim)
nus far wc have learned about major tools for modeling thc dynamic behavior of a random process in continuous time, and how onc can (and cannot) take derivatives and intcgrals under tllese circumstances. nesc tools were not discusscd for their own sake, Rather, they were discusscd because of their uscfulness in pricing various dcrivative instrw ments in linancial markets. Far from being mere theoretical developments, these tools are practical mcthods that can be used by market professionals. ln fact, because of somc special characteristics of derivative products, abstract theoretical modcls in this area are much mol'c n menable to practical applications than in other areas of finance, Modern inancc has developed two major methods of pricing derivativc Products. ne first of these leads to thc utilization of partial differcntial equations, wbich are the subject of this chapter. The second requircs transffrming underlying processes into martingales. This necessitatcs utilization Of equivalent martingale measures, which is the topic of Chapter 14. ln Principle, botb methods sholzltl gve the same answer. However, depending 0n the problem at hand, one method may be morc convenient or cheaper to use than the other. ne mathematical tools behind these two pricing methods are, bowcver, very different. Firkt, we will brielly discuss the logic behind the method of pricing seCulities that leads to the use of PDES. These results will be utilizcd in Chapter 13.
27$
J(, .!.
$
C H A PT ER
276
.
12
.J :
Pricing Derivativq Products
.j
,t
2 Forming Risk-Free Portfollos
'
instnzments are contracts wzitten on other securities, and these have flrtite maturitics. At the time of maturity denoted by T, the contrads derivative contract should depend solely on thc value of the of the price Fr Sp, the timc T, and nothing elsc: security underlying
:
tive of FS f /) with respect to f,1 Our maln interest is in the price of the derivative product, and how this price changes. nus, we begin by positing a model that determines thc dynamics of the underlying sset St, and from there we try to determine how Fl-f f ) behaves. Accordingly, we assume that the stochastic diffcrcntial dSt obeys the SDE ,
'
r
jy :E' ,g
L
,
. !!
This implies that at expiration, we know the exact form of thc function Fsv, F). We assume that the same relatiooship is true for timcs other than T> and that the price of the derivative product can be writtcn as
,)
dS,
:.
!'k
i.,'
(2)
r)-
l
,
(1)
Fq. = Fl5'w, F).
,
important remark about notation. dFt should again be read as the change total in the derivative price Fut, tj during an intenral dt. 'This should not be confused with Ft, which we reserve for the partial derivaIAII
'j
Derivative
FS'
Forming Risk-Free Pordblios
f
'
auh, t ) dt + ajutb tj #H(.
=
(:2
dFt
s''
. j.
dFt
,
,t,
,
(3) T
t) + tst,
.Ju
;;
'
=
(4)
d1 dFt + 0z tk'r.
,'
i
;.
subscript ; general, , % will val'y over time and hence will carly a time Sucb dependence. ln this equation, both d% .t YS WC11. At this Point wc ignore thr . due unpredictable to component that have increments an an d # are '
eL
ln
k
t
innovation term #5F; in dst.
:
i weightg Ortly wben the portfolio Strictly speaking, this stvhastic dferenial is correct Will qlliff #OiIH be Will bc fMIXICF right. lhcR 'J. O11 tflrrkhi tile T1()1 dcpcfld OD Othenvisc! rclevant when we discuss the Black-scholes framework below. ! ..f db
1
-1
Yl1i:
l
'j.
Ft dt +
=
1
2
(6)
Fsso't dt + Fx dst.
(5),and
obtain the SDE for tbe derivative
1 Fsat + jFa., o'ta + Ft dt + F, o d l1(
.
,
where p1, 01 are th e quan tities of the derivative instrument and the undorlying security purchased. They represent potfolio weights. ' The value of this portfolio changes as time t passcs because of chang/s 1 , t?), % as constant, we can write this change as in F(Stb r) and St. Takmg .
d#,
(5)
Note that we simplised the notation by witing at for the drift and et for the difhlsionparameter. If we knew the form of the fundion Flut, l), we could calculatethe corresponding partial derivatives, F,, F,,, Ft, and then obtnin explicitlythis SDE that governs the dynamics of the inancial derivative. The functional form of F(St /). however, is not known. We can use the followingsteps to determine it. We fimt see that tbe SDE in (7)describing the dynamics of dFt is driven by the same wicnerincrement JHSthat drives te St. One should, in prindple, be able to use one of these SDES to eliminate the randomness in the Oter. ln forming risk-free portfolios, tMs is in fact what is done. We now show how this is accomplished. First note that it is the market Pmicipant who selects the portfolio weights pl, %. Second, the latter can always be set such that the dPt is independent of the innovation term (sH( and hence is complete% predictable. ne reason is as follows. Given that dFt and dSt have the same unpredidable component, nd given that pj, % can be set as desired, one can always climinate the dl'pkcomponent from Eq. (4).To do this. consider again
.>
et-st,
=
We substitute for dSt using Eq. asset price:
'i
-
I0,x).
Using this, we clm apply lto's l-emma to hnd dl?
.7
Thc increments in this price witl be denoted by dl. At thc outset, a market .k participant will not lnow the functional form of Fst, r) at times other than ;:, cxpiration. nis function needs to be found. .i.. 'Fhis suggests that jf we have a 1aw of motion for thc St process-i.e-, if .;L. determined-thcn we can E we have an equation describing the way dut is would Lb and #5'/ dFt that But this obtain dlt. means lto's Lemma tf use /: namely be incrcments that have the same sourcc of underlying uncertainty, example, the irmovation part in dSt In oher words, at least in the present j, innovation term. E we have fwtp increments, dFf and dst, that depend on one continuoY ( Such dependence makes it possible to form risk-freeportfolios in fs timc Let Pt dollars be invcsted in a combination of F(St /) and St:
l
t e
dP3
.
=
0j dFt + dz dSt
(8)
r'
'j (
'
S
c HA
altdsubstitute
for dFl using dPt
=
P T ER
12
*
7
Pricing Derivacive Products
(6):2
dt terms are common to all factors, they can be obtain a partial differential cquation:
t
:
(9)
2 *1 ez the
0
r(F(-% f) - Fauh) ,
(wereplace
...,
(10)
1
,
and
-
#(f) in
rF + rFsct + F/ + - Fssg'/2
2
(11) !
-F,.
=
to
'
0,
-a
sm
g.a ,
(1:)
.
We remite 0
:
St
Eq. 0
,
2:
t
(16)as s
7)
rnte
:1
'
',
'
=
j.
y,t +
where the de rivative asset price Fs t, tj is denoted simply by the Ietter F for notational convenience. We have an additional piece of information. der jvative product will have an exp sationdate F. and the relationship between the price of the ffl derlying asset and that of the derivative asset will in general, be known exac tly at expiration, nat is, we know at expiration that the price of the derivative product is given by
:': 'aese partic-ular values for potfolio weights will Iead to cancellation of tbe e tcrms involving dst in (9) and reduces it to ,: 1 'E F t dt + -F rt2 dt. dP' (12) 2 Clearly, given the informat ion set 1t, in this expression therc is no randeterministic increment foT dom term. The dP is a completcly#rcc/fzsle, aIl times f. This means that the port folio Pt is risk-free/ PI, its appreciation must equal the earnings of since tliere is no r isk in f isk-free westmtnt durixtg an interval dt in order to avoid arbitrage. As3' exped od t) sumingthat tbe (constant)risk-free intercst rate is given by r, the ( =
=
(15)by its components.l
,
ez
'eliminatcd''
'
:
, way we wish. Suppose we ln this equation we are free to set ignore for a minute that F depends on St and select =
279
ssncethe
J
1 z dt + 01 dst. t?l Ft dt + F, dSt + ,-Fsset .
? FO rming Risk-Free Portfolios
,
).
..
':)
'
:274
F(.V, F)
,
,
ip th e case where St pays no
#t
and must equal
rp ;
dt
-
Gs
(13) ;
Rdividends,''
(14) # .
i
(r
in the case where St pays dividen ds of per unit time. In the latter cmY, the capital gains in (14)plus t he dividends carned will equal the risk-fl'o
rate.4
,
a
s dt
L
t !
utilizingthe case with no dividends, Eqs, r#ydt
=
Ff dt +
(12)and (13)yield
..)(
?.uj
.
t
.
%-'
2
) l
E 11 y if t)1, e2 do not depond on S,. nliaear'' F will or n tizne. value at vary of over pout: Note this mportant ezset -t e o no opuons, the zl will be a funcuon ot ot. C Producs such as options, or stnlcttlres conta ining yet it wll gi!* that tlae rksk-fyee portfoo metbo d is not satisactory mathematicay, PDE. the f*Ce ie: 4xoc the role of Jl. Some innitesima l tlme must pass in order to earn interest orearnin#i the interest lnteres! ra tes r, dividends.If no t 9mepasses, regardless of the lcvel of E W i1lbe zero. The same is tr'ue for dividcnd earninp. zlkecall that this WII be correct mathematica
3
means
.)
'
'vvk;o
.
.'xw-,
,
''corrcct''
u
'
r)
=
maxlsy
-
K, ()j.
(1p)
-
u/rrlftrgfe,
'
r:
T
d
'
1 '-Fsso z dt.
(18)
According to this equation, if at cxpiration the stock price is below the strike price w K will be negative and the call option will not be exercised, lt will be worthless. Othenvise, the option will have a price equal to the differential betwccn the stock and the strike price. Equation (17)is known as apartial #l//reliftzl equation (PDE). Equation (18)is an associated boundary condition. The reason this method works an d eliminates the innovation term from Eq. (4) is that #(.) represents a price of a derivative instrument, and bence has the same inherent unpredictable commnent #1z#; St. as nus, by combining these hvo assets, it becomes possible to eliminate their common llnpredictable movements. A-q a result, Pt becomes a risk-free investment, since ts hlture path will be known with certginty. This cotlstructon of a risk-free portfolio is heuristic. From a mathematical point of view, it is not satisfadory. In flrmal approach, a one should form self-unancing portfolios using completeness of markets with respect to a class of trading and using the implied ynthctic e quivalents of the assets under consideration. Jarrow (1996)is an excellent source on tjnese concepts. Next section d jsmzsses t jus p ojnt jn more detail.
'
,'
F),
j s a known junction of %. and F. For example, in the case of a call option, G(.), tbe expiration price of the call with a strike pzice K is
capital gains must equal
r
Gsv,
whereG(.)
:.
dt
=
'
>'
f'
''
r'
C H A P T ER
280
*
Pricing Ixrivative Products
12
3 Accuracy
':
3 Accuracy of tlze Method
Method
281
the third derivative of F is there becausc we are applying Ito's Lcmma to the F already diferentiated with rcspect to St. After replacing the differential dst, and arrangingt
wbere
'
1: ;
E
The previous section illustrated the mcthod of risk-frce portfolios iri obtaining the PDE'S corresponding to the arbitrage-free price Fst tj of a derivative asset writtcn on St. Rccall that the idea ws to form a risk-free portfolio by com bining the underlying asset and, say, a call optitm written on it:
of the
'',
,
Jzv (st, t) '
5.
,
obdlj-t,tj
1
Fsssszf
jjjjj'
;i7
'
Pt
'j
(21)
+ (hdst.
i
ysspstzvjt
dt + y
jI
g'xjlyyj.
.T;2T2
E:
,
=
+
Thus, the fonnal diffcrential of
where 0L 0) are the portfolio wcights. Then we took the diferential during an ininiteslmal time period dt by letting: dP,
tsrt/u)
+
F t + Fnmh +
=
(20)
OLFIVS:!) + %St,
=
j
pgtdt + yssgstat
't
.:
Pt
=
when
.;:
.: :1
% is equal dPt
Mathcmatically spe-aking, this equation treated the :1, % as if they are l ctmstants, because thcy were not differcntiated. Up to this point, there is .E' really nothing wrong with the risk-free port folio method, But consider what ' ; Izappen, when we sclect the portfolio weights. seleded thc portfolio weights as: ) .,
+ Fsdst)
.
-
,
03F(St, t) +
hut,
(25)
to -Fs will be given by:
tdt
-
=
S
Fsdst
-
1 Fst +' Fssla.st + jF.zsslrlsl
dt +. F ss tykztlj/i
t
.
.
we
clearly,this portfolio
aavc;
.(.( ''
0j
=
%
1,
=
(22)
-Fs.
S,
dpt
tGunpredictable''
in the sense that it eliminates the makes the portfolio risk-frec, but unformnately it and component random violates assumption that 0, pz are constant, In fact, the % is now the also general, F, is a function of St and f. Thus, in because, dependenton st Erst replacing the :), 0z with their selected values, and then taking tE differentialshould give a very different result, Writlg the dependence of F on St explicitly:
rlnhisselection
,
will not be self-hnancing
Ttworks''
.j
..j
Pt
(F,df + Fgdstj - Fvdst
:
F st + F ss r. t +
q
..t
Using this equation eliminates a,e ssll Ieft wfo,
(24)..
utdl. -
.
and,.,% .at we now have a thira term simce the ,, is epen uenton s, is timc dependem aad stochasuc. In geiwral, tisis term wul not van- .1. which is a . ish. In fact, we can usc Ito's Lemma and calculate the dFs, functionof and r. nis is equivalent to taking the stochastic differenn .s1 of the derivative's DELTA: )' yj '. Fssdst 2sldt Fstdt + Fssso+ dFsut. t ) t
d
xote hence,
,
vt
dn
hdst.
,
.2
lbat + p-ds-)
1
j
r)d/)
-
0,
=
F s6s
g.2,42 /
+
(x2y' ss
st
most of the unwanted
- F-.ds-
''
-
,,
(,,,.,(.-
jtzapi + fg
whiclxwill not hold in general.
,
.
-
sly t
:
)
(M
rjstdo
=
(). terms in
2.
. .
.r;w)
.(),
(26).But we
+ ,;-.,.s,-d.;.
Thus, in order to make the portfolio Pt self-linarlcing we need
'.j
=
-
write
.j
..
=
/)
wuch will, in general, not be the case. ln order to scc this, note that differentiatjng the Blackvcholes PDE in (17)with respect to St again, we can
:
'rhen, differentiating yields:
dpt
e
&FxxIGY11(+
(23) ',.
.
,
..
;'
..
Fstk r) - Fsut, t l-f
dyst
in general, since we do not
On the right-hand side there are extra terms, and these exlra tenns will not equal zero unlcss we have:
.'
,
=
=
(26)
(2a)
:
.)
C H A PT E R
282
)(
PricingDerivative Products
12
.
'
4 Partial Differential Equations
j.
3.1 An Intevpvtation
J
from some underlying assince derivative securities are always setts), the formation of such arbitrage-free portfolios is in general quite straightfonvard. On the other hand, the boundary conditions as well as the implied PDES may get more complicated depending on the derivative product one is working with. But, overall, the method will ccnter on the solution to a PDE. nis concept should be discussed in detail, We discuss partial differential equations in several steps. d<derived''
Although, formally spcaking, the rsk-frec portfolio method is not satisfactory and, in general, makes one work with portfolios that require 111't' fusions of cash or leave some capital gains, thc method still givcs us the result? this correct PDE. How can we interpret The answer is in the additional term, S1tFxstc'dM+ (p, rjjdt. Wis term has nonzero expectation under the true probability P. But once wc switch to a risk-frec measure # and deEne a new Wiencr process H(* undcr this probability, we can write: '
.
'
'
-
'
VZY
(GdWl ''F (X
=
rjdt
-
5'
).
(-$-/
-
:
: '.
l
:.
t ?
4 Partial Differential Equations
4.2 Whut Is the Beunxo
(17)in a general form, using
k
=
0,
0
::
St, 0
:;
l
(29)
t .S T;
.
''
.'
with the boundary condition
.1
F(qV,T)
GCST,
=
(30)
F)>
k
G(.) bcing a known function/ nc method of forming such risk-frce portfolios in order to obtnn PDESarbitrage- free pr ices foz derivative instruments will always lead to 5In the literature, thc PDE notation is diffcrent than what is adoped the I'DE, in (29)would be writtcn as exnmple, aFCX,
f) +
azlx,
l)-Y + iI'FtCX, t) + /hFvthat rcprcsents the th value for thc yth value for f. nese values for & and t will be selected from their in'' to Fst, I). ne result is written respec tjve axes and then aS g.. carry on this calculation, we need to change the partial differential equat jon to a difference equation by replacing aIl differentials by appro-
';
.2o
0
Fl GU RE
5.2 Numericul
=
'
d
i
20
Fij
y
.
!
a
-1()
tj
4. The
Practice
*
! )
-
in
5 Assuming that for small but noninfmitesimal & and tt the same PDE is valid, the values of Fvh, t ) at the grid points should be d e.termined.
:
'
PDF.S
Q.8
,
1.1
(28)
Boundar.y
surm
; ::J
1.
=1
'
0.2
ror
c
etl
.. . x
100
B0
s FI G t; R E
:&
Boundae for t
'-----
o:
E
5
1x
lu(;
:
'.
:g
(2 H A PTE R
308
.
13
309
bnclusions
The Black-scholes PDE ::
Priate differences. nere are various methods of doing this, cach wit.h a differentdegrec of accuracy. Here, we usc the simplest methodlt' IF p. A.F' LF 1 f rF + - o-lSl + r. (30) LS2 2 / ; where the lirst-order partial derivatives arc approximated by the correspondingdifferences. For tirst partials we can use the backward differences !' : Fij - Fi,j-j LF
.
'
k
'
For St that is very low, we let St sjsy
=
and s'min jj
rnin'
tg6j
(j '
'
A.SY
,.
:
ht
f
LF
Fq rsj
rs AS
(31)
-
!.
'', (32) t t 5'f. '
F-L,)
=
r)
Fsv,
,
-,
AS which is : '1 or we can usc forwarddifferences, an cxample of F 1 ) Fq LF rs 15 = rvh .f-85 (aa) J For the second-order partials, we use the approximations Fjj tjIF Ft-j,j F2+.1,/- Fij 1. 1 (34) 2: 1, Au kN 15-2 'T'he wherc i and 1 N. parametel's N and n determinc .; n j 1, Of the which decided number Points at to calculate the sudace Ffut, fz..j g'k we . ' 5 and N 22. Hence. excludmg For example, in Figurc 5 we can 1et n calcula/ of dots the values, have total 80 the points on boundary to we a on the surface. Thesc values can be calculated by solving recursiveb the ) : (system 09 equations in (30). The recursivc nature of the problem is due to the existence of thoundazy conditions. next section deals with these.
l''
-
,
=
maxgm.
(37)
K, ()j.
-
nese give the boundary values for Fq. ln Figure 5, these boundary regjons arc shown explicitiy. Using these boktndary values in Eq. (30),we can solvc for the remaining unknown Fq.
.
,
.
-
is an extremely low price. ncrc is almost no chance In this case, t jaat the option will expire in the money. ne rcsulting call prcmium is cjose to zero. For ( r, we know exactly that qsmin
;
:.J:
6 Concluslons
-
.k
,
.
=
,
.
.
.
,
This chapter discussed some examples of PDES faced in pricing dezivativc assets. We illustrated the dlfhcu jtjes of introducing a second random dement jyj prkcjxg can options. Wc also discussed some exotic derivatives and tjac way poss would change, Onc important point was the geometry of the Black-sc jwjes surfaces. We saw t jaat random trajectories for the underlp'ng assets price led to random Paths on this surface. nis is shown in Figure 6.
,'
=
.
.
.
,
.
.,
=
=
,
.
rrhe
,
)
Conditinns ) 5.2.1 rtlpgrlff/r.p Now, some of the Fij are known because of endpoint conditions. FG example, we always krlow thc value of the option as a function of St at 7 . expiration. For extreme valucs of St, we can use some approxl 'mations that l r arc valid in the limit. In paticular'. t , and 'i St that is high, let we very g . For 5, Ke-fl-t) (35) Fv h
.
.i,jy
eo
-
*
11:1i:5 is a price chosen so that the ca 11 pltz IlzllLl;k Here, smax the expiration date payoff.
is 17y ;
..'.
.
j:
.
(St,') 'WN. -...
()
Epfll!d:p 14:,
jr,s.s
lj,y tftlyk.. g
oqjy
i
0
'
:
'
-.-'''
.-
().12
. .... ''''' --------
,
j4a
' :,
().z$
.....
st
...........
12n
(s
100
:
xts will be seen bclow, elem@n pj. are ignorilv i, j subscripts for notatonal convenience. $: exists each tberc o'ne equation such ms (>)< i, For j equation depe,n j. this digerence d on of
'Mwc
:
cv
..
. (.:'Efiy
2o l
'F
'k'
yyyyy . E,:
co
=
umax
,
4o
':.:':.:'1
'
.
,iytn
50
ks'm.x
max
...
yk: kktjjk.-
st
--..
ao
o.a
so
1
,
llwe can
L'
also use centered differences.
F 1G U R E
i(t j't :
...
.
6
rj..1
:
C H A P T ER
.
The Black-scholes PDE
13
LLL :.
References
(b) More precisely, remember
,
E
.:
lngersoll (1987) prtwides several examplcs of PDES from asset pricing, Otzr of this topic is clearly intended to provide examples for a simple treatment introduction to PDES. An interested readcr should ccmsult other sources if information beyond a simple introdudion is needed. One good introdtzction This bok illustrates thc basics using MAPLE. to PDES is Betounes (1998),
:
l
J@
s
/!
0, and nex't by calculating the integral of tlle normal dens inten'al region in question: over the
?. 5' 'f
c/uares
of this random variable is given by the well-
E
P J
'z
'! .E
1 2
-
-.
z;
%+11
'!
where rt is the known risk-free ratc of treasury bonds with comparable maturity,and where E g.1cxpresses the expectation over possible states of
,
=
That is, on the average, thc risky asset will appreciate faster than the growth of a risk-free investment, nis equality can be rewqitten as
E
(20)
.
.E'/E.%+1 1> (1+
;
2.1.2 Example 2 We cao illustrate this method for changing the means of random varil'elevant GICS Using 2 example from nance. mtfc WC Of tfiple-Arrated corporate bond Rt with fkxcd f will have the a /cld expected value =
,
:
(19)
E'gS;1
vt,
'.
(17)
1.
J 2.1.3 .hxtlznJ7/tr be 1, 2, 'rhe example is dismzssed in discrete time 'Iirst. Let t (bserved the price of some financial asset that pays no dividends. The St is over discrete times r, t -I- 1, Lct rt be the rate of risk-free return, A typical risky asset & must offer a rate of return Rt greater than rt, since othemise there will be no reason to hold it, This means that, using F;g.1, the expectation operator conditional on information available as of time / satisses
'.(.
F,(1 + R,1
=
whichsays that the expected remrn free return approximately by Jt:
:'
(1+
r,)(1 + p),
of the risky asset must exceed the risk-
f,EA,1 21 r/ + y.,
:
&
(27)
(28)
F:
7
'
:
:
E .
Di
..:
. .
'k
.)
'@
in the case where rt and p. arc small enough that the cross-product term tlan be ignored. Under thcsc conditions, #, is the risk premium for htlding the asset for tme period, and (-f.)rt ) is the risk-free discount factor.
C H A PT E R
Pricing Derivative Products
*
analyst who wants to obtain Now consider the problcm of a snancial market valuc of this asset today. That is, the analyst would like to the fair calcmlate.St. Onc way to do this is to exploit the relation
1
Et
(1+
2 Changing Means
;
task at the outset. the second method for changing means does precisely this,
:
5.
2.2 Method 2: Operutfng crt Probtbiitcs
'
'b't
(29)
-
-'r+1
Rtj
''
second way of changing the mean of a random variable is to leave but transform the corresponding probabilit.v the random variablc neasure that governs z/. We introduce this method using a series of examples that gct more and more complicated. At the end we provide the Girsantw theorcm, which extends the method to continuous-time stochastic processes. nc idea may be counterinmitive, but is in fact quite simple, as Example 1 will show. 'rhe
bycalculating tl,e expectation on the left-hand sidel But doing this requires a knowledge of the distribution of R,. which rebefore Quiresknowing the risk premium >.4 Yet knowing the risk pre market value will nowhere the fair is Utilization of rare. knowing (29) go in ternas of calculating the mean of Rt without on the other hand, if one could work.s method might the the Another to use #,, way of transforming having of distribution Rt be found. must thc If a new expectation using a different probability distribution / vields an such as exp ression 1 E9 5'/+1 t (31) jj. -I- %j (
i'intacq''
'. ,'
'mium
,'',
'
.,
:'
-,.5
)
''transtbrm''
2.2.1 F-mmple 1 Consider the rst exnmplc of the previous section, with Z dehned as a f'unction of rolling a dic:
',
-
=
k
'
')
'
(zaverage
'onlybyknowingscan
down.
aa,
:E. :'
'L .'::
a-his
srther cxymplitxates
fBecause
1
(1 +
.571
R()
#
1 (1 + -SrA,)
''
1z
: .'.
'tgctting
')
k. .
5
k':.
E , (Z1 :
r
1
-->
=
3 1
-
-
--->
6)
=
-
1 3
--->
(r
3
? (Gettine *
or 4)
122
,.
.L.
= '
=
%'
1c2
a 1 or 2b =
429
'
tceuing:! or 4) / (Getting 5
r 6)
-
=
-
22 39 5 33
-
-.
(36) (3,)
.
(38)
Xote that the new probabilities are designated by 13. Now calculate the mean under thcse new probabilities!
-'
.
1
-g1()-2j
=
,3
Ftgctting
.1:*
the oucuutions.
the mzan of the distribution of R( could be made equal to r
ygz-sjzjjz
*c
'..'.
(30)
.
=
fatzettinz c 1 or 2
,
and the disvibution of R, .epinnea
(34)
2,
=
1 1 98 a+ z z +.-(-3-21 -(-1-21 3 (35) 3 3 3 Suppose we want to transfonm this random variable so that its mean becomes one, while Ieaving the variamce unchanged. considcrthc following transformation of the original probabilities associatcd with rolling thc die:
vartz)
j
mriod'sP11,
(33)
,
and a variance
,
srl-horc s an additional difliculty. The term tm the left-hand sidc of (29) is a zwalz-'vzr . of R(. Hencc, wc cannot simply move the expectation operator i.u front of Rt: function E,
E'lzj
.
3nis relafion is just th< desnititm of the yield Rt. If we discoun! the next by '1 + Rt, wc naturally recover today's value. becakuuted,
-
.k
The trick here is to accomplish this transformation in the mean without having to use the value of Jt cxplicitly. Even though ths seems an impossble
1
rOl1 of 1 tr 2 roll of 3 or 4 rojj of 5 or 6
with a prcvioesly calculated mean of
(32) ,
%.
-3
=
'
v,,
ele meau
2
.,
of rt. (kllown) wouldprovide an estimate of St. what would F,z:g.1and rt represent in this particular case? r/ will be the risk-freerate. ne cxpcctation operator would be given by the risk-neutral probabilities.By making these transformations, we would be eliminating the risk premium from Rt'. =
z
..j.
be veo, useful for calculating In fact, one could exploit this thiscan by using a modd that describes the dynamics &+.j, ''forecasting'' equality and then discounting the forecast'' by the This
I't
10
;.E J
! ,
,s,.
X/ -
321
.
':
=
..
42N
22
5
1101+ jj (-31 + jj L-11 1. =
(39)
:'
E
C H A PT E R
322
*
The mean is indeed one. Calculate the variance'. 5 22 122 2 11z + 11z + E >gz) (10 39 33 429 -
-
f ''
-
2
11
=
98 3
-.
..;
(40)
transformation of probabilities shown in ne variance has not changcd. (38) accomplishes exactly what the Nrst method did, Yet this second method operated on the probability measure P(Z), rather than on the values of z itself. lt is worth emphasizing that tbese new probabilities do not relate to the true odds of the expcriment. The true pro jlatljjjtjjx agsoctatcd WZ rolling the die are still given by the original numbers, P. ne reader may havc noticed the notatkn wc adopted. In fact, we need FPg.1, rather than F(.l. The probto write the new expectation operator as abilities used in calculating the averages are no longer the same as #, and the use of F(.1 will be misleading. When tbis method is used, special care should be given to designating the probability distribution utilized in calculating expectations under consideration.? 'rhe
4:
((
$>
EE
)
.
'q :
11
p@
'
) '
E :
'.,
Girsanov Theorem
323
?
3.1 A Normully Distrilnzted Rundtmz Wdcb!e
'
i. .
3
ne
Accordingly, if we have to calculate an expectation, and if this expectation is easier tta calculate with an cquivalent measure, then it may be worth switching probabilities, althoug,h the new measure may not be the one that governs tbe tnle states of nature. After all, the purpose is not to make a statement about the odds of vuilus states of nature. nc purpose is to calculatc a quantity in a convenient fashion. The general method can be summn rized as follows: (1) We have an ex-pectation to calculate. (2)We transform the original probability measure so that the expectation becomes easier to calculate. (3)We calculate the expectation under the new probability. (4)Once tlle result is calculated and if desired, we transform this probability back to the original distribution. We now discuss such probability transformations in more complex settings. The Girsanov theorem will bc introduced using special cascs with growing complexity. Then we provide the general theorem and discuss its assumptions and implications.
.:.
-(-3
-(-1
=
3
Pricing Derivative Products
14
The Glrsanov Theotem
Fix t and consider a normally distributed random
,T
zt
; 1.
,
.:
##(z;)
!''
dequivalent''
.:
1
=
f(zd)= When we multiply (zt) by dpztj, be seen from the following:
'
:
'..
=
1.
Csome
'
g/(z,)
'
;
(43)
.
1 2*
e-
jsz yyz ,. )(z/)+.sz,-
After grouping the terms in the exponent, we obtain the exprcssion !. .sjz .. ) dpzt e- z gz tz 2* Clearly dlhztj is a Fzew probability measure, desned by t
=
new
2
(
2 11
Zr-
&
we obtain a new probability. This can
E##(z,)lE;(z,)1
;
nc
(4l )
.N0, 1),
e- jjolz dz J. (42) 2= ln this example, the state spacc is continuous, although we are still working with a single random variable, instead of a random process, Next, dclinc thc function
'
''
partitllr readers may wonder how we found hc, new probabilities /(Z). In tbis conclitions to used unknowns and threc probabilities the as casz,i: was easy. We copsidered condition is tbat the probabilities sum to one, nc second is that the ftrst them. for ne so1vc third is that the variance equals 985. mean is one.
zl'.
Denote the density function of zt l3yfzt) and thc implied probability measurc by # such that
Th e examples just discussed werc clearly simplifed, Fir'st we dealt with random variables that were allowed to assume a fmitc number of valuesE. thc state space was fmite, Sccond, we dealt with a single random variable instead of using a random process. The Girsanov theorcm provides the general framework for transforming measure in more com- i one probability mcasure into another plicated cages. The theorem covers the case of Brownian motion. Hencc, the state space is continuous, and the trangformations are extended to continuous-time stochastic processes. t because, as we The probabilities so trarisformed are called VII see in more detail later in this chapter, they assign positive probabili- J ties to the same domains. nus, although the two probability distributions are diferent, with appzopriate transformations one can always recover one t measure from the other. Since such recoverics are always possible, we may and then, if :, Wan t to I1se the Oonvenient'' distributitm for Our calculations, 1 >itch desircd, back to the original distribution. idequivalent''
'x.
variable
.
=
dpztjqzt).
,
(44)
(45) (46)
324
C H A P T ER
*
14
Pricing Derivative Products
3 The Girsanov Tlaeorem
325
1
By simply reading from the density in (45),we see that #(zz) is the probability associated with a normally distributed random variable mean t and variance 1, lt turns out that by multiplying dpzf) by the function ;(zr), and then in changmg the mean of zt. Note that in switching to #, we sueeded this particular casc, thc multiplication by 4(z;) presen'ed the shape of tbe probability mcasure, In fact, (45)is still a bell-shaped, Gaussian curve with the sme variancc, But Pztj and #(z/) are different memsures. They have diffcrent means and they assign t/s/fcrcnf weigbts to inteaals on the z-axis. Undcr thc measure Pzt), the random variable zt has mean zero, EP EP Ez/) 0, and variancc EF j 1. However, tmder the new probability measure #(zr), zt has mean gzfq la,. ne variance is unchanged. What we have just shown is that there exists a function kzt ) such that if we multiply a probability measure by this function, we get a new probability. The resulting random variable is again nonnal but has a differcnt me-anFinally, the transformation of measures,
(z2, =
=
=
##(z,) f(z,) J#(z,),
(47)
=
which changed the mean of the random variable z?, is revenible:
(ztl-3dlhzt)
(48)
dpzt).
=
Thc transformation leaves the variance of zr uncbanged, and is unique, given p, and e. We can now summarize the two methods of changing means:8 * Method 1: Subtraction of means. Given a random variable z
Z-g =
-
1
z
#
=
Z 8We simplify the nomtion
glhtly.
'w
#
=
(50) variable
Z with
(51)
Nm, 1),
N(t), 1).
,
-
-
-
>
.
3.2 A Nornztlly Dlstdbuted Vector The previous example showed how the mean of a normally distributed random variable could be changed by multiplying the corresponding probability measure by a function zt ). The transfonued measure was shown another that be probability assigned to a different mean to z/, although the variance remained the same. Can we proceed in a similar way if we are given a vector of nonaally distributed variables? The answer is yes. For simplicity we show the bivariate case. Extension to an rl-variate Gaussian vector is analogous. With fxed 1, suppose we are givcn the random variables z:;, ht, jointb distributed as normal. The corresponding densiT will be
fzt,
z2,)
1
=
2.
g.2 9 ) (zz,-?z.z)1 - i E(zl,--pz., tr12 cr..ie 1
where fl is the variancc covariance matrx of D with ojl, i zI,, zcr. The
by f(Z) and
0b-
(52)
g.
'
If1!
2 tl'j
=
tn 2G2
2
=
(z1: (z'a'-
,,1 ) #.cl
c'12 Gl 2
Gl 2
r
-1
-
(53)
(zjr,zazl, (54)
,
la 2 denoting thc variances and lf1lrepresents the determinant: .61
N(0, 1).
transform the probabilities dP through multiplication tain a new probability # such that
Zl
-2-2
>
I
Will
.w.
Z1
=
(49)
1),
have a zero mean, Method 2: USZg equivalent measureg. Given a random Wobability #, WCII
*
Ng,
k l)y transforming Z
dehne a new random variable
2'
ew
Thc next question is whether we can accomplish the same transformations if we are given a sequence of nonually distributed random variables,
g'la
the covariance
between
2
n z, .
Finally, y.j, tz are the means corresponding to zl, and zzt. The joint probability measure can be dehned using .Ptzl/, z2r)
J(z1r,z2/) dz,, dzzt.
=
(56)
This expression is the probability mass associated with a small rectangle dzlzq centered at a particular value for the pair zlt zzt. lt gives tlle Probabllity that zLt, z2; will fall in that particular rectangle jointlj'. Hence the term joint density functbn. Suppose we want to changc the means of zy), ht from tj tn to zero, while leaving the variances unchanged. Can we accomplish this by transforming thc probabilty dpht, ntj just as in the previous example, namely, by multiplying by a fundion f(zl/ zzj )? ,
,
,
r'
.:I
. with respect to P is given by (zt). Such derivatives ae called Itadon-Nikodym derivatives, and fztj can be regarded as the denaity of the probability measure # with respect to the measure P. According to this, if the Radon-Nikodym derivative of # with respect to P exists, tben we can use the resulting density ijztj to transform the mean of zt by leaving its variance structtzre unchanged.
(.
J(zt)
(68)
''deriva-
h'
which in the scalar casc became
/(z;) from
apzt)
,;
,
(oyl
.
(ztjdpztj.
=
#(o)
'
e
,,2
Or, dividing both sides by dpzt),
with future discussion in mind, we would like to emphasize one regu- Ikir : laritythat the readcr may already have obsenred. E:'E univariate of length k, or simply as a Thirik of z/ as represent ing a vector t randomvariable. In transforming the probability measures Pzt) into tzt ), ( the fundion f(zl) was utilized. This function had the following structtlre, .) 'i' -z;f)-.s+.lJz'!)-1s =
1:9
=
obtaining the new probability measure
,
' ',
f(z,)
c'
e -sz,+ 4
=
:
.WNote
Kzt) with
(z/)
':t
,
(64)
.
is what determines the functional form of (zt). Multiplying the original probability measure by (z;) accomplishes this transformation in the exponcnt of e. Given this, a reader may wonder if we could attach a deeper interpretat oyjto what the f(z,) really represents. Thc next section discusses this
,.
.
lyzsa
:(?
nis
:
.b
dzgj.
(6a)
,
.n
we need to add
.'L
a
1 (z,)2
j
-
(58) t
j j j.l)',
'z
327
...,
>tzjt, zct) can bc obtained by multiplying expression (53) by jht, zw), shownin (57).Thc product of these two expessions gives -)(z,,
neorem
.''..
.
l'szkt. zo) by a new probabuity measure
zz,) dl-hzkt,
Gjlsanov
'yhe
'(
The answer is yes. Consider the function dcfined by z2,lg
3
'F
Nlncidentally,
,,
thc function
4(z,)
a
subtrvgctv a mean from z;, whereas
E
'
.!
tlle hmction -1
J'(z,)
;
:. .. '.. .
e-wzt,t
=
-
e
.z,-
.)
.2
(6,6)
CHA PT ER
*
14
Pricing Derivative Products
Clearly, such a trlmsformation is very useful for a financial markct particwltile ipant, because the risk premiums of asset prices can be leaving the volatility structure intact, In the case of options, for exmple, the option price does not depcnd on the mean growth of the underlying asset price, whereas the volatility of the latter is a fundamental determinant. In such circumstances, transforming original probability distributions using 4(z;) would be very convenicnt, ln Figure 4, we show onc cxample of this function J(zy).
4 Statement
CONDITION:
dielintinated''
3.4 Eqltlulcnt
Girsanov Theorem
Given an interval dzt, the probabilities P and 13 satisfy /(#z)
>
if and only if
0
dlhzt) =
(zfldpztt
(71)
note that in order to write the ratio
J#(zr)
(72)
##lz,)
meaningfully, we need the probability mass in the denominator to be differTo perform the inverse transformatitm, we need the numen from zero, ent different f'rom zero. But the numerator and the denominator are be to ator probabilitiesassigned to infinitesimal intcrvals dz. Hence, in order for the Radon-Nikodym derivative to exist, whcn # assigns a nonzero probability to #z, so must #, and vice versa. In other words:
25
(z) z?bz 1
l'(z/) drzt)
=
(74)
I(z/)-' dlhztj.
4 Statement of the Girsanov Theorem In applications of continuous-time hnance, the cxnmples provided thus far will bc of limited use. Contittuous-time Enance dcals wit.h continuous or right continutms stochastic processes, whereas the transformations thus far involved only a jinite sequence of random variables, ne Girsanov theorem provides the conditions under which the Radon-Nikodym derivative J(z;) exists for cases whcre zt is a continuous stochastic process. Transfonuations of probability measures in continuous hnancc use this theorem. We first state the formal version of the Oirsantw theorem. A motivating discussion follows afterwards. The setting of the Girsanov theorem is the following. We are given a family of information sets (32).over a period I0,T1. T is linite.'o Ch'cr this intenral, we deline a random process (t? (/(; dBi- J j,' A du) (t t (E e xu =
jtl wj, ,
,
(,y(p
where Xt is an ft-measurablc process.l 1 The )P) is a Wiener process with probability distribution #. Wc impose an additional condition on Xt. should not vary
=
=
(73)
(70)
able to perform transformations such as
30
0.
nis means that for all practical purposes, the two measures are equivalent. Hcnce, they arc called equivalent probability measures.
derivative,
dlhzt) f(z,), d#(z,) = =
>
and apzt)
exist?That is, whcn would we be dlhzt)
#(Jz)
If this condititm is satisfied, then (z/) would exist, and we can always go back and forth bctween the two measures / and P using the relations
Metlles
When would the Radon-Nikodym
ln hcuristic terms,
of the
1
iltoo
20
mt1Ch''*
15
sgcl', zaj .< ..-5
1() 5 .3
-2
-1
0
FlG U R E
1
4
2
! zy
3
x
t s
p. zj.
ltdNote,that this is not a ven' serious restricton in the case of Enancial derivatives.A1l'ntst all linancial derivatives have Ilnite cxpiration dates. Oflzn, the maturity of the dcrivative ilqrumznt is less than one yeaz. Clnat
is, giveu the informafon sct
6, the
value of X is known euctly.
j'
:'; .j'
C H A P T ER
330
Pricing Derivative Products
14
*
5 A Discussion of
;
Gtnew''
.(i
get-l
JWFx
=
-
dh
,.: ;
=
3 #!
d(t
Jl'P).
/xr
=
of t
Also, we see by simple substitution
(79)
;i '
t) in (76).
=
(79),we
r';
Q
1+
(81)
Jl1Z;.
-l
t)
.'
3
). 1.
: (82) ..:5
#H$
f-
('t
:
is a stochastic integral with respect to a Wiener process. Also, the term : Lxs is f/-adapted and does not move rapidly, All these imply, as shown i.q Chapter 6, that tlw integral ig a (squareintegrable) martingale, r: E where u < martingale.
f
.
u
fX .1p
t)
.$
t/l4$ lu
I
(81),this
Due to
=
0
(sxs
implies that
fTT;,
(83)
(1 is a (squareintegrable)
Z WiCRCF
RWZSWC
PFOCCSS
f'n
UVCS
Witll
XSMCCt
to
bjF
h
ZRd
Wi $11XSPCCt
F1'I1a z 1 dctermined by zz and Pz.( ) .?4
an event with being of tlae event. funetion ..4
4/
whichis similar to importantpoints;
.j
:
'
:.
()
=
,
(88)
Js.
Then, taking the integrals in the exponent in a straightforward fashion, and emembering that P?() %
p
/
(0,T1,
=
p,
=
If the process (t dclined by (76)is a martingale witb resped information sets It, and the probability #, then C, defined by to
te
(g..y.)
,
Xu
...)
THEOREM:
=
-!zrj;xwdw;.-!.h'.-,1 zdlzq
Suppose the Xu was constant and equaled
.l
We are now ready to state the Girsanov thcorem.
11, H,: l
e
.7,
-
vkdu,
=
where we explicitly factored out the (constant)c2 term from the integrals, Alternatively, this term can be incorporated in Xlg.
:!.
l
,
Il1 this section, we go over the notation and assumptions used in the Girsanov theorem systenzatically, and rclate them to previously discussed examples. We also show their relevance to concepts in financial models. We begin with thc function 6
E.
But the term l
(86)
5 A Dlscussion of the Girsanov Theorem
obtain
l =
Xt dt.
-
''
180)
by taking the stochastic integral of
t/1,:
=
That is, I'I; is obtained by subtracting an J/-adaptcd drift from H(. Thc main condition for performing such transformations is that (t is a martingalc with E L'G 1 l We now discuss the notation and the assumptions of the Girsanov theorem in detail. The proof of the theroem can be found in Liptser and Shilyayev (1977).
;
which reduces to
is
Girsanov Theorem
ln heuristic termsa this theorcm states that if wc are given a Wiener HJ;, then, multiplying the probability distributitm of this process by proccss #t, wc can obtain a A?'w Wicncr process 12 with probability distribution #. The two processes are relatcd to each other through
:1
'rhis means that Xt cannot incrcse (ordecrease) rapidly over time. Equa- E1' tion (77)is known as the Novikov condition. property that turns out ) In continuous time, the density (t has a condition is satisficd, Novikov important. It if the that out turns very be to ( then t will be a square integrable matingale. We ftrst show this explicitly. ( Using Ito's Lemma, calculate the differential ll zllj d ' Xu A'J l d: l j.t r d gjjj (7g)
nus,
the
1
(84)
E
;
;i1 '
..
lx being the indicator
-1sI!sB'',-
.z
E
:
(jwj
,
Hzt) discussed
earlier. This shows the following
=
.:
)s2l1
(dmean''
.:l.
(85)
e
1. The symbol Xt used in thc Girsanov theorem plays the same role Jt played in simpler settings. It measures how much the original will be changed. 2, In earlier examples, g was time indepcndent. Herc, Xt may depend on any random quantity, as long as this random quantity is known by time r. nat is the meaning of making Xt Ji-adapted. Hence, much more complicated drift transformations are allowed for. 3, The f/ is a martingale with E (lj) 1,
'3.
to tilc Wpbzbiliv
the
=
.
.
F
' .3 t
.( .'l
C HA PT ER
Pr ic ing Derivative Products
11
*
E
J h Discussion of
the
Girsanov Theorem
333
.k '
Next consider the Wiener process l There is something counterintuitive about this process. lt t'urns out that both )'P2and Hz;are standard Wiener processes. Thus, they do not have any drift. Yet hey relate to each other by
JI'; J)); - Xt #f, whichmeans that at least one of these processes
..: ..;
E y
What is the meaning of lx ? How can we motivate this relation? lx is simply a function that has value 1 if occurs. In fact, we the preceding equation as .4
In the case where
./gl
=
TEla(w1
=
is an infinitesimal g/z
which is similar simpler settings.
to
=
:;
5.1 Applieation to
p,
tEr
a
0, Or,
(q5)
.
Jz dt
is norlzero. Recall
:rF;,
y,t +
=
t c
(0,x).
(96)
O.p;.
(qg)
write lgksrt+s
t .
jusij
+ x) + vsjplos
.stf
=
St +
-
u,ijyj .j. tpp;
(,:)
(99)
Jtx,
,'r
since IB(+x
j
-
Wzij is unpredictable
given
v%t.
Thus, for #,
,i.
:,
,
?
,
,j
'
SDES
0:
t.j(sj
=
st -
(j()1)
gt.
h,
,.
,,,
.)
>
Then l will be a martingale. One disadvantage of this transformation is that in order to obtain one would need to kriow Jt. But y, incomorates any Hsk prernium that the risky stock return has. In general, such Hsk premiums are not known before one filds the fair market value of the asset. The second method to convert St into a martingale is much more promising. Usn the Girsanov theorem, we could easily switch to y. an equivalent measure #, so that the drift of S is zero. do tus, wc have to come up with a function (St4, and multiply it bY the original probability measure associated with S S 2 may be a t CYmartingale under P,
t
in mtlch
l
-
i ')
.$
q%
',
:;
0,
Sf is not a martingale. Yet we can easily convert into a martingale tly eliminating its drift. one method, discussed earlier, was to subtract an appropriate mean from st aad detine
.j'
f---tn
>
> sg,sy.j.yj.rj s:.
:
(93) seen earlier
=
5
inten/al, this means
the probability transformations
WC Can
L
y, dy
Jjji
'
,
ds +
st
:
(92)
4z dl a4
&
t =
',
rewrite
l-Al
With
.'
(91)
fwl.
St
.
t
g-j(p;):
Clearly, St cannot bc a martingale if the drift term that
,
,-zkdt,
1
.
2'Jr/
i
.
The point 1, 11: has zero drift under #, whereas JKhas zero drift under P. Hence, 'j can be used to represent unpredictable enors in dynnmic systemsgiven that we switch the probability measures from P to #. using it as an error term in Iieu Also !' because V, contains a term of W4wolzld reduce the drift of the original SDE under consideration exactly by -xtdt. If the Xt is interpreted as the time-dependent risk premium, the transformation would make all risky assets grow at a risk-free rate. Finally, consider the relation FFlla
tjpqyj
j
'nft
point?
=
H( is assumed to havc the probability distribution P
j
.:
if must have nonzero d Xt is not identical to zero. How can we explain this seemingly contradictory
#w(.d)
ne
CE
(90)
=
0.12
with H() = with
.
i ,''
'ro
We give a heuristic example. i Let dSt denote incremental changes in a stock price. Msume t hat theR c hanges are driven by infnitesimal shocks that have a nlrmal distributiony g equation drien so that we can represent St using the stochastic differential by the Wiener process 'F; t ' x), tfp;, f (94) + e I0, d.% v-at
.
'
.'
E zy
IJ/j gkroy
lznis
=
: ( .
:
(1():)
formulation agaiu perruits ncgative prices a: positive probability, we use it because it is no:ationally comrenient. In any casu, tllo geometric sD'E will tye dealt witla in the ne.xt chapter.
' '.
L
.%t,
>
.
. ' ' ..
'y
JE C HA P T ER
334
j!Ei 1-
'
'
I
,-.
ara
L/+.vI/j
-1
zr cl
zra
yr,k
..1
qu
tltv)
'f.
=
j
1
-) s: -.w)l e2,w0-2t nis defncs the pzobability measure P. We would like to s'witch to a new probability / such that under becomes a martingale. Dehne Ez-f 1211 e- b - ) =
.
t
,
(&)
=
Multiply
/ .$ by this
d#st)
(S =
f
.
y
l )!
:
(jjjj,j /, St
(105)
.
t-
,''
J
.
the exponents =
i
(
:
1
.TE(-',)z dsf.
-
2,:rc2/
e
C
(107)
E
l .
But this is a probability measure associated with a normaliy distributed write the increprocess with zero drift and difzsion (m. 'Ihat means we can 11,/: t:f ments of Sf in terms a new driving term
ds/
=
cdbr
:.
(108) .
.
Such an St process was shown to be a martingale. is defined with respect to probability
'rhe
Weiner process
.j
Vf
t
.
'q: t'
6 Which Probabilities?
? 'v!
The role Played by the synthetic probabilities / appears central to pricin: disOf linancial securitics even at this level of discussion. According to the and discrete no-arbitrage in of cussion in Chapte 2, under thc condition a liquid markets in that trades security of pricc setting, the any timc willbe given by the martingale equality:
d'straightforward''
.),
:
::
.J '.
z')! ,
A market practtioner would then need to take the folsteps in order to price the derivative contractz
* First, the probability distribution # necds to be selected. This is, in general, done indirectly by selecting the Erst- and second-order moments of the underlying processes, as implied by the fundamental theorem of finance. For example, in case the security does not have any payouts and there is no foreign cun-enc'y involved, we lct for a small l > 0: F Sg5 d+a
S -.
'
('
t
with known F( lowing
,
E 'oc! ( t v j
' (.
(106)
=
= F / (D Fs z,
'.
:..
skt st-lu? f /,s e-
=
Ct Ej
get
ip..%-18,.2211 lnvlt
where t < F and tl:e Dt is a discount factor, known or random, depending on the normalization adopted. ln case tlzere are no foreipl currcncies or payouts, and in case sangs accoutt normalization is utilized, Dt will be a function of t-lzerisk-free rate rt. If rs r is constant, thc Dt will be known and will factor out of the expectation operator. The fad that the Dt and the probability # are known makes Eq. (109)a very useful analytical tool, because for all derivative assets there will exist an expiration time F, such that the dependence tf the derivative asset's price, Cr, on is contractually specihed. Hence. using Ct FCS: f ) we write.: can uw
:
t#(.,)
(stt
= c-J
Or, rearranging
) to
.
335
=
As usual, the superscript of the .E'(.1.1operator represents the probability expectation. measurcused to evaluate the transformation, this perform order a (Sg) function nceds to be to ln calculatcd. First recall that the density of St is givcn by f,
6 q/hich Probabilities?
J
#:
but it will bc a martingalc under E
)
Pricing Derivative Products
14
*
12
rth.
(110)
nis determines the arbitrage-free dynamics of the postulated Stochastic Differential Equation. * Second, the market practitioner needs to calibrate the SDES volatility parameterts). This nontrival task is often based on the existenc.e of liquid options, or caps/ioors markets, that provide direct volatility quotes. But, even then calibration needs to bc done carefully, . Once the underlying synthetic probability and the dynamic are determined, the task reduces to one of calculating in (109) thc expedation itself. This can be done either by calculating tl'ke implied closedform solution, or by numerical evaleation of the cxpectation. In case of closed-form solution, one would atake'' the integral, which gives the EPIFCSP, F)1: cxpectat ion
''fair''
sf
=
ElhDtsvj,
j''
(109)1, .
Smox
.Ft&., Nzxia
vldlhsv),
(1 l 1)
k'
E IJ J
C H A P T ER
.
14
Pricing Derivative Product.s
'i
A Method for Generating Equivakent Probabilities J
.
where the # Sv) is the martingalc probability associated with that particular insnitesimal variation in ,Sv. The Smin uLsxis the range of possible movements in In case of GiMonte Carlo'' evaluation, one would use the approximatitlxi:
i will be obtained probabilities pi
'fhat is,
J
truc used the
,
,sz.
.
F
vf
by multiplying the possible values by the that con-espond to the possible states.ls Clearly, if one in place of pt', the resulting forecast
.
f
=
1 E /' FlS r T)j 2 N -
,
j (F(Sv, F)),
j= 1
(112)
=
,
.
,
y IE $
) :.
t
:g
) .
j'
=
1
C
sw + '
j=l
.
.
# s.w.
.
+
PM qy%T
.
+
psfgb'kv
(115) (116)
(117)
=
:: .
and t
0 S Zt,
:
p
(J 13)
(118)
for a1l 1, under a probability P. We show that such Z can be very useful in generating new probabilities. Consider a set in the real line R, and define its indicator function as 1a: -4
kr,
p
.
As secn in Girsaaov Theorem, there is an lteresting way one can use martingales to generate probabilities. For example, assumc that we deEne a random process Zt that assumes tmly nonncgative values. Suppose wc select a random process Z that has the following properties: E'#(z,j l
..: .
rrhe
=
.
7 A Method for Generating Equlvalent Probabilities
IE
'rhen,
s, t
+
would be quite an inaccurate rellection of whcre St would be within an intcn'al A, because under # the Sf would grow at the rate (inacmzrate) rti rather tllan the trtte expected growth ((r; + p,)) that incomorates the risk prcmium g. Having misrepresented the possible growth in St, the mmingale probability # could certally not gencrate satisfactory forecasts. Yct, the # is useful in the process of pricing. For forecasting exercises, a dccision maker should clearly use thc real-world probability and apply thc operator F#g.1.
,
The role played by the ? in these calculations is clearly very important. Thanks to the use of the madingale probability, the pricing carl proceed without having to model the true probability distribution P of the process Stb or for that mattcr without haviog to model the rk premiam. Both of 14 which require difficult and subjective modeling decisions. 'lajs brings us to the main question that we want to discuss. Martingale probability P appeal-s to be an imporant tool to a market praditioner. ls it also as important to, say, an econometrician? In general, not at all. Suppose the econometrician's objectivc is to obtain the use of # would yield miscrable results. the best prediction of Sv. In order to see this, suppose the world at time T has M possible states. denoted by St,will then bc given by'. ubest'' forecast of
T
T
.E...
k
,1
XV V y=1
=
,
E.:
1, N is an index that represcnts trajectories of Sp where the j randomly selected from the arbitrage-free distribution #. This and similar proccdures aTe called Monte Carlo methods. ne law of largc numbers guarantee that, if the randomness ig correctlv modeled in the se/ lection of Sp, and if the number of paths N goes to infinity, the above average will converge to the true expectation. Hence, thc approximation can be made arbitrarily good.i3 . ne last step is simple. In case the discount factor Dt is known, one divides by D/ to express thc value in current dollars. If the Dt is itself random, then its random behavior needs to be taken into accotmt jointly, with the Sp within the expectation operator.
1
M
(y-t'r(Bfr-W;)+1tF(T-I)j.&' -
y
(jj
j
(jj
v 0jj
.l'n
,
(155) (156)
E' .2
c H A PT
ER
14
*
.
'
!
>
Pricing Derivative Produet.s
10 Exercises
.
i
343
,
.;
for some proba bility# delined by: /(.d)
=
10 Exercises
i E
1, Consider a random variable correspondingprobabilities:
:'
(zj j, ,1
:
Notc that. by switching to the probability l t h e term represented by z r : and the expectation is easier to calculate. ln the has simply optionsa transformations that use this method turn s exotic of case pr icing Essentially, we j out to be convcnient ways of obtaining pricing formulas. expectations involving geometric proccsses will contain implicitly that see by Slzch Zv. lt then immediately becomes terms that can be represented tbis section. ne E Possible to change measures using the trick discussed in resulting expectations may be easier to evaluatc, ,
(ax l2lr
,
fddisappeared''
,
'
jax
,;
=
=
1, ptax
1)
=
-0.5, px ,2,
=
with the following values and the
.x
ptax
.?j,
.
-0.5)
= .2)
.2J,
=
,5/.
=
=
(a) Calculate the mean and the variance of this (b) Change the mean of this random variable to
,
,
appropriate
.
constant from Ax,
rrhat
random variable. .05
is, calculate
by subtracting an
':L m
..
8 conclusions
! '.'
.s
such that ay l'as mean Has tlie variance claanged? (c) do the same transformation using a change in probabilities, Now (d) that again the variance remains constant. so Have the values of k;r changed? (e)
,
.
As conclusions, we review some of the important steps of transforming tlwr martingale process. ) 2. into a
s
a
.05.
,
was done by switching the distribution of St fromj: - The transformation # by using a new crror term (, P to nis was accomplished '; still had thc same variance. . Thjs new error term representation (108)from (94) is that the me . What distinguisbes while the altered, presen'ing O f l is zero mcan propcrty of the err j accomplished cbanging the distributions, rather t by terms. This was ' random variable. underlying subtracting a constant from the example, the transformation was used to conr .j,4/..1 . More importantly, in this martingale. ln linancial models, one may want to appv' vert St into a tl/ the transformation to e '-ru t rather than St. e-rtst would reprcsent discounted valuc of the assct price, where th e discount is done with redefm spect to the (risk-free) rate r. The jstj function has to be l : in order to accomp lish this (.
g, Assume that the return Rt of a stock has the following log-normal distribution for fixed t:
,
'
,
'
.
logtlr)
.
x(Jt,
.x.
a. c ).
.
Suppose we let the derlsity of 1og(&) be denoted 1nyfRt) that Jt We further estimate thc variance as G2 ,17.
,
=
(a) yud
a function kRt) such that under the density, (RtjfRtj, jjas a mean equal to the risk-free rate r (,b)Find a (Rt) such that Rf has mean zero, (c) u n der wltich probability is it feasier'' to calculate
,
sjacj( ?
L . . r:
Transform jng stochastic processes into martingales through the use calculus. The so the Girsanov tbeorem is a deeper topic in stochastic method will all be at an that prov ide the technical background of this onc of thc more intui vancedlevcl. Karatzas and Shrevc (1991)provides comprehensive refereno. Shiryayev (1977)is a Liptser
discussions.
an d
Rt
.05.
=
':
9 R efetences
and hypothesize
,09.
=
(d) Is the
variance different under these probabilities?
.
3, The long rate R and the short rate r are known to have a jointly normal distribution with vaxiance-covariance matrix 5: and mean y,. moments are given by
E ...
'rhese
'
.
.
v,
Z
r
,E r,
'j .( .)E
.
' %. '.
' ':
=
.5
.1
.1
.9
.%
:
344
C H A P T E IR
*
14
E)
Pricing Derivative Products
E:
and
' .q'
M=
07 .05
. .,
,)
joint density be denoted by f (R, r), (a) Using Mathematica or Maple plot this joint density.
Let the corrcsponding
'
Equivalent Martingale 4easures
'
(b) Find a function QCR, r) such that the interest rates have zero mean under thc probability'. rldRdr. (R, rlfl, dP
'
'
't
..w
=
(c) Plot the 4.(A, r) and the new density. matrix of interest (d) Has the variance-covariance changed?
Appliccton.s
:
rate
vector
,
t :: .,: :
..
1
lntroductlon
In this chapter, wc show how the method of equivalent matingale measures can bc applied. We usc option pricing to do this. We know that there are two ways of calculating thc arbitrage-free pricc of a European call option Ct written on a stock that does not pay any dividends.
..%
J
:'jk
vh
....( j'
1. The originat Black-scholes approach, where; (1) a riskless portfolio is formcd, (2) a partial differential equation in F(St, f) is obtained, and (3) the PDE is stalved cither directly or numcrically. 2. The martingale methods, wherc one linds a probability # under which St becomes a martingale. One then calculates
E
'j
i. )
usynthetic''
' j
Ei.
.
-
c/
.j .'
.t.
=
s/r-rtr-/lgmaxtyw -
K, ()))
(1)
again, either analytically or numerically.
:1
The first major topic of this chapter is a step-by-step treatment of the mar-
tingaleapproach. we begin with the assumptions set by Black and Scholes and show how to convert the (discounted) a-sset prices into martingales. This is done by Ending an equivalent martingale measure #. nis application does not use the Girsanov theorem directly.
$.' . .1 ..'j '.?:
.
The Girsanov theorem is applied explicitly in tlte second half of the con-espondence between two approaches to asset pricIn particular, we show that converting call (discounted)
:.'
fhapter,where the ingis also discussed.
.
)
.L
J:
i
. .
. '.!
345
j'
346
!H A PT ER
.
:'
Equivalent Martingale Measures
15
.
2 A Martingale Measure
prices into martingalcs is equivalent to forcing the Ftus'j t) to satisfy a particular partial differential equation, which turns out to be the Black-scholes PDE introduced earlier. We conclude that the PDE and the martingale approacbes are closely related.
2./../ cakulation usingthe distribution
,
C
in (2),FgcFrj can be calculated explicitly. Substituting from thc definition in (4),we can mite * eF'A 1 1 jlkusj l'r (/yl ?. Sgc
: 2
A Martingale Measure
, The method of forming risk-free portfolios and using the resulting PDES ! was discussed in Chapter 12, although a step-by-step derivation of the , Black-scholes formula was not provided there. adopts matingale diferent method of equivalent ne measures way i a tedious but is points, derivation at obtaining the formula. 'I'he rests of same on straightfomard mathematics and consequently is conceptually very simple. we will provide a step-bpstep derivation of the Black-scholcs fonnula ; using this approach. , First, some intermediate results need to be discussed. These results are .( important in their own right, since they occur routinely in asset pridng
sjeyrj
c
'
nen
:
s'gdyj
X
!f.
process, j Nll't' tO/),
.
(2) :;
with Fo given. We defme St as the geometric process
X
x
i.
'
1
yypf.
oj
.j.
qy)
,
s) e - )
(.r.c-.3.12-2 +u-(.4m+!
x.
A: r)
.
g.zr'l
;
1 e 2./&22
-;
(j-, . (y,?+...z,)j2 J
dj.r;.
(j?)
=
esr-Fjtr
,
2
r z
(1c)
.
The moment-generating function is a useful tool in statistics. If its kth delivative with respect to is calculated and evaluated at 0, one linds the #th moment of te random variable in question. For exampje, the first moment of F by taking the J can be calculated derrative of with : respect to (10) =
:
:
q
#A,f J
!
FJ.
,.z
e(A#z+1.r
xgta+l(r
, :
obep
z-avzt
-cx)
xv(A)
.
called a generawea wicnerprocess, becase it a lxhasis asome-es and has a variance not neceoaruy equal to one. meau, nonzero ion, 24)may be random, as long ms it ia ndemndent of
luy.,
cy,,
-lsi+rscjzlgtasrq.js/z)
,
is an arbitrary parameter. The cxplicit form of this moment-t' irnportantlp' genera ting function is useful in asset pricing formulas. More momentz? t he types of calculations one has to go over to obtain the stochastii Operations in generat Z g fundion illustrate some standard , Calculus. The following section is useful in this respect as well.
(5)
.
But the integr:zl on the right-hand side of this cxpression is the area under the density of a normally distributed random variable. Hence, it sums to oae. We obtain
xsflis the initial point of St and is given exogenously.z We wcmid like to i obtain the moment-generating fundion of X. The moment-geoerating function denoted by M() is a specihc expecta- ; tion involving Fr, : ljer/j, oj.(A) (4) B' here
(n.s,g
:-J
-
-x
7
=
,
(6)becomes
=
sgrrrj
'
(S)
.
j
=
E:'
SLdY'
=
z
dv t. (a) -c.o g.n.a.c t ne cxptynent of the second exponential function can now be completed nto a square. The terms that do not depcnd on 1$ can be factorcd out. Dolg this, wc get
':.
''-
the equality in .
g
Now let F2 be a continuous-time
2,pv2/
ln this expression, the exponent is not a perfect square, but can be completcd into one by multiplying the right-hand side by
:.
Functm
e-
=
thc exponents:
:'
2..1 The Memenf-oeneraffng
j
..x exprcssion inside the integral can be simplified by grouping together
rne
formulas.
347
Now substitute
normal dism-bun
0 for
=
lgt +
g.
z tkje
jzj-yjga/
c
(1j)
in thig formula to get
.
j.
'. .):' .(
ou
'
:
1E. ..
J
=0
=
#t-
(12)
:'
!
C H A PTE R
348
Equivalenc Martingale Measures
)5
@
.5 Converting Asset Prices into Martingales
349
.1
For the second moment, we take the second dezivative and set
equal to
Using these, we can calculate the conditional expectation of a geomctric Brownian motion. Begin with
).
Ztz.1'O :
t?2
u
o
A=()
JA 2
(13)
(rkt.
=
'
Thesc are useful propcrties. But they az'c of secondary importance in asset formula in asset pricing pricing. The usefulness of the moment-gcnerating relatinship the cxploit Eq. tied We is to (10). FJ
.E'(c
j
tr/z/e.l fWl2
=
'
s'ae'''
=
f (E (O, x),
,
y)
2
tr f
'j
:
t,
Or, multiplying both sides by
):
E (-, Isu u
t
,
it
(15)...(, I
(21)
I
xs'al
=
F(cAF'1.
%,
cM.('-.)+l
(22)
(10)evaluated
fpt-s)
-
(l-X+
l t'f z l z(:-:)
.t.
Observed values of st will occur according to the probabilities given by P. But this does not mean that a hnancial analyst wouid lind this distribution most convenient to work with. Io fact, according to the discussion in Chapter 14, one may be able to obtain an equivalent probability # under which pricing assets becomes much easier. nis will especially be the case if we work witll probability measures that convert asset prices into
S.
S
Note that, by the dehnition of gencralized Wiener processes,
'l'
.$.1,
I
nis formula gives the conditional expedation of a geometric process. lt is routinely used in asset pricing theoly and will be utilized durirlg the following discussion.
J
(16)
).
#
Defme
'
J
$ =
e-gc-rfl
.
By dehnition, it is always true that F,
F(t, ay
=
.7
Nmt,
'w
1 .'
whcre F; again had the distribution
t
satislies the martingale condition. Define p as
.:
&:
(36)
tsing the probability given in (36).ln fact, the formktla for such a conditional expectation was derived earlier in .Eq. (25).We have
E9.
(3n)
e-rt $27
=
(35)
w'here the drift parameter p is arbitrary and is the only difference between the two measures P and P. Both probabilities have the same variance parameter. Now we can evaluate the conditional expectation < /), E #le-rt-us ,
;.
ln fact, because of the existence of a risk premium, in general, we have Epfe-rt,S t ISu u < fl > e-r'ls u (29)
N(Jtf, a2t).
-x.
Nlpt,
.:
.
,
# by
Now, deline a ncw probability .
(28)
&'
(34)
ct has the distribution denoted by P:
,
EPLe-''sl I
ljc''
=
.
351
e-ru.
:
-
(33)
/
j
x
Ll .
1
tr
r. u
. ? whatis its form?
(38)): Elktt-rqt-s
,L E,
a) +
-
1
uj
(f -
fj
=
=
so .
0.
(40) (41)
'
E'
i .
L.
This is the martingale
tingaletmder /. :,j
2: )
':.
'-'ts
(c
,
condition.
Isu,
u
.
-rr
gg
x
e
=
-rv
-.
maxt
sz .K
mauxls w .K
(jyg
(taj
t)jvjp,
(()g;
,
,
,
''
where we also have
'
..
Sv
't
under
, (e
-r(r-;)c
Ct
=
Et
r. ),
(s:)
these substituting
?,
=
'
-
K , 01
-x
y.r.
(64) e-iz!-ftrr-tr-ltr
2,77-G27*
)' ) dv z.
(65)
'Ib eliminate the max function from inside the integrai, we change tl'le limits of intcgration. we note that, after taking logarithms, the condition XCY-T
'
=
...
:
he
(63),
c-rl' maxl-ei'r
C'u
of thc call option. llerer > t is the expiration date option's payoff will be Sv -K if Sp > K. expiration, tlle that at we know tl,e call option expires with zero value. This permits one to write otherwise' theboundary condition C T maxlsz - K. 01. (5X W
in
=
t K
j
(5a)
.rz
Ctl
r(F - /) + fO(T tj dk (54) E' T r tz : Jn these expressions, T is the expiration date of the call option, r ks the g. risk-free interest rate, K is the strike pricc, and f;r is the volatility. ne f'unction N(x) is the probability that a standard normal rarldom variable (, :k N(Jll by given example, is less than x. For ln(u%/r)
=
maxtyz
=
,.
ith
-r(z-j)
(d
Let t 0 and calculate the option price as of time zero. Accordingly, the current information set It becomes &. nis way, instead of using conditional expectations, we can use the unconditional cxpectation operator E p (.1.
'
(53)
I
j!
implies
We now proceed with the step-by-step dcrivation of the Black-scholes formula by directly evaluating
F'
K, 01,
>
Et
=
1.
Irl order to dcrivc thc Black-scholes formula, this expectation will be calculated explicitly, nc derivation is straightfomard, yet involves lengthy cxpressitms. lt is bcst to simplify the notation. We make the following simplilicat j ons;
'
Under these conditions, the Black-scholes formula can bc obtained by solvJ ing the following PDE analytically:
W
355
i11 I ! j g I
(66)
j
'
Ej : ! '
11
!
I
I
p
1 '
1
J.
'
.
:
356
C H A P T ER
Equivalent Martingalc Measure.s
15
.
'u
is equivalent to
.''
-
Using this in
K
=
-
mcjry
-
'
') y,
(6g)
,
-(r-l/)r)2
Using the transformations in
:
x
Ke-rT
L
(69)
separately.
g
!
First we apply a transformation tlzat simpliEes the notation further. We define a new variable Z by z
=
:'
E
'X
1
) 2 .w1:4.%
2
'n'r
tN:p
=Q'
r
Ke-rT 1n(
)-(r-
3-
e-
-rcz . /
J
Jr
t,
( y'z -(r-)cr2)z)2
1 c-lz 277-
2
dz.
lnlv/ulj
) lntS /X') =
,
(
-
a,v
e
-rr
-
sinceSt is a
dXt
(89) . :;
e-rtf;
.
?
:1J :)
But, we can use the Girsanov theorem to convert e-rS t into a martingale. . Wc go over various stcps in detail, because this is a fundamcntal application E' of the Girsanov theorem in fmance, 2 'T'he Girsanov theorem says that we can find an f/-adapted process Xt h and a new Wicner process ( such that .i tvf dXf + #F). (92) ) $
=
=
where the f, is defined
)
F.
(93) 7
,
'
as
:
.
(-,
e
=
jt()
.:t-
.
Jyp
-
;
ay'j
du
(q4)
.
Jgc-rl,rl
=
. i y''
r
:
k$t 1
=
e-rlgpwl
-
rsj dt
5Here r5'; dt is an incrcmental eaming if
ut
-
e-rtn
tfa't-, + e-rtc
g)'fzt l
.
dollars were kept in the riskdree
aruset, a nd
#@c-r;1F+ e-rl #E
=
(99)
=
e-rlf-rFdtj + F ds i -h . Z/; or Ft < Lq, the buyer of a i y # aid-in-arrears receives, at tne (+1, the sum ;
t
'',
Receipt
=
''
- Ltj)5
N-'t
Q
.j
,
'
N gF,
!
$
j
1 i
:J
E E'
if it is positivea or pays
t
L
Conact
N (Fi - Lt J, if it is negative. 'The F.RA rate Ft is selected so that the time l price ot':h !J the FRA contract equals zero, This situation is shown in Figure 1. ln case'..t umelt of FRAS traded in adual markets, often the payment is made at the time the Lt is observed. Hence, it has to be discounted bv tl+.Z,f :). n
';.
r!
I
'
6 References
.%t
.
l
'
jE
hto'j,
1
j '
chapter is simply a brief summary and cannot be considered an introduction to interest-sensitive sccurities. It has, however, the bare minimum necessary for understanding the tools discusscd in the remaining chapters.
vL.
Q
E l
'
'rls
vnsc.ju,; o..o'---'..
, ,
;
.
)
.
.
.
i j
.
r))#f + a'rt. t)d J1't ,
qk
.
; :
''tricking''
:',
.
' requ ire: of interest rate dynnmlc.s
The modications dh
t dt
! i
ii
someof these complications can be handled within a Black-scholes fzamework by either making small modillcations in the assltrnptions or by them in some ingenious way, But the early exercise possibility of interest rate derivat-ives and stochastic intercst rates are tw'o modicatitans that have to be incorporated in derivative assct pricing using new mathel-lft iCa l tools ne following chapters are intendcd to do this
,,
=
!
,
Clcarly, this very broad class of interest rate derivatives cannot bc treated using the assumptions of the Black-scholes environment,
!
W?; d Jjzr -
i
j
I
stocks.
,,!'r
d
! E
,
,
.'
j
r
rrhe
,i,
=
I
'rhe
?E
+ eutdkk,
j.
i '
p
Another complication is the coexistence of many interest rates, Note that within the simple Black-scholes world. there is one underlying asset St. Yet within the ixcd-income sector, there arc many interest rates implied by diferent maturities. Moreover, these intcrest rates cannot follow very diferent dynamics from each othcr because they relate, after all, to similar instruments. Thus, in contrast to the Black-scholes casc for interest rates, one would deal with a vector of random proccsses that must obey complex interrelaresulting k-dimensional dynamics tions due to arbitrage possibilities. arc bound to be more complicated. Note th a t in case of a classical Black-scholcs environment, modeling the risk-free dynamics of the underlying asset mclmt modeling a single SDE, where over-time arbitrage rcstrictions on a single variablc had to he taken into account. But in the case of interest rates, the samc overtime restrictions need to be modeled for l-variables. There is more. Now, arbitragc restrictions across variables nccd to be specihcd as well. Last but not least, therc is the modeling of volatilities. volatility of a b0n d has to vary over time. After alI thc bond matures at some specific date. Hence, these volatilities carmot be assumed constant as in te case ot
) ,'
4.1 Dri# Adfxstvnent
gvhdt
j
4.2 Tenn Strucfuze
g.
'.
=
;
(), lid > 9) are state prices, Thc hrst row represents the payoffs of risk-free lending the payoffs of the stock St, and and yoaowing, the second row fepresents t jae tjajrlj row remesents payoffs of the option Ct.2 According to the Fundamental neorem of Finance, the (#u # J oxist there wgl be positive if and w are no arbitragc possibilities given d). exist and arc positive, reverse is also tnle. If the (r, qj. then there will be no arbitrage opportunity at the prices shown on the Iefshand sjde. e r jsk-free probability # was obtained from the lirst row of this matrix,
.
j.ja
E
E
'rheorem
J
!
'
:
.
i
'
$ '
ff
.
,
,.:
.jjj
rnw
(#Li,
ut,
.
.
'
.
rj.jy
fcorrect''
'
.,
.
1
.
W jautl
prices
lone, can also ask tho following question, Given that we want to convcrt IISK t choose martngalesby modifying the true pro babilitydistribution, is there a way we c.arl way? synthelicmeasure irl some xyyesl',
inW
2wz
;
:
+. (1 + .L4d
,
(jejyrjjug =
=
( 1 + ralyyu (1 +. rhl
ty
make slight mtxlificatiorls in the notation compared to the simple model used in Chapter 2. In particular we introduce iudcxing by u and d, which stand for the two-statcs.
:
: .'
rjpu
(1 +
pd
.
-
J
js
=
.
a, '! : i
;
' .
j
E q.
382
C H A PT ER
Arbitrage Theorem
17
'
2 A Model for New Instrtlments
r
383
2,E.
nus, the probability P expected returns of the risky ssets so that all expected returns became equal to the risk-free rate r. Hence, the term isk-neutral measure or probability. In the next section we extend this framework in two ways. First, we add another time period so that the effects of random Puctuations in the spot Second, we change the types of instruments rate can be taken into aount. considered and introduce interest-sensitive securities.
gave
iTmodified''
l'bd
1 #? + =
The conditions
prices t) u ,
,
#
0
/u, 0
W
l
N
,
.
U
tl 17
.:
.
j
l
E' I
h
v#d
,
,
=
1+ r
=
,
;g
T'11
h'j:
ru
tz
rd
wx
utl
-.
du
..--
d
%
rd '
tl
C 2+A
Ct
l 1
--
:
,%
E>
--'-
I
s''z+ #du+
uzd
+. rj 1 )(1+ r)g 2)
(j +
.
2.3 StymeUndesfruble P'ropeufes The probabilities f' were genera tcd using the
1
y
y,
.
'
'
(.1
+ rt
x
.-
E
=
--
24.1-f- rtg)
1
#
(1 +
x
g
GJ(1 +
%)
Lt 2
.
(30)
Rearranging. we obtain a pricing fonnula which gives the arbitrage-free
5.
FRA rate Ft : 1
..
Now consider thc details of how these probabilities were used in pricing the FRA contract. First, note that pricing the FRA. means determining an F Jausesuch that the time 1: value of the contract is zero, This is the case lj, all F-'lAs are traded at a price of zero and this we consider as the be arbitrage-free price. The task is to determine the arbitrage-free F implie by this price. From the second row of the system in (6)we have ,
'
Ft
1
=
-
E>
1
-
j )(I+-r,a
(1+rq
)1
E
1
.
(1
.j.
r'
)(1+ o ) z,
tg
2
.
(3j)
'.
rjju expression clds a formula to deternne the contractual rate Ft using the risk-free probability 1. But, urilike the case of option valuation with constant interest rates, we immediately see some undesirable properties of the representation.
.
0
=
(F/I
-
Lt,u )4 uu + (F 2,
-
Lwt, lal
)#JJ. + (Fj , - Ldral''z + Ft 1 - Ldtz
(26) .,' .
E
(
'
;.
.!
cHh
394
p TE R
17
.
2 A Model ftarNew Instruments
Arbitrage Theorem
.395
.'
First, in general F, is not an usbiased Ft,
#
estimate of Lt, :
lf-,al
E
'
(32)
.
'
1
1 =
y;
After canceling we
jr
1
EI 3
;jj
-
(1 + rr)(1 + rtg)
jj
( yoy (.s
E
tfsjnj .
cv
.
The only time this will be the casc is when the rl and the Lt are statiatically separately: nerl, tlae expectations can be taken indepertdent. Ft
Proceeding in a similar fashion for the Libor derivative same argument. The pricing equation will be given by:
:
wjth :
:
.
?
t
g:c
EF
jsyj a
.
t!
jg4;
' Under this extreme assumption the fonvard rate becomes an unbiased es- 'E timator of the corresponding Libor process. Blt, il practice, can we really maturity lbor rates are statistically g say that the short rates and the longer independent? This will be a difhcult assumption to maintain. As ; Consider the sectmd drawback of using the risk-neutral measure with taken expectations we noticed earlier, the spot-rate trms inside the )'' dt'l nt factor Ollt. ln contrast to thc simplc model of Chapter respcct to 2, where r was copstant across states, wc now have an rt, that depen on :' stcchastica thc state u, #. Hence, the denominator terms in Eq. (31) are and stay inside the expectation, i Third, the pricing formula for the FRA in (31)is not linear. This prop- :') sight, can be quite a damaging aspect of the crty, althoug,h harmless at first major inconvenicnces for the mar- .. use of risk-ncutral measure. It creates tlle ket practitioner. In fact, when we tr.!rto dctermine the FIIA rate F,j or namely model fwf.l need processes, to pricc of the derivative Ct, we now . the rt and Lt, instcad of one, the Lr. Worse, thcse two processes are correevaluating y te of task The complicated way. lated with cach other in some expressions. corresponding expectations can be arduous with nlzmlinear .. in a denminated Ftt is the dehnition, A linal ctmmcnt. Note that, by Iiskvalue that will be settled in period la. Now consider how the Curre n jy l Pricing Eq. (31).The Pricing formula wit neutral mcasure / tperate: WW works by Iirst discounting to present a value that belongs to time f3. L: tlies to ; after taking the averagc via the expcctation operator, the formula that is ( this discounted term in tfme la dollars, simply because rcevress ; eventually how the contract is settlcd. freo; arbitrageClearly, this is not a ver.y efhcient way of calculating the altoget jwr discounting the with fotward rate. In fact, one can dispensc d because btlth the Ftt and Ltg are measured in time ts o gars!
ts.
'lahe
we now consider an alternativc way of obtaining martingale probabil tfes, pthin the same setup as in (6)and with the same 4/, we can utilize thc thiy.dequation to write; B;
'
Dividing by B; 1
'
'd
=
=
1
puu+
s) .;
1
ud
sf
+
z1
1
j
v
(36)
1
sjr,
$
du
+
1 RqI:,
.....
/
aa .
;;q
.
(37) (38)
,
.
1
this equaton becomes j
nuu + npd +
=
'
Because thc
'y
:
fj
yyu
o nad.
arc positive under the ctndition of no-arbitrage,
.
zr
fj
>
() '
/> y
.s,
g
(yq) we have
jyj)l
This means that the a new set of synthetic marprobabilitieg. ney yeld a new set of martingale relationships, We call the Wj (hc fonvara measurc. Before wc consider the advantages of the fomard measure, over thc risk-neutral measure, #, wc make a few Comments on the new normalization.
:,
m-i.i could Iyc used as
tingale
u
'n'il
.
,
.
.
E. :
ffd
+
and labeling,
.'
:
du
+
,
.
E:
4*
+
1
1
.
:
.35)
2.4 A New Nonntynwjfx
.:t
.
(
'
Clearly, using the moncy market account to define the probabilities # as done Z Eq, (24.), crcates complicatitms Which were not prcsent in Chapter g. jujow we wjll see that a judciolls chok.e of syntlletic probabjlitieg can get around these problems in a very convenient and elegant way.
.
y
c t,
securitics.
;
gktt
- + r) (1+ rt, )(1
make the
A ain jj the ct is an jnterest-sensitive derivative, the same problems g #s will be random discount factor cannot be factored present. otlt of the expectation and the spot rate wll in all Iikelihood be correjated wjth the optio'n payoff Cts if the latter s written on interest-sensitive
:
(33)
1
E
=
'
'.
G, we
'
.
'*'
396
C H A PT E R
@
Arbitrage Theorem
17
2 A Model for New lnstroments
j
i: .
nil,
First, to move from cquations written in terms of state plices to those expressed in terms of =, we need to multiply a11state-dependent values by which is a value determined at time Hence, this term is independent B'% I of the states at future dates, and will not carry a state superscript. means that it will factor out of expectations evaluated under Second, note that we can define a new forward measure for cvery default-frce zero-coupon bond with different maturity. Thus, it may be more appropriate to put a timc subscript on the mcasure, say, wy, indicating te maturity, 71 associated with that particular bond. Given a delivative written on interest-sensitive sccurities, it is clearly more appropriate to work with a fotward measure that is obtained from a bond that matures at the same time that the derivative expires. Fla, note how normalization is done here. To introduce te prob.. abilities in pricing equationsl we multiply and divide each (J' by the Bt /B; as x'iib this n motmts to multiplyina, i'n the maAfter relabeling the ! each trtx Eq. (6), assct price by the corrcsponding entry of thc short bond by the BtL.s Bst Hence. we say that we are (: normalizing rrhese and relatcd issues will be discussed in more detail below.
:! ';
.
!'
'nis
',
t) A;j =
E '.
.n..
!
a*
*'
Frj ;; i
$'
.
evaluating the similar cxpectation undcr the measure m To do this, we take the second row in system (6) and multiply every element by the ratio B;1 //.D1 which obviouslv equals one;
.
.j ;
;'
.'
'i
)
0
=
(F,k
-
'b
,
u Zlz)
+(F ,, -
-'
'
Bs11
sj1
$ uu + (Frl
B5h
LdJa )
sj
-
l
&11
sr f
1
/
UJ
-1-(y /1
-
kd
f)
)
Bq;1
(44)
+
(45)
of the Libor process Lt, 'This means that we
qvii.
gfsraj
E=
=
,
(46)
,
=
(f,,+zaJ
Et=-a
(47)
,
-
.
8-,''r(.1
. .?,
.
)x
sr
1,
with the lt being the information set available at time /. ln this particular case, it consists of the current and past prices of a1l assets under consideration. Nexq we recall the recursive property of conditional expectation operators that was used earlier:
:
dV
Ar$Tg.lf,
=
E E:
g,
*1
Et*
(42) .:
ldd
.
where the subscript of thc E7f operator indicatcs that the expectation is now takcn with respect to information available at time t + Tlmt is,
,
.
xdljj + Adra
.1
''
'*'
.J.u/2 )
+
F, h
:
Now consider
'
m'du
Thus we obtained an important result. Although the F/( is, in general, a biased estimator of Ltj, under the classical risk-neutral measure it becomes arl unbiased estimator tf Lt, under thc new forwardmeasure m Why is tbis rclevant? How can it bc used in pradice? Consider the following gcneral case and revert back to using the k instead of the ti notation. Let the lbor rate for time t + 21 be Sven by Ldq-c, thc current forward rate be Ft, and consider its future value Ft-a with s > 0,9 we can utilize tlAe mcasure = and write:
.
.'
t,
+
FtL
:
(z, j
I'WJ,
+
Lut,m-ud Ldttavdd z-d/a-rrfl'zj gz-/z-rri'u
,,
'''
#
=
(
2.4. 1 J'zo/pdr/d. of the Normalation We now discuss some of the important results of using thc new probab ility measure = instead of #. We proceed in steps. First, recall that within the setup in this chapter, the use of te risk-neutral measurc, #, leads to an equation where the F; is a biased estimator of the Libor process Lt. In fact, we had
(41)
fa'r'2
w'here the right-hand side is clearly the expectation evaluated using the new martingale probabilities now have:
*
-
'n>$
.
,
.
E#
'n'ud
gfvu/a+
11
5
.
and that tlley sum to one, we
*1
/
ry
Ft,
g>) -
'zri/,
Note that here the B( has conveniently factored out becatzse it is constant givcn thc observed, arbitrage-free price Bf Canceling aad rearranging:
;.
'
397
are in fact the corrcsponding elements of obtainafter factoring out the Ft, :
,'
whichsays that the castsnow.lo
t
Recognizing that the ratios
abest''
EfI''QE11 EI E1 '
=
'
,
(48)
forecasts of future forecasts, are simply tbe fore-
:
gnus F is the FRA rate ebsenrod bsewedwithin a short interval of time
,1'
fk'
# 71
.'
(4g)
Lytgan,
.
;
J
'.
) jk
'.
the
x, the forwazd rate dynamics will become
q:,
E .
,'
: :
1
:) .
''
dt
=
g.F/tl1,r,, g F, dt + s
,.
;.
'
':
theoremwherc the p,* is the new riskmdjuated drift implied by the Girsanov E' Black-scboles unlike the Under / this drift is not known at the outset. So, replaced by the known stock price is l casc where the drift of the underlying unknown to . difqcult with a (and constant) spot rate r, we now end up dctermine. Consider what happcns to the forward rate dynamics if we use the k w1t11 instead. Under the forward measure obtained forward measure will be a & T time for Tl-normalization, the folward rate Ft delined l BLtb 13 martingale. Hence w: can write.
.
E
'lv
'-
,'
'
'/ tj =
,,
';r
1
#
Bt
i)
(65)
.
:i
-
A11prices that maturc at time F would then be matingales oncc they are nonualized by the Bt. Note that the ratio
,
.
';
#F t
=
eF f #H(c
,
:
f'
A very convenient property () is a Wiener process under where the unbiased of this SDE is that tlie drift is equal to zero and the Ft is an I'F;A
=.
1
i'
7r /
,'?
--J B
1.
ia' N--
cs timator of Lr-s:
) Ff
E'
=
,!
can be uscd to write:
::'
(fvz-al ,
E.
f*lm diffictty of determining an unknown drm. We nere in Mt' Black-scholes type argument and price this caplet J go ahead with afashion.l4 ) straightfonvard
'J
.,
is no additional
?
13Itis important to rcaliz, that under Will not be a martingale.
: Elo need to discllznt tile OP Black-k,lei . 14Aremazing difference is in the units used hre. 'There is unlike the namics. rate forward nis the if use we present the payoffto cxpressed in time dollars. ' whcre the stock price dynamics dS( are i' envirorlment
:
'
fji =
-,
'&
i.l
-
B/
&
.
l
This way one can go from one measure to another. Would such adjustmeats be any use to us in pricing interest rate sensitive Securites'? The answer is again yes. When we deal with an instrument that depends on more than onc Lv with different tenors F, we can Erst start with one forward measure, but then by taking the derivative with respect to the other. we can obtain the proper terms'' that need to bc introduced. tlcorrection
-'
(67)
'
.
:LL ::%.
C H A PT E R
404
*
17
r
Arbitrage Thcorem
3,
. The annual volatility is
!
and forward In this chapter we introduced the notions of normalization role in priclg delivative securities measure. nese tools play an important in a convenient fashion, More than just theoretical conccpts, they should be regarded as impolant tools in pricing assets in real world markets. ney are especially useful fgr any derivative whose settlement is done at a hzture
g'
!
i
.
, '
g
,
;
yt
iTup''
E
(Tdown''
2. Suppose at time t 0, you are given four default-free zcro-coupon bond prices #tf, T) with maturities from 1 to 4 yeal's:
:',
=
2 '
'
/40, 1)
:%
.94,
P(0, 2)
=
.8(9, 3)
.92,
=
.P(0, 4)
,87,
=
,80
=
.
(a) How can you a spot-rate tree to these bond prices? Discuss. (b) Obtain a tree consistent with the term stnzcture given above, (c) What are the differences, if any, betwecn the tree approaches in Ouestions (a) and (b)? dt''
y .')-
''
r:( E.
4 References
3, Select tcn standard, normal random numbers using Mathematica, Maplc, or Matlab. Suppose interest rates follow the SDE:
)
ne book by Musiela and Rutkowski (1997)is an excellent source for a reader with a strong quantitative background. Although it is much more demanding mathematically than the present tcxt, the results are well worth the efforts, Another possible source is the last chapter in Pliska (19W). Pliska trcats these notions in discrete time, but our treatment was also in discrete time.
h
drt
:'t';
dt +
.02r/
=
.06r/
:11s.
ytssume that the current spot rate is 6%.
. ,.
(a) Discretizc the SDE given above. (b) Calcmlate all estimate for the followirlg expectation interval ,04,
'.(
5 Exerclses
E
j-
'
csdsmaxyj
(ljj
.06, -
,
and the random numbers you selected. Assume that thc expectation is taken with respect to the fme probability. (c) Calculate the sample average fo
1E
on the spot rate rt'. I ,.!
..) .' T
#rt + e'% J?l';.
E
gc-V
''daj
$
* Thc annual drift is
and then multiply this by the sample average for:
r
E
gmaxlr.
Do we obtain the same result? (d) Which approach is correct? :
.t j
using a time
=
'
* 'T'he rt follows: =
.
(a) Suppose instruments are to be priced over a year, Determinc an appropriate time interval A, sth that binomial trees have five steps. (b) Wlaat would be the implied u and d in this case? (c) Determine the tree for the spot rate rt. (d) What arc the and probabilitics implied by the tree?
1
t
s'
normalized by tho * The price of all assets considered here, once of becomes F), bond Bt, arbitrage-free price of a zero-coupon a martingale under Then one can short the bond during :hc pcriod gJ, F), and invest e -r(T-,) of the proceeds to risk-free lending. At t ime T the short (bond)position is worth -$1. But the risk-free lending will return +$1. Hence, at time T the net cash flow will be zero. But, at t the investor is still left with some cash in the pocket because
Also, note that for stochastic differential' s such as dRt, T), Ito's LentmaE needsto be used due to involved wienercomponents. we are now ready a fundamental pricing equation that wiIl be uied tbroughou'z Et to introducc second part of this book, namely, the bond pricing equation. the '
.
'
;
:;hi' ':
sj/
ln tbis section we start discussing thc rst substantial issue of this chapter. wc derive an equation that gives the arbitrage-free price taf a default-free zcro-coupon bond #(f, F), maturing at timc F. We go in steps. We flst bcgin with a simplifed case where the instantaneous spkt rate is constanta and then move to a stochastic risk-free rate. This way of proceeding makes . . it easicr to undel-stand the underlying arbitragc arguments. !,
,
gn)
''
.
:
-
-,(w-,)
e
'.
-E.
. s
Thus we 'hrst let the spot rate rt be constant:
r, 5
Bt, F)
e.
r(r-t) .
yqt p)
:
'( .
.,.
r
Consider the rationale behind this formula. ne r is the continuously coml r c-r(T-/) pounded instantaneous interest ratc, ne ftmc tion p lays the role L.i exponential f function F the of a discount factor at time t. At eqlmj s a1I times. the otlter bond, At same as the maturity value of the ,1 whicb is Side of right-hand 1. the < F, the exponential Hence, fador is lcss than t f ; discounted to time-r dollar, Eq. (9) represents the present va 1u e of one :( at a constant, continuously compounded rate r, Now an investor who faces thcse instruments has the following cboices. , can invest c-r(F-0 dollars in a risk-frce savings account now, He or sbe worth will T be $1 Or the investor can buy tbe r-mattllitr and at time this reium 1 and F) dollars now. This investment will also discountbond pay B 'f i risk, default anu T. Clearly we havc two instruments, w ith no $1 at time with the same payoff at time F, nere are no interim payouts either. lf interest rates are constant and if there is no default risk, any bond that promisesto pay one dollar at t ime T will have to have the game prke as the initial investment of et-r('r-') to risk- fr ee lending. That is. we must have.. y
=
i
I
!
! 1
.().
such arbitrage
j
opporttmities is
'
=
r-r('r-))
'
(1))
q
'
i
.
I
3'2 Stochzzstic Spot Rztcs
:t'
','
earned during the innnitesimal inten'al gf,l + #fl. Thus, rt is known at t, but its future values :uctuatc randomly as time passes. The Fundamental of Finance can bc applied to obtain an arbitrage relation between and 7.) tjw stochastic spot rates, rt, t l gI. r1. s(/ tilizc the methodology introduced in Chapter 17. We take the curwe u rent bond price Bt, Tj and normalize by the current value of the savings account, which is $1. Next, we take the maturity value of the bond, which is $1 aud normalize it by the value of the savings account. 'Fhis value is cqual
:
,1
:'
.
,
.1'
.
r
#(f,
F)
=
e
-r(T-l) .
(10)Ek' .
': ..
.'.;.
'
! : ' '
.
;
i:
h
.
.
1
erheorem
,
,
:
whenthe instantaneous spot rate rs becomes stochastic, the pzicing formula in (9)wili have to change, Suppose rt represents the risk-free rate
.
,
!
'
Hcncc, this relationship is not a dehnititn ., or an assumption. It is a restriction imposed on bond prices and savings accounts by the requircment that therc are no arbitrage opportunities. Notice that n obtaining this equation we did not use the Fundamental Thetarem of Finance, but doing this would havc given exactly the same result.S
E
(9)
,
'
''bond
>
.
=
,
(
Then the price of a dcfault-free pure discount bond paying $1 at time F will be given by:
I
()
>
sjj vj
-
The only condition that would eliminate when thc pricing equation'' holds:
:
(8)
n
=
e-r(r-/)
rjxex
'
3.1 eonsfunf
-
.
.
x
Raze
E
1 !
The other possibility is Bt, F) < e-rl r-f) at time t, one would -r(z-J) dojjars, and buy a bond at a price of #(1, F). When the borrow e (jate arrives the net cash Qow will again be zero. The +$1 received ma (uyjty from t jw bond can be used to pay the loan off. But at time /, there will be a nct gain!
,
'pot
,
,
i.
3 A Bond Priclng Equation
E '
il ,
. .
(
3see
F-xercise 2 at tlx end of tbis chapter.
.' :
.j
!
j
j
I
T 416
CHA PT ER
*
18
Modeling Tenn Saucture and Related Concepts
.3 A Bond Pricing Equation
'
'
.r
rsds because it is the rcturn at time T' to $1 rolled-over at the instanto eh Dividing $1 by this value of rate rs for tbe entire period s e (f, taneous r. a e-lk ' savings account, we get the
.
i
.
According to chapter17, this normalized bond price must be a martingalc under the risk-neutra! mcasure /. Thus, we must havc Bt, T) e-f
wherethe term appliedto the par
Gd,
value
EI) =
gr-.lfrszj
E'
t
(1a)
,
ij
('
.
)
i,
''.
I''
can also be interpretcd as a random discount factor
I
'
.
$1.
.
Some further comments about this formula are in order. First, the bond ?.: priceformula givcn in (12)has another important implication. Bond prices dependon the whole spectrum of future short rates rsmt < < r. In other words,we can Iook at it this way: the yield curve at time t contains a11the information concerning future short rates.4 relevant Second, therc is the issue of wliich probability measure is used to calc'u- t. latethese expectations. One may think that with the class of Treasury bonds . being risk-free assets, there is no tikpremium to eliminate, and hencc, there ! is no need to use the equivalent martingalc measure. This is, in general, incorrect. As interest rates become stochastic, prices of Treasury bonds will i ; risk.'' contain amarket ney depend on the future behavior of spot rates ). E and this behavior is stochatic. To eliminate thc risk-premium associated r. With Cisksy eqtlivalent martingale WC nCCQI SIICU to tlse measures in evaluatill CXPIVSSiODS 2S ill 12) ( We now discuss this formula using discrete intervals of size 0 < 'This ' will show the passage to continuous time, and explain thc mechanics of the )( : bond pricing formula bettcr.
lI
('
-
',
WC n0W
':.
,
Sctting.
(
.
'
(1+
'
j
'
,J
1
(1+ G)(1
+ r,+aA)(1 + rr.j-zaA)
,
:
' :'. t
l t
.':.. ;
(13)
y :
;'
.
: .
'
L.''
e
+r,+-aa)
-->.
(14)separatcly,
.-rr,
e
-r!
I
,SA
.e .'
-r; ,aaA
to
t.16)
e-
p
..
tl
g-rl-roa...-r.aja j
0:
.
'
provide the optimal forecasls in the sense of minin um mean square error givcn an irdbrmation set It
(15)
zzu
or as
.
exmctations
+ r-+.aa)...(1
. ;
)
,.
Now, recall
'
j
(1 +r)(1t
( j4)
'
.
,
;
.,
.
Next, apply this to each ratio on thc right-hand sidc of obtain the approximation .g,= '
:
r, is the known current spot rate on loans that begin at time t and and r,+a, at timc f + are unknown spot rates for thc two future o-yca peods, urilikethe case of continuous timc, tbese are simple interest an by market convention, are multiplied by a.
e-rj
(1 V rjA)
t
3.2.1 Diacrete Tzr/e Consider the special case of a three-period bond in discrctc time. lf A represents some time intezval less than one year and if f is the
xplaced by the exponential function. Second, instead of discrete forward
. E :E .:
''
s E
.
rates we need to use instantaneous fonvard rates, Bccause instantaneous fomard rates may be different, an integral has to bc used in the exponent. Again, note that there is no expectation operator in this equation because y) a jj s(/, are quantities known at time t.
ne
formula:
:
B(t. %.
'
'
.
i
g1 +
F( t, t t +
lXl
,
Now use the approximation has
.
(1 +
.
(n
F( t, t +
F(I,
-
1)
,
t +
nt
T U) and
)1
(2g)
T U),J
z
,.r,4
x =
:
--F(/,d,J+AJA '
s(/, z)
.
..
Bt
'
-
--F(l,/+rI-1)A,/+.r;AJA
.
-
-
r LC-
-
-
'
'
yjvslds
,.(,
=
Take logarithms of these equations
log Bt,
;
(,34)
r) -
log Blt, T + 1)
v =
-
=
1*
F(J, sjds.
(37)
T
l' OW, suppose is small so that the F(/, F) can be considered during the small time intewal (z:z'+ aj. we can write:
:
',',:
,
,t
'r
s(/
=
,y,)
,
xjjm
a-+0
log Bt, F) - log#(/, F + A) .&.
'
.
,L
7nis
will facilitate the derivations of HJM arbitrage conditions
latcr.
yy
(38)
.
That is, the instantaneous fomard rate Flr, F) is closely related derivative of the logarithm of thc discount cun'c.
.)
:
constant
#4 jj F + ) F( t, r)A. - log This equation becomes exact, aftcr takng the limit:
:
-'>'
u:
BLjj F)
jog
.
,
'rhus,
(J6)
.
:.
which means that we can let 0 and increase the number of intorvals to obtain the continuous version of the relation between instamaneou: E forward rates and bond prices, . e-L , #(;'yV# Blt, T) (32);( , is given that the recurring technical conditions are all satished. The F(f, l
woa
F(/, sjds +
t
;) '
'
and subtract:
.F
z'qnN
1
z+a sjas ) e-S syj,
T+
.
:p'
'
'
.;'
2 ar ILC'
e-.L
=
and
:.
(a9)
,
--Ft/,/+.&,l+2A-I
'
-r
i
.
1 1t--'u/ , ? I.t:r But products of exponential terms can be simpliEed by adding the exNnents. So l,,+a)a-,(/,,+a,,+za)a...-,(,,,+(u-1)a.,+sa)a Bt, t + nhl z e(31) Fqt, /+(@- 1)A7t-izlt = (t - E2.1 1
ncous fomar tj yates. We can also go in the opposite diredion and write F(l, 71U) as a function of bond prices. We prefer to do this for the maturities r and U = F + L.' Thus, consider two bonds, #(f, T) and Bt, T + ), whose matu. rities differ only by a small timc inten'al > 0. Thcn writing the formula l
:.
are small, one
fr(,,z;s,)a
g-
prices as a function of instanta-
(t!2)twice:
(2
1
(3:3)
5
..'j'
the Bt, Tj as
Write
s(/
.
that when thc F(/,
(1- + and
.
F(l,')W:
.
g
L .
?'
-
.
';
1
r
e-
=
gives prices of default-free zero-coupon
=
=
F
''
.
.
Bt, Tj
42 1
'
:':g
(39) to the
C 422
c H A PT
ER
18
*
Modeling Term Structure
and
Related Concepts
By going through a similar argument, we can derive a similar expression for the noninstantaneous, but continuously compounded fomard rate f7);S #(/, ZJ logStf, F) logAtl, U) U-F
5 Conclusions: Relevance of
Relationships
the
once discotmted by the instantaneous interest rate rt. The bond Blt, T) paid $1 at matuty, and tbe discounted value of this was
(d-
-
F(1, T) Uj
=
,
(40)
where F(J, 6 U) is the continuously compounded fonvard rate on a loan that begins at time F < U and ends at time U. The contract is written at time t. & we get thc instantaneots folward rate F(f, F); Clearly, by letting F ljm F(l, T; U).
=
I-U
(41)
It is obvious from these arguments that the existencc of F(l, F) assumes that the discount cun'e, that is, the continuum of bond prices. is differentiable with respcct to F, te maturity date. Using Eq, (39) and assuming that some teclmical conditions are satished, we see that F(f, 1)
=
rt
(42)
.
Bt, T)
5 Conclusions: Relevance
It is time to review what we have obtained so far. We have basically derived three relationships between the bond prices Bt, F), the bond yields #(/, F), the fomard rates F(, 7; &), and the spot ratcs rt. The tirst relation was simply delinitional. Given thc bond pricc, we delincd the continuously compounded yicld to maturity Rt, T) as; Rt, T)
=
-
log Bt, Tj F - t
,
(43)
The second relationship was the result of applying the same principle that was used in the fist part of the book to bond prices; namely, that the expectation under thc risk-neutral measure 13 of payoffs of a linancial derivative would equal the current arbitrage-frce price of the instnzment, Izsed as a SEarlicr in this seuion whcn dixussing the discretu time case, F(l, E &) was t)f symlml for simple forward rates. In moving to contilzuous time. and swilcbing to tt) use compounded A continuouslgv rates. symbo! function, te same now denotcs the exmnential symbols for Ihe two differcnt would perlzaps be of prgceeding to use appropriate way gre concepts. But the notation of lhis ehapter is already too ctmplicaled.
j.
El)
=
gc-ftd'j
op-
(44)
.
Thus, this second relationship is based on thc no-arbitrage condition and as such is a pricing equation. That is, given a proper model for r/, it can be used to obtain the acorrect'' market price for the bond #(f, T). The tlrd relationship was derived in the prcvious section. Using again prices of the bonds an arbitrage argument we saw that thc (arbitrage-free) B(t, F), #(f, U) with & > F, and continuously compounded forward rate F(l, 71U) were related according to!
nat is, thc instantaneous fomard rate for a loan that begins at the current time f is simply the spot rate r/. of the Relationships
7* ds rk
The spot rate rt being random, we apply the (conditional) expectation under risk-neutral the relation: obtain #, the to measurc erator
->
Ft, F)
423
Ft,
T; U4
1ogBlt, F) - log Bt, Uj
=
U-T
(45)
,
This can also be used as a pricing equation, except that if we are given a Ft, T; U) we will have one equation and fwo unknowns to determin here, namely the Bs, F), #(1, &). nus, before we can use this as a pricing equation we need to know at least onc of the Bt. F), Blt, Uj. ne addition of other forward rates wottld not help much because each forward rate equation would come with an additional unknown bond price.g To sum up, thc hrst relation is simply a dehnition. It cannot be used for picing. But the other two arc based on arbitrage principles and would hold in liquid and well-functiorting markets. ncy form the basis of the two broad approaches to pricing interest-sensitive instruments. Thc so-called classica1 approach uses the second relation, whereas the recent Heath-larrowMbrton, HJM approach, uses the third. We will study these in the nex't
ehapter.
gsuppose wc brought in unother equation
contairlintr
Blt. U):
1ogBtn U) - log Blt, F(/, U, 5 U We will have two equations, but three uaknowrjs, namely the Bt, Again, an additional piece of information is needed. -)
-%)
=
(46)
.
,
-
7N),
z?(/,&)
and B((,
.).
424
C HA PT ER
and
Modeling Term Srmcrure
18
.
Related Concepa
6 References
Exercist!s 2, Consider a world with two time periods and tw'o possible states at each time t 0, 1, 2. Tlnere are only two assets to invest. One is risk-free borrowing and lcnding at the risk-free rate r,., i 0, 1 The other is to buy a two period bond with current price Bv. ne bond pays $1 at time t 2 whcn it matures, =
can consult this exccllent book for discrete-time lkxed income is one source that contains a good Rebonato (1998) models, Jarrow (1996). of interest rate models, thc other good source i and comprehensivc review For a sunrey of recent issues, see Iegadeesh the publication by Risk (1996). and 'Tbckman (2000). The reader
7 Exercises 1. Considcr the SDE for the spot drt Suppose the parametcrs Wiencr process.
a,
=
a
(/z-
z,
tEr
rate rl rtj dt +
(47)
trtM.
are known, and that, as usual, I'I'; is a
(a) Show that E
Erlrdl
p, + (G
=
pr (r,.Ir,l
=
#,) c-tzt'-o
-
c zf'-
(1-
(48)
c-M@-'))
(49)
,
(b) What do thesc two equations imply for the ctmditional mean and x? variance of spot rate as s (c) Suppose the market price of interest rate risk is constant at A G). Using (i.e.. the Girsanov transformation adjusts the drift by thift and show the that the bond price function given in the text, given diffusion parameters for a bond that matures at time s are by -->
Jt
B
=
U.B
VA 1 r , + -a
(
=
z
-
a
1
-
-
e
e-as-
)
-a(,-,)
,)
)
(50) (51)
.
(d) What happens to bond price volatility as maturity this expected? What happens to the drift coeficient as maturity (e)
approaches?
Is
approaches?
Ls
tbis txpected? (9 Finally, what is te drift and diffusion parameter for a bond with --+ oo'? very long maturity, s
=
.
=
(a) Set up a 2 x 4 system with state prices /. i, j u, d that gives tite arbitrage-free priccs of a savings account and of the bond S. (b) Show how one can get risk-neutral probabilities, #, in this setting. (c) Show that if one adopts a savings account normalization, tllc arbitrage-frcc price of the bond will be givcn by =
B
=
E i
1 (1 + rf))(- 1 +
n)
.
2 The Classical Approach
427
That rtb or for the instantaneous fonvard rates, will also bc is. they will be valid under the risk-neutral measure /. The so-callcd classical approach uses the first arbitrage relation and tries to extract from the 1A(!, F)) a risk-adjusted model for the spot rate rt. This will involve modeling the dri.ft of the spot rate dynamics, as well as calibration to observed vtlatilitics. An assumption on the Markovness of rt is used along the way. The Heath-larrow-Morton (HJM) approach, on the othcr hand, uses the second arbitrage condition and obtains arbitrage-free dynamics of k-dimensional instantaneous fprward rates F(f, F). It involvcs no drift modeling, but volatilities need to be calibrated. lt is more gencral, and, usually, less pactical to use in practice. ne HJM approach does not need spot-rate modcling. Yet, it alst demonstrates t'hat the spot rate rt is in general not Markov. ln this chaptcr we provide a discussitm of these methods used by practitioners in pricing interest-sensitivc securities. Ciiven our lrnited scope, numerical issues and details of the pricing computations will be omitted. lnterested readers can consult several cxcellent texts on thcsc, Our focus is on the understanding of these tw'o fundamcntally different approaches. Glrisk-adjusted.''
Classical and HJM Approaches to Fixed lncome
1 lntroduction Market
practice in pricing intcrcst-sensitive securities can proceed in two
differentwap depending on which of the two arbitrage relations dcveloped in the previous chapter is taken as a starting point. ln fact, Chapter 18 discussedin detail the bond pricing equation Bt, F)
X gd-
?lF rstdj
=
,
which gave arbitrage-free prices of default-free discount bonds #(/, F) under the risk-neutral measurc #, nis was a relation between spot rates rt and bond prices #(f, F) that hcld only when there were no arbitrage possibilities. The second arbitrage relation of Chapter 18 was between instantaneous fomard rates F(f, F) and bond prices: .B(t, F)
:=
!''-
-F
?r
F(! .lJ. '
-
Obviously, both relations can be exploited to calculate arbitrage-free prices of intcrest-sensitive securities. nc market practice is to start with a set of bond priccs qBLt, T)J that above of eithcr the argued arbitrage-free. one to be can rcasonably be nen mcdel for determine rt oz' used and a relations can be to go rclations taf Because F1). thc i two forward rates (#(/, for the set s (f. conditions, the modcl that one obtains for above hold under no-arbtrage Ibacltwardss'
'),
426
2 The Classical Approach The relationship
behveen bond prices and instantaneous spot rates, #(/, F)
=
y'J
jg-
Et/3
cs,'j ,
can bc exploited in (atlcast) two different ways by market practitioners. First, if an accurate and arbitrage-free discount curve (Bt, T) exists, and try to obtain onc can use these in Equation (1), go an arbitragc-free model for the spot rate rr. Onc can then exploit the arbitage-frcc characteristic of this spot-rate modcl to price interest ratc derivatives other than bonds. Second, one may go the othcr way around. lf thcrc are no reliable data on the discount cul've Bs, T), one may first posit an appropriate arbitage-free model for the spot ratc r/, estimate it using historical data on interest ratcs, and thcn use Equation market prices ftar (1)in getting illiquid bonds and other interest-sensitive dcrivatives. Both of thcse will be called thc classical tzp/zmlc to pricing interest rate derivatives. Wc will see that. one way or another, the classical approach is based on modeling tlze instantancous interest rate r!. in the firsl case, by starting from a Set of bond prices jBt. F))., and in the sccond case, from data available tm rt process itsclf. Gtbackwardsj''
Rfair''
1 lim 21 ,&-..a
gc(/,T +
lim
A->t1
1
z
))2
#(1, T +
T)Bt, T))
tzf,
=
,
-
ptr(/, T;Bt,
g
op ))
gtrtf,F + A, #(f, F +
a
/9t7'41, T; Blt,
T)S(f, T))2j
rlt,
tr(/,
6
bFjt,
Bltb F)))
get the corresponding SDE for the
lim JF(r, T; T + A)
=
--yt)
tF(f, F).
Or, dF(t, T)
ct,
=
z?tEr(/,
+
o.t. TJBt, F)) d( pr
T; BLt,F))
T)Bt, JT
r))
tjs
y
(21)
3.3 Interpretafa'en The HJM approach is based on imposing the no-arbitragc restridions directly on the fomard ratcs. Fil-st, a relation between forward rates and bond prices is obtained using an arbitrage argument. Then arbitrage-free dynrlmics are written for Blt, T). Given thc SDES foT bond prices, a SDE that an instantaneous forward rate should satisfy is obtained. To sec the real meaning of this, suppose we postulate a general SDE for the instantaneus fozward rate F(f, F); =
J(F(f, F),
/)#f +
:(F(l, F), f)JH(,
tw(/,
T Bt, Tjj
ty(7'tf,T;#(f, T)) JT
.
(23)
F),
/)
Jfr(I,
T Bt, F))
(24) JF Hence, the previous section derived the amcf no-arbitrage restrictions on the drift coefscient for instantaneous forward rate dynamics. This is similar to the Black-scholes approach that was seen several times in the first part of tbe book. There, the drift term p. of the SDE for a stock price Sf was replaced by thc risk-free interest rate r under the condition that there were no-arbitrage possibilities. Here, the drifl is replaced not by r, but by a somewhat more complicated term that depends on the volatilities of thc bonds under considcration. But, in principle, the drift is detcrmined by arbitrage arguments and will hold only under the condition that there are no-arbitrage possibilities between the forward loan markets and bond prices. nrouglzout this process no rate modeling'' was done. It is worth emphasizing that the risk-adjusted drift of instantaneous forward rates depends only on the volatility para' meters. This is again similar to the Black-scholes cnvironment where there was no need to model the expected rate tf return on the tlnderlying stock, but modeling or calibrating thc volatility wwu needed. lt is in this sense that the HJM approach can be regarded as a true extension of the Black-scholcs methodolor to flxed income sector. =
.
Iforward
where the (r(.) are the bond price volatilities. Wc have several cornmcnts to make on tMs result,
#F(l, F)
-->
The difhlsion parameter will be given by:
.
(20)we
44l
A rcader may wonder how one would obtain these risk-adjusted parameters that are valid under the condition of no-arbitrage. Well, the prcvious section just established that under narbitrage, risk-adjusted drift can be replaced by:
.r))
j
Term Stnzctuze
to
4(F(f, F), f)
F))
JF
Putting these together in iytstantaneoua fomard ratez
-
(m
3 The HJM Approach
(22)
where the aFjt, F), !) and h(F(f, F), f) are supposed to bc the riskadjusted drift and tlle diffusion parameters, and the J#: is the zisk-neutral probability.
3.4 Tl
rt fn the
Approuch
Further, note that in the HJM approach there is no need to model any short-rate process. In pmicular, an exact model for tlle spot rate r; is not needed. Yet, suppose there is a spot ratc in the market. What would the Sbp,s obtained for the forward rates F(l, T) imply for tbis spot rate? The question is relevant becaesc the smt rate corrcsponds to the nearest insnitesimal forward Ioan, the one that starts at time f R-hus, realizing tlmt ,
F(/, t) (25) for a1l f, we can in fact derive an equation for the spot rate starting from the sDEs for fozward rateg. Before wc start, we simplify the notation and rt
=
C H A PTFR
*2
*
Classical and HJM Approaches rta Fixed lncome
19
bs, t) in (24).Then, write the integral equation for write: bFls, F), f) F(l, F) using the new !z(.) notation! =
s(/,w)
-
F'(o,
w)+
jjTbs,
' bs,
whcrewe used (23)and tation for the spot rate r/: =
F(0,
1) +
Next, select F
bls, ujdu
l
=
l
#.$+
0
y
bs, /)#Hrt,
(26)
where the bls, tj is the volatility of the Fs, tj. The srst important result that we obtain from this equation is that tlle fomard rates are biased estimators of the future spot rates under the riskfree measure. ln fact, consider taking the conditional expcctation of some future spot rate rv with initial point t < 'r:
Et/
Erzl =
.E? l j/-tj
+,
glj
,
bs,
Et
,-)
1.
bs,
T
bs
.
/
F(f,
Et>(z'zl
'r)
(28)
-
The second major implication of the SDE for r/ has to do with tho non-Markovness of the spot rate. To see this, note tbat the rt given l>y Equation (2$ depends on te term: J
n
1) !7(-v,
pt
j
bs, ujdu
ds,
(29)
all past forward rate volatilithat, in general, will be a complex function of aaccumulation'' simply of past chanjes not is particular, this term an ties. In the way a typical drift or diffusion term would lead to
.2',
M(G, ds
t
t)
brs, -)J1K.-
J(j
(30) (31)
>(J,
uldu
ds,
(32)
$
and would not be captred by a state variable. The difference between (29) and (32)will depend on interest rates obscrk'ed before t would make the intercst rate non-Markov in general. Next we see an example. rrhis
-
.
3.4.1 Constant Forward f'latilities Suppose al1 fonvard rates F(/, T) have volatilities that are constant at b. Then for each one of thcse fomard rates the equation under no-arbitrage will be given by:
The dynet'mics
Here, the fomard rate in the first expedation is known at tue /; bence it light-hand comes out of the expectation sign. The third expectation on thc side is zero because it is taken with respect to a Wiener process. But the second term is in general positive and does not vanish. Hence we have:
r-- ?k
bls, t 1) -
r)
#F(f,
(27)
'rlp:
,
--2
t
uldujsj
'
Et>
+
g -
'
Term Struc:ure
to
In fact, the new term in the equation for rt is more like a cross product. Hence. the similar term for an interest rate obselved period before the rt would be
a to get a represen-
t
t
bs, tj
rldws, j'bls,
+
x
in
/
rt
ult/ajts
.J, (24) (21). z')
3 The HJM Approach
=
blT
of the bond pricc will be
tjdt + 5#46.
rtBlt, F)Jl + bT tlBt, F)#F). From thesc wc can derive the equation for the spot rate by taking
dBt, T)
integrals in
=
(26):
rj
=
F(0, t) +
1
jb a,t
2
+ /)'r'6,
(33) (34) thc
(35)
which gives the SDE drt where the F/(0,
f
=
(G((),f ) + bldt
+ JJJF;.
(36)
) is given by Ft (0, f )
=
pF(0, t) .
t?f
Note that according to this model, the spot rate has a time-dcpendent drift and a constant volatility.
3.5 Anotltcv Advuntuge oj the HJM Appwouch The I'IJM approach exploited rates and bond prices to impose stantaneous forward rates directly. model the expeded rate of change
the arbitrage
relation betsvccn fonvard restrictiorls on the dynamics of thc inBy doing this it eliminatcd the need to of the spot rate.
:'
:.; :
.
444
C H A P T ER
*
19
Classical and HJM Approaches
to
.:1
Fixed income
'i .
.
i
tt
But the approach has other advantages as well. As was seen in ean lier chapters, a k-dimensional Markov process would in general yield nonMarkov univariate models. Hence, within the HJM framework one could in principle impose Markovness on the bchavior of a set of forward rateg and in a multivariate sense this would bc a reasonable approximation. Yet, in a univariate sense when we modci the spot rate, the latter would still behave in a non-Markovian fasition, nis point is important because currcnt empirical work indicates that spot rate behavior in reality may fail to be Markovian. Hence. from this angle, the HJM approach provides an important :exibility to market prac-
,'
''
.:
r
.
r'
I
.
7 i EE:
E !
i
to
lnitial Term Structure
,
,
.
,
,
drt
E !
'S;
't
''
'
E
The HJM approach is clearly the more appropriate philosophy to adopt from te point of view of arbitrage-free pricing. lt incorporates arbitrage restrictions directly into the model and is more iexiblc. Howcver, it appears that market practice still prcfers the classical ap- C'. proach and continues to use spot-rate modeling one way or another. How y' can we explain this discrepany? As discussed in Musiela and Rutkowski (1997),modeling the irlstanta- 1 / . neous spot rate has its own difculties. W'hen one imposes a Gaussian struc- ?j ture to SDES that govern thc dynamics of the JFtla F) and when one useg t, constant percentagc volatilities, tlle processes under consideration explodo .' in hnite timc. nis is clearly not a very desirable propcrty of a dynamic j model. It can introduce major instabilities in the pricing effort. 'F lt is also true that there are signmcant resources investe d in spot-rate is, again, a great dcal of famodels both tinancially and time-wise. miliarity with the spot-rate models, and it may be that they provide good g'l t.', alnproximatiml to arbitrage-free pricts anmay. 0The recent models that exploit the fonvardmeasure secm tt be an SWCC to Problems of instantaneous fomard-rate modeling, and should lx considered as a promising alternative.
.
drt
,
( t j ''i-. ? ,
'
:
.
.,
l .
4 !
.
'
.)
'
(
tE:
to Initial Term Structure
:2
rt
=
rt-x +
(ztx
-
his hOw this could be dcme in practice. 'IYY WC IWVCI' ShoWed QDSCLkSSiOIV t numerical issues book tries to keep to a minimltm, but there arc some casG of pricing metho ds facilitateg the understanding practical where a dscussion
../
trdl
Euler scheme:l3
rtnjz
+
t7.lllzl
-
H/;-a1,
(38)
where is the discretization intcrval. nc rcmaining part of the calibration excrcisc dcpcnds on thc method adopted. We discuss some simple examples, ,
1
Suppose we know that incrcmcnts IH'I Wt-al are independent and are normally distributcd with mean zero and variance A. Suppose we have also calibrated the volatility parameter tr and the speed of mean reversion a. Hence, there ig tnly one unknown parameter x. Finally, we also have the initial spot rate ra. Consider the following exercise. Seled M standard normal random variables using some random number generator. Multiply each random number by V&Start with a historical estimate of ar and obtain the firxt Monte Carlo trajectory for rrl starting with rll and using Equation (38) recursivcly. Repeat this N times to obtain N such spot-ratc trajectorics:
l frltl'Illr,' ,
:E.
)dt+
- rt
4. 1 Mtmte Cflo
'p
;.
a(x
and then discretize this using the straightforward
1i
r
'rhcre
=
C'..
k
rt
bLrt,fl#H(
to this term strtture. How can this be done in practice? Several methods arc open to us. ney a1l start by positing a class of plausible spot-rate models and then continue by discretizing it. Thus, we can let rt follow the Vasicek model:
''
.
Pgtzctice
4 How to Fit
tldt +
art,
=
')
.
3.6
Fit rt
to
of the conceptual issues. Some simple examples of how an arbitrage-free spot-rate model can be obtained fall into this category. We discuss this briedy at the end of the chapter. Suppose we are given an arbitrage-free family of a bond prices #(f, Tf), i 1 n. Supppose also that we decided to use the classical approach to price interest-sensitive securities. Assuming a one factor model, we first need to lit a risk-adjusted spot-rate model =
;
f
E
titifmers.
Mlrket
How
'''
-
.
-
,
11 lr''vr -
;
.
'f
J'
Ef
l3Euler scheme replas difrerentiaks by qrst differences. It is a first-order approximation that may end up causing signitleant cumulative errors.
r' 446
C HA PT ER
*
19
Classical and HJM Approaches
to
'rhen calculate the prices by using the sample equivalent pricing formula:
(t rj) ,
=
a'skr
1
p
6 References
of the bond
where the ri/ are t he f'th clement of the y'th tree trajectory and Nk is the number of tree trajectories for a bond that matures after Tk steps. These trajectories depend on the ui #j, and hence, these equations can be used to determine the latter. To do this we need to impose enough restrictions such that the total numbcr of unknown parameters in the tree becomes equal to the number of equations, The tree parameters can then be obtained from these equations. ne tree will fit the initial term structure exactly, An example to this way of proceeding is in Black, Derman, and Toy (1984).
g
r: . jy;?.s , a
j
,
where M may be diferent for cach bond, depending on the maturity. Now, because l was selected arbitrarily, the 6.) will not be arbitrage-free. But, we also have the obscn'ed term structure, which is known to be arbitrage-free. So, we can try to adjust the & in a way to minimize the distance:
tf,
'rmax e
i uuc 1
447
,
e-
j=1
Fixed lncome
2 I#(r,T') - Bt, Ff) I
4.3 Closed-Fbnn
'sollanu
Supposc we can analytically calculatc thc cxpcctation: csdxj Bt, Tj Etz3
This way we fmd a value for K such that the calculated term structure is as close as possible to thc obscrved term structure. Once such a K is arbitrage-free, in the determined, the rt dynnmics bccomes (approximately) sense that using the modcl parameters, and this new % one can obtain bond prices that come to the obsen'ed term structure. 'sclosc''
=
'rlhtlx
min
slodeu
K
.
E=
ne prcvious appzoach used a single paramctcr x to make calculated bond prices come as close as possible to an obsewed term structure. fit was not perfect because the distance bctween the two term stnzctures was not reduced to zero, although it was minimized. By adopting a general the it. tree approach one can Once we considcr a binomial model ftr movements in r, wc can choose the arbitrage-free the relevant paramctcrs so that the tree trajectories term structurc and the relevant volatilities. For examplc, we can assume tllat we have N arbitrage-free bond prices. Suppose wc also know the volatilities o'i of each bond S(f, L). Let the up and down movements in r; at stage i bc dcnoted by ui, di, such that: rrhe
K'improve''
'lit''
uidi
eumple
:n) I.a(r,
of obtaining
G(r/, Ff
,
an
x)
1 .
arbitrage-free (approximately)
5 Concluslons This chapter has brielly summarized the fw'o major approaches to pricing devative securities that depend on interest rates. The classical approach was shtawn to be an effort in spot-rate modeling. The arbitrage restrictions were incorporated indirectly through a process of an initial tlunre.'s The HJM approach on the other hand was ari extertsion of the Black-scholcs formula to interest-sensitive securities. ftfitting
.
,
1 Nk
This is another model for rt.
1
1
=
Given this restriction, the tree will be recombining and at evcry stage we pill have i unknown parameters, ne nex't task will be to dctermine thee ui #j by using the equality: S((), Tk ) =
.(''
and get a closed-form solution for the S(/, F), as will be discussed in the ncxt chapter. Suppose this results in the function: Bt, T) Grt, T; K4. nen, we can minimize tbe distance between tlle closed-form solution and the observed arbitrage-free yield curve by choosing K in some optimal sense: )
4.2 Trce
j-
=
-
n 1 rf'a e - )Z;
-.
=
1
,
6
References
'lhe best source on tliese issues is Musiela and Rutkowski (1998). Of course, this source is quite technical, btlt we recomrnend that readers who are scriously interested in lhcd-income sector put in the necessary effort and become more familiar with it, The excellent discrete time treatment, Jarrow (1996), should also be mentioned here.
448
C H A PT E R
*
Classical and HJM Approaches
19
to
Fixed Income
1. Consider the equation below that gives interest rate dynamics in a settingwhere the time axis (0,F1 is subdivided into n equal intervals, each of lengt.h 1: rl+a
where the random
r: +
+'
tr?
trltl#;ia
W?))-f- frztH'l
-
-
Wzkal,
r/-ha R !+A
(W$+a
=
-
awsx j,
M)
-(all .
=
Rt +
r,
tz 21
a 22
Rt
H''jia 1
+
,
a+2/+.(
Suppose
R'J lii6 #rl + *1(V6+.& ) + t6( -
3. Suppose at time t = 0. we are given four zero-coupon bond prices (Sj, Bz, #3 A4Jthat mature at times l = 1. 2, 3, 4. Titis forms the term strudure of interest rates. We also have one-period fomard rates (, , fi is h , ./'aJ,where each = f 0 on a loan that begins at timc f = i and the rate contracted at tne ends at time f = i + 1, ln other words, if a borrowcr borrows $1 at timc t = i, he or shc will pay back N(1 + hj at time f i + l ne spot rate is denoted by ri. By defnition we have ,
(a) Explain the structure of the enor terms in this cquation. In particular, do you lind it plausible that M-a may cnter the dynami of observed interest rates? (b) Can you write a stochastic differential equation that will bc the analog of this in continuous time? What is the difficulty? (c) Now suppose you krlow, in addition, tlmt long-term intcrest rates, Rt, move according to a dynamic givcn by A/+.a
a12
(a) Derive a univariate representation for the short rate rt. (b) According to this representation, is rt a Markov process? (c) Under what conditions, if any, would the urtivariate process rt be Markov?
are distributed normally as -
=
fz1 1
where the error term is jointlynormal and serially uncorrelated. rt is a short rate, while Rt is a Iong rate.
error tcrms
Ws
09
has the following dynamics,
7 Exercises
=
7 Exercises
l'V-A),
-
where we also know the covariance:
=
rtl The
.
all fomard ioans are default-free. each At time peziod there are fwc possiblc states of the world, denoted ,
=
a
(#JJand
by (?.f/di
A;IAIPZJI'Jp.
=
.
:
=
1, ?, 3, 4J.
(a) Looked at from time i 0. how many possible states of the world are there at time i 3? (b) Suppose =
=
for thc vector process
Can you write a represcntation Xt
(#1
rt
=
Rt
(vector)Markov X
process Xt, r/
=
R:
.87,
=
%
B4
.82,
=
.75)
=
and
(./)
=
8%,
/1
=
9%,
h
=
10%,
fs
=
18%1..
Forrn tliree arbitrage portfolios that will guarantee a net positive return at times i 1, 2, 3 with no risk. (c) Form three arbitrage portfolios that will guarantee a net return at time i 0 with no risk. (d) Given a default-free zero-coupon bond. Bn, that matures at time t n, and alI the forward rates (f, s-1J, obtain a formula that cxpresses Bn as a function of fi. =
=
=
,
Bz
,
such that X is a first-ordcr Markov? r (d) Can you wnte a continuous time equivalent of this system? @) Suppose short or long rates are individually non-Markov. ls it possible that they are jointlyso? 2. Suppose the
.9,
=
.
.
.
.
CHA PT FR
*
(e) Now consider to the system:
19
Classical and HJM Approaches
to
Fixed lncome
the Fundamental Theorem of Finance as applied
B1 B1
s' 2
/1
#3
sy
Bd3
42
B4
,2
ad4
Bz =
.
Can al1 Bi be determined indcpendently? (9 In the system above can aIl the (/fJ be dete=ined independcntly? (g) Can we claim that all h are normally distributed? Prove your
Classical PDE Analysis for lnterest Rate Derivatives
a'llsWe'r.
4. Consider again the setup of Ouestion 1. Supposc we want to price thrce European style call options written on one period (spot)Libor rates 0. 1, 2, 3, as in the above case. Let these option prices be L with i denoted by G. F-ach option has the payoff: =
c
where N is a notional general.
=
N mxgz.j
-
K, 0j,
amount that we set equal to one without loss of arly
price such an option? (a) How can mu (b) Suppose we assume the following: (i) Each h is a current observation on the future unknown value of Li. (ii) Eacb h is normally distributed with mean zero and constant variance cri (iii) We can use the Black formula to price the calls. .
(c) Would thesc assumpticms be appropriate under the risk-neutral measure obtained using money market normalization? Fwxplain. (d) How would the use of the forwardmeasure that corresponds to each Li improve the situation? (e) In fact, cao you obtain the forward measeres for times f 1, 2? (9 Price the call optitan for time t 2 using the fomard measure. =
=
1 lntroduction The reader is already fa rniliar with various derivations of the Black-scholes formula, one of which is thc partial differential equations (PDE) method. ln pmicular, Chaptcr 12 showed how risk-free borrowing and lcnding, the underlying instrumcnt, and the corresponding options can bc combined to obtain risk-free portfolios. Over time, these potfolios behaved in such a way that small random perturbations in the positions taken canceled each other, and the portfolio return became deterministic. As a result, with no dcfault risk the portfolio had to yield thc same return as the risk-free spot ratc r, which was assumed to bc constant. Otherwise, there would be arbitrage opportunities. The application of Ito's Lemma within this contcxt
rcsulted in the fundamcntal Black-scholes PDE. The Black-scholcs PDE was of the form: 1
rvtl - F'F + Ft +
2
+ - r.r 2
ut
zFss = 0,
(1)
with the boundary condition: Ftv
,
F)
=
max
(.% -
K, 01
.
'The r is the constant risk-free instantaneous spot rate, the S, is the price of a stock that paid no dividends. the F is the time f price of a European call option written on the stock. The K and the F are the strike price and the expiration date of the call, respectively. ln Chapter 15 it was also 45I
7 452
n-
H A P T ER
.
20
Classical PDE Analsis for lnterest Rate Derivatives
mentioned that the solution of this PDE corresponded to the conditional expectation Fst,
tj
.......-
E') gc-'fr-/)F(,,.,
r)j
(3)
,
calculated with the risk-neutral probability #. Given that we are now dealing with derivatives written on interestsensitive securities, we can now ask (at least) two questions; PDES in the casc of interest rate derivatives? For * Do we get similar example, considering the simplest case, what type of a PDE would the price of a default-free discount bond satisfy? @ Given a PDE involving an interest ratc derivativea can we obtain its solution as a conditional expectation similar to (3):?
ncse questions can be answcred in fww different ways. First, we can fol1ow the same approach as in Chapter 12 and obtain a PDE for discount bond prices along the lines similar to the derivation of tlle Black-scholes PDE. In particular, we can form a portfolio and equate its deterministic rcturn to that of a risk-free instantaneous investment in a savings account. Application of lto's Lemma should yield the desired PDE.I ne second way of obtairiing PDES for intcrest-sensitive securities is by exploiting the martingale equalities and the so-called Feynman-Knc results directly. In fact, when we westigate the relationship between a certnin class of expectations and PDES, we are led to an interesting mathematical regularity, It turns out that thcre is a very close connection between a representation such as:
=
e-rT-tle'l '!'
gFty:r,. r)j
(2)
.
At other times, the pricing was discussed using PDE methods. For example, in the previous chapter, using the method of risk-free portfolios we derivcd the PDE that a default-frec discount-bond price #(f, T) must satisly' under the condition of no-arbitrage! Br (J(r,,
/)
-
,?,(r,>
r))
+
1 .%+ j.Brrbrt, 467
t) c
- z.,s
=
0,
(3)
e'
468
C H A PT E R
.
Relating Conditional Expccratiorts to PDL'
21
with the boundary condition
B(T T)
=
(4)
1.
Similarly, under the Biack-scholcs assumptions with constant spot rate r. Black-scholes PDE for a call option we earlicr obtained the fundamental writtcn on St: with strike price K and cxpiration T, 1 0. (5) bhlvsr+ Ft + jlqrlrt z - rF =
The boundary condition was
(6) max gts'z &), 01 approaches Thus, the pricing effort went back and forth between PDE and approaches that used conditional expectations. Yet, both of these methods are supposcd to givc the same arbitrage-free price Fut, t ). This sugbetwcen conditional gests that thezc may be some dccper correspondence PDES that are shown in (3)or expectations, such as in (1) or (2),and the rcspectively. (5), tj is given by ln fact, suppose wc sbowed that whcn a function FLS: FSy, T)
=
1)
s',
-
/)
-
469
Fvh, /). lf one could establish a PDE that corresponds to such expectations, this could give a fastcr, more accurate, or simply a more practical numerical method for obtaining the fair market price F(x%, 1) of a hnancial derivative written on Sf.3 Alternatively, a markct practitioner can be given a PDE that he or she does not know how to solvc. lf thc conditional expectation in (8) is shown to be a solution for this PDE, thcn this may yield a convenient way of for Fuh. t). Again, thc correspondence will be very uscful. In this chapter we discuss the mechanics of obtaining such correspondences and thc tools that are associated with them. Llsolving''
,
r)q Le-br-dsl-s-r, ,
(7)
1) would automatically whcre Ft5'j, f) is twice differcntiable, tbe same Flvt, satisfy a specific PDE. And supposc we derived the general form of this PDE. TMs would be very convenient. We discuss some examples. A1l interest rat derivatives have to assume that instantancous spot rates of Finance wotlid are random. At the same time, thc Fundamcntal Thcorem always permit one to write the dcrivatives' price Fvh, t ) as
Fls,
PDL
to
-
,
Fs,
2 From Conditional Expectations
wlj
ge-zrrsas-t-r,
zti'
(8)
expectations under the risk-neutral measure. As a result, such conditional especially thc case for interarise naturally in derivative pricing. nis is assumed constant, and bc cannot est rate dcrivatives, where tbe spot rate random. hcnce, thc discount factors will have to be twaluate. The But thcse conditionai expectations are not always easy to jndecd. Ofttn, task complex makc this a very stochastic behavior of rt can used. there is no closed-form solution and numerici methods need to bc numerically, speed and acEven when such eeedations ean be evaluatcd methods. Thus, it may l>e alternative necessitate curacy considerations may rcprescntation tliat avoids the quite uscful to have an alternative the arbitrage-free price calculating expectations conditional in cvaluation of corresponds to tlie conFt/, f ). ln particular, if we can obtain a PDE that numerical schemes to calctllato expedations ditional (1) or (2),we can use
tdirectl
2 From Conditional Expectations to
PDES
In this section we establish a correspondenc between a class of conditional expectations and PDES. Using simple examples, we illustrate that starting with a function detincd via a certain class of conditional expcctations, we can always obtain a corresponding PDE satislied by this fundion, as long as some nontrivial conditions are satished. ne main condition necessary for such a correspondence to exist is Markovness of the processcs under consideration. Our discussion will begin with a simple example that is not directly uscful to a market participant. But this will facilitate the understanding of thc derivations. Also, we gradually complicate these examples and show how thc methods discessed here can be utilized in practical derivatives pricing as well, 2.1 Cse
1: Cmutcnt Dlscount Ft-s
Consider the function Fxt) of a by the conditional expectationl F(x,)
=
l-andom X
E,P l
e-p''#(x
process xr e N
)ds
,
g0,x),
defned
(9)
where p > 0 represents a constant instantaneous discount rate, #(.) is some continuous payout that depends on the value assumed by the random proEPt cess xt. (.1is the expectation tmder the probability # and conditional on thc information set h, botb of which are left unspeciEed at this point. 'I'he process xt obeys the SDE: dx3 ydt + t7WH(, (10) where pz, c' are known constants. =
1For evample, in dealing wit American-style derivatives, it will in general be more convenient to work with ntlmerieal PDE metizod irlstead of evaluaizlg 1he conditional expectations through Monte Carlo.
..
pa . ,.,.y
(--H A P T E R
470
Relating Conditional Expectations
21
.
to PDF..S
2 From Conditional Expectations
PDL'
to
4-11
''
This FLxt)can be interpreted as thc expccted value of some discounted random variable future cash tlowg(.x,) that depends on an fy-measurable deterministic. < xs. The discount factor 0 p is markets will, in general, in snancial of interest the cash Clearly, tlows is especially the case for be discounted by random discount factors. thc momcnt. A1I we interest rate derivatives, but we will leave this asidc at to wdnt to accomplish at this point is to obtain a DE that will lead ltentl that the stcps study in detail We to in the expectation (9). this, random discount factors can to this PDE. Once we learn how to do introduced, easily bc expectation (9) in several We now obtain a PDE that corresponds to applied to more complicated stcps. These steps are general and can be expectations than the one in (9). We proceed in a mechanical way. to ilthat the initial iustrate the derivation. To simplify thc notaon wc assume point is given by t 0. and split the pcriod (0,x) First, consider a small time inten'al 0 < represented by the intenral i9, j, in two. One being the immediate future, and tlze othcr representcd by (A, x),
7.
: :
Itcorresponds''
Flxnj
c-*g(;r.)JJ
EP
=
+
A
0
:.
)('
F(x
,.
k
:L
,::'
-ki
6
.j J-
:
.
-
:
.'EE t.. : .
.
.!(i ) .fr' .(
;$L Er',
y,P 0
=
;
'7
.
jA'P t'
replacing the T hus, (.11.2 we get..
This ptrmits
EI
lF'Pa
Er 0
x -&
c
-*
# (xs )ds
2Re,cal1that at time
=
'
EP
(.1)
=
l
.
eJ(.) operator
.j
E;
e-J'AF# a
(12)
by the operato:
:' .
J
e-ptl-)
# (.':)ds
.
=
,
We
WZ have more irlformation than at time t
=
(I-
=
(17)
0.
divide all terms by
,
and
e-/k,gtxll,& +
(c-VA
-
1)F(-u) + (F(.u)
-
F(xo)j
(18)
0.
=
=
lim
-..
A
1
:
e-n'%
# (-x)ds
.
,.
=
gx
o
).
(20)
The third term, on thc other hand, involves the expectation of a stochastic differential and hence rcquires the application of Ito's Lemma. First, we approximate using Taylor series and m'ite;
.'
.
''
'zrk,-lxal ) -
c
'f
F(xs)
-
=
,
(1* ? '$''
.
.jt,--ofq
zzgz--txa -
x.)1 +
(zt)
.
'
B'Here lhe F(xz) is te value of F(.) observed at tirne z 0. It is condititmal on Thc otl the other hand, is the value lhat will be obsen'cd arter a time interval of length at t A. It *1 be ndililal on
E. ',
l
jo
,-fx-)1
co
+ e-nFxxj
--.
.
:.
jo
0 of each term on the left-hand As the Iast step, we take the limit as k side. ne second term is, in fact, a standard derivative of eex evaluated at x 0: 1 e-' a lim -p. - 1) (19) a-+pX The first term is the derivative with respect to the upper Iimit of a Riemann integral!
'j .'.
.j
...
a e-psgnlds
side and moving them inside the
.
(
'.
.
in
(16)
.
.
(131.'t'
(.1
E;
Qu
expectations. ' The third step will apply the recursive property of conditional :' nested, it is the expecAs seen earlier, when conditional expectations are Thus, ff wc ; tation with respect to the smallcr information set that matters. y have l't 1s. wc can write: EP
.
..
'
;.5
(12) t': $ 1
# (xs jds
+ c-#AF(.u)
tr-/gtxsldx
feafrangel
t..-..tjjjli,1
).'1
g-pl
E;
=
As the fourth step, we add and subtract F(.u),
4.Jz
.;!;k. ..
,'j
.
ds
E;
)).:..
.
.
...
C -ps #(A:)
)
Grouping all terms on the right-hand expectation operator we obtain:
$) !
t,.:
.n'
A
U
.
'
P
(11):
A
,
r
Z'
(15)
.
'
':
1nThe second step wolves some elemcntazy transformatitns that are b this right-hand of side tended to introduce a future value of #(.) to thc .. rewrit- J expression. In fact, note tbat te second term in the brackets can bc L. e-'# as! ten aftcr multiplying and dividing by g-ptx-hl
E; j-pFtaulj
=
a
r
,7:'
(11)
.
e-#'.(x,)Jx
i 11.
.5
'
e-pglxajds
:
.
:q
=
*
EoP
nis Iast expression can now be utilized in
t
1
X
. :
r, 1. . ..:.'.: g
But we can recognize the term inside the inner brackets on the right-hand side as the F(.u) and write/
/.)
't
rfhis
z
'
L,
=
,t
F(.v),
..
7! . :;
:
.ta,
=
'
.
.r:.
C H A PT ER
472 Then let
k
-+
*
Relating Conditional Expectations
21
0 and take the eoectation 1 p lim u.F: gF(.u)
-
F(x.)j
to
PDF.s
1 2 Fxp, + jFxxt:r
,
A*0
(22)
the formula as a where Jz is the drift of the random process x that enters xoj. result of applying the expcctation operator to (.r reach Replacing the Iimits obtained in (19)-(22)in expression (18),we the desired PDE:
#(f, F)
1Fxxe 2 -
j
JIF + #
=
(j
,
(23)
where the &, Fxx. 15 and g are all functions of x. behveen the condiOne may wonder what causes this correspondence conccpts scemed tional expectation (9)and this PDE? After all, these two A heuristic arkswer to this question is to be quite unrelated at the outset. the following. value'' of cash 'rhe PDE corresponds to the expectation of the conditional the F(.) by given is 0ow stream ('(.r,)). If this present value of xo and funetion arbitrary expectation shown above, then it cannot be an expected the constraints due to its behavior over time must satisfy somc PDE. the constraints lead to fmure behavior of x. These Fxo) is the result of an optimal forefunction the precisely, More projecting ways in which Fxt) may cast. nis optimal forecast requires variable x3, determinischange over time. Expected changes in the random and thc second order lto gxt), tic changcs in the time variable f, payouts F(.). The optimal pTechangcs in corredion all cause various predictable The PDE that corresponds diction should take these changcs into aount. obtained in sucb a way that the to the conditional expectation operators are and its variance expected value of the prediction error is set cqultl to zero, consideratioll. 4 taken ztt'l is minnized, once tese predictablc changes are Ktpresent
correspondence between a We now sec a more relevant example to the Sm PDES. fact, In we now apply the class of conditional expectations and prices. dgscount bond derivation to obtain a PDE for default-free pure wif.h default-free purc discount bond spot Consider the prjce Bt, T) of a that the instantaneous maturity T in a no-arbitrage setting, Msume 4In fact, note that in obtahling Zef 0.
the PDE we replaced
the Wiener complnent of the
-r
wit.b
Et>
473
(e-
jr czdaj
BT F)
(24)
,
1.
=
Here the expectation is taken w1t.11respect to the risk-neutral measure P and with respect to the conditioning set available at time f, namely the It. This is assumed to include the current observation on the spot rate rt. lf rt is a Markov process #(f, F) will depend only on thc latest observation of jJ, r;. Because we are in the risk-neutral world, as dictated by the use of will the follow the rt the dynamics given by SDE: drt
=
(tI(G, t)
r#trj,
-
f)1
dt
+
blrt, /)#H(.
where H( is a Wiener process under the risk-neutral thc market price of interegt rate risk delled by t
=
/L -
measure
rt
(25) A 'Ihe
A., is
(26)
with t, g' being the short-hand notation for the drift and diffusion components of the bond price dynamics: dB
y,B, tlBdt + =B,
=
/)z14/1,9).
Thus, we again have a conditional expectation and a process that is driving it, just as in the preous case. nis means that wc can apply the same steps used there and obtain a PDE that con-esponds to Blt, T), Yet. in the present case, this PDE may also have some practical use in pricing bonds. lt can be solved numerically, or if a closed-form solution exists, analytically. The snme steps will be applied in a mechanical way, without discussing the details. First, split the intewal if, F1 into two parts to wrile: Bt,
2.2 Cse 2: Bond Pricing
=
with
-
F g + x
PDES'
to
rate h is a Markov process and write the price of the bond with par value $1, using the familiar formula:
to obtain: =
2 From Conditiona Expectations
'r)
=
.E:
Second, try to introduce the
(2g) gte-.?7-rxd.l te-.?itrxl'lj future price .
of the bond, Bt +. ?, F), in
this expression. In fact, the second exponential on the right-hand sidc can easilybe recognized as B(t + F) once we use thc recursive property of conditionalcxpectations. Using .
E:
we can write Bt,
r)
=
(.jj gl,#+a Bt a, g(e-.$'-r,z&)
r.)
EI)
=
s,>
,
+
Flj
(28) (29)
474
2H A PT E R
Relating Conditional Expectatiolzs to
71
*
PDF-S
2.3 Ctzse 3: A
because Bt +
T)
,
Eil.-h
=
?-'''-
'-.:1.91
gc-
(30)
-
add and ln the third step, group a1l terms inside the expectaton sign, : subtract Bt +. T), and divide by ,
1s/ f
2 From Conditional Expectations
gj-./7+&r>J: -
1)
s(f +
r) + (stf +
,
,
r) -
T)11
Bt,
=
0.
(31)
T) S(f, T)1 to the leftNote that this introduces the increment (S(l + applying Ito's Lernrna. hand side. This will bc used for 0 of the first term in this equation:s Fourth, takc the limit as k
Qenexalfztlhtm
F(r/
,
--
I(
-
ts
s(, .y.a. z)j
j)
-.('-ar
'
-
Then, apply lto's Lernma to the second term in tion:
1 lim Et (#(1+ i -..()
F)
,
= Bt + lrl/tn,
1)
take the expecta-
(31)and
Afh(r/, f)1 + jBrrblrl,
/) z ,
(33)
1) where the drift and the difhlsion of the spot-rate process art, t), !7(rl, used.ti are obtain the ln the Nnal step, replace these limits in expression (31) to conditional expectation (24): PDE that corresponds to the
- rtB + Bt +
BrLart 1) ,
-
htbrt,
1
+ jBrrbrt t ()1
,
1) z
=
0,
5(71T)
1.
=
E
$
F f
(ch'
r'j
gruldu
.
(36)
This F(.) would represent the price of all instrument that makes interest rate dependent payments at times u (E (f,F1, and hence needs to bc cvaluated using the random discount factor Du at cach u: Ds
=
rd, e ju
(g,y)
,
time u.7
Vrious instruments and interest rate derivatives, such as coupcm bonds, financial futures that are markcd to market, and index-linkcd dcrivatives fall into this category where the arbitragc-free price will be given by conditional expectations such as in (36).nus thc methods that were digcussed in the last two sections can be applied to tind thc implied PDE if the processtes) that drive these cxpectations are Markov. Thc corresmnding PDES may bc exploited for real-lifc pricing of these complex instptrnents.
(34) 2.4 iome Cltyd/icutimz
with, of course. the usual boundary condition'. =
f)
lt is interesting to note that the expectation of this Du is notbing other than the time / pricc of a default-free pure discount bond that pays $1 at
1
-
(g2)
w)j
A(l,
-
z).
.-x,s(f,
475
We have scen in dctail two cases where the existence of a ccrtain typc of conditional expectation led to a corrcsponding PDE. In thc ftrst case there was a random cash llow stream dcpending on an underlying process xt but the discoum rate wms constant. ln the second case, the instrument paid a single, flxcd cash flow at maturity, yet the discount factor was random. Ckarly, one can combinc these tw'o basic examples to obtain the PDE that corresponds to instruments that make spot-rate dependent pamuts glrt) and that need to be discountcd by random discount factors:
,
1 iim-. A o
PDL
to
(35)
of a This is a PDE that must be satisEed by an arbitrage-free price risk. In Chapter 20, the same PDE was pure discotmt bond with no-default obtained using the method of risk-free portfolios, of limit sHere we are assuming that 111e,technical conditions permitting the iptcrehange and expcctation operators are satisfied. depends on t as well as on rt. ftrnlike the previous example, here the Blt, T) functionbefore. did not exst Hence the,re will be an addilonal B, term that
We need to comment on some issucs that may be confusing at the first reading. zl-lere we cannot directiy apply the frP(.1 operator to D. because the glru) will be corzelatcd with the D.. If such correlation did not exist, and if #(.) depended (m an independent random variable, say xu only, then we could take expectations scparately and simply multfply the payout by the t'orresponding discount bond price Bu ttl discount it: r r 15 hu fl Ez #(%r,)(y & s u g r l#(x x lz/uj ys..u
=
J
z
t
assumirlg tlzat the neccssary intercharlge of the olycrators is allowed. On t,he other hand, equation (38) can always be applied if we uscd the fonvard measure as discussed in Chaptcr 17.
.'
j
(-aH A P T E R
476
*
21
Relaring Conditimal Expecrations
to
PDES
2.4.1 F/zd Importance of Markovness The derivation used hcre in obtaining the PDE that corresponds to the class of conditional expectations is valid only if the undcrlyinj stochastic processes are Markov. lt may be worthwhilc to see exactly where this assumption of Markovness was used in the prcceding discussion. During th9 derivation of the PDE, we uscd the conditional expectation operators Z7rg.l that we now express in the expanded form, showing the conditioning information set cxplicitly: E
r
?
c-
)-*r 3 ds gxutdu '
t
Ifl
=
E
W
f>
v;d
r ! ds
-
c
gxujdu
t
-
Ft,
Irt
(39)
(40)
rt).
These cmerations are valid tmly when the rf process is Markov, lf thi/ assumption is not true, then thc conditional expcctations that we considerod would depend on more than just the rt. ln fact, pst spot rates (rs,s < would also be determining factors of the price of the instrument. In other words, the latter price could no longer be written as F(l, r,), a ftmction that depended on rt and t only. Thc rest of thc derivation would not follow itl general. Hencc, we see that the assumption of Markovness plays a central role in the choice of pricing methtads that one uses for interest rate dezivativcs.
z.5 Wlclx Dvift? One may also wonder which parameter should be used as tbe drift of straightforwaTd, the random process in such PDE derivations. answer is but it may be worthwhile to repeat it. The ccmditional expectations under study are obtaincd with respect to probability distribution. For examplc, when we write the some (conditional) arbitrage-frce plice of a bond as: 'lnhe
S(f, F)
=
E(
j-
?'rfGdsj ,
(41)
risk-neutral probability we take the expectation with respect to #, the consideration under is rt, this choice of Given that the random process risk-adjusted drift for rt the risk-neutral probabili rcquires that we use and write the corresponding SDE as (42) drt (/?(r;,1) - ktbrt, tj) dt + blrt, l)dM,
2 Fttam Conditionak Expeccations to PtlEs
LKreal
world'' SDE:
drt
=
art,
where the 1*7 is a Wiener pross
tjdt + brt, f)#H;7, with resped
to real-world
(43) probability #.
'
Hence, within the present ccmtext, while using Ito's Lemma, whenever a tbrt, drift substitetion for d% is needed wc have to use art, /) t)j and not art, l). This was the case, for exnmple, in obtaining the limit in (33). Will the nonadjusted drift ever be used? question is intercsting because it teaches us something about pricing approaches that use other than the risk-neutral measure; formulas that, in principle, should give the same answer, but may nevertheless not be very practical. In other words, the question will show the power of thc martingalc approach, lndeed, during tlle same derivation. instead ff using the risk-adjusted drift we can indeed use the original drift of thc spot-rate process. But this requires that the conditional expectatitm under consideration be evaluated using the rcal-world probability P, instead of the risk-neutral probability. However, we know that an expression such as
t( '
'rhe
Bt,
w)
je-
Jr r,j
.P
=
:J
(
(44)
I
cannot hold in gcneral if the Bt, T) is arbitrage-free, and if the expectation is taken with respect to real-world probability #. If one insists on using the real-world probability then the formula for the arbitrage-free pice will instead be given by:
ge..('r
#(f, F)
=
Et.p
rs6uej'
E(r,,.),p7 -
jztrsqsl?tslj ,
:
!
I ''
!
1,
(4s)
where a1l symbols are as in (42)and (43). One can in fact obtain thc same PDE as in (34) by departing from this conditional expectation and using exactly the samc steps as before. The only major diffcrence will be at the stage when one calculates the limit corrcsponding to (33).There, one would substitute the eal-world drift art, t) instcad of the risk-adjusted dn'ft.
j '
!
t(
(
2.6 Aaothe'r Btmd Pdce Forrnltlz ne main focus of this chapter is the correspondence between PDES and conditional expectations, But, in passing, it may be appropriate to discuss an application of equivalent martingale measures to bond pricing. The preceding section considered /:$,5 bond pricing formulas, One used the martingale measure l and gave the compact eApression: #(/, T)
=
instead of thc
l
477
=
Et9
j-
LC
G'j
1
! .
l
(46)
.
The other used the rcal-world probability P and resultcd in Bt, T)
=
E;
ge-f'
!
rsJ>cf'I2(r,.,J)J%*-l(r.,s)2J'1j .
g
(
C H A P T ER
478
Relating Conditionai Expectations
21
.
to
PDF.S
Of course, the two Bt, F) would be identical, except for the way they are charaderized and calculated. 'rhe question that we touch on brie*y here is how to go from one bond price formula to the other. nis provides a good example of the use ol Girsanov theorem. First, we remind the readcr that within the context of Chapter lf two probabilities # and # are equivalent if they are rclated by ,
dh
(48)
(tdptb where the Radon-Nikodym derivative (t was given by (Au#W'l - l sil ('t e L =
=
(49)
,
where A.t is an f/-measurable prccess,8 We now show how to get pricing formula (47)startg from (46),assuming that aIl technical conditions of Girsanov theorem are satisfied. Stazt with the bond pzicing equation: Bt, F)
Et>
=
?7*rsdaj
gd -
(50)
,
Write the same pxpression using thc definition of the conditional expectaEtP'. tion operator 'j3
j-
./;F
r,dzj
r,dy)
(g.('
=
(1
dp-z
(51)
range at which future rt will take values. Now, shown in (48)to substitute for d? in tfse the equivalence betwcen / and # this equation:
where the l is thc relevant
j
j-
.s> l
substitutng for (,r .(''
Et>
rspyj
(g/,F
rxaj .
(52)
j.vdp
we get the desired equivalcnce:
ge-
.
= Etz
rsdxj
(e.r
rsf/'j
ejo''la.dH?z
-
)
..'csJ;e?-,'E(r.v.,):Fe-)
je-
Jdj
2s
(53)
dp
lrs-.spda'lj .
(54)
probability ob-
This is, indeed, the bond pricing formula witb real-world tained earlier. of default-free Thus, the connection betwcen the two characterizations tbeorem simple once te pure discount btnd pyices becomes very that the show did not derivation, abovc thc utilized. in we Of coursc, is clearly is risk. Btlt it a drift term ; is the market pzice for interest rate equation. adjustment to the interest rate stochastic differential 'iirsanov
8ln tls partictllar
caqc,
AJ
wili be tize rnarket
price of spot interest rate risk.
3 From PDF.S
to
Conditional Expectations
479
2.7 Which Formulu? Expressions (46) and (54) give two different characterizatitms for #(/, F), But the second formula, derived with respect to real-world probability, seems to be messier because it is a function of r whereas characvterization (50) does not contain this vaziable. Hence one may be tempted to conclude that if one is utilizing Monte Carlo approach to calculate bond prices, or the prices of related derivatives, the formula in (50) is the one that should be used, lt does not require the knowledge of t. The appearances are unfortunately deceiving in tltis particular cse. Whether tme uses (46) or (54),as long as one stays within the boundaries of the classical approach, Monte Carlo pricing of bonds or other interest-sensitive securities would necessitate a calibration of 2. ln the casc of (54)this is obvious, the r is in the pricing formula. ln the case of (50),some numerical estimate of the A, will also be needed in generating random paths for the rt through the corresponding SDE under the zvz/rriag?leprobability A
drt
=
art,
/)
-
/#(r/,
/))
dt + bt,
f)#M.
(55)
Obviously, this cquation becomes usable only if some numerical estimate for is plugged in. Thus, in tane case, the intcgral contains the / but not the SDR In the other case. the 2 is in the SDE btd does not show up in the integral. But in Monte Carlo pricinga the market participant has to usc both the 'itegral and thc SDE. That is why t-le approacb outlined here is still thc approach and rcquires, one way or another, modeling underlying drifts. HJM approach avoids this difficul. i.t
'Kclassical''
'rhe
3
From
PDES
to Conditional Expectations
Up to this point we showed that if the underlying proccsses are Marktw and if some technical conditions are satished, then the arbitragc-free prices cbaracterized as conditional expectations witb respect to some appropriate measure would satisfy a PDE. That is. given a class of conditional cxpectatiens, we obtain a corresponding PDE. ln this section we investigate going in the opposite direction. Suppose by an asset price Fsi, tj. Can we go from we are givcn a PDE gatished there to conditional expcctations as a possible solution class? We discuss thig within a special case. We let the F(BI, tj be the price of derivative that is written on the Wiener process J#; dchned with a snancial
480
C HA PT ER
Relating Condititanal Expecrations
21
*
to
PDL
respect to probability. The choicc of a H') as the driving pross may not seem to be vezy realistic but it can easily bc generalized. Furhera it permits the use of a known PDE called the heat equation in cngineering literature, Suppose this price F(H( f) of thc derivative was known to satisfy the ftllowing PDE:
3 From PDES
Ff +
Fu';g
(56)
0
=
and that we have the following boundary condition at expiration, F( M$, F) ,
=
l
=
F(I#;
1 a2F
j g-sf zkp,j PF
dy
-(
+
Ft +
-
F(M':, 1)
JF dt + ppszz!,Tr
j asza jl dt
(58)
+ Fw,aI,#).
T
J,
Y
EG(114/.)1 -
JF
#u,
dl'l
.
(62)
Thus, if we can show that the sccond expectation on thc right-hand side is
function F(.) can be determined by t'elking the zero, then the (unknown) expectation of the known function G(.). But this rcquircs that: T
Et>
pz-
tuj
(63)
().
=
dllz-
To show that this is the case, we invoke an important propcrty of Ito integrals with respect to Wiener processes. From Chapter 10 we know that if (Ht) is a nonanticipative function with respect to an information sct h, and with respect to the probability #, then the expectation of integrals with respect to J;fziwill vanish: l
E;
lr(H$)dH(;
0.
=
(64)
Lct us repeat why this is so. The H( is a Wiener process, Its increments, #r#;, do not dcpcnd on the past, including the immediate past. But if (J#;) is nonanticipative, then /,(1) will not depend on the eithcr. So, in (56)we have the expectation of a product where the individual terms are independent of one another. Also, one of tbese, namely the #14t, has KGfuturen
1' JF
=
pl.F
1'
F! +.
/
z'
Jl1$ +
r(
F, +
ds.
EFP'P'
,
I
j'sx
ds
(59)
(60)
0.
=
Using this and taking the expectation with respcct to
/
of the two sides of
(59),we can write: Et?'(Ftp' z,,
X
=
(57)
Recall that the partial derivatives Ft and Fp.pr are themselves functions of k;' and s. Now, wc know something about the integrals on the right-hand sidc. As second integral a matter of fact, using the PDE in (56),we know that the equals zero:
Equation
f)
l
where we use the fact that the Wiener process has a drift parameter that cquals zero and a diffusion parameter that equals one. nis stochastic differential equation shows how F(F;, tj evolves over time. The next step is integrating hoth sides of this cquality from t to W: F(W'w,T)
,
(J(H''w)
,
481 =
F)
for some known function G(.). We show that the solution of this PDE can be represented as a ctanditional expectation. To do this, we hrst assume that all technical conditions f): arc satisied and start by applying lto's Lcmma to F(H( -
Conditional Expectations
Now. F(Hi, T) is the value of F(.) at the boundary f T, so we can repiace it by the known fundion G( l#'z). Doing this and rearranging:
,
1
to
niean
ZCrO.
Going back to equality namcly the
(62),we F
.E'/ #
see that the term we equate (JF
dpz
t
J'r,.'q
is exactly of this type. It is an integral of a nonanticipative f'unction with respect to the Wiener process. This means that its expectation is zerox given that F( satisfies some tecllnical conditions. .)
E:
gj'
l7 d.;j
=
F(J#:,
/) +
T
F,>
PF ,?sz
/
#l,F;
.
(61)
(66)
()
-
t
Thus wc obtai' ned:
#'(p;2) ,
=
EP l
lclp-wl! ,
(67)
of the price F( I/PI/) as a conditional expectawhich is a characterization of condition the G(J#y) and the probability #. This funcboundaly tion tion is algo the solution f the heat equation. ln fact, beginning with a ,
'r)1
(65)
,
.
to zero,
482
C H A PT E R
*
21
Relating Conditional Expectations to
PDF-S
PDE involving an unknown function F(/, W ), we determined the solution probability, w1f.14 as dn euectation of a known function with respect to a respect to which H( is a Wiener process. :
4.2 A4arkov Propcrty This property was seen before. Ltt SDE: dut
4 Generatots, Feynman-Kac Formula, and Other Tools Given the importance of the issues discussed above, it is not very surprising that the theory of stochastic processes developed some sptematic tools and concepts to facilitate tbe treatment of similar problems. Many of these tools simplify the notation and make the derivations mechanical. nis is the case with the notion of a generator, which is the formal equivalent of obtaining limits such as in (33),and the Feynman-lac theorem, which gives the probabilistic solution for a class of PDES, We complete this cbapter by formalizing these concepts utilizcd implicitly during the earlier discussion. 4.1 1to Diffuslo'ns
=
ast,
tldt + rvh,
/)#H(,
f (E
(0,x).
(68)
only.g We now assume that the drift and diffusion parameters depend on St The SDE cari bc written as: dst
=
=
alutjdt
xv
be an 1to diffusion satisfying the
+ tr(5'f)#H(,
t
(E
(70)
(0,cr).
Let /(.) be any bounded function, and suppose that the information set It contains a1l Su, u :i f until time t. 'Then we say that St satisfes the Marknv Jrcwilr/.pif ; E
I .J,1 If(.5+h)
=
E
1 E/'(-%+) .%1
,
>
0-
(71)
nat is, fumre movements in St, given what we obsen'ed until time 1, are likely to be the same as starting the proccss at time t. ln other words. te observations on St from the distant past do not belp to improve forecasts, given the St.
4.3 qenevatov /J tzn Ito Di/hlio?z I-et St be the lto diffusion given in (70).Let fvtj be a twice differentiable function of and suppose the process St has reached a pmicular value st as of time f We may wondcr how f (&) may move startirlg from the cunvnt state h. We dene an operator to represent this movement. We let the operator bc dcfined as the expected rate of c/ztlagc for fSf) as: ,%,
A continuous stochastic process St that has hnite first- and second-order moments was shown to follow the general SDE! dSt
483
4 Generators, Feynman-Kac Formula, and Other Tools
asflds
+ trt.5'rllT#;,
t
6
(0,x),
(69)
where the J(.) and c(.) are the drift and diffusion parameters. Processes that have thfs characteristic are called time-homogenous Ito ffu-orl-.'The and (11results below apply to those processes whose instantaneous drift (z4.) and fusion are not dependent on f diredly. Usual conditions apply to fast.'' tr(.), in that they arc not supposed to vary diflsions. of properties 1to discuss We can two tTtoo
SDES utilized in 9In almost all case,s of interest where there are no jumps involved, thc latter is especially popular practice arc either of geometrie, or of mean rcverting type. widely believcd to havc a mean reverting wilh intcrest rate derivatfvcs lxlatlse the shct't ratc is would be a functien of character. Under thesc conditions, the drift and diffusion parameers initial tenn stnlcture. rpatch ailowed the timc is to dopendencc on -%only. Howevcr, often rrhe
,
-.4
((&+a)If(&)) flstl
E
lim ( A a-a Hcrc thc small case letter st indicates an already obsen'ed value for The numerator of the expression on the right-hand side measures expected change in f(St). As we divide this by z', thc operator becomes a rate of change. ln the theory of stochastic proccsses zd is called the generator of the lto diffusion St. Some readers may wonder how we can desne a rate of change for fv), which indiredly is a function of a Wiener process. A rate of change is like a derivativc and we have shown that Wiener processes are not diffcrcntiable, So, how can we justly'thc existence of an operator such as one may ask, The answcr to this question is simple. does not deal with the actual represents an opected rate of change. rate of change in f(&). Instead, Although the Wiener process may be too erratic and nondifferentiable, note that expected changes in fst) will bc a smoothcr function and, under some conditions, a limit can be defined.lo Afh)
=
-u)
.
.%.
azl
.4,
..4
.,4
oEvery expectation titmlar values.
represents
an average.
B.ydefmiton, averagcs are smoothcr
than par-
3
484
C H A PT ER
4.4 A
*
for
RepT'cserlftztier
Relating Conditional Expectatiorus to PDL
21
4 Generators, Feynman-Kac Formula, and Othex Ttots
..4
The corrcsponding
First note that is an expected rate of change in the Iimit. That 1, we consider the immediate future with an infinitesimal change of tlme. nen, it is obvious that such a change would relate directly to lto's Lemrna. In fact, in the present case where St is a univariate stochastic process:
the operator
,4
astjdt
=
+ fr(-$,)#I,#),
t e' (0,txz),
(73)
is given by: Af
af
=
pf
1 u JV , o't M z
+
y
d.r(,sl
1./?2TjJ,+
gtk/,?.y M+
,
os,
Hence, thc difference between the operator Lemma is at two points;
(74)
ntld;. os
(75)
E=1
+
fyxj
J a,
1
f jtt7.rov )ij J-'ll''-q oyoyj
(78)
,
=1
/=1
where the term hafj represents the f/th clement of the matrx crtthln). 'rhe difference between the univariatc case and this multivariate formula is the existence of cross-product tcrms. Othenvise, the extension is immediate. In most advanced books on stochastic calculus, it is this multivariate form of that is introduced. The expressitm in (78)is known as the inhnitesimal generator of .(.).
4.5 Kolmogorx's Bcckwurd Eqwzzer Suppose we are given the lto diffalsion Also, assume that wc have a functlon of St denoted by f St). Consider the cxpectation: ut.
8%
and the application
..d
J'-!ap
k
k
JJ
..4
,
lt is worthwhile to compare this with what lto's Lemma would give. Applying Ito's Lemma tt futj with Si given by (73): -
--I.J =
will then be given by
operator
k
,4
dSt
.,zl
485
f (-
of Ito's
These two differences are consistent with the definition of A.smenticmed calculates an apected rate of change starting from the immediate above. state st. -d.
.p1.
,
(.(.%)I 1
(79)
--
.
,
1) represents
.-
..4
,
Jf
.4y.
(80)
=
-J/ Remembering the definition of
4.4.1 Multivariate Case For completion, we should provide the multivariate case for Let Xt be a k-dimensional lto diffusion given by the (vector)SDE',
E
-
where fSthe forecasted value and S- is the latest value observed before time t. Hcuristically speaking, is the immediate past, Then, using the t) may operator, wc can characterize how the fSchange ovcr time. This evolution of the forecast is given by Kolmogorov's backward rblftzlon: ,
1. The (fHz)term in lto's formula is replaced by its drift, which is zero. 2. Next, the remairting part of Ito's formula is divided by dt.
/)
-.
ad.'.
-.4.
dz'Lt :
a j, :
..=
dxkt
,.1
l
*'''t
d /' .j-.
*-'
vk 1 *'''t
akt
d HG
.lk
t
:
rrk k f
*'''
(76)
#W/r12
where the ait are the diffusion coeocients dcpending on Xt and the tp are the diffusion coecients possibly depending on Xt as well. nis equaticm is written in the symbolic form; dXt where al
.)
=
atdt +
tz'/##),
is a k x 1 vcctor and the
oj
/
is a k
e
I0,x),
x k matrix-
(77)
ga
m
1 z z? f jyts yj DSa It is easy to see that the equality in (81)is none other than the PDE: 1 c atfs + hrt J.u. Af -
=
at
lf
+
.
nx
a
=
(81)
(82)
nus, we again see the important correspondence betwcen conditional expectations such as and the PDE in different ways: @
rrhe
=
Effst)
I
,-1
(83)
(81).As before. this correspondence
h.%-f) >
s--/)
satisfies the PDE in Equation
can be statcd in two
(81).
k
.E ! 'k :L'.'. '.
C H A PT E R
486
.
*
Relating Con ditionalExpectacions to
21
eiventhe PDE in Equation (81)we can tind an
thePD E is satisflcd. ...s.
...
fs -
,
,'
h
.
* ' ',ii.:,
PDF-S
'
.'
8 Exercises
4
,j;j
i.. j..
t'' 'Y. 4(. (
5 Feynman-lfac Formula
X'.E
vkJp.
'j
t'i.
.r
y.y
uence,
.
ne
ykynman-xac formula is an extension of Kolmogorov's backward equation as well as being a formalization of the issues discussed earlier in this chapter. The formula provides a probabilistic solution f that corresponds to a given PDE.
1': :
j'j %L
,
(:
rt.
.%
,
4(tt,,
ijt
:,.
L
E (;!> k9-g (
;
Feynman-xhc Formula: Given
,lu.lsf.'
4.5.1 E'turzw/e
jtf, rt )
nl
'
Consider the function!
:;
.
t,
kl) ,
!)
1
=
e-
(.5.-&))?2:
,
liat
,.'-
=
:;
.
p
.( .
w jwre tjw operator
.1
ot
(85) ))j
dW6'
.,
know that
twice-
m
Af
=
=
'
;':;
?.;.
a
at
=
case, we have: 0
(r, j
ft-
-
-.fx..
=
2
:.
: ..
,g
,
4
',
:
'
''
()ne
.'
References
Several intercsting cases using this corrcspondence are found in Kushner (1995). nis sourcc also gives practical ways of calculating the implied poss,
E
:
is suc h ftmction . lt turns out that the conditional density #4-$,, &, witb resped to t and thc f.- To see this, take the hrst part ial derivative substitute in (89). The equation 111 sccondpar tial with respect to and E bc satislied. According to this result, thc condition al density fun-tion of a (generab PDE Kolmogorov 's backward equation, Ths ized) Wiener process satisies associated tf St will value with a pa rticular tells us how tllc probability . initial point the given &. passes, as t f)
Orrespindcncc between poss and some conditional expectations is useful in practical asset pricing. Given an instrument with special charvery acteristics,a market practitioner can use this correspondence and derive the implied PDES. These can then be numerically evaluated,
Hence, the Feynman-Kac formula provides conditional PDES. a solution that corresponds to a certain class of
q
7
-
&
(85),in this particular
or
a
c72 1 f + -% a :
'
,?
,
-
at
p /
'
density. Wc a we apply Kolmogorov 's formula to this satisfy backward Kolmogorov's would f S function f (.) o differentiable equation. . 1 2 (86) ftm f? / f. + 2 U't x.
(91)
is given by:
'
=
...
.:
A.f - qrljf,
=
-
,
.
ds /
(90)
,
a
..
'.'
eter
jd-tkuvtrsldxytrx j
we have
'
-
'
Bet according to
E;
=
.
... .....Jt. (g4) .L. 2rr/ .k . this is the conditional density function of a wiener inspection shows t f 0 and moves over time with zero Processthat starts from S at time #'E drift and variancc t. equation for this pro)' j If we wcre to write down a stochastic differential the drift parameter as zero an d tbe diffusion paramCe y, s we would choosewould satisfy; :: as one. The %
pt
axn
487
'E'.'..
that
,) such
.: /. j/ j Soltltitm fOr thc P,-'x-. ra..ja xn result means that f(S- 1) iS a Of correspondence bethe Kolmogorov 's backward equation is an example PDES in tlis earlier seen tween an expecta tion of a stochastic process and chapter. ryjs
..
'
:
8 Exerclses
ut
'
1
,
'
'
:
.
evolve
;
bond prie.c /J(/, T) satishes the following PDE!
supposethe -
ime
'
rt
s
+. lt +.
i
a srlt - xy s ) +. -s,,fy. 2
=
(j
(:9)
488
c
H A PT E R
.
21
Relating Conditiona: Expectations to A(T; T)
Deline the variable Pr(u) as )tmrr ds j'd( tr, P'(u) e e-
..):/8.k -
,
=
where
s
is tbe market price of intercst
(94)
1.
=
l'ate
PDF-S
)A(&sx)2J
(p.5;
,
,
lisk.
(a) Let #(f, F) be the bond price, Calculatc the #(#F). (b) Use thc PDE in (93)to get an cxpression for dB(t, F). lnd take expectations with (c) Integrate this expression from t to T respect to mmingale cquality to obtain the bond pricing formula: ,J)2J'lj j't' ruds ,.)J)K- l trs Etp (96) #(/, T).
gc
Stopping Times and American-Type Securities
gFltrr
-
=
where thc expectation is conditional on tbe currcnt assumed to be known.
,
rt which is
1
lntroduction
Options considered in this book can be dividcd into hvo catcgories. The hrst group was characterized using a pricing equation that depended on the currcnt value of thc underlying asscts St and on thc tne 1. For example, the price of a plain-vanilla call optitm at tne t was written as: C,
F(299 mrtingale approach to, 354-358 cornparison to PDE approach, 358-366 oveniew of, 296-299. 451-452 pricing and, 299-301, 353-366 references on, 366 solulons of closcd-form,301-305 numerical, 306-309 underlying assumptions, 299-3, 368, 369
452,
HJM methodology, and, 441 market prico of equity risk, 456
compared to classical pricing of interest-sensilivesecurities, 434-435 compared to PDE analysis tf nterest-senshivc secures, 459-460
arbitrage tbeorem on, 391-392 equivalent martingale measures 477-479 partial differcntial equalion for,
and, 126,
130-132
deflned, 177-178 Girsanov theorem and, 322
Bond pricing
trees. 30, 116, 183n,!495-499; Lattice models also see Black-scholes equation, 78, 79-80 conditions, 301309, 451, boundary
Black-scholes
515
Subject lndex
47-52
fundamental theorem of, 205 integrals, 5%.64 cc J/m Integrals) integration by parts, 65-66 Call options, 279., sce also American-style options; Options on bonds, 374-375