Financial Risk Management 2010-11 Topics
T1
Stock index futures Duration, Convexity, Immunization
T2
Repo and revers...

Author:
Dr Dennis Philip

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Financial Risk Management 2010-11 Topics

T1

Stock index futures Duration, Convexity, Immunization

T2

Repo and reverse repo Futures on T-bills Futures on T-bonds Delta, Gamma, Vega hedging

T3

Portfolio insurance Implied volatility and volatility smiles

T4

Modelling stock prices using GBM Interest rate derivatives (Bond options, Caps, Floors, Swaptions)

T5

Value at Risk

T6

Value at Risk: statistical issues Monte Carlo Simulations Principal Component Analysis Other VaR measures

T7

Parametric volatility models (GARCH type models) Non-parametric volatility models (Range and high frequency models) Multivariate volatility models (Dynamic Conditional Correlation DCC models)

T8

Credit Risk Measures (credit metrics, KMV, Credit Risk Plus, CPV)

T9

Credit derivatives (credit options, total return swaps, credit default swaps) Asset Backed Securitization Collateralized Debt Obligations (CDO)

* This file provides you an indication of the range of topics that is planned to be covered in the module. However, please note that the topic plans might be subject to change.

Topics

Financial Risk Management

Futures Contract: Speculation, arbitrage, and hedging

Topic 1 Managing risk using Futures Reading: CN(2001) chapter 3

Stock Index Futures Contract: Hedging (minimum variance hedge

ratio) Hedging market risks

Futures Contract Agreement to buy or sell “something” in the future at

a price agreed today. (It provides Leverage.) Speculation with Futures: Buy low, sell high Futures (unlike Forwards) can be closed anytime by taking an opposite position Arbitrage with Futures: Spot and Futures are linked

by actions of arbitragers. So they move one for one. Hedging with Futures: Example: In January, a farmer

wants to lock in the sale price of his hogs which will be “fat and pretty” in September. Sell live hog Futures contract in Jan with maturity in Sept

Speculation with Futures

Speculation with Futures

Speculation with Futures

Purchase at F0 = 100 Hope to sell at higher price later F1 = 110

Profit/Loss per contract

Close-out position before delivery date.

Long future

Obtain Leverage (i.e. initial margin is ‘low’)

$10

Long 61,000 Nikkei-225 index futures (underlying

value = $7bn). Nikkei fell and he lost money (lots of it) - he was supposed to be doing riskless ‘index arbitrage’ not speculating

0

-$10

F1 = 90 F0 = 100

Example: Example: Nick Leeson: Feb 1995

F1 = 110

Futures price

Short future

Speculation with Futures Profit payoff (direction vectors) F increase then profit increases Profit/Loss

F increase then profit decrease Profit/Loss

Arbitrage with Futures

-1

+1 Underlying,S

+1

or Futures, F

-1

Long Futures

Short Futures

or, Long Spot

or, Short Spot

Arbitrage with Futures At expiry (T), FT = ST . Else we can make riskless

profit (Arbitrage). Forward price approaches spot price at maturity Forward price, F Forward price ‘at a premium’ when : F > S (contango)

Arbitrage with Futures General formula for non-income paying security:

F0 = S0erT

or

F0 = S0(1+r)T

Futures price = spot price + cost of carry For stock paying dividends, we reduce the ‘cost of

carry’ by amount of dividend payments (d) 0

Stock price, St T

At T, ST = FT

F0 = S0e(r-d)T

For commodity futures, storage costs (v or V) is

negative income Forward price ‘at a discount’, when : F < S (backwardation)

Arbitrage with Futures For currency futures, the ‘cost of carry’ will be

reduced by the riskless rate of the foreign currency (rf) F0 = S0e(r-rf)T

For stock index futures, the cost of carry will be

reduced by the dividend yield F0 = S0

e(r-d)T

F0 = S0e(r+v)T or F0 = (S0+V)erT

Arbitrage with Futures Arbitrage at t S0erT then buy the asset and short the futures contract If F0 < S0erT then short the asset and buy the futures contract Example of ‘Cash and Carry’ arbitrage: S=£100,

r=4%p.a., F=£102 for delivery in 3 months. 0.04×0.25 = 101 £ We see Fɶ = 100 × e Since Futures is over priced, time = Now •Sell Futures contract at £102

time = in 3 months •Pay loan back (£101)

•Borrow £100 for 3 months and buy stock •Deliver stock and get agreed price of £102

Hedging with Futures F and S are positively correlated To hedge, we need a negative correlation. So we

Hedging with Futures

long one and short the other. Hedge = long underlying + short Futures

Hedging with Futures

Hedging with Futures

Simple Hedging Example: You long a stock and you fear falling prices over the next 2 months, when you want to sell. Today (say January), you observe S0=£100 and F0=£101 for April delivery. so r is 4% Today: you sell one futures contract In March: say prices fell to £90 (S1=£90). So F1=S1e0.04x(1/12)=£90.3. You close out on Futures.

F1 value would have been different if r had changed.

Profit on Futures: 101 – 90.3 = £10.7 Loss on stock value: 100 – 90 =£10

Net Position is +0.7 profit. Value of hedged portfolio

= S1+ (F0 - F1) = 90 + 10.7 = 100.7

This is Basis Risk (b1 = S1 – F1) Final Value

= S1 + (F0 - F1 ) = £100.7 = (S1 - F1 ) + F0 = b1 + F0 where “Final basis” b1 = S1 - F1 At maturity of the futures contract the basis is zero

(since S1 = F1 ). In general, when contract is closed out prior to maturity b1 = S1 - F1 may not be zero. However, b1 will usually be small in relation to F0.

Stock Index Futures Contract Stock Index Futures contract can be used to

eliminate market risk from a portfolio of stocks

Stock Index Futures Contract Hedging with SIFs

Hedging with Stock Index Futures Example: A portfolio manager wishes to hedge her

portfolio of $1.4m held in diversified equity and S&P500 index Total value of spot position, TVS0=$1.4m S0 = 1400 index point Number of stocks, Ns = TVS0/S0 = $1.4m/1400 =1000 units We want to hedge Δ(TVSt)= Ns . Δ(St)

F0 = S0 × e( r − d )T If this equality does not hold then index arbitrage (program trading) would generate riskless profits. Risk free rate is usually greater than dividend yield

(r>d) so F>S

Hedging with Stock Index Futures The required number of Stock Index Futures contract

to short will be 3 TVS 0 $1, 400, 000 NF = − = − = − 3.73 $375, 000 FVF0 In the above example, we have assumed that S and F have correlation +1 (i.e. ∆ S = ∆ F ) In reality this is not the case and so we need

Use Stock Index Futures, F0=1500 index point, z=

contract multiplier = $250 FVF0 = z F0 = $250 ( 1500 ) = $375,000

minimum variance hedge ratio

Hedging with Stock Index Futures Minimum Variance Hedge Ratio

To obtain minimum, we differentiate with respect to Nf

∆V = change in spot market position + change in Index Futures position + Nf . (F1 - F0) z = Ns . (S1-S0) Ns S0. ∆S /S0 TVS0 . ∆S /S0

= =

+ +

Hedging with Stock Index Futures 2

(∂σ V

Nf F0. (∆ ∆F /F0) z Nf . FVF0 . (∆ ∆F /F0)

where, z = contract multiple for futures ($250 for S&P 500 Futures); ∆S = S1 - S0, ∆F = F1 - F0

/ ∂N

2

2

2

2

f

( F V F0 ) 2 σ ∆2 F / F

N f

= −

2

TVS 0 FVF0 . σ ∆S / S , ∆F / F

2 ∂σV / ∂N = 0 f

TVS 0 .β p Nf = − FVF0

(∆S

0

⋅ F V F ⋅ σ ∆ S / S ,∆ F / F 0

( σ ∆ S / S ,∆ F / F

2

σ ∆F / F

)

/ S ) = α 0 + β ∆S / ∆F ( ∆ F / F ) + ε

Hedging with Stock Index Futures Application: Changing beta of your portfolio: “Market Timing Strategy” TVS

implies

Value of Spot Position = − FaceValue of futures at t = 0

TVS0 F V F0

= −TVS

where Ns = TVS0/S0 and beta is regression coefficient of the regression

Hedging with Stock Index Futures SUMMARY

and set to zero

TVS0 =− β ∆ S / S ,∆ F / F F V F0

σ V = (TVS 0 ) σ ∆S / S + ( N f ) ( FVF ) σ ∆F / F 0 + 2N

= 0)

N f

The variance of the hedged portfolio is

2

f

Nf =

0

FVF0

.( β h − β p )

Example: βp (=say 0.8) is your current ‘spot/cash’ portfolio of stocks

βp

But

• You are more optimistic about ‘bull market’ and desire a higher exposure of βh (=say, 1.3) • It’s ‘expensive’ to sell low-beta shares and purchase high-beta shares

If correlation = 1, the beta will be 1 and we just have

TVS0 Nf = − FVF0

• Instead ‘go long’

more Nf Stock Index Futures contracts

Note: If βh= 0, then

Nf = - (TVS0 / FVF0) βp

Hedging with Stock Index Futures

Hedging with Stock Index Futures Application: Stock Picking and hedging market risk

If you hold stock portfolio, selling futures will place a hedge and reduce the beta of your stock portfolio. If you want to increase your portfolio beta, go long futures. Example: Suppose β = 0.8 and Nf = -6 contracts would make β = 0. If you short 3 (-3) contracts instead, then β = 0.4 If you long 3 (+3) contracts instead, then β = 0.8+0.4 = 1.2

You hold (or purchase) 1000 undervalued shares of Sven plc V(Sven) = $110

(e.g. Using Gordon Growth model)

P(Sven) = $100 (say) Sven plc are underpriced by 10%.

Therefore you believe Sven will rise 10% more than the market over the next 3 months. But you also think that the market as a whole may fall by 3%. The beta of Sven plc (when regressed with the market return) is 2.0

Hedging with Stock Index Futures

Hedging with Stock Index Futures

Can you ‘protect’ yourself against the general fall in the market and hence any ‘knock on’ effect on Sven plc ?

Application: Future stock purchase and hedging market risk

Yes .

You want to purchase 1000 stocks of takeover target with βp = 2, in 1 month’s time when you will have the cash.

Sell Nf index futures, using:

N

f

= −

TVS FVF

0

.β

You fear a general rise in stock prices. p

0

If the market falls 3% then Sven plc will only change by about

10% - (2x3%) = +4%

But the profit from the short position in Nf index futures, will give you an additional return of around 6%, making your total return around 10%.

Go long Stock Index Futures (SIF) contracts, so that gain on the futures will offset the higher cost of these particular shares in 1 month’s time.

N

f

=

TVS FVF

0

.β

p

0

SIF will protect you from market risk (ie. General rise in prices) but not from specific risk. For example if the information that you are trying to takeover the firm ‘leaks out’ , then price of ‘takeover target’ will move more than that given by its ‘beta’ (i.e. the futures only hedges market risk)

Topics

Financial Risk Management

Duration, immunization, convexity Repo (Sale and Repurchase agreement)

Topic 2 Managing interest rate risks Reference: Hull(2009), Luenberger (1997), and CN(2001)

and Reverse Repo Hedging using interest rate Futures Futures on T-bills Futures on T-bonds

Readings Books Hull(2009) chapters 6 CN(2001) chapters 5, 6 Luenberger (1997) chapters 3 Journal Article Fooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis”

Managerial Finance,Vol 25, no. 3

Hedging Interest rate risks: Duration

Duration

Duration (also called Macaulay Duration)

Duration measures sensitivity of price changes (volatility) with changes in interest rates 1 Lower the coupons

n

B =

2 Greater the time to

D =

T PB = ∑ C t t + ParValue T (1+ r ) t =1 (1+ r )

T PB = ∑ C t t + ParValue T (1+ r ) t =1 (1+ r )

approximate response of bond prices to change in yields. A better approximation could be convexity of the bond .

for a given time to maturity, greater change in price to change in interest rates

T

T

Duration of the bond is a measure that summarizes

maturity with a given coupon, greater change in price to change in interest rates

3 For a given percentage change in yield, the actual price increase is

∑

c i e − y ti

i =1 n

∑

i =1

weight

t i ⋅ c i e − y ti B

=

n

∑

i =1

c e − y ti ti i B

Duration is weighted average of the times when payments

are made. The weight is equal to proportion of bond’s total present value received in cash flow at time ti. Duration is “how long” bondholder has to wait for cash flows

greater than a price decrease

Macaulay Duration For a small change in yields ∆ y / d y

Evaluating d B :

dy

dB ∆B = ∆y dy n ∆ B = − ∑ t i c i e − y ti ∆ y i =1 = −B ⋅ D ⋅∆y

∆B = −D ⋅∆y B D measures sensitivity of percentage change in bond prices to (small) changes in yields Note negative relationship between Price (B) and yields (Y)

Modified Duration and Dollar Duration For Macaulay Duration, y is expressed in continuous

compounding. When we have discrete compounding, we have Modified Duration (with these small modifications) If y is expressed as compounding m times a year, we divide D D by (1+y/m) ∆B = − B ⋅ ⋅ ∆y

(1 + y / m)

∆B = − B ⋅ D* ⋅ ∆y Dollar Duration, D$ = B.D That is, D$ = Bond Price x Duration (Macaulay or Modified)

∆B = − D$ ⋅ ∆y

So D$

is like Options Delta

D$ = −

∆B ∆y

Duration

Duration -example

Example: Consider a trader who has $1 million in

bond with modified duration of 5. This means for every 1 bp (i.e. 0.01%) change in yield, the value of the bond portfolio will change by $500. ∆B = − ( $1, 000, 000 × 5 ) ⋅ 0.01% = −$500

Example: Consider a 7% bond with 3 years to maturity. Assume that the bond is selling at 8% yield.

A Year

A zero coupon bond with maturity of n years has a

Duration = n A coupon-bearing bond with maturity of n years will have Duration < n Duration of a bond portfolio is weighted average of the durations of individual bonds D

p o r tfo lio

=

∑ (B

i

/ B )⋅ D i

B

0.5 1.0 1.5 2.0 2.5 3.0 Sum

C

D

E

Present value Weight = Payment Discount A× E =B× C D/Price factor 8% 3.5 0.962 3.365 0.035 0.017 3.5 0.925 3.236 0.033 0.033 3.5 0.889 3.111 0.032 0.048 3.5 0.855 2.992 0.031 0.061 3.5 0.822 2.877 0.030 0.074 103.5 0.79 81.798 0.840 2.520 Price = 97.379 Duration = 2.753

Here, yield to maturity = 0.08, m = 2, y = 0.04, n = 6, Face value = 100.

i

Qualitative properties of duration

Properties of duration

Duration of bonds with 5% yield as a function of

maturity and coupon rate. Coupon rate

Years to maturity 1 2 5 10 25 50 100 Infinity

1%

2%

5%

10%

0.997 1.984 4.875 9.416 20.164 26.666 22.572 20.500

0.995 1.969 4.763 8.950 17.715 22.284 21.200 20.500

0.988 1.928 4.485 7.989 14.536 18.765 20.363 20.500

0.977 1.868 4.156 7.107 12.754 17.384 20.067 20.500

1. Duration of a coupon paying bond is always less than its maturity. Duration decreases with the increase of coupon rate. Duration equals bond maturity for noncoupon paying bond. 2. As the time to maturity increases to infinity, the duration do not increase to infinity but tend to a finite limit independent of the coupon rate. 1 + mλ

where λ is the yield to maturity λ per annum, and m is the number of coupon

Actually, D →

payments per year.

Properties of Duration 3. Durations are not quite sensitive to increase in coupon rate (for bonds with fixed yield). They don’t vary huge amount since yield is held constant and it cancels out the influence of coupons. 4. When the coupon rate is lower than the yield, the duration first increases with maturity to some maximum value then decreases to the asymptotic limit value. 5. Very long durations can be achieved by bonds with very long maturities and very low coupons.

Immunization (or Duration matching) This is widely implemented by Fixed Income Practitioners.

time 0 0

time 1

time 2

time 3

pay $

pay $

pay $

You want to safeguard against interest rate increases.

Changing Portfolio Duration Changing Duration of your portfolio: If prices are rising (yields are falling), a bond trader might want to switch from shorter duration bonds to longer duration bonds as longer duration bonds have larger price changes. Alternatively, you can leverage shorter maturities. Effective portfolio duration = ordinary duration x leverage ratio.

Immunization Matching present values (PV) of portfolio and obligations This means that you will meet your obligations with the cash

from the portfolio. If yields don’t change, then you are fine. If yields change, then the portfolio value and PV will both change by varied amounts. So we match also Duration (interest rate risk)

PV1 + PV2 = PVobligation

A few ideas: 1. Buy zero coupon bond with maturities matching timing of

cash flows (*Not available) [Rolling hedge has reinv. risk] 2. Keep portfolio of assets and sell parts of it when cash is needed & reinvest in more assets when surplus (* difficult as Δ value of in portfolio and Δ value of obligations will not identical) 3. Immunization - matching duration and present values of portfolio and obligations (*YES)

Matching duration Here both portfolio and obligations have the same sensitivity to

interest rate changes. If yields increase then PV of portfolio will decrease (so will the PV of the obligation streams) If yields decrease then PV of portfolio will increase (so will the PV of the obligation streams)

D1 PV1 + D 2 PV2 = Dobligation PVobligation

Immunization

Immunization

Example

Suppose only the following bonds are available for its choice.

Suppose Company A has an obligation to pay $1 million in 10 years. How to invest in bonds now so as to meet the future obligation? • An obvious solution is the purchase of a simple zero-coupon bond with maturity 10 years. * This example is from Leunberger (1998) page 64-65. The numbers are rounded up by the author so replication would give different numbers.

Bond 1 Bond 2 Bond 3

coupon rate 6% 11% 9%

maturity 30 yr 10 yr 20 yr

yield 9% 9% 9%

duration 11.44 6.54 9.61

•

Present value of obligation at 9% yield is $414,642.86.

•

Since Bonds 2 and 3 have durations shorter than 10 years, it is not possible to attain a portfolio with duration 10 years using these two bonds.

Suppose we use Bond 1 and Bond 2 of amounts V1 & V2, V1 + V2 = PV P1V1 + D2V2 = 10 × PV giving V1 = $292,788.64,

Immunization

price 69.04 113.01 100.00

V2 = $121,854.78.

Immunization

Yield 9.0 Bond 1 Price Shares Value

8.0

10.0

69.04 77.38 62.14 4241 4241 4241 292798.64 328168.58 263535.74

Bond 2 Price 113.01 120.39 106.23 Shares 1078 1078 1078 Value 121824.78 129780.42 114515.94 Obligation value 414642.86 456386.95 376889.48 Surplus -19.44 1562.05 1162.20

Observation: At different yields (8% and 10%), the value of the portfolio almost agrees with that of the obligation.

Difficulties with immunization procedure 1. It is necessary to rebalance or re-immunize the portfolio from time to time since the duration depends on yield. 2. The immunization method assumes that all yields are equal (not quite realistic to have bonds with different maturities to have the same yield). 3. When the prevailing interest rate changes, it is unlikely that the yields on all bonds change by the same amount.

Duration for term structure We want to measure sensitivity to parallel shifts in the spot

rate curve For continuous compounding, duration is called FisherFisher-Weil duration. duration If x0, x1,…, xn is cash flow sequence and spot curve is st where t = t0,…,tn then present value of cash flow is PV =

n

∑ i=0

xti ⋅ e

1 PV

(

changes to sti + ∆y

Consider parallel shift in term structure: st

i

Then PV becomes

P ( ∆y ) =

n

∑x i=0

ti

⋅e

(

)

− sti + ∆ y ⋅ti

Taking differential w.r.t ∆y in the point ∆y=0 we get n dP ( ∆ y ) − s ⋅t | ∆ y = 0 = − ∑ t i x t i ⋅ e ti i d ∆y i=0

− sti ⋅ti

So we find relative price sensitivity is given by DFW

The Fisher-Weil duration is

D FW =

Duration for term structure

n

∑t i=0

i

⋅ x ti ⋅ e

1 dP (0) ⋅ = − D FW P (0) d ∆ y

− sti ⋅ti

Convexity Duration applies to only small changes in y Two bonds with same duration can have different

change in value of their portfolio (for large changes in yields)

Convexity Convexity for a bond is n

1 d 2B C = = B dy 2

∑

i =1

t i2 ⋅ c i e − y t i B

=

c e − y ti t i2 i B

n

∑

i =1

Convexity is the weighted average of the ‘times squared’

when payments are made. From Taylor series expansion

First order approximation cannot capture this, so we

take second order approximation (convexity)

1 d 2B dB ∆ y + dy 2 dy 2 1 = − D ⋅ ∆ y + C ⋅ 2

∆ B =

(∆

y

)

∆ B B

(∆

y

)

2

2

So Dollar convexity is like Gamma measure in

options.

)

Short term risk management using Repo

REPO and REVERSE REPO

Repo is where a security is sold with agreement to buy it back at a later date (at the price agreed now) Difference in prices is the interest earned (called repo rate) rate It is form of collateralized short term borrowing (mostly overnight) Example: a trader buys a bond and repo it overnight. The money from repo is used to pay for the bond. The cost of this deal is repo rate but trader may earn increase in bond prices and any coupon payments on the bond. There is credit risk of the borrower. Lender may ask for

margin costs (called haircut) to provide default protection. Example: A 1% haircut would mean only 99% of the value of

collateral is lend in cash. Additional ‘margin calls’ are made if market value of collateral falls below some level.

Short term risk management using Repo Hedge funds usually speculate on bond price differentials

using REPO and REVERSE REPO Example: Assume two bonds A and B with different prices (say price(A)<price(B)) but

similar characteristics. Hedge Fund (HF) would like to buy A and sell B simultaneously.This can be financed with repo as follows: (Long position) Buy Bond A and repo it. The cash obtained is used to pay for

the bond. At repo termination date, sell the bond and with the cash buy bond back (simultaneously). HF would benefit from the price increase in bond and low repo rate (short position) Enter into reverse repo by borrowing the Bond B (as collateral for money lend) and simultaneously sell Bond B in the market. At repo termination date, buy bond back and get your loan back (+ repo rate). HF would benefit from the high repo rate and a decrease in price of the bond.

Interest Rate Futures (Futures on T-Bills)

Interest Rate Futures In this section we will look at how Futures contract written on a Treasury Bill (T-Bill) help in hedging interest rate risks Review - What is T-Bill? T-Bills are issued by government, and quoted at a discount Prices are quoted using a discount rate (interest earned as % of face value) Example: 90-day T-Bill is quoted at 0.08. 0.08 This means annualized return is 8% of FV. So we can work out the price, as we know FV. d 90 P = F V 1 − 100 360 Day Counts convention (in US) 1. Actual/Actual (for treasury bonds) 2. 30/360 (for corporate and municipal bonds) 3. Actual/360 (for other instruments such as LIBOR)

Hedge decisions When do we use these futures contract to hedge? Examples: 1) You hold 3m T-Bills to sell in 1-month’s time ~ fear price fall ~ sell/short T-Bill futures 2) You will receive $10m in 3m time and wish to place it on a Eurodollar bank deposit for 90 days ~ fear a fall in interest rates ~ go long a Eurodollar futures contract 3) Have to issue $100m of 180-day Commercial Paper in 3 months time (I.e. borrow money) ~ fear a rise in interest rates ~ sell/short a T-bill futures contract as there is no commercial bill futures contract (cross hedge)

Interest Rate Futures So what is a 3-month T-Bill Futures contract? At expiry, (T), which may be in say 2 months time the (long) futures delivers a T-Bill which matures at T+90 days, with face value M=$100. As we shall see, this allows you to ‘lock in’ at t=0, the forward rate, f12

T-Bill Futures prices are quoted in terms of quoted index, Q

(unlike discount rate for underlying) Q = $100 – futures discount rate (df) So we can work out the price as

d f 90 F = F V 1 − 100 360

Cross Hedge: US T-Bill Futures Example: Today is May. Funds of $1m will be available in August to invest for further 6 months in bank deposit (or commercial bills) ~ spot asset is a 6-month interest rate Fear a fall in spot interest rates before August, so today BUY Tbill futures Assume parallel shift in the yield curve. (Hence all interest rates move by the same amount.) ~ BUT the futures price will move less than the price of the higher the maturity, more commercial bill - this is duration at work! sensitive are changes in prices to interest rates

Use Sept ‘3m T-bill’ Futures, ‘nearby’ contract

~ underlying this futures contract is a 3-month interest rate

Cross Hedge: US T-Bill Futures 3 month exposure period

Cross Hedge: US T-Bill Futures Question: How many T-bill futures contract should I purchase?

Desired investment/protection period = 6-months

We should take into account the fact that:

to hedge exposure of 3 months, we have used T-bill futures with 4 months time-to-maturity 2. the Futures and spot prices may not move one-to-one 1.

May

Aug.

Dec.

Sept.

Feb.

Maturity of ‘Underlying’ in Futures contract

Purchase T-Bill Known $1m Maturity date of Sept. future with Sept. cash receipts T-Bill futures contract delivery date

We could use the minimum variance hedge ratio:

Nf =

TVS0 .β p FVF0

However, we can link price changes to interest rate

changes using Duration based hedge ratio

Question: How many T-bill futures contract should I purchase?

Duration based hedge ratio Using duration formulae for spot rates and futures:

∆S = − DS ⋅ ∆ys S

∆F = − DF ⋅ ∆yF F

So we can say volatility is proportional to Duration:

∆S 2 2 = DS ⋅ σ ( ∆ys ) S

σ2

∆F 2 2 = DF ⋅ σ ( ∆yF ) F

σ2

∆S ∆F Cov , = Ε ( − DS ⋅ ∆ys )( − DF ⋅ ∆yF ) S F = DS ⋅ DF ⋅ σ ( ∆ys ∆yF )

Duration based hedge ratio Expressing Beta in terms of Duration:

TVS0 Nf = .β p FVF 0 ∆S ∆F Cov , TVS0 S F = FVF0 σ 2 ∆F F TVS0 Ds σ ( ∆ys ∆yF ) = 2 ∆ FVF D y σ ( ) 0 F F

We can obtain last term by regressing ∆yS = α0 + βy∆yF + ε

Duration based hedge ratio Summary:

TVS0 Ds βy Nf = . FVF0 DF

Cross Hedge: US T-Bill Futures Example REVISITED

3 month exposure period

Aug.

May

where beta is obtained from the regression of yields

Desired investment/protection period = 6-months

Dec.

Sept.

∆yS = α0 + β y ∆yF + ε

Feb.

Maturity of ‘Underlying’ in Futures contract

Purchase T-Bill Known $1m Maturity date of Sept. future with Sept. cash receipts T-Bill futures contract delivery date Question: How many T-bill futures contract should I purchase?

Cross Hedge: US T-Bill Futures May (Today). Funds of $1m accrue in August to be invested for 6- months in bank deposit or commercial bills( Ds = 6 )

Cross Hedge: US T-Bill Futures Suppose now we are in August: 3 month US T-Bill Futures : Sept Maturity

Use Sept ‘3m T-bill’ Futures ‘nearby’ contract ( DF = 3) Cross-hedge. Here assume parallel shift in the yield curve

Spot Market(May)

CME Index

Futures Price, F

Face Value of $1m

(T-Bill yields)

Quote Qf

(per $100)

Contract, FVF

May

y0 (6m) = 11%

Qf,0 = 89.2

97.30

$973,000

August

y1(6m) = 9.6%

Qf,1 = 90.3

97.58

$975,750

Change

-1.4%

1.10 (110 ticks)

0.28

$2,750 (per contract)

Qf = 89.2 (per $100 nominal) hence: F0 = 100 – (10.8 / 4) = 97.30 F

FVF0 = $1m (F0/100)

= $973,000

Durations are : Ds = 0.5, Df = 0.25 Amount to be hedged = $1m. No. of contracts held = 2

Key figure is F1 = 97.575 (rounded 97.58) Gain on the futures position

Nf = (TVS0 / FVF0) (Ds / DF ) = ($1m / 973,000) ( 0.5 / 0.25) = 2.05 (=2)

= TVS0 (F1 - F0) NF = $1m (0.97575 – 0.973) 2 = $5,500

Cross Hedge: US T-Bill Futures Invest this profit of $5500 for 6 months (Aug-Feb) at y1=9.6%:

= $5500 + (0.096/2) = $5764 Loss of interest in 6-month spot market (y0=11%, y1=9.6%)

= $1m x [0.11 – 0.096] x (1/2) = $7000 Net Loss on hedged position $7000 - $5764 = $1236

(so the company lost $1236 than $7000 without the hedge)

Interest Rate Futures (Futures on T-Bonds)

Potential Problems with this hedge: 1. Margin calls may be required 2. Nearby contracts may be maturing before September. So we may have to roll over the hedge 3. Cross hedge instrument may have different driving factors of risk

US T-Bond Futures Contract specifications of US T-Bond Futures at CBOT: Contract size

$100,000 nominal, notional US Treasury bond with 8% coupon

Delivery months

March, June, September, December

Quotation

Per $100 nominal

Tick size (value)

1/32 ($31.25)

Last trading day

7 working days prior to last business day in expiry month

Delivery day

Any business day in delivery month (seller’s choice)

Settlement

Any US Treasury bond maturing at least 15 years from the contract month (or not callable for 15 years)

US T-Bond Futures Conversion Factor (CF): (CF): CF adjusts price of actual bond to be

delivered by assuming it has a 8% yield (matching the bond to the notional bond specified in the futures contract) Price = (most recent settlement price x CF) + accrued interest Example: Possible bond for delivery is a 10% coupon (semi-

annual) T-bond with maturity 20 years. The theoretical price (say, r=8%): 40

Notional is 8% coupon bond. However, Short can choose to

deliver any other bond. So Conversion Factor adjusts “delivery price” to reflect type of bond delivered T-bond must have at least 15 years time-to-maturity Quote ‘98‘98-14’ means 98.(14/32)=$98.4375 per $100 nominal

5 100 + = 119.794 i 1.0440 i =1 1.04

P=∑

Dividing by Face Value, CF = 119.794/100 = 1.19794 (per

$100 nominal) If Coupon rate > 8% then CF>1

If Coupon rate < 8% then CF 0 ∂S ∂P ∆ put = = N ( d1 ) − 1 < 0 ∂S

(for long positions)

If we have lots of options (on same underlying) then

delta of portfolio is

∆ portfolio = ∑ N k ⋅ ∆ k k

where Nk is the number of options held. Nk > 0 if long Call/Put and Nk < 0 if short Call/Put

Delta So if we use delta hedging for a short call position, we

must keep a long position of N(d1) shares What about put options? The higher the call’s delta, the more likely it is that the option ends up in the money: Deep out-of-the-money: Δ ≈ 0 At-the-money: In-the-money:

Δ ≈ 0.5 Δ≈1

Intuition: if the trader had written deep OTM calls, it

would not take so many shares to hedge - unlikely the calls would end up in-the-money

Theta The rate of change of the value of an option with

respect to time Also called the time decay of the option For a European call on a non-dividend-paying stock, Θ=−

S0 N '(d1 )σ 2T

− rKe

− rT

1 N (d 2 ) where N '( x) = e 2π

x2 − 2

Related to the square root of time, so the relationship is

not linear

Theta Theta is negative: as maturity approaches, the option

tends to become less valuable The close to the expiration date, the faster the value of

the option falls (to its intrinsic value) Theta isn’t the same kind of parameter as delta The passage of time is certain, so it doesn’t make

any sense to hedge against it!!! Many traders still see theta as a useful descriptive statistic

because in a delta-neutral portfolio it can proxy for Gamma

Gamma The rate of change of delta with respect to the 2 ∂ f share price: ∂S 2 Calculated as Γ = N '(d1 ) S0σ T Sometimes referred to as an option’s curvature If delta changes slowly → gamma small → adjustments

to keep portfolio delta-neutral not often needed

Gamma If delta changes quickly → gamma large → risky to

leave an originally delta-neutral portfolio unchanged for long periods: Option price

C'' C' C S

S'

Stock price

Gamma Making a Position GammaGamma-Neutral We must make a portfolio initially gamma-neutral as well as delta-neutral

if we want a lasting hedge But a position in the underlying share can’t alter the portfolio gamma since the share has a gamma of zero So we need to take out another position in an option that isn’t linearly dependent on the underlying share If a delta-neutral portfolio starts with gamma Γ, and we buy wT options each with gamma ΓT, then the portfolio now has gamma Γ + wT Γ T We want this new gamma to = 0:

−Γ Rearranging, wT = ΓT

Γ + wT Γ T = 0

Delta-Theta-Gamma For any derivative dependent on a non-dividend-paying stock, Δ , θ, and Г are related The standard Black-Scholes differential equation is ∂f ∂f 1 2 2 ∂ 2 f + rS + σ S = rf 2 ∂t ∂S 2 ∂S

where f is the call price, S is the price of the underlying share and r is the risk-free rate 2 ∂ f ∂ f ∂ f But Θ = , ∆= and Γ = ∂t

∂S

∂S 2

1 2 2 Θ + rS ∆ + Θ S Γ = rf So 2 So if Θ is large and positive, Γ tends to be large and negative, and vice-versa This is why you can use Θ as a proxy for Γ in a delta-neutral portfolio

Vega NOT a letter in the Greek alphabet! Vega measures, the sensitivity of an option’s

price to volatility: volatility ∂f = ∂σ

υ

υ = S0 T N '(d1 )

High vega → portfolio value very sensitive to

small changes in volatility Like in the case of gamma, if we add in a traded

option we should take a position of – υ/υT to make the portfolio vega-neutral

Rho The rate of change of the value of a portfolio of

options with respect to the interest rate ∂f ρ= ρ = KTe− rT N (d 2 ) ∂r Rho for European Calls is always positive and Rho for European Puts is always negative (since as interest rates rise, forward value of stock increases). Not very important to stock options with a life of a few

months if for example the interest rate moves by ¼% More relevant for which class of options?

Delta Hedging Value of portfolio = no of calls x call price + no of stocks x

stock price V = NC C + NS S

∂V = N ∂S

∂C + N S ⋅1 = 0 C ⋅ ∂S ∂C NS = −NC ⋅ ∂S N S = − N C ⋅ ∆ c a ll

So if we sold 1 call option then NC = -1. Then no of stocks to

buy will be NS = ∆call So if ∆call = 0.6368 then buy 0.63 stocks per call option

Delta Hedging Example: As a trader, you have just sold (written)

100 call options to a pension fund (and earned a nice little brokerage fee and charged a little more than Black-Scholes price). You are worried that share prices might RISE, RISE hence the call premium RISE, hence showing a loss on your position. Suppose ∆ of the call is 0.4. Since you are short,

your ∆ = -0.4 (When S increases by +$1 (e.g. from 100 to 101), then C decrease by $0.4 (e.g. from 10 to 9.6)).

Delta Hedging Your 100 written (sold) call option (at C0 = 10 each option) You now buy 40-shares Suppose S FALLS by $1 over the next month THEN fall in C is 0.4 ( = “delta” of the call) So C falls to C1 = 9.6 To close out you must now buy back at C1 = 9.6 (a GAIN of $0.4)

Loss on 40 shares Gain on calls

= $40 = 100 (C0 - C1 )= 100(0.4)

= $40

Delta hedging your 100 written calls with 40 shares means that the value of your ‘portfolio is unchanged.

Delta Hedging Call Premium

∆ = 0.5 B

.

∆ = 0.4

.

A

0 100

110

Stock Price

As S changes then so does ‘delta’ , so you have to rebalance your portfolio. E.g. ‘delta’ = 0.5, then you now have to hold 50 stocks for every written call. This brings us to ‘Dynamic Hedging’, over many periods. Buying and selling shares can be expensive so instead we can maintain the hedge by buying and selling options.

(Dynamic) Delta Hedging You’ve written a call option and earned C0 =10.45

(with K=100,

σ = 20%, r=5%, T=1) At t = 0: Current price S0 = $100. We calculate ∆ 0 = N(d1)= 0.6368. So we buy ∆0 = 0.6368 shares at S0 = $100 by borrowing debt. Debt, D0 = ∆0 x S0 = $63.68

At t = 0.01: stock price rise S1 = $100.1. We calculate ∆ 1 = 0.6381. So buy extra (∆ 1 – ∆ 0) =0.0013 no of shares at $100.1.

Debt, D1 = D0 ert + (∆ 1 – ∆ 0) S1 = $63.84 So as you rebalance, you either accumulate or reduce debt

levels.

Delta Hedging At t=T, if option ends up well “in the money” Say ST = 163.3499. Then ∆ T = 1 (hold 1 share for 1 call). Our final debt amount DT = 111.29 (copied from Textbook page 247) The option is exercised. We get strike $100 for the share. Our Net Cost: NCT = DT – K = 111.29 – 100 = $11.29

How have we done with this hedging? At t = 0, 0 we received $10.45 and at t = T we owe $11.29 % Net cost of hedge, % NCT = [ (DT – K )-C0 ] / C0 = 8% (8% is close to 5% riskless rate)

Delta Hedging One way to view the hedge: The delta hedge is supposed to be riskless (i.e. no change in value of portfolio of “One written call + holding ∆ shares” , over any very small time interval ) Hence for a perfect hedge we require: dV = If we choose NS = ∆

NS dS + (NC ) dC ≈ NS dS + (-1) [ ∆ dS ] ≈ 0 then we will obtain a near perfect hedge

(ie. for only small changes in S, or equivalently over small time intervals)

Delta Hedging Another way to view the hedge: The delta hedge is supposed to be riskless, so any money we borrow (receive) at t=0 which is delta hedged over t to T , should have a cost of r Hence: For a perfect hedge we expect: NDT / C0 = erT

so,

NDT e-r T - C0 ≈ 0

If we repeat the delta hedge a large number of times then: % Hedge Performance, HP =

stdv( NDT e-r T - C0) / C0

HP will be smaller the more frequently we rebalance the portfolio (i.e. buy or sell stocks) although frequent rebalancing leads to higher ‘transactions costs’ (Kuriel and Roncalli (1998))

Gamma and Vega Hedging ∂2 f Γ = ∂S 2

∂f υ = ∂σ

υ Short Call/Put have negative Γ and υ Long Call/Put have positive Γ and

Gamma /Vega Neutral: Stocks and futures have

Γ ,υ = 0

So to change Gamma/Vega of an existing options portfolio, we have to take positions in further (new) options.

Delta-Gamma Neutral Example: Suppose we have an existing portfolio of options, with a value of

Γ = - 300 (and a ∆ = 0)

Note:

Γ = Σi ( Ni Γi )

Can we remove the risk to changes in S (for even large changes in S ? ) Create a “Gamma-Neutral” Portfolio Let ΓZ = gamma of some “new” option (same ‘underlying’)

For Γport = NZ ΓZ + Γ = 0 we require: NZ = - Γ / ΓZ “new” options

Delta-Gamma Neutral Suppose a Call option “Z” with the same underlying (e.g. stock) has a delta =

0.62 and gamma of 1.5 How can you use Z to make the overall portfolio gamma and delta neutral? We require:

Nz Γz + Γ = 0 Nz = - Γ / Γz = -(-300)/1.5 = 200

implies 200 long contracts in Z (ie buy 200 Z-options) The delta of this ‘new’ portfolio is now ∆ = Nz.∆z = 200(0.62) = 124 Hence to maintain delta neutrality you must short 124 units of the underlying -

this will not change the ‘gamma’ of your portfolio (since gamma of stock is zero).

Delta-Gamma-Vega Neutral Example:You hold a portfolio with

∆ port = − 500, Γ port = − 5000, υ port = − 4000 We need at least 2 options to achieve Gamma and Vega neutrality. Then

we rebalance to achieve Delta neutrality of the ‘new’ Gamma-Vega neutral portfolio. Suppose there is available 2 types of options: υ Z = 0.8 Option Y with ∆ Y = 0.6, Γ Y = 0.3, υ Y = 0.4 Option Z with ∆ Z = 0.5, Γ Z = 1.5,

We need

N Z υ Z + N Y υ Y + υ port = 0 N Z Γ Z + N Y Γ Y + Γ port = 0

Delta-Gamma-Vega Neutral So

N Z ( 0.8 ) + N Y ( 0.4 ) − 4000 = 0 N Z (1.5 ) + N Y ( 0.3 ) − 5000 = 0

Solution:

N Z = 2222.2 N Y = 5555.5

Go long 2222.2 units of option Z and long 5555.5 units of option Y to

attain Gamma-Vega neutrality. New portfolio Delta will be:

2222.2 × ∆ Z + 5555.5 × ∆ Y + ∆ port = 3944.4 Therefore go short 3944 units of stock to attain Delta neutrality

Portfolio Insurance

Portfolio Insurance You hold a portfolio and want insurance against

market declines. Answer: Buy Put options From put-call parity: Stocks + Puts = Calls + T-bills Stock+Put = {+1, +1} + {-1, 0} = {0, +1} = ‘Call payoff’ This is called Static Portfolio Insurance. Alternatively replicate ‘Stocks+Puts’ portfolio price movements

with ‘Stocks+T-bills’ or ‘Stocks+Futures’. [called Dynamic Portfolio Insurance] Why replicate? Because it’s cheaper!

Dynamic Portfolio Insurance Stock+Put (i.e. the position you wish to replicate) N0 = V0 /(S0 +P0) (hold 1 Put for 1 Stock) N0 is fixed throughout the hedge: At t > 0 ‘Stock+Put’ portfolio: Vs,p = N0 (S + P) Hence, change in value:

∂Vs, p ∂ P = N0 (1+ ∆ p ) = N0 1+ ∂S ∂S

This is what we wish to replicate

Dynamic Portfolio Insurance Replicate with (N0*) Stocks + (Nf) Futures: N0* = V0 / S0 (# of index units held in shares) N0* is also held fixed throughout the hedge. Note: position in futures costs nothing (ignore interest cost on margin funds.) At t > 0:

VS,F = N0* S + Nf (F zf)

Hence:

∂ VS , F = N 0* + z f N f ∂S

F = S ⋅ e r (T − t )

∂ F ∂S

∂ F F r (T − t ) =e ∂S

Equating dV of (Stock+Put) with dV(Stock+Futures) to get Nf :

Nft

= [N (1 + ∆ ) − N ] 0

p t

* 0

e − r (T − t ) zf

Dynamic Portfolio Insurance Replicate with ‘Stock+T-Bill’ VS,B = NS S + NB B ∂ VS , B ∂S

=

NB,t =

Ns

(V s , p ) t − ( N S ) t S t Bt

Equate dV of (Stock+Put) with dV(Stock+T-bill) ( N s ) t = N 0 (1 + ∆

p

)

t

=

N 0 (∆ c ) t

Dynamic Portfolio Insurance Example: Value of stock portfolio S&P500 index Maturity of Derivatives Risk free rate Compound\Discount Factor Standard deviation S&P

V0 = $560,000 S0 = 280 T - t = 0.10 r = 0.10 p.a. (10%) er (T – t) = 1.01 σ = 0.12

Put Premium Strike Price Put delta (Call delta)

P0 = 2.97 (index units) K = 280 ∆p = -0.38888 (∆c = 1 + ∆p = 0.6112)

Futures Price (t=0) Price of T-Bill

F0 = S0 er(T – t ) = 282.814 B = Me-rT = 99.0

Dynamic Portfolio Insurance Hedge Positions Number of units of the index held in stocks = V0 /S0 = 2,000 index units Stock-Put Insurance N0 = V0 / (S0 + P0) = 1979 index units Stock-Futures Insurance Nf = [(1979) (0.6112) - 2,000] (0.99/500) = - 1.56 (short futures) Stock+T-Bill Insurance No. stocks = N0 ∆c = 1979 (0.612) = 1,209.6 (index units) NB = 2,235.3 (T-bills)

Dynamic Portfolio Insurance 1) Stock+Put Portfolio Gain on Stocks = N0.dS = 1979 ( -1) Gain on Puts = N0 dP = 1979 ( 0.388) Net Gain

= -1,979 = 790.3 = -1,209.6

2) Stock + Futures: Dynamic Replicatin Gain on Stocks = Ns,o dS = 2000 (-1) = -2,000 Gain on Futures = Nf.dF.zf = (-1.56) (-1.01) 500 = +790.3 Net Gain = -1,209.6

Dynamic Portfolio Insurance 3) Stock + T-Bill: Dynamic Replication Gain on Stocks = Ns dS = 1209.6 (-1) Gain on T-Bills (No change in T-bill price) Net Gain

= -1,209.6 = 0

= -1,209.6

The loss on the replication portfolios is very close to that on the stock-put portfolio (over the infinitesimally small time period). Note:We are only “delta replicating” and hence, if there are large changes in S or changes in σ, then our calculations will be inaccurate When there are large market falls, liquidity may “dry up” and it may not be possible to trade quickly enough in ‘stocks+futures’ at quoted prices (or at any price ! e.g. 1987 crash).

Financial Risk Management Topic 3b Option’s Implied Volatility

Topics Option’s Implied Volatility VIX Volatility Smiles

Readings Books Hull(2009) chapter 18 VIX http://www.cboe.com/micro/vix/vixwhite.pdf

Journal Articles Bakshi, Cao and Chen (1997) “Empirical Performance of Alternative

Option Pricing Models”, Journal of Finance, 52, 2003-2049.

Options Implied Volatility

Estimating Volatility Itō’s Lemma: The Lognormal Property If the stock price S follows a GBM (like in the BS model),

then ln( ln(ST/S0) is normally distributed. σ2 ln S T − ln S 0 = ln( S T / S T ) ≈ φ µ − 2

2 T , σ T

The volatility is the standard deviation of the

continuously compounded rate of return in 1 year The standard deviation of the return in time ∆t is σ ∆t Estimating Volatility: Historical & Implied – How?

Estimating Volatility from Historical Data Take observations S0, S1, . . . , Sn at intervals of t years

(e.g. t = 1/12 for monthly) Calculate the continuously compounded return in each

interval as:

u i = ln( S i / S i −1 )

Calculate the standard deviation, s , of the ui´s

s=

1 n 2 ( u − u ) ∑ i n − 1 i =1

The variable s is therefore an estimate for So:

σˆ = s / τ

σ ∆t

Estimating Volatility from Historical Data For volatility estimation (usually) we assume that there are 252 trading days within one year mean

-0.13%

stdev (s)

3.5%

τ

1/252

σ(yearly)

s / sqrt(ττ) = 55.56%

Back or forward looking volatility measure? 7

Date 03/11/2008 04/11/2008 05/11/2008 06/11/2008 07/11/2008 10/11/2008 11/11/2008 12/11/2008 13/11/2008 14/11/2008 17/11/2008 18/11/2008 19/11/2008 20/11/2008 21/11/2008 24/11/2008 25/11/2008 26/11/2008 27/11/2008 28/11/2008 01/12/2008 02/12/2008 03/12/2008 04/12/2008 05/12/2008 08/12/2008

Close 4443.3 4639.5 4530.7 4272.4 4365 4403.9 4246.7 4182 4169.2 4233 4132.2 4208.5 4005.7 3875 3781 4153 4171.3 4152.7 4226.1 4288 4065.5 4122.9 4170 4163.6 4049.4 4300.1

Price Relative St/St-1

Daily Return ln(St/St-1)

1.0442 0.9765 0.9430 1.0217 1.0089 0.9643 0.9848 0.9969 1.0153 0.9762 1.0185 0.9518 0.9674 0.9757 1.0984 1.0044 0.9955 1.0177 1.0146 0.9481 1.0141 1.0114 0.9985 0.9726 1.0619

0.0432 -0.0237 -0.0587 0.0214 0.0089 -0.0363 -0.0154 -0.0031 0.0152 -0.0241 0.0183 -0.0494 -0.0332 -0.0246 0.0938 0.0044 -0.0045 0.0175 0.0145 -0.0533 0.0140 0.0114 -0.0015 -0.0278 0.0601

Implied Volatility BS Parameters Observed Parameters: S: underlying index value

Unobserved Parameters: Black and Scholes

X: options strike price

σ: volatility

T: time to maturity r: risk-free rate q: dividend yield

•

Traders and brokers often quote implied volatilities rather than dollar prices

How to estimate it?

Implied Volatility The implied volatility of an option is the volatility

for which the Black-Scholes price equals (=) the market price There is a one-to-one correspondence between prices and implied volatilities (BS price is monotonically increasing in volatility) Implied volatilities are forward looking and price traded options with more accuracy Example: If IV of put option is 22%, this means that pbs = pmkt when a volatility of 22% is used in the Black-Scholes model. 9

Implied Volatility Assume c is the call price, f is an option pricing

model/function that depends on volatility σ and other inputs: c = f (S , K , r , T , σ )

Then implied volatility can be extracted by inverting the

formula:

σ = f −1 (S , K , r , T , c mrk )

where cmrk is the market price for a call option. The BS does not have a closed-form solution for its inverse function, so to extract the implied volatility we use rootfinding techniques (iterative algorithms) like NewtonNewtonRaphson method 10

f (S , K , r , T , σ ) − c mrk = 0

Volatility Index - VIX In 1993, CBOE published the first implied

11

volatility index and several more indices later on. VIX: VIX 1-month IV from 30-day options on S&P VXN: VXN 3-month IV from 90-day options on S&P VXD: VXD volatility index of CBOE DJIA VXN: VXN volatility index of NASDAQ100 MVX: MVX Montreal exchange vol index based on iShares of the CDN S&P/TSX 60 Fund VDAX: VDAX German Futures and options exchange vol index based on DAX30 index options Others: VXI, VX6, VSMI, VAEX, VBEL, VCAC

Volatility Smile

Volatility Smile What is a Volatility Smile? It is the relationship between implied

volatility and strike price for options with a certain maturity The volatility smile for European call options should be exactly the same as that for European put options

13

Volatility Smile Put-call parity p +S0e-qT = c +Ke–r T holds for market

prices (pmkt and cmkt) and for Black-Scholes prices (pbs and cbs) It follows that the pricing errors for puts and calls

are the same: pmkt−pbs=cmkt−cbs When pbs=pmkt, it must be true that cbs=cmkt It follows that the implied volatility calculated from a

European call option should be the same as that calculated from a European put option when both have the same strike price and maturity 14

Volatility Term Structure In addition to calculating a volatility

smile, traders also calculate a volatility term structure This shows the variation of implied volatility with the time to maturity of the option for a particular strike

15

IV Surface

16

IV Surface

17

IV Surface

Also known as: Volatility smirk 18 Volatility skew

Volatility Smile Implied Volatility Surface (Smile) from Empirical Studies (Equity/Index)

19

Bakshi, Cao and Chen (1997) “Empirical Performance of Alternative Option Pricing Models ”, Journal of Finance, 52, 2003-2049.

Volatility Smile Implied vs Lognormal Distribution

20

Volatility Smile In practice, the left tail is heavier and the right tail is less

heavy than the lognormal distribution What are the possible causes of the Volatility Smile anomaly? Enormous number of empirical and theoretical papers

to answer this …

21

Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes

(e.g. S&P 500 index)

22

Date

Open

High

Low

Close

Volume

Adj Close

Return

04/01/2000

1455.22

1455.22

1397.43

1399.42

1.01E+09

1399.42

-3.91%

18/02/2000

1388.26

1388.59

1345.32

1346.09

1.04E+09

1346.09

-3.08%

20/12/2000

1305.6

1305.6

1261.16

1264.74

1.42E+09

1264.74

-3.18%

12/03/2001

1233.42

1233.42

1176.78

1180.16

1.23E+09

1180.16

-4.41%

03/04/2001

1145.87

1145.87

1100.19

1106.46

1.39E+09

1106.46

-3.50%

10/09/2001

1085.78

1096.94

1073.15

1092.54

1.28E+09

1092.54

0.62%

17/09/2001

1092.54

1092.54

1037.46

1038.77

2.33E+09

1038.77

-5.05%

16/03/00

1392.15

1458.47

1392.15

1458.47

1.48E+09

1458.47

4.65%

15/10/02

841.44

881.27

841.44

881.27

1.96E+09

881.27

4.62%

05/04/01

1103.25

1151.47

1103.25

1151.44

1.37E+09

1151.44

4.28%

14/08/02

884.21

920.21

876.2

919.62

1.53E+09

919.62

3.93%

01/10/02

815.28

847.93

812.82

847.91

1.78E+09

847.91

3.92%

11/10/02

803.92

843.27

803.92

835.32

1.85E+09

835.32

3.83%

24/09/01

965.8

1008.44

965.8

1003.45

1.75E+09

1003.45

3.82%

-ve Price jumps

Trading was suspended

+ve price jumps

Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes Volatility for asset price is stochastic In the case of equities volatility is negatively related to stock

prices because of the impact of leverage. This is consistent with the skew (i.e., volatility smile) that is observed in practice

23

Volatility Smile

24

Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes Volatility for asset price is stochastic In the case of equities volatility is negatively related to stock

prices because of the impact of leverage. This is consistent with the skew that is observed in practice Combinations of jumps and stochastic volatility

25

Volatility Smile Alternatives to Geometric Brownian Motion Accounting for negative skewness and excess kurtosis by

generalizing the GBM Constant Elasticity of Variance Mixed Jump diffusion Stochastic Volatility Stochastic Volatility and Jump

Other models (less complex → ad-hoc) The Deterministic Volatility Functions (i.e., practitioners Black

and Scholes) (See chapter 26 (sections 26.1, 26.2, 26.3) of Hull for these alternative specifications to Black-Scholes) 26

Topic # 4: Modelling stock prices, Interest rate derivatives Financial Risk Management 2010-11 February 7, 2011

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

1

2

Modelling stock prices Modelling the evolution of stock prices is about introducing a process that will explain the random movements in prices. This randomness is explained in the E¢ cient Market Hypothesis (EMH) that can be summarized in two assumptions: 1. Past history is re‡ected in present price 2. Markets respond immediately to any new information about the asset So we need to model arrival of new information that a¤ects price (or much more returns). If asset price is S. Suppose price changes to S + dS in a small time interval (say dt). into deThen we can decompose returns dS S terministic/anticipated part and a random part where prices changed due to some external unanticipated news. dS = dt + dW S

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

3

The randomness in the random part is explained by a Brownian Motion process and scaled by the volatility of returns.

We can introduce time subscripts and rearrange to get dSt = St dt + St dWt This process is called the Geometric Brownian Motion.

Why have we used Brownian Motion process to explain randomness? – In practice, we see that stock prices behave, atleast for long stretches of time, like random walks with small and frequent jumps – In statistics, random walk, being the simplest form, have limiting distributions and since BM is a limit of the random walk, we can easily understand the statistics of BM (use of CLT) FRM

c Dennis PHILIP 2011

1 Modelling stock prices

4

Next we see, what is this W (and in turn what is dW)? Brownian motion is a continuous time (rescaled) random walk.

Consider the iid sequence "1 ; "2 ; ::: with mean and variance 2 : Consider the rescaled random walk model 1 X Wn (t) = p "j n 1 j nt

The interval length t is divided into nt equal subintervals of length 1=n and the displacements / jumps "j ; j = 1; 2; :::; nt in nt steps are mutually independent random variables.

Then for large n; according to Central Limit Theorem: W (t) N ( t; 2 t) : FRM

c Dennis PHILIP 2011

1 Modelling stock prices

5

Special cases: Standard Brownian Motion arises when we have = 0; and = 1.

W is a Standard Brownian Motion if 1. W (0) = 0 2. W has stationary (for 0 s t; Wt Ws and Wt s have the same distribD ution. That is, Wt Ws = Wt s N (0; t s)) 3. W has independent increments (for s t; Wt Ws is independent of past history of W until time s) 4. Wt

FRM

N (0; t)

c Dennis PHILIP 2011

1 Modelling stock prices

6

For a Brownian Motion only the present value of the variable is relevant for predicting the future [also called Markov property]. Therefore BM is a markov process. It does not matter how much you zoom in, it just looks the same. That is, the randomness does not smooth out when we zoom in. BM …ts the characteristics of the share price. Imagine a heavy particle (share price) that is jarred around by lighter particles (trades). Trades a¤ect the price movement. what is this dW? Consider a small increment in W W (t + where "(t +

FRM

t) = W (t) + "(t + t)

t)

iidN (0; t) [Std BM].

c Dennis PHILIP 2011

1 Modelling stock prices

Taking limit as is dWt = =

7

! 0; the change in W (t) lim W (t + dt)

dt!0

W (t)

lim "(t + dt)

dt!0

iidN (0; dt) So in the di¤erential form, we can write the Standard Brownian motion process as p dWt = et dt where et N (0; 1)

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

8

Stochastic processes used in Finance Arithmetic Brownian Motion for a share price A stock price does not generally have a mean zero and atleast would grow on average with the rate of in‡ation. Therefore we can write dSt = (St ; t) dt + (St ; t)dWt = drif t term + dif f usion term = E(dS) + Stddev(dS) When the drift function (St ; t) = and di¤usion function (St ; t) = ; both constants, we have the Arithmetic BM. dSt = =

dt + dWt p dt + et dt

In the case of ABM, S may be positive or negative. Since prices cannot be negative, we generally use the Geometric BM for asset prices and made the drift and volatility as functions of the stock price. FRM

c Dennis PHILIP 2011

1 Modelling stock prices

9

Geometric Brownian Motion dSt = St dt + St dWt If S starts at a positive value, then it will remain positive. The solution of the SDE St is an exponential function which is always positive. Also, note that S will be lognormally distributed. GBM is related to ABM according to dSt = dt + dWt St where is the instantaneous share price volatility, and is the expected rate of return

The Hull and White (1987) Model uses GBM.

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

10

Ornstein-Uhlenbeck (OU) Process The Arithmetic Ornstein-Uhlenbeck process is given by dSt =

(

St ) dt + dWt

where is the long run mean and ( > 0) is the rate of mean reversion. The drift term is the mean reversion component, in that the di¤erence between the long run mean and the current price decides the upward or downward movement of the stock price towards the long run mean : Over time, the price process drifts towards its mean and the speed of mean reversion is determined by :

This is an important process to model interest rates that show mean reversion where prices are pulled back to some long-run average level over time. The Vasicek Model uses this kind of process. FRM

c Dennis PHILIP 2011

1 Modelling stock prices

11

A special case is when the mean is zero. Then we can write the OU process as dSt =

St dt + dWt

In the OU process, the stock price can be negative. Therefore we can introduce the Geometric OU process

The Geometric OU process is given by dSt =

(

St ) St dt + St dWt

where the asset prices St would always be positive.

So we can model asset prices using the Geometric OU process and their log returns will then follow an Arithmetic OU process. dSt = dSt = St FRM

(

St ) St dt + St dWt

(

St ) dt + dWt c Dennis PHILIP 2011

1 Modelling stock prices

12

Square Root Process A square root process satis…es the SDE p dSt = St dt + St dWt This type of process generates positive prices and used for asset prices whose volatility does not increase too much when St increases. Cox-Ingersoll-Ross (CIR) process The CIR combines mean reversion and square root process and satis…es the SDE p St dWt dSt = ( St ) dt + This process was introduced in the Hull and White (1988), and Heston (1993) stochastic volatility models. This class of models generated strictly non-negative volatility and accounted for the clustering e¤ect and mean reversion observed in volatility. FRM

c Dennis PHILIP 2011

1 Modelling stock prices

13

Also used to model short rates features positive interest rates, mean reversion, and absolute variance of interest rates increases with interest rates itself.

Solving the Stochastic Di¤erential Equations Consider the GBM dSt = St dt + St dWt In the integral form ZT 0

dSt =

ZT

St dt +

0

ST = S0 +

ZT

St dWt

0

ZT 0

St dt +

ZT

St dWt

0

= reimann integ + It^ o integ So we have to solve the intergrals to get a closed form solutions to this SDE. FRM

c Dennis PHILIP 2011

1 Modelling stock prices

14

We use Ito-lemma to solve this problem. Not all SDE’s have closed form solutions. When there are no solutions, we have to do numerical approximations for these integrals. Examples: Geometric Brownian Motion dSt = St dt + St dWt has the solution 1 2

St = S0 e(

2

)t+

Wt

Ornstein-Uhlenbeck (OU) Process dSt =

(

St ) dt + dWt

has the solution S t = S0 e

t

+

1

e

t

+

Zt

e

(t s)

dWs

0

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

15

Consider the following process dSt = St dt + dWt has the solution St = S 0 e t +

Zt

e

(t s)

dWs

0

Simulating Geometric Brownian Motion 1 2 We can write St = S0 e( 2 )t+ Wt in discrete time intervals and substituting for Wt as p 1 2 St = St 1 e( 2 ) t+ et t

where et

N (0; 1)

So we randomly draw et and …nd the value of St

FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

2

16

Interest Rate Derivatives The payo¤ of interest rate derivatives would depend on the future level of interest rates. The main challenge in valuing these derivatives are that interest rates are used both for discounting and for de…ning payo¤s. For valuation, we will need a model to describe the behavior of the entire yield curve. Black’s Model to price European Options Consider a call option on a variable whose value is V: To calculate expected payo¤, the model asssumes: 1. VT has lognormal distribution with V ar(lnVT ) =

2

T

2. E(VT ) = F0 FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

The payo¤ is max(VT

17

K; 0) at time T .

We discount the expected payo¤ at time T using the risk-free rate given by P (0; T )

We will use the key result that you know from Derivatives: If V is lognormally distributed and standard deviation on ln(V ) is s, then

E[max ( V K; 0)] = E[ V ] N (d1 ) KN (d2 ) where d1 =

ln(E[ V ]=K) + s2 =2 s

d2 =

ln(E[ V ]=K) s

s2 =2

Therefore, value of the call option is given by c = P (0; T ) [F0 N (d1 ) FRM

KN (d2 )]

c Dennis PHILIP 2011

2 Interest Rate Derivatives

18

where d1 =

ln(F0 =K) + p T

2

T =2

and d2 =

ln(F0 =K) p

2

T =2

T

= d1

p

T

where – F is forward price of V for a contract with maturity T – F0 is value of F at time zero – K is strike of the option –

is volatility of forward contract

Similarly, for a put option p = P (0; T ) [KN ( d2 )

FRM

F0 N ( d1 )]

c Dennis PHILIP 2011

2 Interest Rate Derivatives

19

European Bond Options Bond option is an option to buy or sell a particular bond by a certain date for a particular price. Callable bonds and Puttable bonds are examples of embedded bond options. The payo¤ is given by max(BT a call option.

K; 0) for

To price an European Bond Option: – we assume bond price at maturity of option is lognormal – we de…ne such that p standard deviaT tion of ln(BT ) = – F0 can be calculated as F0 =

B0 I P (0; T )

where B0 is bond (dirty) price at time zero and I is the present value of coupons that will be paid during the life of option FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

20

– Then using Black’s model we price of a bond option

Interest Rate Caps and Floors An interest rate Cap provides insurance against the rate of interest on a ‡oating-rate note rising above a certain level (called Cap rate). Example: Principal amount = $10 million Tenor = 3 months (payments made every quarter) Life of Cap = 5 years Cap rate = 8% If the ‡oating-rate exceeds 8 %, then you get cash of the di¤erence. Suppose at a reset date, 3-month LIBOR is 9%, the ‡oating-rate note would have to pay 0:25 FRM

0:09

$10million = $225; 000 c Dennis PHILIP 2011

2 Interest Rate Derivatives

21

and with the Cap rate at 8%, the payment would be 0:25

0:08

$10million = $200; 000

Therefore the Cap provides a payo¤ of $25,000 to the holder. Consider a Cap with total life of Tn ; a Principal of L, Cap rate of RK based on a reference rate (say, on LIBOR) with a month maturity denoted by R(t) at date t. The contract follows the schedule: t T0 T1 T2 C1 C2 T0 is the starting date. For all j = 1; :::; n, we assume a constant tenor Tj Tj 1 = On each date Tj ; the Cap holder receives a cash ‡ow of Cj Cj = L

max [R(Tj 1 )

RK ; 0]

The Cap is a portfolio of n such options and each call option is known as the caplets. FRM

c Dennis PHILIP 2011

Tn Cn

2 Interest Rate Derivatives

22

Lets now consider a Floor with the same characteristics. On each date Tj ; the Floor holder receives a cash ‡ow of Fj Fj = L

max [RK

R(Tj 1 ); 0]

The Floor is a portfolio of n such options and each put option is known as the ‡oorlets.

FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

23

Interest rate Caps can be regarded as a portfolio of European put options on zero-coupon bonds. Put-Call parity relation: Consider a Cap and Floor with same strike price RK . Consider a Swap to receive ‡oating and pay a …xed rate of RK , with no exchange payments on the …rst reset date. The Put-Call parity states: Cap price = Floor price + value of Swap

FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

24

Collar A Collar is designed to guarantee that the interest rate on the underlying ‡oating-rate note always lie between two levels. Collar = long position in Cap + short position in Floor It is usually constructed so that the price of Cap is equal to price of the ‡oor. Then the cost of entering into a Collar is zero. Valuation of Caps and Floors

If the rate R(Tj ) is assumed to be lognormal with volatility j , the value of the caplet today (t) for maturity Tj is given by Caplett = L

P (t; Tj ) FTj

1 ;Tj

N (d1 )

RK N (d2 )

where d1 = FRM

ln(FTj

1 ;Tj

=RK ) + 2j (Tj p (Tj 1 t)

1

t) =2

c Dennis PHILIP 2011

2 Interest Rate Derivatives

and d2 = d1

j

q

(Tj

25

1

t)

where FTj 1 ;Tj is the forward rate underlying the Caplet from Tj 1 to Tj : Similarly, F loorlett = L

FRM

P (t; Tj ) RK N ( d2 )

FTj

1 ;Tj

N ( d1 )

c Dennis PHILIP 2011

Financial Risk Management Lecture 5 Value at Risk Readings: CN(2001) chapters 22,23; Hull_RM chp 8

1

Topics Value at Risk (VaR) Forecasting volatility Back-testing Risk Grades VaR: Mapping cash flows

2

Value at Risk Example:

If at 4.15pm the reported daily VaR is $10m (calculated at 5% tolerance level) then: I expect to lose more than $10m only 1 day in every 20 days (ie. ie. 5% of the time) The VaR of $10m assumes my portfolio of assets is fixed Exactly how much will I lose on any one day? Unknown !!!

3

Value at Risk Statement (how bad can things get?):

“We are x% certain that we will not loose more than V dollars in the next N days” V dollars = f(x%, N days)

Suppose asset returns is niid, then risk can be measured by

variance/S.D. From Normal Distribution critical values table, we can work out the VaR. Example: For 90% certainty, we can expect actual return to be

between the range { µ − 1 .6 5 σ , µ + 1 .6 5 σ }

4

Value at Risk Normal Distribution (N(0,σ)) Probability

Mean = 0

5% of the area 5% of the area

-1.65σ

0.0

+1.65σ

Only 5% of the time will the actual % return R be below: “ R = µ - 1.65 σ1” where µ = Mean (Daily) Return. 5

If we assume µ=0, VaR = $V (1.65 σ1)

Return

VaR for single asset Example: Mean return = 0 %. Let σ1 = 0.02 (per day) Only 5% of the time will the loss be more than 3.3% (=1.65 x 2%) VaR of a single asset (Initial Position V0 =$200m in equities)

VaR = V0 (1.65 σ1 ) = 200 ( 0.033) = $6.6m

That is “(dollar) VaR is 3.3% of $200m” = $6.6m VaR is reported as a positive number (even though it’s a loss)

6

Are Daily Returns Normally Distributed? - NO • Fat tails (excess kurtosis), peak is higher and narrower, negative skewness, small (positive) autocorrelations, squared returns have strong autocorrelation, ARCH. • But niid is a (reasonable) approx for portfolios of equities, long term bonds, spot FX , and futures (but not for short term interest rates or options)

VaR for portfolio of assets

7

VaR for portfolio of assets

8

VaR for portfolio of assets

9

VaR for portfolio of assets

10

VaR for portfolio of assets Summary: Variance – Covariance method

If Vp is the market value of your portfolio of n assets and wi is the proportionate weight in each asset i then

VaR p = V p [ zCz ']

1/2

where

z = w1 (1.65σ 1 ) , w2 (1.65σ 2 ) ,… , wn (1.65σ n ) 1 ρ C = 21 ⋮ ρ n1

11

ρ12 … ρ1n ⋱

1

Forecasting

12

Forecasting σ Simple Moving Average ( Assume Mean Return = 0 )

σ2 t+1|t = (1/n) Σi R2t-i Exponentially Weighted Moving Average EWMA

σ2 t+1|t = Σi wi R2t-i

wi = (1-λ) λi

It can be shown that this may be re-written:

σ2t+1|t = λ σ2t| t-1 + (1- λ) Rt2 Longer Horizons:

13

T -rule σΤ = T

- for iid returns. σ

Forecasting σ Exponentially Weighted Moving Average (EWMA)

σ2t+1|t = λ σ2t| t-1 + (1- λ) Rt2 How to estimate λ? 1. Use GARCH models to estimate λ 2. Minimize forecast error Σ (Rt+12 – σ2 t+1|t) where the sum is

over all assets, and say 100 days 3. λ = 0.94 as by JPMorgan Suppose λ = 0.94 then weights decline as 0.94, 0.88, 0.83,….

and past observations are given less weight than current forecast of variance. 14

Back-testing In back-testing, we compare our (changing) daily

forecast of VaR with actual profit or loss over some historic period. Example: For a portfolio of assets, • forecast all the individual VaRi = Vi1.65 σt+1|t ,

• calculate portfolio VaR for each day:

VaRp =

[Z C Z’]1/2

• then see if actual portfolio losses exceed this only 5% of the time (over some historic period, e.g. 100 days). 15

Back-testing Daily $m profit/loss

= forecast = actual

Days

16

Only 6 violations out of 100 = just ‘OK’

VaR and Capital Adequacy-Basle Basle uses a more ‘conservative’ measure of VaR than J. P. Morgan Calc VaR for worst 1% of losses over 10 days Use at least 1-year of daily data to estimate σt+1|t

VaRi = 2.33 10 σ

( 2.33 = 1% left tail critical value, σ = daily vol )

Internal Models approach

Capital Charge KC KC = Max ( Avg. of previous 60-days VaR x M, previous day’s VaR) M = multiplier (min = 3)

17

Pre-commitment approach

• KC set equal to max. forecast loss over 20 day horizon = preannounced $VaR • If losses exceed VaR, more than 1 day in 20, then impose a penalty.

VaR and Coherent Risk Measures Risk measures that satisfy all the following 4 conditions are called as a

Coherent Risk Measure. Monotonicity: X 1 ≤ X 2 ⇒ R ( X 1 ) ≤ R ( X 2 ) (higher the riskiness of the portfolio, higher should be risk capital) Translation invariance:

R ( X + k ) = R ( X ) − k ∀k ∈ ℝ

(if cash k is added to portfolio, risk should go down by k) Homogeneity:

R ( λ X ) = λ R ( X ) ∀λ ≥ 0

(if you change portfolio by a factor of λ, risk is proportionally increased) Subadditivity: 18

R ( X + Y ) ≤ R ( X ) + R (Y )

(diversification leads to less risk)

VaR and Coherent Risk Measures VaR violates the subadditivity condition and therefore not

coherent. VaR cannot capture the benefits of diversification. VaR can actually show negative diversification benefit! VaR only captures the frequency of default but not the size

of default. Even if the largest loss is doubled, the VaR figure could remain the same. Other measures such as Expected Shortfall are coherent

measures. 19

Risk Grades

20

Risk Grades RG helps to calculate changing forecasts of risks (volatilities) RG quantifies volatility/risk (similar to variance, std. deviation,

beta, etc) RG can range from 0 to over 1000, where 100 corresponds to

the average risk of a diversified market-cap weighted index of global equities. So if two portfolio’s have RG1 = 100 and RG2 = 400, portfolio 2 is four times riskier than portfolio 1 RG scales all assets to a common scale and so it is able to compare risk across all asset classes.

21

Risk Grades RG of a single asset

σi 252 σi 252 RG = σ ×100 = ×100 i 0.20 base

σi is the DAILY standard deviation σbase is fixed at 20% per annum (= 5 yr. av. for international portfolio of stocks) Formula looks complex but RG is just a “scaled” daily standard deviation e.g. If RG = 100% then asset has 20% p.a. risk RG of a portfolio of assets 22

RG 2 = ∑ w i2 RG i2 + ∑ ∑ w i w j ρ RG i RG j P

Risk Grades Risk Grades in 2009

www.riskgrades.com 23

Risk Grades Risk Grades in 2009 of indices heating up and cooling off

24

VaR Mapping (VaR for different assets)

25

VaR for different assets PROBLEMS STOCKS : Too many covariances [= n(n-1)/2 ] FOREIGN ASSETS : Need VaR

in “home currency”

BONDS: Many different coupons paid at different times DERIVATIVES: Options payoffs can be highly nonlinear (ie. NOT normally distributed)

26

SOLUTIONS = “Mapping” (RiskMetricsTM produce volatility & correlations for various assets across 35 countries and useful for “Mapping”)

VaR for different assets STOCKS

Within each country use “single index model” SIM FOREIGN ASSETS

Treat one asset in foreign country as = “local currency risk”+ spot FX risk (like 2-assets, with equal weight) BONDS

Consider each bond as a series of “zeros” OTHER ASSETS

Forward-FX, FRA’s, Swaps: decompose into ‘constituent parts’ 27

DERIVATIVES(non-linear)

Mapping Stocks Consider ‘p’ = portfolio of stocks held in one country with (Rm , σm) (for e.g. S&P500 in US) Problem : Too many covariances to estimate Soln. All n(n-1)/2 covariances “collapse or mapped” into σm

and the asset betas (“n” of them)

Single Index Model:

Ri = ai + bi Rm + εi Rk = ak + bk Rm + εk 28

assume Eεi εk = 0 and cov (Rm , ε ) = 0

All the systematic variation in Ri AND Rk is due to Rm

Mapping Stocks 1) In a portfolio idiosyncratic risk εi is diversified away = 0 2) Then each return-i depends only on the market return (and beta). Hence ALL variances and covariances also only depend on these 2 factors. It can be shown that

σp = bp σm

(i.e. Calculation of portfolio beta requires, only n-beta’s and σm ) 3) Also, ρ = 1 because (in a well diversified portfolio) each return moves only with Rm 4) We end up with or equivalently 29

VaRp = VP 1.65 ( bP σm ) VaRp = (Z C Z’ )1/2

where Z = [ VaR1, VaR2 …. }

C is the unit matrix

Mapping Foreign Assets (Mapping foreign stocks into domestic currencyVaR)

30

Mapping Foreign assets Example: US resident holds a diversified portfolio of German stocks equivalent to German stocks + Euro-USD, FX risk

31

Use SIM to obtain stdv of foreign (German) portfolio returns, σG

Then treat ‘foreign portfolio’ as (two) equally weighted assets: = $V in German asset + $V foreign currency position

Then use standard VaR formula for 2-assets

Mapping Foreign assets US based investor:

with

€100m in a German stock portfolio

σG = βP σDAX Sources of risk: a) Stdv of the German portfolio (‘local currency’ portfolio) b) Stdv of €/$ exchange rate ( σFX ) c) one covariance/correlation coefficient ρ (between DAX and FX rate) e.g. Suppose when German stock market falls then the € also falls ‘double whammy’ for the US investor, from this positive correlation, so foreign assets are very risky (in terms of their USD ‘payoff’) Let : S = 1.2 $/ € Dollar initial value Vo$ = 100m x 1.2 = $120m Linear 32

dVP = V0$ (RG + RFX)

above implies wi = Vi / V0$ = 1

Mapping Foreign assets

Dollar-VaRp σp =

(

2 σG

= Vo$ 1.65 σp

2 + σ FX

+ 2 ρσ Gσ FX

)

1

2

No ‘relative weights’ appear in the formula Matrix Representation: Dollar VaR

Let

Z = [ V0,$ 1.65 =[

VaR1

σG , V0,$ 1.65 σFX ] ,

VaR2 ]

V0,$ = $120m for both entries in the Z-vector (i.e. equal amounts)

Then 33

VaRp = (Z C Z’ )1/2

Mapping Bonds (Mapping coupon paying bonds)

34

Mapping Coupon paying Bonds Example: Coupons paid at t=5 and t=7 Treat each coupon as a separate zero coupon bond

100 100 P= + 5 (1 + y5 ) (1 + y7 )7

P = V5 + V7 P is linear in the ‘price’ of the zeros, V5 and V7 We require two variances of “prices” V5 and V7 and covariance between these prices.

35

Note: σ5(dV5 / V5) = D σ(dy5) but Risk Metrics provides the price volatilities, σ5(dV5 / V5)

Mapping Coupon paying Bonds Treat each coupon as a zero Calculate: price of zero, e.g. V5 = 100 / (1+y5)5 VaR5 = V5 (1.65 σ5)

VaR7 = V7 (1.65 σ7) VaR (both coupon payments): VaRp = (Z C Z’ )1/2 =

[ VaR + VaR + 2 ρ VaR 5VaR 7 ] 2 5

2 7

r = correlation: bond prices at t=5 and t=7 (approx 0.95 - 0.99 ) 36

1/ 2

Mapping FRA

37

Mapping FRA To calculate VaR for a FRA, we break down cash flows into

equivalent synthetic FRA and use spot rates only (since we do not know the forward volatilities) Example: Consider an FRA on a notional of $1m that involves

lending $1m in 6 months time for a future of 6 months. Receipt of $1m + Interest

0

6m

12m

Lend $1m

Let y6 = 6.39%, y12 = 6.96% and there are 182 days in the first

leg and 183 days in the second leg (day count: actual/365). The implied f6,12 = 7.294% and therefore the 12 month investment will give $1,036,572 return (with round off error). 38

Mapping FRA The original FRA

0

Receipt of $1,036,572

6m

12m

Lend $1m

Synthetic FRA

Receipt of $1,036,572 from 12 month lending

Borrow at 6 month rate

0 Lend at 12 month rate

6m

12m

Repay 6 month loan of $1m

So at time 0, we borrow $969,121 [=1m / 1+(y6*182/365)] and lend

39

this money at a 12 month rate leading to $1,036,572 [=$969,121*(1+y12)]

Mapping FRA Suppose the standard deviation of the prices for 6 month asset is

0.1302% and for 12 month asset is 0.2916%. Suppose ρ = 0.7 To calulate the VaR for each of these positions: VaR6 = $969,121 (1.65) (0.1302%) = $2082 VaR12 = $969,121 (1.65) (0.2916%) = $4663

VaR = [ VaR + VaR + 2 ρ ( −VaR 6 ) VaR12 ] 2 6

= $3534

40

2 12

1/ 2

Mapping FX Forwards

41

Mapping FX Forwards Consider a US resident who holds a long forward contract to

purchase €10million in 1 year. What is the VaR for this contract? We map Forward into two spot rates and one spot FX rate. Then we calculate VaR from the VaR of each individual mapped

asset. Mapping a forward contract

42

Mapping FX Forwards

43

Mapping FX Forwards

44

Mapping Options

45

Mapping Options

46

Mapping Options

47

Mapping Options

48

Mapping Options

49

Financial Risk Management Topic 6 Statistical issues in VaR Readings: CN(2001) chapters 24, Hull_RM chp 8, Barone-Adesi et al (2000) RiskMetrics Technical Document (optional)

1

Topics Value at Risk for options Monte Carlo Simulation Historical Simulation Bootstrapping Principal component analysis Other related VaR measures Marginal VaR, Incremental VaR, ES 2

MCS - VaR of Call option Option premia are non-linear (convex) function of

underlying Distribution of gains/losses is not normally distributed Therefore dangerous to use Var-Cov method Assets Held: One call option on stock

Here, Black-Scholes is used to price the option during the Monte Carlo Simulation (MCS). Problem

Find the VaR over a 5-day horizon 3

MCS - VaR of Call option If V is price of the option (call or put) and P is price of

underlying asset in the option contract (stock)

V will change from minute to minute as P changes. For an

4

equal change in P of +1 or -1 ,the change in call premia are NOT equal: this gives “non-normality” in distribution of the change in call premium

MCS - VaR of Call option To find VaR over a 5-day horizon:

1) Given P0 calculate the option price, V0 = BS(P0, K, T0 …..) This is fixed throughout the MCS 2) MCS = Simulate the stock price and calculate P5 3) Calculate the new option price, V5= BS(P5, K, T0 – 5/365,

…..)

4) Calculate change in option premium ∆V(1) = V5 – V0 5) Repeat steps 2-4, 1000 times and plot a histogram of the change in the call premium. We can then find the 5% lower cut-off point for the change in value of the call (i.e. it’s VaR). 5

MCS - VaR of Call option 20 18

5% of area

16

Frequency

14 12 10 8 6 4 2 0 -10

-8

-6

-4

-2

0

2

4

6

8

10

More

$-VaR is $5m Change in Call Premium 6

MCS - VaR of Call option Simulate stock prices using

P(t+1) = P(t) exp[ (µ - σ2/2) ∆t + σ (∆t)1/2 εt+1 ] To generate 5-day prices from daily prices, you can use the

root-T rule

∆t = 5/365 (or 5/252) ( ie. five day) Alternatively, we can generate Tt+5 directly.

If P0 is today’s known price. Use a ‘do-loop’ over 5 ‘periods’ to get P5 (using 5 - ‘draws’ of εt+1 )

7

MCS - VaR of Call option Stock price paths generated

8

Monte Carlo Simulation for Two Assets (two call options on different underlying assets)

9

Simulating correlated random variables

10

Simulating correlated random variables

11

Simulating correlated random variables

12

MCS and VaR: two asset example

13

MCS and VaR: two asset example

14

MCS and VaR: two asset example

15

MCS and VaR: two asset example

16

MCS and VaR: two asset example

17

MCS and VaR: two asset example

18

MCS and VaR: two asset example

19

Comparing VaR forecasts

20

Comparing VaR forecasts

21

Historical Simulation (Historical simulation + bootstrapping)

22

Historical Simulation (HS) Suppose you currently hold $100 in each of 2 assets Day =

1

2

3

4

5

6

…1000

R1(%)

+2

+1

+4

-3

-2

-1

+2

R2(%)

+1

+2

0

-1

-5

-6

-5

_____________________________________________________ ∆Vp($)

+3

+3

+4

-4

-7

-7

-5

Order ∆Vp in ascending order (of 1000 numbers) e.g. -12, -11, -11 -10, -9, -9, -8, -7, -7, -6 | -5, -4, -4, …. +8, ……. +14 VaR forecast for tomorrow at 1% tail (10th most negative) = -$6 23

Above is equivalent to the histogram

Historical Simulation (HS) This is a non parametric method since we do not estimate any

variances or covariances or assume normality. We merely use the historic returns, so our VaR estimates encapsulate

whatever distribution the returns might embody ( e.g. Student’s t) as well as any autocorrelation in individual returns. Also, the historic data “contain” the correlations between the returns

on the different assets, their ‘own volatility’ and their own autocorrelation over time It does rely on ‘tomorrow’ being like ‘the past’. 24

HS + Bootstrapping Problems:

Is data >3 years ago useful for forecasting tomorrow? Use most recent data - say last 100 days ? 1% tail: Has only one number in this tail, for the actual data !

Extreme case ! Actual data might have largest negative (for 100 days ago) of minus 50% - this would be your forecast VaR for tomorrow using historic simulation approach. Is this okay or not? 25

HS + Bootstrapping You have “historic” daily data on each of 10 stock returns (i.e.

your portfolio ) But only use last 100 days of historic daily returns, So we have

a data matrix of 10 x 100. We require VaR at the 1st percentile (1% cut off) We sample “with replacement” from these 100 observations,

26

giving equal probability to each ‘day’ , when we sample. This allows any one day’s returns to be randomly chosen more than once (or not at all). It is as if we are randomly ‘replaying’ the last 100 days of history, giving each day equal probability

HS + Bootstrapping The Bootstrap Draw randomly from a uniform distribution with an equal probability of drawing any number between 1 and 100. If you draw “20” then take the 10-returns in column 20 and revalue the

portfolio. Call this $-value, ∆VP(1)

Repeat above for 10,000 “trials/runs” (with replacement), obtaining

10,000 possible (alternative) values ∆VP (i)

“With replacement” means that in the 10,000 runs you will “choose” some of

the 100 columns more than once. Plot a histogram of the 10,000 values of ∆VP(i) - some of which will be

negative

Read off the “1% cut off” value (=100th most negative value). This is 27

VaRp

Filtered Historical Simulation (FHS)

28

FHS HS assumes risk factors are i.i.d however this is usually not the case. HS assumes that distribution of returns are stable and that the past and

present moments of the density function of returns are constant and equal. The probability of having a large loss is not equal across different days.

There are periods of high volatility and periods of low volatility (volatility clustering). In FHS, historical returns are first standardized by volatility estimated on

that particular day (hence the word Filtered). The filtering process yields approximately i.i.d returns (residuals) that are

suited for historical simulation. 29

read Barone-Adesi et al (2000) paper in DUO on FHS

Principal Component Analysis (Estimating risk factors using PCA)

30

Estimating risk factors using PCA

31

Estimating risk factors using PCA

32

Estimating risk factors using PCA

33

Estimating risk factors using PCA

34

Estimating risk factors using PCA

35

Estimating risk factors using PCA

36

Estimating risk factors using PCA

37

PCA and risk management

38

PCA and risk management

39

Other related VaR measures

40

Other related VaR measures

41

Other related VaR measures

42

Other related VaR measures

43

Other related VaR measures

44

Other related VaR measures

45

Other related VaR measures

46

Other related VaR measures

47

Topic # 7: Univariate and Multivariate Volatility Estimation Financial Risk Management 2010-11 February 28, 2011

FRM

c Dennis PHILIP 2011

1 Volatility modelling

1

2

Volatility modelling Volatility refers to the spread of all likely outcomes of an uncertain variable. It can be measured by sample standard deviation v u T u 1 X (rt )2 ^=t T 1 t=1

where rt is the return on day t, and is the average return over the T day period. But this statistic only measures the spread of a distribution and not the shape of a distribution (except normal and lognormal). Black Scholes model assumes that asset prices are lognormal (which implies that returns are normally distributed). In practice, returns are however non-normal and also the return ‡uctuations are time varying.

FRM

c Dennis PHILIP 2011

1 Volatility modelling

3

Example: daily returns of S&P 100 show features of volatility clustering

Therefore Engle (1982) proposed Autoregressive Conditional Heteroscedasticity (ARCH) models for modelling volatility Other characteristics documented in literature FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

4

– Long memory e¤ect of volatility (autocorrelations remain positive for very long lags) – Squared returns proxy volatility – Volatility asymmetry / leverage e¤ect (volatility increases if the previous day returns are negative)

1.1

Parametric volatility models

ARCH model ‘Autoregressive’because high/low volatility tends to persist, ‘Conditional’ means timevarying or with respect to a point in time, and ‘Heteroscedasticity’is a technical jargon for non-constant volatility. Consider previous t day’s squared returns ("2t 1 ; "2t 2 ; :::) that proxy volatility. It makes sence to give more weight to recent data and less weight to far away observations. FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

5

Suppose we are assuming that previous q observations a¤ect today’s returns. So today’s volatility can be 2 t

=

q X

2 j "t j

j=1

where

i

j and

Pq

j=1

j

=1

Also we can include a long run average variance that should be given some weight as well q X 2 2 j "t j t = V0 + j=1

where V0 is average variance rate and Pq j=1 j = 1

+

The weights are unknown and needs to be estimated. This is the ARCH model introduced by Engle (1982)

FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

6

ARCH (1) An ARCH(1) model is given by rt =

+ "t 2 t

=

"t 0

+

2

N 0; 2 1 "t 1

Since 2t is variance and has to be positive, we impose the condition 0

0 and

1

0

Generalization: ARCH(q) model 2 t

=

0

+

2 1 "t 1

+ ::: +

2 q "t q

where shocks up to q periods ago a¤ect the current volatility of the process.

FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

7

EWMA Exponentially weighted moving average (EWMA) model is the same as the ARCH models but the weights decrease exponentially as you move back through time. The model can be written as 2 t

where

2 t 1

=

)u2t

+ (1

1

is the constant decay rate, say 0.94.

To see that the weights cause an exponential decay, we substitute for 2t 1 :

2 t

= = (1

2 t 2

)u2t

+ (1

) u2t 1

+ (1

)u2t

+ (1

2

) u2t 2

+

1 2 2 t 2

With =0.94

2 t

= (0:06) u2t

1

+ (0:056) u2t

Substituting for FRM

2 t 2

2

+ (0:883)

2 t 2

now: c Dennis PHILIP 2011

1.1

2 t

Parametric volatility models

= (1

) u2t

= (1

) u2t 1

1

+ (1

) u2t

+ (1

) u2t 2

2

8

+

2

+

2

2 t 3

(1

+ (1

)u2t 3

+

)u2t

3 3 2 t 3

With =0.94

2 t

= (0:06) u2t 1 + (0:056) u2t 2 + (0:053) u2t 3 + (0:83) Risk Metrics uses EWMA model estimates for volatility with = 0.94.

FRM

c Dennis PHILIP 2011

2 t 3

1.1

Parametric volatility models

9

Generalized ARCH (GARCH) model This model generalizes the ARCH speci…cation. As one increase the q lags in an ARCH model for capturing the higher order ARCH e¤ects present in data, we loose parsimonity. Bollerslev (1986) proposed GARCH(p; q) 2 t

=

0

+

q X

2 j "t j

j=1

+

p X

j

2 t j

j=1

where the weights 0 0; j 0 and j 0: Further, for stationarity of this autoregressive model, we need the condition ! q p X X 2 > 3 > ::: > 1 FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

Also assume a geometric decline in that k k = a where 0

0; + 0; and 1 + 0. Hence, T is allowed to 1 + T be negative provided + 1 > j Tj: FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

15

Exponential GARCH (EGARCH) Recall that in the case of (G)ARCH models, we provided certain coe¢ cient restrictions in order to ensure 2t (conditional variance of "t ) is non-negative with probability one. An alternative way of ensuring positivity is specifying an EGARCH framework for 2t : Another characteristic of the model is that it allows for positive and negative shocks to have di¤erent e¤ects on conditional variances (unlike GARCH). The model re‡ects the fact that …nancial markets respond asymmetrically to good news and bad news. EGARCH(p,q) model is ln

2 t

=

0

+

q X

"t j

t j

j=1

+

p X

j

j

ln

+

q X

"t j

j=1

2 t j

j=1

FRM

c Dennis PHILIP 2011

j

t j

1.1

Parametric volatility models

where if theh error q "t = i terms then = E "tt = 2

t

16

N (0; 1)

Remarks:

– We can use other fat failed distributions such as student t; or GED in the case of non-normal errors. In this case, will take other forms. – Specifying the model as a logarithm ensures positivity of 2t : Therefore the leverage e¤ect is exponential rather than quadratic. – We divide the errors by the conditional standard deviations, "tt : Therefore we standardize (scale) the shocks. –

"t

captures the relative size of the shocks t and j captures the sign of the relative shocks

– The magnitude is captured by the variable that substracts the mean from the absolute value of the scaled shocks. Example:

FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

17

– Suppose we specify a EGARCH(0,1) 2 t

=

where

t 1

ln

0

+

1 t 1

= "t 1 =

+

1

[j

t 1j

]

t 1:

– Consider the estimated component: ^1

t 1

+ ^ 1 [j

t 1j

]

where ^ 1 = 0:3; ^ 1 = 0:6 and

= 0:85:

– Case 1: impact of positive scaled shock +1:0 0:3 (1) + 0:6 [j1j

0:85] = 0:39

Case 2: impact of negative scaled shock 1:0 0:3 ( 1) + 0:6 [j 1j

0:85] =

0:21

We see that positive shock has a greater impact than negative shock for ^ 1 positive. If we have ^ 1 negative, say ^ 1 = 0:3; a +1:0 shock will have an impact of 0:21 and a 1:0 shock will have an impact of 0.39.

FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

18

Thus, ^ 1 allows for the sign of the shock to have an impact on the conditional volatility; over and above the magnitude captured by ^ 1:

1.2

Non-parametric volatility models

Range-based estimators Suppose log prices of assets follow a Geometric Brownian Motion (GBM). The various variance estimators have been proposed in literature. Notation: –

volatility to be estimated

– Ct closing price on date t – Ot opening price on date t – Ht high price on date t – Lt low price on date t FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

– ct = ln Ct ing price

19

ln Ot , the normalized clos-

– ot = ln Ot ln Ct 1 , the normalized opening price – ht = ln Ht price

ln Ot , the normalized high

– lt = ln Lt price

ln Ot , the normalized low

The classical sample variance estimator of variance 2 is 1

2

^ =

T

1

T X

[(oi + ci )

2

(o + c)]

i=1

where (o + c) =

T 1X (oi + ci ) T i=1

and T is the total number of days considered. So this is the average volatility over T days. FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

20

Parkinson (1980) introduced a range estimator of daily volatility based on the highest and lowest prices on a particular day. He used the range of log prices to de…ne ^ 2t =

1 (ht 4 ln 2

lt )2

since it can be shown that E (ht 4 ln(2) 2t

lt )2 =

Garman and Klass (1980) extended Parkinson’s estimator where information about opening and closing prices are incorporated as follows: ^ 2t = 0:5 (ht

lt )2

[2 ln 2

1] c2t

Parkinson (1980) and Garman and Klass (1980) assume that the log-price follows a GBM with no drift term. This means that the average return is assumed to be equal to zero. Rogers and Satchell (1991) relaxes this assumption by using daily opening, highest, lowest, and closing prices into estimating volatility. FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

21

Rogers and Satchell (1991) estimator is given by ^ 2t = ht (ht ct ) + lt (lt ct ) This estimator performs better than the estimators proposed by Parkinson (1980) and Garman and Klass (1980). Yang and Zhang (2000) proposed a re…nement to Rogers and Satchell (1991) estimator for the presence of opening price jumps. Due to overnight volatility, the opening price and the previous day closing price are mostly not the same. Estimators that do not incorporate opening price jumps underestimate volatility. Yang and Zhang (2000) estimator is given by ^ 2 = ^ 2open + k^ 2close + (1

k)^ 2RS

where ^ 2open and ^ 2close are the classical sample variance estimators with the use of daily opening and closing prices, respectively. ^ 2RS is the average variance estimator introduced by Rogers and Satchell (1991). FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

22

The constant k is set to be k=

0:34 1:34 + (T + 1)=(T

1)

where T is the number of days.

Realized Volatility Realized volatility is referred to volatility estimates calculated using intraday squared returns at short intervals such as 5 or 15 minutes. For a series that has zero mean and no jumps, the realized volatility converges to the continuous time volatility. Consider a continuous time martigale process for asset prices dpt =

t dWt

where dWt is a standard brownian motion.

FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

23

Then the conditional variance for one-period returns, rt+1 pt+1 pt is Z t+1 2 s ds t

which is called the integrated volatility (or the true volatility) over the period t to t + 1: 2 t

We don’t know what it.

is. So we estimate

Let m be the sampling frequency such that there are m continuously compounded returns in one unit of time (say, one day). The j th return is given by rt+j=m

pt+j=m

pt+(j

1)=m

The realized volatility (in one unit of time) can be de…ned as X 2 RVt+1 = rt+j=m j=1;:::;m

FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

24

Then from the theory of quadratic variation, if sample returns are uncorrelated, ! Z t+1 X 2 2 p lim rt+j=m =0 s ds m!1

t

j=1;:::;m

As we increase sampling frequency, we get a consistent estimate of volatility. In the presence of jumps, RV is no longer a consistent estimator of volatility. An extension to this estimator is the standardized Realized Bipower Variation measure de…ned as [a;b] BVt+1

=

1 m

[1 (a+b)=2]

m X

rt+j=m

a

rt+(j

j=1

for a; b > 0: When jumps are large but rare, the simplest case where a = b = 1 captures the jumps well.

FRM

c Dennis PHILIP 2011

b 1)=m

1.2

Non-parametric volatility models

25

High frequency returns measured below 5 minutes are a¤ected by market microstructure e¤ects including nonsynchronous trading, discrete price observations, intraday periodic volatility patterns and bid–ask bounce.

FRM

c Dennis PHILIP 2011

2 Multivariate Volatility Models

2

26

Multivariate Volatility Models Multivariate modelling of volatilities enable us to study movements across markets and across assets (co-volatilities). Applications in …nance: asset pricing and portfolio selection, market linkages and integration between markets, hedging and risk management, etc. Consider a n-dimensional process fyt g : If we denote as the …nite vector of parameters, we can write yt = where and

t

t

( ) + "t

( ) is the conditional mean vector 1=2

"t = H t 1=2 Ht

where matrix. The N

( ) is an N

N positive de…nite

1 vector zt is such that zt

FRM

( ) zt

iidD (0; IN ) c Dennis PHILIP 2011

2 Multivariate Volatility Models

27

where IN is the identity matrix of order N: The matrix Ht is the conditional variance matrix of yt How do we parameterize Ht ? Vech Representation Bollerslev, Engle and Wooldridge (1988) propose a natural multivariate extension of the univariate GARCH(p; q) models where q X 0 vech (Ht ) = W + Ai vech "t i "t i=1

i

p X + j=1

where vech is the vector-half operator, which stacks the lower triangular elements of an N N matrix into a [N (N + 1) =2] 1 vector. The challenge in this parameterization is to ensure Ht is positive de…nite covariance matrix. Also, as the number of assets N increase, the number of parameters to be estimated is very large. FRM

c Dennis PHILIP 2011

j

vech (Ht j )

2 Multivariate Volatility Models

28

f"t g is covariance stationary if all the eigenvalues of A and B are less than 1 in modulus. Bollerslev, Engle and Wooldridge (1988) proposed a "Diagonal vech" representation where Ai and j are diagonal matrices. Example: For N = 2 assets and a period lag model (p = q = 1), 2 3 2 3 2 h11;t w1 a11 4 h21;t 5 = 4 w2 5 + 4 a22 h22;t w3 32 2 b11 54 b22 +4 b33

single-

a33 h11;t h21;t h22;t

32

"21;t 1 5 4 "2;t 1 ; "1;t "22;t 1 3 1 1 1

The diagonal restriction reduces the number of parameters but the model is not allowed to capture the interactions in variances among assets (copersistence, causality relations, asymmetries) Pq The diagonal vech is stationary i¤ i=1 aii + Pp b < 1 j=1 jj FRM

c Dennis PHILIP 2011

5

1

3 5

2 Multivariate Volatility Models

29

BEKK Representation Engle and Kroner (1995) propose a BEKK representation where q X Ht = cc + Ai 0

0

"t i " t

i

p X

A0i +

i=1

0 j

j Ht j

j=1

where c is a lower triangular matrix and therefore cc0 will be positive de…nite. Also, by estimating A and B rather than A and B ; we ensure positive de…niteness. In the case of 2 assets:

h11;t h12;t h21;t h22;t

= cc0 +

a11 a12 a21 a22 h11;t h21;t

"21;t 1 "2;t 1 ; "1;t

a11 a12 a21 a22

1 1

0

1

b11 b12 b21 b22

+

h12;t h22;t

"1;t 1 ; "2;t "22;t 1

1 1

b11 b12 b21 b22

0

To reduce the number of parameters to be estimated, we can impose a "diagonal BEKK" model where Ai and Bj are diagonal. FRM

c Dennis PHILIP 2011

1

2 Multivariate Volatility Models

30

Alternatively, we can have Ai and Bj as scalar times a matrix of ones. In this case, we will have a "scalar-BEKK" model. Diagonal BEKK andP Scalar-BEKK P are covariance stationary if qi=1 a2nn;i + pj=1 b2nn;j < P P 1 8n = 1; 2; :::; N and qi=1 a2i + pj=1 b2j < 1 respectively. Constant Conditional Correlation (CCC) Model Bollerslev (1990), assuming conditional correlations constant, proposed that conditional covariances (Ht ) can be parameterized as a product of corresponding conditional standard deviations. Ht = Dt RDt p = hii;t hjj;t ij

2 p h11;t 6 .. where Dt = 4 . FRM

p

hN N;t

3

7 5; R =

c Dennis PHILIP 2011

2 Multivariate Volatility Models 2 6 6 6 4

1

12

21

1

.. .

N1

1N

.. .

..

. 1

31

3 7 7 7 5

Each conditional standard deviations can be in turn de…ned as any univariate GARCH model such as GARCH(1,1) hii;t = wi + i "2i;t 1 + i hii;t

1

i = 1; 2; :::; N

Ht is positive de…nite i¤ all N conditional covariances are positive and R is positive de…nite. In most empirical applications, the conditional correlations are not constant. Therefore Engle (2002) and Tse and Tsui(2002) propose a generalization of the CCC model by allowing for conditional correlation matrix to be time-varying. This is the DCC model.

FRM

c Dennis PHILIP 2011

2 Multivariate Volatility Models

32

Tests for Costant Correlations Tse (2000) proposes testing the null that p hijt = ij hiit hjjt against the alternative that p hijt = ijt hiit hjjt

where the conditional variances, hiit and hjjt are GARCH-type models. The test statistic is an LM statistic which is asymptotically 2 (N (N 1) =2) : Engle and Sheppard (2001) propose another test with the null hypothesis H0 : Rt = R for all t against the alternative H1 : vech (Rt ) = vech R + ::: + p vech (Rt p )

1 vech (Rt 1 )+

The test statistic employed is again chi-squared distributed.

FRM

c Dennis PHILIP 2011

2 Multivariate Volatility Models

33

Dynamic Conditional Correlation (DCC) model Tse (2000) and Engle and Sheppard (2001) propose tests of constant conditional correlation hypothesis. In most applications, we see the hypothesis of constant conditional correlation is rejected. Engle (2002) propose the DCC framework, Ht = Dt Rt Dt where Dt is the matrix of standard deviations (as de…ned in the case of CCC), hii;t can be any univariate GARCH model and Rt is the conditional correlation matrix. We then standardize each return by the dynamic standard deviations to get standardized returns. Let i 1 h p h11t hN N t ut = "t diag FRM

c Dennis PHILIP 2011

2 Multivariate Volatility Models

34

be the vector of standardized residuals of N GARCH models. These variables now have standard deviations of one. We now model the conditional correlations of raw returns ("t ) by modelling conditional covariances of standardized returns (ut ). We de…ne Rt as 1=2

Rt = diag (Qt )

Qt diag (Qt )

1=2

where Qt is an N N symmetric positive de…nite matrix given by ! q p q X X X 0 Qt = 1 Q+ i i ut i ut j i=1

j=1

i=1

where –

i

0;

j

0;

Pq

i=1

i

+

Pp

j=1

j

T1

Stock index futures Duration, Convexity, Immunization

T2

Repo and reverse repo Futures on T-bills Futures on T-bonds Delta, Gamma, Vega hedging

T3

Portfolio insurance Implied volatility and volatility smiles

T4

Modelling stock prices using GBM Interest rate derivatives (Bond options, Caps, Floors, Swaptions)

T5

Value at Risk

T6

Value at Risk: statistical issues Monte Carlo Simulations Principal Component Analysis Other VaR measures

T7

Parametric volatility models (GARCH type models) Non-parametric volatility models (Range and high frequency models) Multivariate volatility models (Dynamic Conditional Correlation DCC models)

T8

Credit Risk Measures (credit metrics, KMV, Credit Risk Plus, CPV)

T9

Credit derivatives (credit options, total return swaps, credit default swaps) Asset Backed Securitization Collateralized Debt Obligations (CDO)

* This file provides you an indication of the range of topics that is planned to be covered in the module. However, please note that the topic plans might be subject to change.

Topics

Financial Risk Management

Futures Contract: Speculation, arbitrage, and hedging

Topic 1 Managing risk using Futures Reading: CN(2001) chapter 3

Stock Index Futures Contract: Hedging (minimum variance hedge

ratio) Hedging market risks

Futures Contract Agreement to buy or sell “something” in the future at

a price agreed today. (It provides Leverage.) Speculation with Futures: Buy low, sell high Futures (unlike Forwards) can be closed anytime by taking an opposite position Arbitrage with Futures: Spot and Futures are linked

by actions of arbitragers. So they move one for one. Hedging with Futures: Example: In January, a farmer

wants to lock in the sale price of his hogs which will be “fat and pretty” in September. Sell live hog Futures contract in Jan with maturity in Sept

Speculation with Futures

Speculation with Futures

Speculation with Futures

Purchase at F0 = 100 Hope to sell at higher price later F1 = 110

Profit/Loss per contract

Close-out position before delivery date.

Long future

Obtain Leverage (i.e. initial margin is ‘low’)

$10

Long 61,000 Nikkei-225 index futures (underlying

value = $7bn). Nikkei fell and he lost money (lots of it) - he was supposed to be doing riskless ‘index arbitrage’ not speculating

0

-$10

F1 = 90 F0 = 100

Example: Example: Nick Leeson: Feb 1995

F1 = 110

Futures price

Short future

Speculation with Futures Profit payoff (direction vectors) F increase then profit increases Profit/Loss

F increase then profit decrease Profit/Loss

Arbitrage with Futures

-1

+1 Underlying,S

+1

or Futures, F

-1

Long Futures

Short Futures

or, Long Spot

or, Short Spot

Arbitrage with Futures At expiry (T), FT = ST . Else we can make riskless

profit (Arbitrage). Forward price approaches spot price at maturity Forward price, F Forward price ‘at a premium’ when : F > S (contango)

Arbitrage with Futures General formula for non-income paying security:

F0 = S0erT

or

F0 = S0(1+r)T

Futures price = spot price + cost of carry For stock paying dividends, we reduce the ‘cost of

carry’ by amount of dividend payments (d) 0

Stock price, St T

At T, ST = FT

F0 = S0e(r-d)T

For commodity futures, storage costs (v or V) is

negative income Forward price ‘at a discount’, when : F < S (backwardation)

Arbitrage with Futures For currency futures, the ‘cost of carry’ will be

reduced by the riskless rate of the foreign currency (rf) F0 = S0e(r-rf)T

For stock index futures, the cost of carry will be

reduced by the dividend yield F0 = S0

e(r-d)T

F0 = S0e(r+v)T or F0 = (S0+V)erT

Arbitrage with Futures Arbitrage at t S0erT then buy the asset and short the futures contract If F0 < S0erT then short the asset and buy the futures contract Example of ‘Cash and Carry’ arbitrage: S=£100,

r=4%p.a., F=£102 for delivery in 3 months. 0.04×0.25 = 101 £ We see Fɶ = 100 × e Since Futures is over priced, time = Now •Sell Futures contract at £102

time = in 3 months •Pay loan back (£101)

•Borrow £100 for 3 months and buy stock •Deliver stock and get agreed price of £102

Hedging with Futures F and S are positively correlated To hedge, we need a negative correlation. So we

Hedging with Futures

long one and short the other. Hedge = long underlying + short Futures

Hedging with Futures

Hedging with Futures

Simple Hedging Example: You long a stock and you fear falling prices over the next 2 months, when you want to sell. Today (say January), you observe S0=£100 and F0=£101 for April delivery. so r is 4% Today: you sell one futures contract In March: say prices fell to £90 (S1=£90). So F1=S1e0.04x(1/12)=£90.3. You close out on Futures.

F1 value would have been different if r had changed.

Profit on Futures: 101 – 90.3 = £10.7 Loss on stock value: 100 – 90 =£10

Net Position is +0.7 profit. Value of hedged portfolio

= S1+ (F0 - F1) = 90 + 10.7 = 100.7

This is Basis Risk (b1 = S1 – F1) Final Value

= S1 + (F0 - F1 ) = £100.7 = (S1 - F1 ) + F0 = b1 + F0 where “Final basis” b1 = S1 - F1 At maturity of the futures contract the basis is zero

(since S1 = F1 ). In general, when contract is closed out prior to maturity b1 = S1 - F1 may not be zero. However, b1 will usually be small in relation to F0.

Stock Index Futures Contract Stock Index Futures contract can be used to

eliminate market risk from a portfolio of stocks

Stock Index Futures Contract Hedging with SIFs

Hedging with Stock Index Futures Example: A portfolio manager wishes to hedge her

portfolio of $1.4m held in diversified equity and S&P500 index Total value of spot position, TVS0=$1.4m S0 = 1400 index point Number of stocks, Ns = TVS0/S0 = $1.4m/1400 =1000 units We want to hedge Δ(TVSt)= Ns . Δ(St)

F0 = S0 × e( r − d )T If this equality does not hold then index arbitrage (program trading) would generate riskless profits. Risk free rate is usually greater than dividend yield

(r>d) so F>S

Hedging with Stock Index Futures The required number of Stock Index Futures contract

to short will be 3 TVS 0 $1, 400, 000 NF = − = − = − 3.73 $375, 000 FVF0 In the above example, we have assumed that S and F have correlation +1 (i.e. ∆ S = ∆ F ) In reality this is not the case and so we need

Use Stock Index Futures, F0=1500 index point, z=

contract multiplier = $250 FVF0 = z F0 = $250 ( 1500 ) = $375,000

minimum variance hedge ratio

Hedging with Stock Index Futures Minimum Variance Hedge Ratio

To obtain minimum, we differentiate with respect to Nf

∆V = change in spot market position + change in Index Futures position + Nf . (F1 - F0) z = Ns . (S1-S0) Ns S0. ∆S /S0 TVS0 . ∆S /S0

= =

+ +

Hedging with Stock Index Futures 2

(∂σ V

Nf F0. (∆ ∆F /F0) z Nf . FVF0 . (∆ ∆F /F0)

where, z = contract multiple for futures ($250 for S&P 500 Futures); ∆S = S1 - S0, ∆F = F1 - F0

/ ∂N

2

2

2

2

f

( F V F0 ) 2 σ ∆2 F / F

N f

= −

2

TVS 0 FVF0 . σ ∆S / S , ∆F / F

2 ∂σV / ∂N = 0 f

TVS 0 .β p Nf = − FVF0

(∆S

0

⋅ F V F ⋅ σ ∆ S / S ,∆ F / F 0

( σ ∆ S / S ,∆ F / F

2

σ ∆F / F

)

/ S ) = α 0 + β ∆S / ∆F ( ∆ F / F ) + ε

Hedging with Stock Index Futures Application: Changing beta of your portfolio: “Market Timing Strategy” TVS

implies

Value of Spot Position = − FaceValue of futures at t = 0

TVS0 F V F0

= −TVS

where Ns = TVS0/S0 and beta is regression coefficient of the regression

Hedging with Stock Index Futures SUMMARY

and set to zero

TVS0 =− β ∆ S / S ,∆ F / F F V F0

σ V = (TVS 0 ) σ ∆S / S + ( N f ) ( FVF ) σ ∆F / F 0 + 2N

= 0)

N f

The variance of the hedged portfolio is

2

f

Nf =

0

FVF0

.( β h − β p )

Example: βp (=say 0.8) is your current ‘spot/cash’ portfolio of stocks

βp

But

• You are more optimistic about ‘bull market’ and desire a higher exposure of βh (=say, 1.3) • It’s ‘expensive’ to sell low-beta shares and purchase high-beta shares

If correlation = 1, the beta will be 1 and we just have

TVS0 Nf = − FVF0

• Instead ‘go long’

more Nf Stock Index Futures contracts

Note: If βh= 0, then

Nf = - (TVS0 / FVF0) βp

Hedging with Stock Index Futures

Hedging with Stock Index Futures Application: Stock Picking and hedging market risk

If you hold stock portfolio, selling futures will place a hedge and reduce the beta of your stock portfolio. If you want to increase your portfolio beta, go long futures. Example: Suppose β = 0.8 and Nf = -6 contracts would make β = 0. If you short 3 (-3) contracts instead, then β = 0.4 If you long 3 (+3) contracts instead, then β = 0.8+0.4 = 1.2

You hold (or purchase) 1000 undervalued shares of Sven plc V(Sven) = $110

(e.g. Using Gordon Growth model)

P(Sven) = $100 (say) Sven plc are underpriced by 10%.

Therefore you believe Sven will rise 10% more than the market over the next 3 months. But you also think that the market as a whole may fall by 3%. The beta of Sven plc (when regressed with the market return) is 2.0

Hedging with Stock Index Futures

Hedging with Stock Index Futures

Can you ‘protect’ yourself against the general fall in the market and hence any ‘knock on’ effect on Sven plc ?

Application: Future stock purchase and hedging market risk

Yes .

You want to purchase 1000 stocks of takeover target with βp = 2, in 1 month’s time when you will have the cash.

Sell Nf index futures, using:

N

f

= −

TVS FVF

0

.β

You fear a general rise in stock prices. p

0

If the market falls 3% then Sven plc will only change by about

10% - (2x3%) = +4%

But the profit from the short position in Nf index futures, will give you an additional return of around 6%, making your total return around 10%.

Go long Stock Index Futures (SIF) contracts, so that gain on the futures will offset the higher cost of these particular shares in 1 month’s time.

N

f

=

TVS FVF

0

.β

p

0

SIF will protect you from market risk (ie. General rise in prices) but not from specific risk. For example if the information that you are trying to takeover the firm ‘leaks out’ , then price of ‘takeover target’ will move more than that given by its ‘beta’ (i.e. the futures only hedges market risk)

Topics

Financial Risk Management

Duration, immunization, convexity Repo (Sale and Repurchase agreement)

Topic 2 Managing interest rate risks Reference: Hull(2009), Luenberger (1997), and CN(2001)

and Reverse Repo Hedging using interest rate Futures Futures on T-bills Futures on T-bonds

Readings Books Hull(2009) chapters 6 CN(2001) chapters 5, 6 Luenberger (1997) chapters 3 Journal Article Fooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis”

Managerial Finance,Vol 25, no. 3

Hedging Interest rate risks: Duration

Duration

Duration (also called Macaulay Duration)

Duration measures sensitivity of price changes (volatility) with changes in interest rates 1 Lower the coupons

n

B =

2 Greater the time to

D =

T PB = ∑ C t t + ParValue T (1+ r ) t =1 (1+ r )

T PB = ∑ C t t + ParValue T (1+ r ) t =1 (1+ r )

approximate response of bond prices to change in yields. A better approximation could be convexity of the bond .

for a given time to maturity, greater change in price to change in interest rates

T

T

Duration of the bond is a measure that summarizes

maturity with a given coupon, greater change in price to change in interest rates

3 For a given percentage change in yield, the actual price increase is

∑

c i e − y ti

i =1 n

∑

i =1

weight

t i ⋅ c i e − y ti B

=

n

∑

i =1

c e − y ti ti i B

Duration is weighted average of the times when payments

are made. The weight is equal to proportion of bond’s total present value received in cash flow at time ti. Duration is “how long” bondholder has to wait for cash flows

greater than a price decrease

Macaulay Duration For a small change in yields ∆ y / d y

Evaluating d B :

dy

dB ∆B = ∆y dy n ∆ B = − ∑ t i c i e − y ti ∆ y i =1 = −B ⋅ D ⋅∆y

∆B = −D ⋅∆y B D measures sensitivity of percentage change in bond prices to (small) changes in yields Note negative relationship between Price (B) and yields (Y)

Modified Duration and Dollar Duration For Macaulay Duration, y is expressed in continuous

compounding. When we have discrete compounding, we have Modified Duration (with these small modifications) If y is expressed as compounding m times a year, we divide D D by (1+y/m) ∆B = − B ⋅ ⋅ ∆y

(1 + y / m)

∆B = − B ⋅ D* ⋅ ∆y Dollar Duration, D$ = B.D That is, D$ = Bond Price x Duration (Macaulay or Modified)

∆B = − D$ ⋅ ∆y

So D$

is like Options Delta

D$ = −

∆B ∆y

Duration

Duration -example

Example: Consider a trader who has $1 million in

bond with modified duration of 5. This means for every 1 bp (i.e. 0.01%) change in yield, the value of the bond portfolio will change by $500. ∆B = − ( $1, 000, 000 × 5 ) ⋅ 0.01% = −$500

Example: Consider a 7% bond with 3 years to maturity. Assume that the bond is selling at 8% yield.

A Year

A zero coupon bond with maturity of n years has a

Duration = n A coupon-bearing bond with maturity of n years will have Duration < n Duration of a bond portfolio is weighted average of the durations of individual bonds D

p o r tfo lio

=

∑ (B

i

/ B )⋅ D i

B

0.5 1.0 1.5 2.0 2.5 3.0 Sum

C

D

E

Present value Weight = Payment Discount A× E =B× C D/Price factor 8% 3.5 0.962 3.365 0.035 0.017 3.5 0.925 3.236 0.033 0.033 3.5 0.889 3.111 0.032 0.048 3.5 0.855 2.992 0.031 0.061 3.5 0.822 2.877 0.030 0.074 103.5 0.79 81.798 0.840 2.520 Price = 97.379 Duration = 2.753

Here, yield to maturity = 0.08, m = 2, y = 0.04, n = 6, Face value = 100.

i

Qualitative properties of duration

Properties of duration

Duration of bonds with 5% yield as a function of

maturity and coupon rate. Coupon rate

Years to maturity 1 2 5 10 25 50 100 Infinity

1%

2%

5%

10%

0.997 1.984 4.875 9.416 20.164 26.666 22.572 20.500

0.995 1.969 4.763 8.950 17.715 22.284 21.200 20.500

0.988 1.928 4.485 7.989 14.536 18.765 20.363 20.500

0.977 1.868 4.156 7.107 12.754 17.384 20.067 20.500

1. Duration of a coupon paying bond is always less than its maturity. Duration decreases with the increase of coupon rate. Duration equals bond maturity for noncoupon paying bond. 2. As the time to maturity increases to infinity, the duration do not increase to infinity but tend to a finite limit independent of the coupon rate. 1 + mλ

where λ is the yield to maturity λ per annum, and m is the number of coupon

Actually, D →

payments per year.

Properties of Duration 3. Durations are not quite sensitive to increase in coupon rate (for bonds with fixed yield). They don’t vary huge amount since yield is held constant and it cancels out the influence of coupons. 4. When the coupon rate is lower than the yield, the duration first increases with maturity to some maximum value then decreases to the asymptotic limit value. 5. Very long durations can be achieved by bonds with very long maturities and very low coupons.

Immunization (or Duration matching) This is widely implemented by Fixed Income Practitioners.

time 0 0

time 1

time 2

time 3

pay $

pay $

pay $

You want to safeguard against interest rate increases.

Changing Portfolio Duration Changing Duration of your portfolio: If prices are rising (yields are falling), a bond trader might want to switch from shorter duration bonds to longer duration bonds as longer duration bonds have larger price changes. Alternatively, you can leverage shorter maturities. Effective portfolio duration = ordinary duration x leverage ratio.

Immunization Matching present values (PV) of portfolio and obligations This means that you will meet your obligations with the cash

from the portfolio. If yields don’t change, then you are fine. If yields change, then the portfolio value and PV will both change by varied amounts. So we match also Duration (interest rate risk)

PV1 + PV2 = PVobligation

A few ideas: 1. Buy zero coupon bond with maturities matching timing of

cash flows (*Not available) [Rolling hedge has reinv. risk] 2. Keep portfolio of assets and sell parts of it when cash is needed & reinvest in more assets when surplus (* difficult as Δ value of in portfolio and Δ value of obligations will not identical) 3. Immunization - matching duration and present values of portfolio and obligations (*YES)

Matching duration Here both portfolio and obligations have the same sensitivity to

interest rate changes. If yields increase then PV of portfolio will decrease (so will the PV of the obligation streams) If yields decrease then PV of portfolio will increase (so will the PV of the obligation streams)

D1 PV1 + D 2 PV2 = Dobligation PVobligation

Immunization

Immunization

Example

Suppose only the following bonds are available for its choice.

Suppose Company A has an obligation to pay $1 million in 10 years. How to invest in bonds now so as to meet the future obligation? • An obvious solution is the purchase of a simple zero-coupon bond with maturity 10 years. * This example is from Leunberger (1998) page 64-65. The numbers are rounded up by the author so replication would give different numbers.

Bond 1 Bond 2 Bond 3

coupon rate 6% 11% 9%

maturity 30 yr 10 yr 20 yr

yield 9% 9% 9%

duration 11.44 6.54 9.61

•

Present value of obligation at 9% yield is $414,642.86.

•

Since Bonds 2 and 3 have durations shorter than 10 years, it is not possible to attain a portfolio with duration 10 years using these two bonds.

Suppose we use Bond 1 and Bond 2 of amounts V1 & V2, V1 + V2 = PV P1V1 + D2V2 = 10 × PV giving V1 = $292,788.64,

Immunization

price 69.04 113.01 100.00

V2 = $121,854.78.

Immunization

Yield 9.0 Bond 1 Price Shares Value

8.0

10.0

69.04 77.38 62.14 4241 4241 4241 292798.64 328168.58 263535.74

Bond 2 Price 113.01 120.39 106.23 Shares 1078 1078 1078 Value 121824.78 129780.42 114515.94 Obligation value 414642.86 456386.95 376889.48 Surplus -19.44 1562.05 1162.20

Observation: At different yields (8% and 10%), the value of the portfolio almost agrees with that of the obligation.

Difficulties with immunization procedure 1. It is necessary to rebalance or re-immunize the portfolio from time to time since the duration depends on yield. 2. The immunization method assumes that all yields are equal (not quite realistic to have bonds with different maturities to have the same yield). 3. When the prevailing interest rate changes, it is unlikely that the yields on all bonds change by the same amount.

Duration for term structure We want to measure sensitivity to parallel shifts in the spot

rate curve For continuous compounding, duration is called FisherFisher-Weil duration. duration If x0, x1,…, xn is cash flow sequence and spot curve is st where t = t0,…,tn then present value of cash flow is PV =

n

∑ i=0

xti ⋅ e

1 PV

(

changes to sti + ∆y

Consider parallel shift in term structure: st

i

Then PV becomes

P ( ∆y ) =

n

∑x i=0

ti

⋅e

(

)

− sti + ∆ y ⋅ti

Taking differential w.r.t ∆y in the point ∆y=0 we get n dP ( ∆ y ) − s ⋅t | ∆ y = 0 = − ∑ t i x t i ⋅ e ti i d ∆y i=0

− sti ⋅ti

So we find relative price sensitivity is given by DFW

The Fisher-Weil duration is

D FW =

Duration for term structure

n

∑t i=0

i

⋅ x ti ⋅ e

1 dP (0) ⋅ = − D FW P (0) d ∆ y

− sti ⋅ti

Convexity Duration applies to only small changes in y Two bonds with same duration can have different

change in value of their portfolio (for large changes in yields)

Convexity Convexity for a bond is n

1 d 2B C = = B dy 2

∑

i =1

t i2 ⋅ c i e − y t i B

=

c e − y ti t i2 i B

n

∑

i =1

Convexity is the weighted average of the ‘times squared’

when payments are made. From Taylor series expansion

First order approximation cannot capture this, so we

take second order approximation (convexity)

1 d 2B dB ∆ y + dy 2 dy 2 1 = − D ⋅ ∆ y + C ⋅ 2

∆ B =

(∆

y

)

∆ B B

(∆

y

)

2

2

So Dollar convexity is like Gamma measure in

options.

)

Short term risk management using Repo

REPO and REVERSE REPO

Repo is where a security is sold with agreement to buy it back at a later date (at the price agreed now) Difference in prices is the interest earned (called repo rate) rate It is form of collateralized short term borrowing (mostly overnight) Example: a trader buys a bond and repo it overnight. The money from repo is used to pay for the bond. The cost of this deal is repo rate but trader may earn increase in bond prices and any coupon payments on the bond. There is credit risk of the borrower. Lender may ask for

margin costs (called haircut) to provide default protection. Example: A 1% haircut would mean only 99% of the value of

collateral is lend in cash. Additional ‘margin calls’ are made if market value of collateral falls below some level.

Short term risk management using Repo Hedge funds usually speculate on bond price differentials

using REPO and REVERSE REPO Example: Assume two bonds A and B with different prices (say price(A)<price(B)) but

similar characteristics. Hedge Fund (HF) would like to buy A and sell B simultaneously.This can be financed with repo as follows: (Long position) Buy Bond A and repo it. The cash obtained is used to pay for

the bond. At repo termination date, sell the bond and with the cash buy bond back (simultaneously). HF would benefit from the price increase in bond and low repo rate (short position) Enter into reverse repo by borrowing the Bond B (as collateral for money lend) and simultaneously sell Bond B in the market. At repo termination date, buy bond back and get your loan back (+ repo rate). HF would benefit from the high repo rate and a decrease in price of the bond.

Interest Rate Futures (Futures on T-Bills)

Interest Rate Futures In this section we will look at how Futures contract written on a Treasury Bill (T-Bill) help in hedging interest rate risks Review - What is T-Bill? T-Bills are issued by government, and quoted at a discount Prices are quoted using a discount rate (interest earned as % of face value) Example: 90-day T-Bill is quoted at 0.08. 0.08 This means annualized return is 8% of FV. So we can work out the price, as we know FV. d 90 P = F V 1 − 100 360 Day Counts convention (in US) 1. Actual/Actual (for treasury bonds) 2. 30/360 (for corporate and municipal bonds) 3. Actual/360 (for other instruments such as LIBOR)

Hedge decisions When do we use these futures contract to hedge? Examples: 1) You hold 3m T-Bills to sell in 1-month’s time ~ fear price fall ~ sell/short T-Bill futures 2) You will receive $10m in 3m time and wish to place it on a Eurodollar bank deposit for 90 days ~ fear a fall in interest rates ~ go long a Eurodollar futures contract 3) Have to issue $100m of 180-day Commercial Paper in 3 months time (I.e. borrow money) ~ fear a rise in interest rates ~ sell/short a T-bill futures contract as there is no commercial bill futures contract (cross hedge)

Interest Rate Futures So what is a 3-month T-Bill Futures contract? At expiry, (T), which may be in say 2 months time the (long) futures delivers a T-Bill which matures at T+90 days, with face value M=$100. As we shall see, this allows you to ‘lock in’ at t=0, the forward rate, f12

T-Bill Futures prices are quoted in terms of quoted index, Q

(unlike discount rate for underlying) Q = $100 – futures discount rate (df) So we can work out the price as

d f 90 F = F V 1 − 100 360

Cross Hedge: US T-Bill Futures Example: Today is May. Funds of $1m will be available in August to invest for further 6 months in bank deposit (or commercial bills) ~ spot asset is a 6-month interest rate Fear a fall in spot interest rates before August, so today BUY Tbill futures Assume parallel shift in the yield curve. (Hence all interest rates move by the same amount.) ~ BUT the futures price will move less than the price of the higher the maturity, more commercial bill - this is duration at work! sensitive are changes in prices to interest rates

Use Sept ‘3m T-bill’ Futures, ‘nearby’ contract

~ underlying this futures contract is a 3-month interest rate

Cross Hedge: US T-Bill Futures 3 month exposure period

Cross Hedge: US T-Bill Futures Question: How many T-bill futures contract should I purchase?

Desired investment/protection period = 6-months

We should take into account the fact that:

to hedge exposure of 3 months, we have used T-bill futures with 4 months time-to-maturity 2. the Futures and spot prices may not move one-to-one 1.

May

Aug.

Dec.

Sept.

Feb.

Maturity of ‘Underlying’ in Futures contract

Purchase T-Bill Known $1m Maturity date of Sept. future with Sept. cash receipts T-Bill futures contract delivery date

We could use the minimum variance hedge ratio:

Nf =

TVS0 .β p FVF0

However, we can link price changes to interest rate

changes using Duration based hedge ratio

Question: How many T-bill futures contract should I purchase?

Duration based hedge ratio Using duration formulae for spot rates and futures:

∆S = − DS ⋅ ∆ys S

∆F = − DF ⋅ ∆yF F

So we can say volatility is proportional to Duration:

∆S 2 2 = DS ⋅ σ ( ∆ys ) S

σ2

∆F 2 2 = DF ⋅ σ ( ∆yF ) F

σ2

∆S ∆F Cov , = Ε ( − DS ⋅ ∆ys )( − DF ⋅ ∆yF ) S F = DS ⋅ DF ⋅ σ ( ∆ys ∆yF )

Duration based hedge ratio Expressing Beta in terms of Duration:

TVS0 Nf = .β p FVF 0 ∆S ∆F Cov , TVS0 S F = FVF0 σ 2 ∆F F TVS0 Ds σ ( ∆ys ∆yF ) = 2 ∆ FVF D y σ ( ) 0 F F

We can obtain last term by regressing ∆yS = α0 + βy∆yF + ε

Duration based hedge ratio Summary:

TVS0 Ds βy Nf = . FVF0 DF

Cross Hedge: US T-Bill Futures Example REVISITED

3 month exposure period

Aug.

May

where beta is obtained from the regression of yields

Desired investment/protection period = 6-months

Dec.

Sept.

∆yS = α0 + β y ∆yF + ε

Feb.

Maturity of ‘Underlying’ in Futures contract

Purchase T-Bill Known $1m Maturity date of Sept. future with Sept. cash receipts T-Bill futures contract delivery date Question: How many T-bill futures contract should I purchase?

Cross Hedge: US T-Bill Futures May (Today). Funds of $1m accrue in August to be invested for 6- months in bank deposit or commercial bills( Ds = 6 )

Cross Hedge: US T-Bill Futures Suppose now we are in August: 3 month US T-Bill Futures : Sept Maturity

Use Sept ‘3m T-bill’ Futures ‘nearby’ contract ( DF = 3) Cross-hedge. Here assume parallel shift in the yield curve

Spot Market(May)

CME Index

Futures Price, F

Face Value of $1m

(T-Bill yields)

Quote Qf

(per $100)

Contract, FVF

May

y0 (6m) = 11%

Qf,0 = 89.2

97.30

$973,000

August

y1(6m) = 9.6%

Qf,1 = 90.3

97.58

$975,750

Change

-1.4%

1.10 (110 ticks)

0.28

$2,750 (per contract)

Qf = 89.2 (per $100 nominal) hence: F0 = 100 – (10.8 / 4) = 97.30 F

FVF0 = $1m (F0/100)

= $973,000

Durations are : Ds = 0.5, Df = 0.25 Amount to be hedged = $1m. No. of contracts held = 2

Key figure is F1 = 97.575 (rounded 97.58) Gain on the futures position

Nf = (TVS0 / FVF0) (Ds / DF ) = ($1m / 973,000) ( 0.5 / 0.25) = 2.05 (=2)

= TVS0 (F1 - F0) NF = $1m (0.97575 – 0.973) 2 = $5,500

Cross Hedge: US T-Bill Futures Invest this profit of $5500 for 6 months (Aug-Feb) at y1=9.6%:

= $5500 + (0.096/2) = $5764 Loss of interest in 6-month spot market (y0=11%, y1=9.6%)

= $1m x [0.11 – 0.096] x (1/2) = $7000 Net Loss on hedged position $7000 - $5764 = $1236

(so the company lost $1236 than $7000 without the hedge)

Interest Rate Futures (Futures on T-Bonds)

Potential Problems with this hedge: 1. Margin calls may be required 2. Nearby contracts may be maturing before September. So we may have to roll over the hedge 3. Cross hedge instrument may have different driving factors of risk

US T-Bond Futures Contract specifications of US T-Bond Futures at CBOT: Contract size

$100,000 nominal, notional US Treasury bond with 8% coupon

Delivery months

March, June, September, December

Quotation

Per $100 nominal

Tick size (value)

1/32 ($31.25)

Last trading day

7 working days prior to last business day in expiry month

Delivery day

Any business day in delivery month (seller’s choice)

Settlement

Any US Treasury bond maturing at least 15 years from the contract month (or not callable for 15 years)

US T-Bond Futures Conversion Factor (CF): (CF): CF adjusts price of actual bond to be

delivered by assuming it has a 8% yield (matching the bond to the notional bond specified in the futures contract) Price = (most recent settlement price x CF) + accrued interest Example: Possible bond for delivery is a 10% coupon (semi-

annual) T-bond with maturity 20 years. The theoretical price (say, r=8%): 40

Notional is 8% coupon bond. However, Short can choose to

deliver any other bond. So Conversion Factor adjusts “delivery price” to reflect type of bond delivered T-bond must have at least 15 years time-to-maturity Quote ‘98‘98-14’ means 98.(14/32)=$98.4375 per $100 nominal

5 100 + = 119.794 i 1.0440 i =1 1.04

P=∑

Dividing by Face Value, CF = 119.794/100 = 1.19794 (per

$100 nominal) If Coupon rate > 8% then CF>1

If Coupon rate < 8% then CF 0 ∂S ∂P ∆ put = = N ( d1 ) − 1 < 0 ∂S

(for long positions)

If we have lots of options (on same underlying) then

delta of portfolio is

∆ portfolio = ∑ N k ⋅ ∆ k k

where Nk is the number of options held. Nk > 0 if long Call/Put and Nk < 0 if short Call/Put

Delta So if we use delta hedging for a short call position, we

must keep a long position of N(d1) shares What about put options? The higher the call’s delta, the more likely it is that the option ends up in the money: Deep out-of-the-money: Δ ≈ 0 At-the-money: In-the-money:

Δ ≈ 0.5 Δ≈1

Intuition: if the trader had written deep OTM calls, it

would not take so many shares to hedge - unlikely the calls would end up in-the-money

Theta The rate of change of the value of an option with

respect to time Also called the time decay of the option For a European call on a non-dividend-paying stock, Θ=−

S0 N '(d1 )σ 2T

− rKe

− rT

1 N (d 2 ) where N '( x) = e 2π

x2 − 2

Related to the square root of time, so the relationship is

not linear

Theta Theta is negative: as maturity approaches, the option

tends to become less valuable The close to the expiration date, the faster the value of

the option falls (to its intrinsic value) Theta isn’t the same kind of parameter as delta The passage of time is certain, so it doesn’t make

any sense to hedge against it!!! Many traders still see theta as a useful descriptive statistic

because in a delta-neutral portfolio it can proxy for Gamma

Gamma The rate of change of delta with respect to the 2 ∂ f share price: ∂S 2 Calculated as Γ = N '(d1 ) S0σ T Sometimes referred to as an option’s curvature If delta changes slowly → gamma small → adjustments

to keep portfolio delta-neutral not often needed

Gamma If delta changes quickly → gamma large → risky to

leave an originally delta-neutral portfolio unchanged for long periods: Option price

C'' C' C S

S'

Stock price

Gamma Making a Position GammaGamma-Neutral We must make a portfolio initially gamma-neutral as well as delta-neutral

if we want a lasting hedge But a position in the underlying share can’t alter the portfolio gamma since the share has a gamma of zero So we need to take out another position in an option that isn’t linearly dependent on the underlying share If a delta-neutral portfolio starts with gamma Γ, and we buy wT options each with gamma ΓT, then the portfolio now has gamma Γ + wT Γ T We want this new gamma to = 0:

−Γ Rearranging, wT = ΓT

Γ + wT Γ T = 0

Delta-Theta-Gamma For any derivative dependent on a non-dividend-paying stock, Δ , θ, and Г are related The standard Black-Scholes differential equation is ∂f ∂f 1 2 2 ∂ 2 f + rS + σ S = rf 2 ∂t ∂S 2 ∂S

where f is the call price, S is the price of the underlying share and r is the risk-free rate 2 ∂ f ∂ f ∂ f But Θ = , ∆= and Γ = ∂t

∂S

∂S 2

1 2 2 Θ + rS ∆ + Θ S Γ = rf So 2 So if Θ is large and positive, Γ tends to be large and negative, and vice-versa This is why you can use Θ as a proxy for Γ in a delta-neutral portfolio

Vega NOT a letter in the Greek alphabet! Vega measures, the sensitivity of an option’s

price to volatility: volatility ∂f = ∂σ

υ

υ = S0 T N '(d1 )

High vega → portfolio value very sensitive to

small changes in volatility Like in the case of gamma, if we add in a traded

option we should take a position of – υ/υT to make the portfolio vega-neutral

Rho The rate of change of the value of a portfolio of

options with respect to the interest rate ∂f ρ= ρ = KTe− rT N (d 2 ) ∂r Rho for European Calls is always positive and Rho for European Puts is always negative (since as interest rates rise, forward value of stock increases). Not very important to stock options with a life of a few

months if for example the interest rate moves by ¼% More relevant for which class of options?

Delta Hedging Value of portfolio = no of calls x call price + no of stocks x

stock price V = NC C + NS S

∂V = N ∂S

∂C + N S ⋅1 = 0 C ⋅ ∂S ∂C NS = −NC ⋅ ∂S N S = − N C ⋅ ∆ c a ll

So if we sold 1 call option then NC = -1. Then no of stocks to

buy will be NS = ∆call So if ∆call = 0.6368 then buy 0.63 stocks per call option

Delta Hedging Example: As a trader, you have just sold (written)

100 call options to a pension fund (and earned a nice little brokerage fee and charged a little more than Black-Scholes price). You are worried that share prices might RISE, RISE hence the call premium RISE, hence showing a loss on your position. Suppose ∆ of the call is 0.4. Since you are short,

your ∆ = -0.4 (When S increases by +$1 (e.g. from 100 to 101), then C decrease by $0.4 (e.g. from 10 to 9.6)).

Delta Hedging Your 100 written (sold) call option (at C0 = 10 each option) You now buy 40-shares Suppose S FALLS by $1 over the next month THEN fall in C is 0.4 ( = “delta” of the call) So C falls to C1 = 9.6 To close out you must now buy back at C1 = 9.6 (a GAIN of $0.4)

Loss on 40 shares Gain on calls

= $40 = 100 (C0 - C1 )= 100(0.4)

= $40

Delta hedging your 100 written calls with 40 shares means that the value of your ‘portfolio is unchanged.

Delta Hedging Call Premium

∆ = 0.5 B

.

∆ = 0.4

.

A

0 100

110

Stock Price

As S changes then so does ‘delta’ , so you have to rebalance your portfolio. E.g. ‘delta’ = 0.5, then you now have to hold 50 stocks for every written call. This brings us to ‘Dynamic Hedging’, over many periods. Buying and selling shares can be expensive so instead we can maintain the hedge by buying and selling options.

(Dynamic) Delta Hedging You’ve written a call option and earned C0 =10.45

(with K=100,

σ = 20%, r=5%, T=1) At t = 0: Current price S0 = $100. We calculate ∆ 0 = N(d1)= 0.6368. So we buy ∆0 = 0.6368 shares at S0 = $100 by borrowing debt. Debt, D0 = ∆0 x S0 = $63.68

At t = 0.01: stock price rise S1 = $100.1. We calculate ∆ 1 = 0.6381. So buy extra (∆ 1 – ∆ 0) =0.0013 no of shares at $100.1.

Debt, D1 = D0 ert + (∆ 1 – ∆ 0) S1 = $63.84 So as you rebalance, you either accumulate or reduce debt

levels.

Delta Hedging At t=T, if option ends up well “in the money” Say ST = 163.3499. Then ∆ T = 1 (hold 1 share for 1 call). Our final debt amount DT = 111.29 (copied from Textbook page 247) The option is exercised. We get strike $100 for the share. Our Net Cost: NCT = DT – K = 111.29 – 100 = $11.29

How have we done with this hedging? At t = 0, 0 we received $10.45 and at t = T we owe $11.29 % Net cost of hedge, % NCT = [ (DT – K )-C0 ] / C0 = 8% (8% is close to 5% riskless rate)

Delta Hedging One way to view the hedge: The delta hedge is supposed to be riskless (i.e. no change in value of portfolio of “One written call + holding ∆ shares” , over any very small time interval ) Hence for a perfect hedge we require: dV = If we choose NS = ∆

NS dS + (NC ) dC ≈ NS dS + (-1) [ ∆ dS ] ≈ 0 then we will obtain a near perfect hedge

(ie. for only small changes in S, or equivalently over small time intervals)

Delta Hedging Another way to view the hedge: The delta hedge is supposed to be riskless, so any money we borrow (receive) at t=0 which is delta hedged over t to T , should have a cost of r Hence: For a perfect hedge we expect: NDT / C0 = erT

so,

NDT e-r T - C0 ≈ 0

If we repeat the delta hedge a large number of times then: % Hedge Performance, HP =

stdv( NDT e-r T - C0) / C0

HP will be smaller the more frequently we rebalance the portfolio (i.e. buy or sell stocks) although frequent rebalancing leads to higher ‘transactions costs’ (Kuriel and Roncalli (1998))

Gamma and Vega Hedging ∂2 f Γ = ∂S 2

∂f υ = ∂σ

υ Short Call/Put have negative Γ and υ Long Call/Put have positive Γ and

Gamma /Vega Neutral: Stocks and futures have

Γ ,υ = 0

So to change Gamma/Vega of an existing options portfolio, we have to take positions in further (new) options.

Delta-Gamma Neutral Example: Suppose we have an existing portfolio of options, with a value of

Γ = - 300 (and a ∆ = 0)

Note:

Γ = Σi ( Ni Γi )

Can we remove the risk to changes in S (for even large changes in S ? ) Create a “Gamma-Neutral” Portfolio Let ΓZ = gamma of some “new” option (same ‘underlying’)

For Γport = NZ ΓZ + Γ = 0 we require: NZ = - Γ / ΓZ “new” options

Delta-Gamma Neutral Suppose a Call option “Z” with the same underlying (e.g. stock) has a delta =

0.62 and gamma of 1.5 How can you use Z to make the overall portfolio gamma and delta neutral? We require:

Nz Γz + Γ = 0 Nz = - Γ / Γz = -(-300)/1.5 = 200

implies 200 long contracts in Z (ie buy 200 Z-options) The delta of this ‘new’ portfolio is now ∆ = Nz.∆z = 200(0.62) = 124 Hence to maintain delta neutrality you must short 124 units of the underlying -

this will not change the ‘gamma’ of your portfolio (since gamma of stock is zero).

Delta-Gamma-Vega Neutral Example:You hold a portfolio with

∆ port = − 500, Γ port = − 5000, υ port = − 4000 We need at least 2 options to achieve Gamma and Vega neutrality. Then

we rebalance to achieve Delta neutrality of the ‘new’ Gamma-Vega neutral portfolio. Suppose there is available 2 types of options: υ Z = 0.8 Option Y with ∆ Y = 0.6, Γ Y = 0.3, υ Y = 0.4 Option Z with ∆ Z = 0.5, Γ Z = 1.5,

We need

N Z υ Z + N Y υ Y + υ port = 0 N Z Γ Z + N Y Γ Y + Γ port = 0

Delta-Gamma-Vega Neutral So

N Z ( 0.8 ) + N Y ( 0.4 ) − 4000 = 0 N Z (1.5 ) + N Y ( 0.3 ) − 5000 = 0

Solution:

N Z = 2222.2 N Y = 5555.5

Go long 2222.2 units of option Z and long 5555.5 units of option Y to

attain Gamma-Vega neutrality. New portfolio Delta will be:

2222.2 × ∆ Z + 5555.5 × ∆ Y + ∆ port = 3944.4 Therefore go short 3944 units of stock to attain Delta neutrality

Portfolio Insurance

Portfolio Insurance You hold a portfolio and want insurance against

market declines. Answer: Buy Put options From put-call parity: Stocks + Puts = Calls + T-bills Stock+Put = {+1, +1} + {-1, 0} = {0, +1} = ‘Call payoff’ This is called Static Portfolio Insurance. Alternatively replicate ‘Stocks+Puts’ portfolio price movements

with ‘Stocks+T-bills’ or ‘Stocks+Futures’. [called Dynamic Portfolio Insurance] Why replicate? Because it’s cheaper!

Dynamic Portfolio Insurance Stock+Put (i.e. the position you wish to replicate) N0 = V0 /(S0 +P0) (hold 1 Put for 1 Stock) N0 is fixed throughout the hedge: At t > 0 ‘Stock+Put’ portfolio: Vs,p = N0 (S + P) Hence, change in value:

∂Vs, p ∂ P = N0 (1+ ∆ p ) = N0 1+ ∂S ∂S

This is what we wish to replicate

Dynamic Portfolio Insurance Replicate with (N0*) Stocks + (Nf) Futures: N0* = V0 / S0 (# of index units held in shares) N0* is also held fixed throughout the hedge. Note: position in futures costs nothing (ignore interest cost on margin funds.) At t > 0:

VS,F = N0* S + Nf (F zf)

Hence:

∂ VS , F = N 0* + z f N f ∂S

F = S ⋅ e r (T − t )

∂ F ∂S

∂ F F r (T − t ) =e ∂S

Equating dV of (Stock+Put) with dV(Stock+Futures) to get Nf :

Nft

= [N (1 + ∆ ) − N ] 0

p t

* 0

e − r (T − t ) zf

Dynamic Portfolio Insurance Replicate with ‘Stock+T-Bill’ VS,B = NS S + NB B ∂ VS , B ∂S

=

NB,t =

Ns

(V s , p ) t − ( N S ) t S t Bt

Equate dV of (Stock+Put) with dV(Stock+T-bill) ( N s ) t = N 0 (1 + ∆

p

)

t

=

N 0 (∆ c ) t

Dynamic Portfolio Insurance Example: Value of stock portfolio S&P500 index Maturity of Derivatives Risk free rate Compound\Discount Factor Standard deviation S&P

V0 = $560,000 S0 = 280 T - t = 0.10 r = 0.10 p.a. (10%) er (T – t) = 1.01 σ = 0.12

Put Premium Strike Price Put delta (Call delta)

P0 = 2.97 (index units) K = 280 ∆p = -0.38888 (∆c = 1 + ∆p = 0.6112)

Futures Price (t=0) Price of T-Bill

F0 = S0 er(T – t ) = 282.814 B = Me-rT = 99.0

Dynamic Portfolio Insurance Hedge Positions Number of units of the index held in stocks = V0 /S0 = 2,000 index units Stock-Put Insurance N0 = V0 / (S0 + P0) = 1979 index units Stock-Futures Insurance Nf = [(1979) (0.6112) - 2,000] (0.99/500) = - 1.56 (short futures) Stock+T-Bill Insurance No. stocks = N0 ∆c = 1979 (0.612) = 1,209.6 (index units) NB = 2,235.3 (T-bills)

Dynamic Portfolio Insurance 1) Stock+Put Portfolio Gain on Stocks = N0.dS = 1979 ( -1) Gain on Puts = N0 dP = 1979 ( 0.388) Net Gain

= -1,979 = 790.3 = -1,209.6

2) Stock + Futures: Dynamic Replicatin Gain on Stocks = Ns,o dS = 2000 (-1) = -2,000 Gain on Futures = Nf.dF.zf = (-1.56) (-1.01) 500 = +790.3 Net Gain = -1,209.6

Dynamic Portfolio Insurance 3) Stock + T-Bill: Dynamic Replication Gain on Stocks = Ns dS = 1209.6 (-1) Gain on T-Bills (No change in T-bill price) Net Gain

= -1,209.6 = 0

= -1,209.6

The loss on the replication portfolios is very close to that on the stock-put portfolio (over the infinitesimally small time period). Note:We are only “delta replicating” and hence, if there are large changes in S or changes in σ, then our calculations will be inaccurate When there are large market falls, liquidity may “dry up” and it may not be possible to trade quickly enough in ‘stocks+futures’ at quoted prices (or at any price ! e.g. 1987 crash).

Financial Risk Management Topic 3b Option’s Implied Volatility

Topics Option’s Implied Volatility VIX Volatility Smiles

Readings Books Hull(2009) chapter 18 VIX http://www.cboe.com/micro/vix/vixwhite.pdf

Journal Articles Bakshi, Cao and Chen (1997) “Empirical Performance of Alternative

Option Pricing Models”, Journal of Finance, 52, 2003-2049.

Options Implied Volatility

Estimating Volatility Itō’s Lemma: The Lognormal Property If the stock price S follows a GBM (like in the BS model),

then ln( ln(ST/S0) is normally distributed. σ2 ln S T − ln S 0 = ln( S T / S T ) ≈ φ µ − 2

2 T , σ T

The volatility is the standard deviation of the

continuously compounded rate of return in 1 year The standard deviation of the return in time ∆t is σ ∆t Estimating Volatility: Historical & Implied – How?

Estimating Volatility from Historical Data Take observations S0, S1, . . . , Sn at intervals of t years

(e.g. t = 1/12 for monthly) Calculate the continuously compounded return in each

interval as:

u i = ln( S i / S i −1 )

Calculate the standard deviation, s , of the ui´s

s=

1 n 2 ( u − u ) ∑ i n − 1 i =1

The variable s is therefore an estimate for So:

σˆ = s / τ

σ ∆t

Estimating Volatility from Historical Data For volatility estimation (usually) we assume that there are 252 trading days within one year mean

-0.13%

stdev (s)

3.5%

τ

1/252

σ(yearly)

s / sqrt(ττ) = 55.56%

Back or forward looking volatility measure? 7

Date 03/11/2008 04/11/2008 05/11/2008 06/11/2008 07/11/2008 10/11/2008 11/11/2008 12/11/2008 13/11/2008 14/11/2008 17/11/2008 18/11/2008 19/11/2008 20/11/2008 21/11/2008 24/11/2008 25/11/2008 26/11/2008 27/11/2008 28/11/2008 01/12/2008 02/12/2008 03/12/2008 04/12/2008 05/12/2008 08/12/2008

Close 4443.3 4639.5 4530.7 4272.4 4365 4403.9 4246.7 4182 4169.2 4233 4132.2 4208.5 4005.7 3875 3781 4153 4171.3 4152.7 4226.1 4288 4065.5 4122.9 4170 4163.6 4049.4 4300.1

Price Relative St/St-1

Daily Return ln(St/St-1)

1.0442 0.9765 0.9430 1.0217 1.0089 0.9643 0.9848 0.9969 1.0153 0.9762 1.0185 0.9518 0.9674 0.9757 1.0984 1.0044 0.9955 1.0177 1.0146 0.9481 1.0141 1.0114 0.9985 0.9726 1.0619

0.0432 -0.0237 -0.0587 0.0214 0.0089 -0.0363 -0.0154 -0.0031 0.0152 -0.0241 0.0183 -0.0494 -0.0332 -0.0246 0.0938 0.0044 -0.0045 0.0175 0.0145 -0.0533 0.0140 0.0114 -0.0015 -0.0278 0.0601

Implied Volatility BS Parameters Observed Parameters: S: underlying index value

Unobserved Parameters: Black and Scholes

X: options strike price

σ: volatility

T: time to maturity r: risk-free rate q: dividend yield

•

Traders and brokers often quote implied volatilities rather than dollar prices

How to estimate it?

Implied Volatility The implied volatility of an option is the volatility

for which the Black-Scholes price equals (=) the market price There is a one-to-one correspondence between prices and implied volatilities (BS price is monotonically increasing in volatility) Implied volatilities are forward looking and price traded options with more accuracy Example: If IV of put option is 22%, this means that pbs = pmkt when a volatility of 22% is used in the Black-Scholes model. 9

Implied Volatility Assume c is the call price, f is an option pricing

model/function that depends on volatility σ and other inputs: c = f (S , K , r , T , σ )

Then implied volatility can be extracted by inverting the

formula:

σ = f −1 (S , K , r , T , c mrk )

where cmrk is the market price for a call option. The BS does not have a closed-form solution for its inverse function, so to extract the implied volatility we use rootfinding techniques (iterative algorithms) like NewtonNewtonRaphson method 10

f (S , K , r , T , σ ) − c mrk = 0

Volatility Index - VIX In 1993, CBOE published the first implied

11

volatility index and several more indices later on. VIX: VIX 1-month IV from 30-day options on S&P VXN: VXN 3-month IV from 90-day options on S&P VXD: VXD volatility index of CBOE DJIA VXN: VXN volatility index of NASDAQ100 MVX: MVX Montreal exchange vol index based on iShares of the CDN S&P/TSX 60 Fund VDAX: VDAX German Futures and options exchange vol index based on DAX30 index options Others: VXI, VX6, VSMI, VAEX, VBEL, VCAC

Volatility Smile

Volatility Smile What is a Volatility Smile? It is the relationship between implied

volatility and strike price for options with a certain maturity The volatility smile for European call options should be exactly the same as that for European put options

13

Volatility Smile Put-call parity p +S0e-qT = c +Ke–r T holds for market

prices (pmkt and cmkt) and for Black-Scholes prices (pbs and cbs) It follows that the pricing errors for puts and calls

are the same: pmkt−pbs=cmkt−cbs When pbs=pmkt, it must be true that cbs=cmkt It follows that the implied volatility calculated from a

European call option should be the same as that calculated from a European put option when both have the same strike price and maturity 14

Volatility Term Structure In addition to calculating a volatility

smile, traders also calculate a volatility term structure This shows the variation of implied volatility with the time to maturity of the option for a particular strike

15

IV Surface

16

IV Surface

17

IV Surface

Also known as: Volatility smirk 18 Volatility skew

Volatility Smile Implied Volatility Surface (Smile) from Empirical Studies (Equity/Index)

19

Bakshi, Cao and Chen (1997) “Empirical Performance of Alternative Option Pricing Models ”, Journal of Finance, 52, 2003-2049.

Volatility Smile Implied vs Lognormal Distribution

20

Volatility Smile In practice, the left tail is heavier and the right tail is less

heavy than the lognormal distribution What are the possible causes of the Volatility Smile anomaly? Enormous number of empirical and theoretical papers

to answer this …

21

Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes

(e.g. S&P 500 index)

22

Date

Open

High

Low

Close

Volume

Adj Close

Return

04/01/2000

1455.22

1455.22

1397.43

1399.42

1.01E+09

1399.42

-3.91%

18/02/2000

1388.26

1388.59

1345.32

1346.09

1.04E+09

1346.09

-3.08%

20/12/2000

1305.6

1305.6

1261.16

1264.74

1.42E+09

1264.74

-3.18%

12/03/2001

1233.42

1233.42

1176.78

1180.16

1.23E+09

1180.16

-4.41%

03/04/2001

1145.87

1145.87

1100.19

1106.46

1.39E+09

1106.46

-3.50%

10/09/2001

1085.78

1096.94

1073.15

1092.54

1.28E+09

1092.54

0.62%

17/09/2001

1092.54

1092.54

1037.46

1038.77

2.33E+09

1038.77

-5.05%

16/03/00

1392.15

1458.47

1392.15

1458.47

1.48E+09

1458.47

4.65%

15/10/02

841.44

881.27

841.44

881.27

1.96E+09

881.27

4.62%

05/04/01

1103.25

1151.47

1103.25

1151.44

1.37E+09

1151.44

4.28%

14/08/02

884.21

920.21

876.2

919.62

1.53E+09

919.62

3.93%

01/10/02

815.28

847.93

812.82

847.91

1.78E+09

847.91

3.92%

11/10/02

803.92

843.27

803.92

835.32

1.85E+09

835.32

3.83%

24/09/01

965.8

1008.44

965.8

1003.45

1.75E+09

1003.45

3.82%

-ve Price jumps

Trading was suspended

+ve price jumps

Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes Volatility for asset price is stochastic In the case of equities volatility is negatively related to stock

prices because of the impact of leverage. This is consistent with the skew (i.e., volatility smile) that is observed in practice

23

Volatility Smile

24

Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes Volatility for asset price is stochastic In the case of equities volatility is negatively related to stock

prices because of the impact of leverage. This is consistent with the skew that is observed in practice Combinations of jumps and stochastic volatility

25

Volatility Smile Alternatives to Geometric Brownian Motion Accounting for negative skewness and excess kurtosis by

generalizing the GBM Constant Elasticity of Variance Mixed Jump diffusion Stochastic Volatility Stochastic Volatility and Jump

Other models (less complex → ad-hoc) The Deterministic Volatility Functions (i.e., practitioners Black

and Scholes) (See chapter 26 (sections 26.1, 26.2, 26.3) of Hull for these alternative specifications to Black-Scholes) 26

Topic # 4: Modelling stock prices, Interest rate derivatives Financial Risk Management 2010-11 February 7, 2011

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

1

2

Modelling stock prices Modelling the evolution of stock prices is about introducing a process that will explain the random movements in prices. This randomness is explained in the E¢ cient Market Hypothesis (EMH) that can be summarized in two assumptions: 1. Past history is re‡ected in present price 2. Markets respond immediately to any new information about the asset So we need to model arrival of new information that a¤ects price (or much more returns). If asset price is S. Suppose price changes to S + dS in a small time interval (say dt). into deThen we can decompose returns dS S terministic/anticipated part and a random part where prices changed due to some external unanticipated news. dS = dt + dW S

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

3

The randomness in the random part is explained by a Brownian Motion process and scaled by the volatility of returns.

We can introduce time subscripts and rearrange to get dSt = St dt + St dWt This process is called the Geometric Brownian Motion.

Why have we used Brownian Motion process to explain randomness? – In practice, we see that stock prices behave, atleast for long stretches of time, like random walks with small and frequent jumps – In statistics, random walk, being the simplest form, have limiting distributions and since BM is a limit of the random walk, we can easily understand the statistics of BM (use of CLT) FRM

c Dennis PHILIP 2011

1 Modelling stock prices

4

Next we see, what is this W (and in turn what is dW)? Brownian motion is a continuous time (rescaled) random walk.

Consider the iid sequence "1 ; "2 ; ::: with mean and variance 2 : Consider the rescaled random walk model 1 X Wn (t) = p "j n 1 j nt

The interval length t is divided into nt equal subintervals of length 1=n and the displacements / jumps "j ; j = 1; 2; :::; nt in nt steps are mutually independent random variables.

Then for large n; according to Central Limit Theorem: W (t) N ( t; 2 t) : FRM

c Dennis PHILIP 2011

1 Modelling stock prices

5

Special cases: Standard Brownian Motion arises when we have = 0; and = 1.

W is a Standard Brownian Motion if 1. W (0) = 0 2. W has stationary (for 0 s t; Wt Ws and Wt s have the same distribD ution. That is, Wt Ws = Wt s N (0; t s)) 3. W has independent increments (for s t; Wt Ws is independent of past history of W until time s) 4. Wt

FRM

N (0; t)

c Dennis PHILIP 2011

1 Modelling stock prices

6

For a Brownian Motion only the present value of the variable is relevant for predicting the future [also called Markov property]. Therefore BM is a markov process. It does not matter how much you zoom in, it just looks the same. That is, the randomness does not smooth out when we zoom in. BM …ts the characteristics of the share price. Imagine a heavy particle (share price) that is jarred around by lighter particles (trades). Trades a¤ect the price movement. what is this dW? Consider a small increment in W W (t + where "(t +

FRM

t) = W (t) + "(t + t)

t)

iidN (0; t) [Std BM].

c Dennis PHILIP 2011

1 Modelling stock prices

Taking limit as is dWt = =

7

! 0; the change in W (t) lim W (t + dt)

dt!0

W (t)

lim "(t + dt)

dt!0

iidN (0; dt) So in the di¤erential form, we can write the Standard Brownian motion process as p dWt = et dt where et N (0; 1)

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

8

Stochastic processes used in Finance Arithmetic Brownian Motion for a share price A stock price does not generally have a mean zero and atleast would grow on average with the rate of in‡ation. Therefore we can write dSt = (St ; t) dt + (St ; t)dWt = drif t term + dif f usion term = E(dS) + Stddev(dS) When the drift function (St ; t) = and di¤usion function (St ; t) = ; both constants, we have the Arithmetic BM. dSt = =

dt + dWt p dt + et dt

In the case of ABM, S may be positive or negative. Since prices cannot be negative, we generally use the Geometric BM for asset prices and made the drift and volatility as functions of the stock price. FRM

c Dennis PHILIP 2011

1 Modelling stock prices

9

Geometric Brownian Motion dSt = St dt + St dWt If S starts at a positive value, then it will remain positive. The solution of the SDE St is an exponential function which is always positive. Also, note that S will be lognormally distributed. GBM is related to ABM according to dSt = dt + dWt St where is the instantaneous share price volatility, and is the expected rate of return

The Hull and White (1987) Model uses GBM.

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

10

Ornstein-Uhlenbeck (OU) Process The Arithmetic Ornstein-Uhlenbeck process is given by dSt =

(

St ) dt + dWt

where is the long run mean and ( > 0) is the rate of mean reversion. The drift term is the mean reversion component, in that the di¤erence between the long run mean and the current price decides the upward or downward movement of the stock price towards the long run mean : Over time, the price process drifts towards its mean and the speed of mean reversion is determined by :

This is an important process to model interest rates that show mean reversion where prices are pulled back to some long-run average level over time. The Vasicek Model uses this kind of process. FRM

c Dennis PHILIP 2011

1 Modelling stock prices

11

A special case is when the mean is zero. Then we can write the OU process as dSt =

St dt + dWt

In the OU process, the stock price can be negative. Therefore we can introduce the Geometric OU process

The Geometric OU process is given by dSt =

(

St ) St dt + St dWt

where the asset prices St would always be positive.

So we can model asset prices using the Geometric OU process and their log returns will then follow an Arithmetic OU process. dSt = dSt = St FRM

(

St ) St dt + St dWt

(

St ) dt + dWt c Dennis PHILIP 2011

1 Modelling stock prices

12

Square Root Process A square root process satis…es the SDE p dSt = St dt + St dWt This type of process generates positive prices and used for asset prices whose volatility does not increase too much when St increases. Cox-Ingersoll-Ross (CIR) process The CIR combines mean reversion and square root process and satis…es the SDE p St dWt dSt = ( St ) dt + This process was introduced in the Hull and White (1988), and Heston (1993) stochastic volatility models. This class of models generated strictly non-negative volatility and accounted for the clustering e¤ect and mean reversion observed in volatility. FRM

c Dennis PHILIP 2011

1 Modelling stock prices

13

Also used to model short rates features positive interest rates, mean reversion, and absolute variance of interest rates increases with interest rates itself.

Solving the Stochastic Di¤erential Equations Consider the GBM dSt = St dt + St dWt In the integral form ZT 0

dSt =

ZT

St dt +

0

ST = S0 +

ZT

St dWt

0

ZT 0

St dt +

ZT

St dWt

0

= reimann integ + It^ o integ So we have to solve the intergrals to get a closed form solutions to this SDE. FRM

c Dennis PHILIP 2011

1 Modelling stock prices

14

We use Ito-lemma to solve this problem. Not all SDE’s have closed form solutions. When there are no solutions, we have to do numerical approximations for these integrals. Examples: Geometric Brownian Motion dSt = St dt + St dWt has the solution 1 2

St = S0 e(

2

)t+

Wt

Ornstein-Uhlenbeck (OU) Process dSt =

(

St ) dt + dWt

has the solution S t = S0 e

t

+

1

e

t

+

Zt

e

(t s)

dWs

0

FRM

c Dennis PHILIP 2011

1 Modelling stock prices

15

Consider the following process dSt = St dt + dWt has the solution St = S 0 e t +

Zt

e

(t s)

dWs

0

Simulating Geometric Brownian Motion 1 2 We can write St = S0 e( 2 )t+ Wt in discrete time intervals and substituting for Wt as p 1 2 St = St 1 e( 2 ) t+ et t

where et

N (0; 1)

So we randomly draw et and …nd the value of St

FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

2

16

Interest Rate Derivatives The payo¤ of interest rate derivatives would depend on the future level of interest rates. The main challenge in valuing these derivatives are that interest rates are used both for discounting and for de…ning payo¤s. For valuation, we will need a model to describe the behavior of the entire yield curve. Black’s Model to price European Options Consider a call option on a variable whose value is V: To calculate expected payo¤, the model asssumes: 1. VT has lognormal distribution with V ar(lnVT ) =

2

T

2. E(VT ) = F0 FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

The payo¤ is max(VT

17

K; 0) at time T .

We discount the expected payo¤ at time T using the risk-free rate given by P (0; T )

We will use the key result that you know from Derivatives: If V is lognormally distributed and standard deviation on ln(V ) is s, then

E[max ( V K; 0)] = E[ V ] N (d1 ) KN (d2 ) where d1 =

ln(E[ V ]=K) + s2 =2 s

d2 =

ln(E[ V ]=K) s

s2 =2

Therefore, value of the call option is given by c = P (0; T ) [F0 N (d1 ) FRM

KN (d2 )]

c Dennis PHILIP 2011

2 Interest Rate Derivatives

18

where d1 =

ln(F0 =K) + p T

2

T =2

and d2 =

ln(F0 =K) p

2

T =2

T

= d1

p

T

where – F is forward price of V for a contract with maturity T – F0 is value of F at time zero – K is strike of the option –

is volatility of forward contract

Similarly, for a put option p = P (0; T ) [KN ( d2 )

FRM

F0 N ( d1 )]

c Dennis PHILIP 2011

2 Interest Rate Derivatives

19

European Bond Options Bond option is an option to buy or sell a particular bond by a certain date for a particular price. Callable bonds and Puttable bonds are examples of embedded bond options. The payo¤ is given by max(BT a call option.

K; 0) for

To price an European Bond Option: – we assume bond price at maturity of option is lognormal – we de…ne such that p standard deviaT tion of ln(BT ) = – F0 can be calculated as F0 =

B0 I P (0; T )

where B0 is bond (dirty) price at time zero and I is the present value of coupons that will be paid during the life of option FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

20

– Then using Black’s model we price of a bond option

Interest Rate Caps and Floors An interest rate Cap provides insurance against the rate of interest on a ‡oating-rate note rising above a certain level (called Cap rate). Example: Principal amount = $10 million Tenor = 3 months (payments made every quarter) Life of Cap = 5 years Cap rate = 8% If the ‡oating-rate exceeds 8 %, then you get cash of the di¤erence. Suppose at a reset date, 3-month LIBOR is 9%, the ‡oating-rate note would have to pay 0:25 FRM

0:09

$10million = $225; 000 c Dennis PHILIP 2011

2 Interest Rate Derivatives

21

and with the Cap rate at 8%, the payment would be 0:25

0:08

$10million = $200; 000

Therefore the Cap provides a payo¤ of $25,000 to the holder. Consider a Cap with total life of Tn ; a Principal of L, Cap rate of RK based on a reference rate (say, on LIBOR) with a month maturity denoted by R(t) at date t. The contract follows the schedule: t T0 T1 T2 C1 C2 T0 is the starting date. For all j = 1; :::; n, we assume a constant tenor Tj Tj 1 = On each date Tj ; the Cap holder receives a cash ‡ow of Cj Cj = L

max [R(Tj 1 )

RK ; 0]

The Cap is a portfolio of n such options and each call option is known as the caplets. FRM

c Dennis PHILIP 2011

Tn Cn

2 Interest Rate Derivatives

22

Lets now consider a Floor with the same characteristics. On each date Tj ; the Floor holder receives a cash ‡ow of Fj Fj = L

max [RK

R(Tj 1 ); 0]

The Floor is a portfolio of n such options and each put option is known as the ‡oorlets.

FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

23

Interest rate Caps can be regarded as a portfolio of European put options on zero-coupon bonds. Put-Call parity relation: Consider a Cap and Floor with same strike price RK . Consider a Swap to receive ‡oating and pay a …xed rate of RK , with no exchange payments on the …rst reset date. The Put-Call parity states: Cap price = Floor price + value of Swap

FRM

c Dennis PHILIP 2011

2 Interest Rate Derivatives

24

Collar A Collar is designed to guarantee that the interest rate on the underlying ‡oating-rate note always lie between two levels. Collar = long position in Cap + short position in Floor It is usually constructed so that the price of Cap is equal to price of the ‡oor. Then the cost of entering into a Collar is zero. Valuation of Caps and Floors

If the rate R(Tj ) is assumed to be lognormal with volatility j , the value of the caplet today (t) for maturity Tj is given by Caplett = L

P (t; Tj ) FTj

1 ;Tj

N (d1 )

RK N (d2 )

where d1 = FRM

ln(FTj

1 ;Tj

=RK ) + 2j (Tj p (Tj 1 t)

1

t) =2

c Dennis PHILIP 2011

2 Interest Rate Derivatives

and d2 = d1

j

q

(Tj

25

1

t)

where FTj 1 ;Tj is the forward rate underlying the Caplet from Tj 1 to Tj : Similarly, F loorlett = L

FRM

P (t; Tj ) RK N ( d2 )

FTj

1 ;Tj

N ( d1 )

c Dennis PHILIP 2011

Financial Risk Management Lecture 5 Value at Risk Readings: CN(2001) chapters 22,23; Hull_RM chp 8

1

Topics Value at Risk (VaR) Forecasting volatility Back-testing Risk Grades VaR: Mapping cash flows

2

Value at Risk Example:

If at 4.15pm the reported daily VaR is $10m (calculated at 5% tolerance level) then: I expect to lose more than $10m only 1 day in every 20 days (ie. ie. 5% of the time) The VaR of $10m assumes my portfolio of assets is fixed Exactly how much will I lose on any one day? Unknown !!!

3

Value at Risk Statement (how bad can things get?):

“We are x% certain that we will not loose more than V dollars in the next N days” V dollars = f(x%, N days)

Suppose asset returns is niid, then risk can be measured by

variance/S.D. From Normal Distribution critical values table, we can work out the VaR. Example: For 90% certainty, we can expect actual return to be

between the range { µ − 1 .6 5 σ , µ + 1 .6 5 σ }

4

Value at Risk Normal Distribution (N(0,σ)) Probability

Mean = 0

5% of the area 5% of the area

-1.65σ

0.0

+1.65σ

Only 5% of the time will the actual % return R be below: “ R = µ - 1.65 σ1” where µ = Mean (Daily) Return. 5

If we assume µ=0, VaR = $V (1.65 σ1)

Return

VaR for single asset Example: Mean return = 0 %. Let σ1 = 0.02 (per day) Only 5% of the time will the loss be more than 3.3% (=1.65 x 2%) VaR of a single asset (Initial Position V0 =$200m in equities)

VaR = V0 (1.65 σ1 ) = 200 ( 0.033) = $6.6m

That is “(dollar) VaR is 3.3% of $200m” = $6.6m VaR is reported as a positive number (even though it’s a loss)

6

Are Daily Returns Normally Distributed? - NO • Fat tails (excess kurtosis), peak is higher and narrower, negative skewness, small (positive) autocorrelations, squared returns have strong autocorrelation, ARCH. • But niid is a (reasonable) approx for portfolios of equities, long term bonds, spot FX , and futures (but not for short term interest rates or options)

VaR for portfolio of assets

7

VaR for portfolio of assets

8

VaR for portfolio of assets

9

VaR for portfolio of assets

10

VaR for portfolio of assets Summary: Variance – Covariance method

If Vp is the market value of your portfolio of n assets and wi is the proportionate weight in each asset i then

VaR p = V p [ zCz ']

1/2

where

z = w1 (1.65σ 1 ) , w2 (1.65σ 2 ) ,… , wn (1.65σ n ) 1 ρ C = 21 ⋮ ρ n1

11

ρ12 … ρ1n ⋱

1

Forecasting

12

Forecasting σ Simple Moving Average ( Assume Mean Return = 0 )

σ2 t+1|t = (1/n) Σi R2t-i Exponentially Weighted Moving Average EWMA

σ2 t+1|t = Σi wi R2t-i

wi = (1-λ) λi

It can be shown that this may be re-written:

σ2t+1|t = λ σ2t| t-1 + (1- λ) Rt2 Longer Horizons:

13

T -rule σΤ = T

- for iid returns. σ

Forecasting σ Exponentially Weighted Moving Average (EWMA)

σ2t+1|t = λ σ2t| t-1 + (1- λ) Rt2 How to estimate λ? 1. Use GARCH models to estimate λ 2. Minimize forecast error Σ (Rt+12 – σ2 t+1|t) where the sum is

over all assets, and say 100 days 3. λ = 0.94 as by JPMorgan Suppose λ = 0.94 then weights decline as 0.94, 0.88, 0.83,….

and past observations are given less weight than current forecast of variance. 14

Back-testing In back-testing, we compare our (changing) daily

forecast of VaR with actual profit or loss over some historic period. Example: For a portfolio of assets, • forecast all the individual VaRi = Vi1.65 σt+1|t ,

• calculate portfolio VaR for each day:

VaRp =

[Z C Z’]1/2

• then see if actual portfolio losses exceed this only 5% of the time (over some historic period, e.g. 100 days). 15

Back-testing Daily $m profit/loss

= forecast = actual

Days

16

Only 6 violations out of 100 = just ‘OK’

VaR and Capital Adequacy-Basle Basle uses a more ‘conservative’ measure of VaR than J. P. Morgan Calc VaR for worst 1% of losses over 10 days Use at least 1-year of daily data to estimate σt+1|t

VaRi = 2.33 10 σ

( 2.33 = 1% left tail critical value, σ = daily vol )

Internal Models approach

Capital Charge KC KC = Max ( Avg. of previous 60-days VaR x M, previous day’s VaR) M = multiplier (min = 3)

17

Pre-commitment approach

• KC set equal to max. forecast loss over 20 day horizon = preannounced $VaR • If losses exceed VaR, more than 1 day in 20, then impose a penalty.

VaR and Coherent Risk Measures Risk measures that satisfy all the following 4 conditions are called as a

Coherent Risk Measure. Monotonicity: X 1 ≤ X 2 ⇒ R ( X 1 ) ≤ R ( X 2 ) (higher the riskiness of the portfolio, higher should be risk capital) Translation invariance:

R ( X + k ) = R ( X ) − k ∀k ∈ ℝ

(if cash k is added to portfolio, risk should go down by k) Homogeneity:

R ( λ X ) = λ R ( X ) ∀λ ≥ 0

(if you change portfolio by a factor of λ, risk is proportionally increased) Subadditivity: 18

R ( X + Y ) ≤ R ( X ) + R (Y )

(diversification leads to less risk)

VaR and Coherent Risk Measures VaR violates the subadditivity condition and therefore not

coherent. VaR cannot capture the benefits of diversification. VaR can actually show negative diversification benefit! VaR only captures the frequency of default but not the size

of default. Even if the largest loss is doubled, the VaR figure could remain the same. Other measures such as Expected Shortfall are coherent

measures. 19

Risk Grades

20

Risk Grades RG helps to calculate changing forecasts of risks (volatilities) RG quantifies volatility/risk (similar to variance, std. deviation,

beta, etc) RG can range from 0 to over 1000, where 100 corresponds to

the average risk of a diversified market-cap weighted index of global equities. So if two portfolio’s have RG1 = 100 and RG2 = 400, portfolio 2 is four times riskier than portfolio 1 RG scales all assets to a common scale and so it is able to compare risk across all asset classes.

21

Risk Grades RG of a single asset

σi 252 σi 252 RG = σ ×100 = ×100 i 0.20 base

σi is the DAILY standard deviation σbase is fixed at 20% per annum (= 5 yr. av. for international portfolio of stocks) Formula looks complex but RG is just a “scaled” daily standard deviation e.g. If RG = 100% then asset has 20% p.a. risk RG of a portfolio of assets 22

RG 2 = ∑ w i2 RG i2 + ∑ ∑ w i w j ρ RG i RG j P

Risk Grades Risk Grades in 2009

www.riskgrades.com 23

Risk Grades Risk Grades in 2009 of indices heating up and cooling off

24

VaR Mapping (VaR for different assets)

25

VaR for different assets PROBLEMS STOCKS : Too many covariances [= n(n-1)/2 ] FOREIGN ASSETS : Need VaR

in “home currency”

BONDS: Many different coupons paid at different times DERIVATIVES: Options payoffs can be highly nonlinear (ie. NOT normally distributed)

26

SOLUTIONS = “Mapping” (RiskMetricsTM produce volatility & correlations for various assets across 35 countries and useful for “Mapping”)

VaR for different assets STOCKS

Within each country use “single index model” SIM FOREIGN ASSETS

Treat one asset in foreign country as = “local currency risk”+ spot FX risk (like 2-assets, with equal weight) BONDS

Consider each bond as a series of “zeros” OTHER ASSETS

Forward-FX, FRA’s, Swaps: decompose into ‘constituent parts’ 27

DERIVATIVES(non-linear)

Mapping Stocks Consider ‘p’ = portfolio of stocks held in one country with (Rm , σm) (for e.g. S&P500 in US) Problem : Too many covariances to estimate Soln. All n(n-1)/2 covariances “collapse or mapped” into σm

and the asset betas (“n” of them)

Single Index Model:

Ri = ai + bi Rm + εi Rk = ak + bk Rm + εk 28

assume Eεi εk = 0 and cov (Rm , ε ) = 0

All the systematic variation in Ri AND Rk is due to Rm

Mapping Stocks 1) In a portfolio idiosyncratic risk εi is diversified away = 0 2) Then each return-i depends only on the market return (and beta). Hence ALL variances and covariances also only depend on these 2 factors. It can be shown that

σp = bp σm

(i.e. Calculation of portfolio beta requires, only n-beta’s and σm ) 3) Also, ρ = 1 because (in a well diversified portfolio) each return moves only with Rm 4) We end up with or equivalently 29

VaRp = VP 1.65 ( bP σm ) VaRp = (Z C Z’ )1/2

where Z = [ VaR1, VaR2 …. }

C is the unit matrix

Mapping Foreign Assets (Mapping foreign stocks into domestic currencyVaR)

30

Mapping Foreign assets Example: US resident holds a diversified portfolio of German stocks equivalent to German stocks + Euro-USD, FX risk

31

Use SIM to obtain stdv of foreign (German) portfolio returns, σG

Then treat ‘foreign portfolio’ as (two) equally weighted assets: = $V in German asset + $V foreign currency position

Then use standard VaR formula for 2-assets

Mapping Foreign assets US based investor:

with

€100m in a German stock portfolio

σG = βP σDAX Sources of risk: a) Stdv of the German portfolio (‘local currency’ portfolio) b) Stdv of €/$ exchange rate ( σFX ) c) one covariance/correlation coefficient ρ (between DAX and FX rate) e.g. Suppose when German stock market falls then the € also falls ‘double whammy’ for the US investor, from this positive correlation, so foreign assets are very risky (in terms of their USD ‘payoff’) Let : S = 1.2 $/ € Dollar initial value Vo$ = 100m x 1.2 = $120m Linear 32

dVP = V0$ (RG + RFX)

above implies wi = Vi / V0$ = 1

Mapping Foreign assets

Dollar-VaRp σp =

(

2 σG

= Vo$ 1.65 σp

2 + σ FX

+ 2 ρσ Gσ FX

)

1

2

No ‘relative weights’ appear in the formula Matrix Representation: Dollar VaR

Let

Z = [ V0,$ 1.65 =[

VaR1

σG , V0,$ 1.65 σFX ] ,

VaR2 ]

V0,$ = $120m for both entries in the Z-vector (i.e. equal amounts)

Then 33

VaRp = (Z C Z’ )1/2

Mapping Bonds (Mapping coupon paying bonds)

34

Mapping Coupon paying Bonds Example: Coupons paid at t=5 and t=7 Treat each coupon as a separate zero coupon bond

100 100 P= + 5 (1 + y5 ) (1 + y7 )7

P = V5 + V7 P is linear in the ‘price’ of the zeros, V5 and V7 We require two variances of “prices” V5 and V7 and covariance between these prices.

35

Note: σ5(dV5 / V5) = D σ(dy5) but Risk Metrics provides the price volatilities, σ5(dV5 / V5)

Mapping Coupon paying Bonds Treat each coupon as a zero Calculate: price of zero, e.g. V5 = 100 / (1+y5)5 VaR5 = V5 (1.65 σ5)

VaR7 = V7 (1.65 σ7) VaR (both coupon payments): VaRp = (Z C Z’ )1/2 =

[ VaR + VaR + 2 ρ VaR 5VaR 7 ] 2 5

2 7

r = correlation: bond prices at t=5 and t=7 (approx 0.95 - 0.99 ) 36

1/ 2

Mapping FRA

37

Mapping FRA To calculate VaR for a FRA, we break down cash flows into

equivalent synthetic FRA and use spot rates only (since we do not know the forward volatilities) Example: Consider an FRA on a notional of $1m that involves

lending $1m in 6 months time for a future of 6 months. Receipt of $1m + Interest

0

6m

12m

Lend $1m

Let y6 = 6.39%, y12 = 6.96% and there are 182 days in the first

leg and 183 days in the second leg (day count: actual/365). The implied f6,12 = 7.294% and therefore the 12 month investment will give $1,036,572 return (with round off error). 38

Mapping FRA The original FRA

0

Receipt of $1,036,572

6m

12m

Lend $1m

Synthetic FRA

Receipt of $1,036,572 from 12 month lending

Borrow at 6 month rate

0 Lend at 12 month rate

6m

12m

Repay 6 month loan of $1m

So at time 0, we borrow $969,121 [=1m / 1+(y6*182/365)] and lend

39

this money at a 12 month rate leading to $1,036,572 [=$969,121*(1+y12)]

Mapping FRA Suppose the standard deviation of the prices for 6 month asset is

0.1302% and for 12 month asset is 0.2916%. Suppose ρ = 0.7 To calulate the VaR for each of these positions: VaR6 = $969,121 (1.65) (0.1302%) = $2082 VaR12 = $969,121 (1.65) (0.2916%) = $4663

VaR = [ VaR + VaR + 2 ρ ( −VaR 6 ) VaR12 ] 2 6

= $3534

40

2 12

1/ 2

Mapping FX Forwards

41

Mapping FX Forwards Consider a US resident who holds a long forward contract to

purchase €10million in 1 year. What is the VaR for this contract? We map Forward into two spot rates and one spot FX rate. Then we calculate VaR from the VaR of each individual mapped

asset. Mapping a forward contract

42

Mapping FX Forwards

43

Mapping FX Forwards

44

Mapping Options

45

Mapping Options

46

Mapping Options

47

Mapping Options

48

Mapping Options

49

Financial Risk Management Topic 6 Statistical issues in VaR Readings: CN(2001) chapters 24, Hull_RM chp 8, Barone-Adesi et al (2000) RiskMetrics Technical Document (optional)

1

Topics Value at Risk for options Monte Carlo Simulation Historical Simulation Bootstrapping Principal component analysis Other related VaR measures Marginal VaR, Incremental VaR, ES 2

MCS - VaR of Call option Option premia are non-linear (convex) function of

underlying Distribution of gains/losses is not normally distributed Therefore dangerous to use Var-Cov method Assets Held: One call option on stock

Here, Black-Scholes is used to price the option during the Monte Carlo Simulation (MCS). Problem

Find the VaR over a 5-day horizon 3

MCS - VaR of Call option If V is price of the option (call or put) and P is price of

underlying asset in the option contract (stock)

V will change from minute to minute as P changes. For an

4

equal change in P of +1 or -1 ,the change in call premia are NOT equal: this gives “non-normality” in distribution of the change in call premium

MCS - VaR of Call option To find VaR over a 5-day horizon:

1) Given P0 calculate the option price, V0 = BS(P0, K, T0 …..) This is fixed throughout the MCS 2) MCS = Simulate the stock price and calculate P5 3) Calculate the new option price, V5= BS(P5, K, T0 – 5/365,

…..)

4) Calculate change in option premium ∆V(1) = V5 – V0 5) Repeat steps 2-4, 1000 times and plot a histogram of the change in the call premium. We can then find the 5% lower cut-off point for the change in value of the call (i.e. it’s VaR). 5

MCS - VaR of Call option 20 18

5% of area

16

Frequency

14 12 10 8 6 4 2 0 -10

-8

-6

-4

-2

0

2

4

6

8

10

More

$-VaR is $5m Change in Call Premium 6

MCS - VaR of Call option Simulate stock prices using

P(t+1) = P(t) exp[ (µ - σ2/2) ∆t + σ (∆t)1/2 εt+1 ] To generate 5-day prices from daily prices, you can use the

root-T rule

∆t = 5/365 (or 5/252) ( ie. five day) Alternatively, we can generate Tt+5 directly.

If P0 is today’s known price. Use a ‘do-loop’ over 5 ‘periods’ to get P5 (using 5 - ‘draws’ of εt+1 )

7

MCS - VaR of Call option Stock price paths generated

8

Monte Carlo Simulation for Two Assets (two call options on different underlying assets)

9

Simulating correlated random variables

10

Simulating correlated random variables

11

Simulating correlated random variables

12

MCS and VaR: two asset example

13

MCS and VaR: two asset example

14

MCS and VaR: two asset example

15

MCS and VaR: two asset example

16

MCS and VaR: two asset example

17

MCS and VaR: two asset example

18

MCS and VaR: two asset example

19

Comparing VaR forecasts

20

Comparing VaR forecasts

21

Historical Simulation (Historical simulation + bootstrapping)

22

Historical Simulation (HS) Suppose you currently hold $100 in each of 2 assets Day =

1

2

3

4

5

6

…1000

R1(%)

+2

+1

+4

-3

-2

-1

+2

R2(%)

+1

+2

0

-1

-5

-6

-5

_____________________________________________________ ∆Vp($)

+3

+3

+4

-4

-7

-7

-5

Order ∆Vp in ascending order (of 1000 numbers) e.g. -12, -11, -11 -10, -9, -9, -8, -7, -7, -6 | -5, -4, -4, …. +8, ……. +14 VaR forecast for tomorrow at 1% tail (10th most negative) = -$6 23

Above is equivalent to the histogram

Historical Simulation (HS) This is a non parametric method since we do not estimate any

variances or covariances or assume normality. We merely use the historic returns, so our VaR estimates encapsulate

whatever distribution the returns might embody ( e.g. Student’s t) as well as any autocorrelation in individual returns. Also, the historic data “contain” the correlations between the returns

on the different assets, their ‘own volatility’ and their own autocorrelation over time It does rely on ‘tomorrow’ being like ‘the past’. 24

HS + Bootstrapping Problems:

Is data >3 years ago useful for forecasting tomorrow? Use most recent data - say last 100 days ? 1% tail: Has only one number in this tail, for the actual data !

Extreme case ! Actual data might have largest negative (for 100 days ago) of minus 50% - this would be your forecast VaR for tomorrow using historic simulation approach. Is this okay or not? 25

HS + Bootstrapping You have “historic” daily data on each of 10 stock returns (i.e.

your portfolio ) But only use last 100 days of historic daily returns, So we have

a data matrix of 10 x 100. We require VaR at the 1st percentile (1% cut off) We sample “with replacement” from these 100 observations,

26

giving equal probability to each ‘day’ , when we sample. This allows any one day’s returns to be randomly chosen more than once (or not at all). It is as if we are randomly ‘replaying’ the last 100 days of history, giving each day equal probability

HS + Bootstrapping The Bootstrap Draw randomly from a uniform distribution with an equal probability of drawing any number between 1 and 100. If you draw “20” then take the 10-returns in column 20 and revalue the

portfolio. Call this $-value, ∆VP(1)

Repeat above for 10,000 “trials/runs” (with replacement), obtaining

10,000 possible (alternative) values ∆VP (i)

“With replacement” means that in the 10,000 runs you will “choose” some of

the 100 columns more than once. Plot a histogram of the 10,000 values of ∆VP(i) - some of which will be

negative

Read off the “1% cut off” value (=100th most negative value). This is 27

VaRp

Filtered Historical Simulation (FHS)

28

FHS HS assumes risk factors are i.i.d however this is usually not the case. HS assumes that distribution of returns are stable and that the past and

present moments of the density function of returns are constant and equal. The probability of having a large loss is not equal across different days.

There are periods of high volatility and periods of low volatility (volatility clustering). In FHS, historical returns are first standardized by volatility estimated on

that particular day (hence the word Filtered). The filtering process yields approximately i.i.d returns (residuals) that are

suited for historical simulation. 29

read Barone-Adesi et al (2000) paper in DUO on FHS

Principal Component Analysis (Estimating risk factors using PCA)

30

Estimating risk factors using PCA

31

Estimating risk factors using PCA

32

Estimating risk factors using PCA

33

Estimating risk factors using PCA

34

Estimating risk factors using PCA

35

Estimating risk factors using PCA

36

Estimating risk factors using PCA

37

PCA and risk management

38

PCA and risk management

39

Other related VaR measures

40

Other related VaR measures

41

Other related VaR measures

42

Other related VaR measures

43

Other related VaR measures

44

Other related VaR measures

45

Other related VaR measures

46

Other related VaR measures

47

Topic # 7: Univariate and Multivariate Volatility Estimation Financial Risk Management 2010-11 February 28, 2011

FRM

c Dennis PHILIP 2011

1 Volatility modelling

1

2

Volatility modelling Volatility refers to the spread of all likely outcomes of an uncertain variable. It can be measured by sample standard deviation v u T u 1 X (rt )2 ^=t T 1 t=1

where rt is the return on day t, and is the average return over the T day period. But this statistic only measures the spread of a distribution and not the shape of a distribution (except normal and lognormal). Black Scholes model assumes that asset prices are lognormal (which implies that returns are normally distributed). In practice, returns are however non-normal and also the return ‡uctuations are time varying.

FRM

c Dennis PHILIP 2011

1 Volatility modelling

3

Example: daily returns of S&P 100 show features of volatility clustering

Therefore Engle (1982) proposed Autoregressive Conditional Heteroscedasticity (ARCH) models for modelling volatility Other characteristics documented in literature FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

4

– Long memory e¤ect of volatility (autocorrelations remain positive for very long lags) – Squared returns proxy volatility – Volatility asymmetry / leverage e¤ect (volatility increases if the previous day returns are negative)

1.1

Parametric volatility models

ARCH model ‘Autoregressive’because high/low volatility tends to persist, ‘Conditional’ means timevarying or with respect to a point in time, and ‘Heteroscedasticity’is a technical jargon for non-constant volatility. Consider previous t day’s squared returns ("2t 1 ; "2t 2 ; :::) that proxy volatility. It makes sence to give more weight to recent data and less weight to far away observations. FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

5

Suppose we are assuming that previous q observations a¤ect today’s returns. So today’s volatility can be 2 t

=

q X

2 j "t j

j=1

where

i

j and

Pq

j=1

j

=1

Also we can include a long run average variance that should be given some weight as well q X 2 2 j "t j t = V0 + j=1

where V0 is average variance rate and Pq j=1 j = 1

+

The weights are unknown and needs to be estimated. This is the ARCH model introduced by Engle (1982)

FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

6

ARCH (1) An ARCH(1) model is given by rt =

+ "t 2 t

=

"t 0

+

2

N 0; 2 1 "t 1

Since 2t is variance and has to be positive, we impose the condition 0

0 and

1

0

Generalization: ARCH(q) model 2 t

=

0

+

2 1 "t 1

+ ::: +

2 q "t q

where shocks up to q periods ago a¤ect the current volatility of the process.

FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

7

EWMA Exponentially weighted moving average (EWMA) model is the same as the ARCH models but the weights decrease exponentially as you move back through time. The model can be written as 2 t

where

2 t 1

=

)u2t

+ (1

1

is the constant decay rate, say 0.94.

To see that the weights cause an exponential decay, we substitute for 2t 1 :

2 t

= = (1

2 t 2

)u2t

+ (1

) u2t 1

+ (1

)u2t

+ (1

2

) u2t 2

+

1 2 2 t 2

With =0.94

2 t

= (0:06) u2t

1

+ (0:056) u2t

Substituting for FRM

2 t 2

2

+ (0:883)

2 t 2

now: c Dennis PHILIP 2011

1.1

2 t

Parametric volatility models

= (1

) u2t

= (1

) u2t 1

1

+ (1

) u2t

+ (1

) u2t 2

2

8

+

2

+

2

2 t 3

(1

+ (1

)u2t 3

+

)u2t

3 3 2 t 3

With =0.94

2 t

= (0:06) u2t 1 + (0:056) u2t 2 + (0:053) u2t 3 + (0:83) Risk Metrics uses EWMA model estimates for volatility with = 0.94.

FRM

c Dennis PHILIP 2011

2 t 3

1.1

Parametric volatility models

9

Generalized ARCH (GARCH) model This model generalizes the ARCH speci…cation. As one increase the q lags in an ARCH model for capturing the higher order ARCH e¤ects present in data, we loose parsimonity. Bollerslev (1986) proposed GARCH(p; q) 2 t

=

0

+

q X

2 j "t j

j=1

+

p X

j

2 t j

j=1

where the weights 0 0; j 0 and j 0: Further, for stationarity of this autoregressive model, we need the condition ! q p X X 2 > 3 > ::: > 1 FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

Also assume a geometric decline in that k k = a where 0

0; + 0; and 1 + 0. Hence, T is allowed to 1 + T be negative provided + 1 > j Tj: FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

15

Exponential GARCH (EGARCH) Recall that in the case of (G)ARCH models, we provided certain coe¢ cient restrictions in order to ensure 2t (conditional variance of "t ) is non-negative with probability one. An alternative way of ensuring positivity is specifying an EGARCH framework for 2t : Another characteristic of the model is that it allows for positive and negative shocks to have di¤erent e¤ects on conditional variances (unlike GARCH). The model re‡ects the fact that …nancial markets respond asymmetrically to good news and bad news. EGARCH(p,q) model is ln

2 t

=

0

+

q X

"t j

t j

j=1

+

p X

j

j

ln

+

q X

"t j

j=1

2 t j

j=1

FRM

c Dennis PHILIP 2011

j

t j

1.1

Parametric volatility models

where if theh error q "t = i terms then = E "tt = 2

t

16

N (0; 1)

Remarks:

– We can use other fat failed distributions such as student t; or GED in the case of non-normal errors. In this case, will take other forms. – Specifying the model as a logarithm ensures positivity of 2t : Therefore the leverage e¤ect is exponential rather than quadratic. – We divide the errors by the conditional standard deviations, "tt : Therefore we standardize (scale) the shocks. –

"t

captures the relative size of the shocks t and j captures the sign of the relative shocks

– The magnitude is captured by the variable that substracts the mean from the absolute value of the scaled shocks. Example:

FRM

c Dennis PHILIP 2011

1.1

Parametric volatility models

17

– Suppose we specify a EGARCH(0,1) 2 t

=

where

t 1

ln

0

+

1 t 1

= "t 1 =

+

1

[j

t 1j

]

t 1:

– Consider the estimated component: ^1

t 1

+ ^ 1 [j

t 1j

]

where ^ 1 = 0:3; ^ 1 = 0:6 and

= 0:85:

– Case 1: impact of positive scaled shock +1:0 0:3 (1) + 0:6 [j1j

0:85] = 0:39

Case 2: impact of negative scaled shock 1:0 0:3 ( 1) + 0:6 [j 1j

0:85] =

0:21

We see that positive shock has a greater impact than negative shock for ^ 1 positive. If we have ^ 1 negative, say ^ 1 = 0:3; a +1:0 shock will have an impact of 0:21 and a 1:0 shock will have an impact of 0.39.

FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

18

Thus, ^ 1 allows for the sign of the shock to have an impact on the conditional volatility; over and above the magnitude captured by ^ 1:

1.2

Non-parametric volatility models

Range-based estimators Suppose log prices of assets follow a Geometric Brownian Motion (GBM). The various variance estimators have been proposed in literature. Notation: –

volatility to be estimated

– Ct closing price on date t – Ot opening price on date t – Ht high price on date t – Lt low price on date t FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

– ct = ln Ct ing price

19

ln Ot , the normalized clos-

– ot = ln Ot ln Ct 1 , the normalized opening price – ht = ln Ht price

ln Ot , the normalized high

– lt = ln Lt price

ln Ot , the normalized low

The classical sample variance estimator of variance 2 is 1

2

^ =

T

1

T X

[(oi + ci )

2

(o + c)]

i=1

where (o + c) =

T 1X (oi + ci ) T i=1

and T is the total number of days considered. So this is the average volatility over T days. FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

20

Parkinson (1980) introduced a range estimator of daily volatility based on the highest and lowest prices on a particular day. He used the range of log prices to de…ne ^ 2t =

1 (ht 4 ln 2

lt )2

since it can be shown that E (ht 4 ln(2) 2t

lt )2 =

Garman and Klass (1980) extended Parkinson’s estimator where information about opening and closing prices are incorporated as follows: ^ 2t = 0:5 (ht

lt )2

[2 ln 2

1] c2t

Parkinson (1980) and Garman and Klass (1980) assume that the log-price follows a GBM with no drift term. This means that the average return is assumed to be equal to zero. Rogers and Satchell (1991) relaxes this assumption by using daily opening, highest, lowest, and closing prices into estimating volatility. FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

21

Rogers and Satchell (1991) estimator is given by ^ 2t = ht (ht ct ) + lt (lt ct ) This estimator performs better than the estimators proposed by Parkinson (1980) and Garman and Klass (1980). Yang and Zhang (2000) proposed a re…nement to Rogers and Satchell (1991) estimator for the presence of opening price jumps. Due to overnight volatility, the opening price and the previous day closing price are mostly not the same. Estimators that do not incorporate opening price jumps underestimate volatility. Yang and Zhang (2000) estimator is given by ^ 2 = ^ 2open + k^ 2close + (1

k)^ 2RS

where ^ 2open and ^ 2close are the classical sample variance estimators with the use of daily opening and closing prices, respectively. ^ 2RS is the average variance estimator introduced by Rogers and Satchell (1991). FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

22

The constant k is set to be k=

0:34 1:34 + (T + 1)=(T

1)

where T is the number of days.

Realized Volatility Realized volatility is referred to volatility estimates calculated using intraday squared returns at short intervals such as 5 or 15 minutes. For a series that has zero mean and no jumps, the realized volatility converges to the continuous time volatility. Consider a continuous time martigale process for asset prices dpt =

t dWt

where dWt is a standard brownian motion.

FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

23

Then the conditional variance for one-period returns, rt+1 pt+1 pt is Z t+1 2 s ds t

which is called the integrated volatility (or the true volatility) over the period t to t + 1: 2 t

We don’t know what it.

is. So we estimate

Let m be the sampling frequency such that there are m continuously compounded returns in one unit of time (say, one day). The j th return is given by rt+j=m

pt+j=m

pt+(j

1)=m

The realized volatility (in one unit of time) can be de…ned as X 2 RVt+1 = rt+j=m j=1;:::;m

FRM

c Dennis PHILIP 2011

1.2

Non-parametric volatility models

24

Then from the theory of quadratic variation, if sample returns are uncorrelated, ! Z t+1 X 2 2 p lim rt+j=m =0 s ds m!1

t

j=1;:::;m

As we increase sampling frequency, we get a consistent estimate of volatility. In the presence of jumps, RV is no longer a consistent estimator of volatility. An extension to this estimator is the standardized Realized Bipower Variation measure de…ned as [a;b] BVt+1

=

1 m

[1 (a+b)=2]

m X

rt+j=m

a

rt+(j

j=1

for a; b > 0: When jumps are large but rare, the simplest case where a = b = 1 captures the jumps well.

FRM

c Dennis PHILIP 2011

b 1)=m

1.2

Non-parametric volatility models

25

High frequency returns measured below 5 minutes are a¤ected by market microstructure e¤ects including nonsynchronous trading, discrete price observations, intraday periodic volatility patterns and bid–ask bounce.

FRM

c Dennis PHILIP 2011

2 Multivariate Volatility Models

2

26

Multivariate Volatility Models Multivariate modelling of volatilities enable us to study movements across markets and across assets (co-volatilities). Applications in …nance: asset pricing and portfolio selection, market linkages and integration between markets, hedging and risk management, etc. Consider a n-dimensional process fyt g : If we denote as the …nite vector of parameters, we can write yt = where and

t

t

( ) + "t

( ) is the conditional mean vector 1=2

"t = H t 1=2 Ht

where matrix. The N

( ) is an N

N positive de…nite

1 vector zt is such that zt

FRM

( ) zt

iidD (0; IN ) c Dennis PHILIP 2011

2 Multivariate Volatility Models

27

where IN is the identity matrix of order N: The matrix Ht is the conditional variance matrix of yt How do we parameterize Ht ? Vech Representation Bollerslev, Engle and Wooldridge (1988) propose a natural multivariate extension of the univariate GARCH(p; q) models where q X 0 vech (Ht ) = W + Ai vech "t i "t i=1

i

p X + j=1

where vech is the vector-half operator, which stacks the lower triangular elements of an N N matrix into a [N (N + 1) =2] 1 vector. The challenge in this parameterization is to ensure Ht is positive de…nite covariance matrix. Also, as the number of assets N increase, the number of parameters to be estimated is very large. FRM

c Dennis PHILIP 2011

j

vech (Ht j )

2 Multivariate Volatility Models

28

f"t g is covariance stationary if all the eigenvalues of A and B are less than 1 in modulus. Bollerslev, Engle and Wooldridge (1988) proposed a "Diagonal vech" representation where Ai and j are diagonal matrices. Example: For N = 2 assets and a period lag model (p = q = 1), 2 3 2 3 2 h11;t w1 a11 4 h21;t 5 = 4 w2 5 + 4 a22 h22;t w3 32 2 b11 54 b22 +4 b33

single-

a33 h11;t h21;t h22;t

32

"21;t 1 5 4 "2;t 1 ; "1;t "22;t 1 3 1 1 1

The diagonal restriction reduces the number of parameters but the model is not allowed to capture the interactions in variances among assets (copersistence, causality relations, asymmetries) Pq The diagonal vech is stationary i¤ i=1 aii + Pp b < 1 j=1 jj FRM

c Dennis PHILIP 2011

5

1

3 5

2 Multivariate Volatility Models

29

BEKK Representation Engle and Kroner (1995) propose a BEKK representation where q X Ht = cc + Ai 0

0

"t i " t

i

p X

A0i +

i=1

0 j

j Ht j

j=1

where c is a lower triangular matrix and therefore cc0 will be positive de…nite. Also, by estimating A and B rather than A and B ; we ensure positive de…niteness. In the case of 2 assets:

h11;t h12;t h21;t h22;t

= cc0 +

a11 a12 a21 a22 h11;t h21;t

"21;t 1 "2;t 1 ; "1;t

a11 a12 a21 a22

1 1

0

1

b11 b12 b21 b22

+

h12;t h22;t

"1;t 1 ; "2;t "22;t 1

1 1

b11 b12 b21 b22

0

To reduce the number of parameters to be estimated, we can impose a "diagonal BEKK" model where Ai and Bj are diagonal. FRM

c Dennis PHILIP 2011

1

2 Multivariate Volatility Models

30

Alternatively, we can have Ai and Bj as scalar times a matrix of ones. In this case, we will have a "scalar-BEKK" model. Diagonal BEKK andP Scalar-BEKK P are covariance stationary if qi=1 a2nn;i + pj=1 b2nn;j < P P 1 8n = 1; 2; :::; N and qi=1 a2i + pj=1 b2j < 1 respectively. Constant Conditional Correlation (CCC) Model Bollerslev (1990), assuming conditional correlations constant, proposed that conditional covariances (Ht ) can be parameterized as a product of corresponding conditional standard deviations. Ht = Dt RDt p = hii;t hjj;t ij

2 p h11;t 6 .. where Dt = 4 . FRM

p

hN N;t

3

7 5; R =

c Dennis PHILIP 2011

2 Multivariate Volatility Models 2 6 6 6 4

1

12

21

1

.. .

N1

1N

.. .

..

. 1

31

3 7 7 7 5

Each conditional standard deviations can be in turn de…ned as any univariate GARCH model such as GARCH(1,1) hii;t = wi + i "2i;t 1 + i hii;t

1

i = 1; 2; :::; N

Ht is positive de…nite i¤ all N conditional covariances are positive and R is positive de…nite. In most empirical applications, the conditional correlations are not constant. Therefore Engle (2002) and Tse and Tsui(2002) propose a generalization of the CCC model by allowing for conditional correlation matrix to be time-varying. This is the DCC model.

FRM

c Dennis PHILIP 2011

2 Multivariate Volatility Models

32

Tests for Costant Correlations Tse (2000) proposes testing the null that p hijt = ij hiit hjjt against the alternative that p hijt = ijt hiit hjjt

where the conditional variances, hiit and hjjt are GARCH-type models. The test statistic is an LM statistic which is asymptotically 2 (N (N 1) =2) : Engle and Sheppard (2001) propose another test with the null hypothesis H0 : Rt = R for all t against the alternative H1 : vech (Rt ) = vech R + ::: + p vech (Rt p )

1 vech (Rt 1 )+

The test statistic employed is again chi-squared distributed.

FRM

c Dennis PHILIP 2011

2 Multivariate Volatility Models

33

Dynamic Conditional Correlation (DCC) model Tse (2000) and Engle and Sheppard (2001) propose tests of constant conditional correlation hypothesis. In most applications, we see the hypothesis of constant conditional correlation is rejected. Engle (2002) propose the DCC framework, Ht = Dt Rt Dt where Dt is the matrix of standard deviations (as de…ned in the case of CCC), hii;t can be any univariate GARCH model and Rt is the conditional correlation matrix. We then standardize each return by the dynamic standard deviations to get standardized returns. Let i 1 h p h11t hN N t ut = "t diag FRM

c Dennis PHILIP 2011

2 Multivariate Volatility Models

34

be the vector of standardized residuals of N GARCH models. These variables now have standard deviations of one. We now model the conditional correlations of raw returns ("t ) by modelling conditional covariances of standardized returns (ut ). We de…ne Rt as 1=2

Rt = diag (Qt )

Qt diag (Qt )

1=2

where Qt is an N N symmetric positive de…nite matrix given by ! q p q X X X 0 Qt = 1 Q+ i i ut i ut j i=1

j=1

i=1

where –

i

0;

j

0;

Pq

i=1

i

+

Pp

j=1

j

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close