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The Greek LettersUniversity of Macedonia

E i MBAExecutive MBA

Financial EngineeringFinancial EngineeringTextbook: Hull, J.C. “Options Futures and Other

Derivatives”, 6e, Prentice Hall, 2006

Achilleas Zapranis

Hedging the Risk of the Issuer

If an issuer can sell an option for more thanIf an issuer can sell an option for more than its worth and then hedge away all the risk for the rest of the option’s life he has locked athe rest of the option s life, he has locked a guaranteed, risk-free profit

Financial EngineeringThe Greek Letters

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Hedging the Risk of the Issuerg gcontinued

A bank has sold for $300,000 a European call option on 100,000 shares of a non dividend-p ,paying stockS0 = 49, K = 50, r = 5%, σ = 20%, 0 , , , ,

T = 20 weeks, μ = 13%The Black-Scholes value of the option isThe Black Scholes value of the option is $240,000How does the bank hedge its risk to lock in aHow does the bank hedge its risk to lock in a $60,000 profit?


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Naked & Covered Positions• Naked position

Take no actionTake no action

C d iti• Covered positionBuy 100,000 shares todayy , y

Both strategies leave the bank exposed toBoth strategies leave the bank exposed to significant risk


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Naked & Covered Positions Payoff Diagrams

Profit/Loss Profit/Loss

S SKSTK STK

Naked position Covered position


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Stop-Loss Strategy

This involves:Buying 100 000 shares as soon asBuying 100,000 shares as soon as price reaches $50S lli 100 000 hSelling 100,000 shares as soon as price falls below $50This deceptively simple hedging strategy does not work well


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Stop-Loss Strategyp gycontinued

Caveats:•The cash flows to h h dthe hedger occur at different times and must beand must be discounted•Purchases and sales cannot be made at exactly th ithe same price


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DeltaDelta (Δ) is the rate of change of theDelta (Δ) is the rate of change of the option price with respect to the underlying

Optionprice

BSlope = Δ

A Stock price


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Delta HedgingSuppose a stock call option with Δ = 0.6Every time that the stock price changes theEvery time that the stock price changes the option price changes by 60%Imagine an investor who has sold 20 call optionImagine an investor who has sold 20 call option contracts – that is options to buy 2,000 sharesIf th i f th ti i $10 h h dIf the price of the option is c = $10 he can hedge his position by buying 0.6 x 2,000 = 1,200 sharesThe gain/loss of the position would then tend to be offset by the loss/gain on the stock position


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Delta HedgingIf the stock price goes up by 1 (producing a gain of $1,200 on the shares purchased), the option price will tend to go up by 0.6 x $1 = $0.60 (producing a loss of $1,200 on the options written)

If the stock price goes down by 1 (producing a loss p g y (p gof $1,200 on the shares purchased), the option price will tend to down by 0.6 x $1 = $0.60 (producing a gain of $1,200 on the options written)


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Delta HedgingThis involves maintaining a delta neutral portfolioThe delta of a European call on a stock paying dividends at rate q ispaying dividends at rate q is

N (d 1)e– qT

The delta of a European put isThe delta of a European put is e– qT [N (d 1) – 1]


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Delta HedgingDelta Hedgingcontinued

The hedge position must be frequently g p q yrebalanced

Delta hedging a written option involves “b hi h ll l ” di la “buy high, sell low” trading rule


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Simulation of Delta hedging g gOption Closes ITM


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Using Futures for Delta Hedging

The delta of a futures contract is e(r-q)T times the delta of a spot contractthe delta of a spot contract

The position required in futures for delta hedging is therefore e-(r-q)T times the position required in the corresponding spot contract


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Delta of a PortfolioIt is the weighted average of the Δ of the individual positions in the portfolio

∑n

Suppose a financial institution in the US has the

∑=

Δ=Δi

iiw1

Suppose a financial institution in the US has the following 3 positions in GBP:

A long position in 100,000 call options with K = 0.55 and g p pexpiration date in 3 months. Δ = 0.533A short position in 200.000 call options with K = 0.56 and expiration date in 5 months. Δ = 0.468pA short position in 50.000 put options with K = 0.56 and expiration date in 2 months. Δ = -0.508


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Making the Portfolio Delta Neutral

Portfolio Δ:100,000x0.533 – 200,000x0.468 – 50,000x(-0.508)

= -14,900

Thi th t th tf li b d d ltThis means that the portfolio can be made delta neutral with a long position of 14,900 GBP

Another alternative is to take a long position on a 6-month futures for buying14,650 GBP (suppose that y g ( ppthe risk-free rate in US is 8% and in UK is 4%

14,900e-(0.08-0.04)x0.5=14,605


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Theta

Theta (Θ) of a derivative (or portfolio of derivatives) is the rate of change of the valuederivatives) is the rate of change of the value with respect to the passage of timeThe theta of a call or put is usually negativeThe theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remainingthe underlying asset and its volatility remaining the same, the value of the option declines


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GammaGamma (Γ) is the rate of change of delta (Δ) withGamma (Γ) is the rate of change of delta (Δ) with respect to the price of the underlying assetFor a European call or put option on a non dividendFor a European call or put option on a non dividend paying stock:

( )dN 1′=Γ

For a European index call or put option with a

TSσΓ

p p pdividend yield q:

( )dN qT−′( )TSedN qT

σ′

=Γ 1


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Gamma


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Gamma Addresses Delta Hedging Errors Caused By Curvature

CallCallprice

C''C'

C

C

SStock price

S'Financial Engineering

The Greek Letters20

Interpretation of GammaF d lt t l tf liFor a delta neutral portfolio,

ΔΠ ≈ Θ Δt + ½ Γ ΔS 2

ΔΠΔΠ

ΔS

ΔS

Positive Gamma Negative GammaFinancial Engineering

The Greek Letters

Positive Gamma Negative Gamma21

Making a Portfolio Gamma Neutral

A position on the underlying or on a future has zero Γhas zero Γ

The only way to change the portfolio Γ is byThe only way to change the portfolio Γ is by taking a position on wT options with ΓΤ

The portfolio Γ then becomes: wTΓΤ + ΓwT ΓΤ Γ


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Making a Portfolio Gamma Neutral (continued)

Hence, the required position on the option in order for the portfolio to become gamma neutral is

-Γ/ ΓΤHowever, including the option in the portfolio changes the portfolio’s Δ, so the position on the

d l i h b h d i i d lunderlying has to be changed to maintain delta neutralityA ti th iti th ti iAs time passes, the position on the option is being adjusted so that the portfolio remains gamma neutral


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gamma neutral

Making a Portfolio Gamma Neutral (Example)

Suppose that portfolio is delta neutral with Γ = -3,000Suppose a call option with Δ = 0.62 and Γ = 1.50The portfolio will become gamma neutral if weThe portfolio will become gamma neutral if we include a long position on 3,000/1.5 = 2000 call optionsHowever, the portfolio Δ will change from 0 to 2,000 x 0.62 = 1,240A quantity 1,240 of the underlying asset must be sold


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Relationship Between Delta, Gamma, and Theta

For a portfolio of derivatives on a stock paying a continuous dividend yield at p y g yrate q

Θ Δ Γ Π+ − + =( )r q S S r1 2 2σΘ Δ Γ Π+ +( )r q S S r2

σ


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Vega

Vega (ν) is the rate of change of the value of a derivatives portfolio with respect toof a derivatives portfolio with respect to volatilityVega tends to be greatest for options that are close to the moneyA position on the underlying asset or on a future written on the underlying has zerofuture written on the underlying has zero vega


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Vega (continued)

•If V is the vega of the portfolio and VT is the vega of a traded option a position of V/V invega of a traded option, a position of–V/VT in the option makes the portfolio vega neutral•Unfortunately a portfolio that is gamma•Unfortunately a portfolio that is gamma neutral will not in general be vega neutral, and vise versa•If a hedger requires a portfolio to be both gamma and vega neutral, at least two traded derivatives dependent on the underlying must be used


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Vega (continued)

Consider a portfolio that is delta neutral, with Γ = -5,000 and V=-8,000Suppose a traded option with Γ=0.5, V=2, Δ=0.6The portfolio can be made vega neutral by including a long position on 8,000/2 = 4,000 optionsThis would increase Δ from 0 to 4,000 x 0.6 = 2 4002,400It is required 2,400 units of the asset to be sold for the portfolio to remain delta neutralthe portfolio to remain delta neutralΓ will change from –5,000 to -5,000 + 0.5 x 4,000 = -3,000


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,

Vega (continued)

To make the portfolio vega and gamma neutral two options are neededoptions are neededSuppose that there is a 2nd option with Γ=0.8,V=1 2 Δ=0 5V=1.2, Δ=0.5If w1 and w2 are the quantities of the two options included in the portfolio then it is required thatincluded in the portfolio, then it is required that

5 000 + 0 5 x w +0 8 x w = 0-5,000 + 0.5 x w1 +0.8 x w2 = 0-8,000 + 2.0 x w1 +1.2 x w2 = 0


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Vega (continued)

The solution to these equation isw1 = 400w1 = 400w2 = 6,000

The Δ of the portfolio will change to

400 x 0.6 + 6,000 x 0.5 = 3,240

H 3 240 it f th d l i t t bHence, 3,240 units of the underlying asset must be sold to maintain delta neutrality


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Managing Delta, Gamma, & Vega

• Δ can be changed by taking a position in g y g pthe underlying

To adjust Γ & ν it is necessary to take aTo adjust Γ & ν it is necessary to take a position in an option or other derivative


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RhoRho is the rate of change of the value of a derivative with respect to the pinterest rateFor currency options there are 2 rhosFor currency options there are 2 rhosFor example, suppose a European put index with Rho = 52 67index with Rho = -52.67 That means that for 1% change in the i k f t ( f 8% t 9%) thrisk-free rate (e.g., from 8% to 9%) the

value of the option increases by 0.5267


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Hedging in Practice

Traders usually ensure that their portfolios are delta-neutral at least once a dayare delta neutral at least once a dayWhenever the opportunity arises, they improve gamma and vegaimprove gamma and vegaAs portfolio becomes larger hedging becomes less expensive


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Scenario Analysis

A scenario analysis involves testing the effect on the value of a portfolio of differenton the value of a portfolio of different assumptions concerning asset prices and their volatilitiestheir volatilities


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Hedging vs. Creation of an Option Synthetically

When we are hedging we take positions that offset Δ, Γ, ν, etc.When we create an optionWhen we create an option synthetically we take positions that

t h Δ Γ & νmatch Δ, Γ, & ν


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Portfolio InsuranceIn October of 1987 many portfolio managers attempted to create a put option g p p pon a portfolio syntheticallyThis involves initially selling enough of theThis involves initially selling enough of the portfolio (or of index futures) to match the Δof the put optionof the put option


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Portfolio Insurancecontinued

As the value of the portfolio increases, the Δof the put becomes less negative and someof the put becomes less negative and some of the original portfolio is repurchasedAs the value of the portfolio decreases the ΔAs the value of the portfolio decreases, the Δof the put becomes more negative and more

f th tf li t b ldof the portfolio must be sold


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Portfolio Insurancecontinued

Th t t did t k ll O t b 19The strategy did not work well on October 19, 1987...


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Assignment QuestionsHull, Chapter 15, p. 371

Try to answer assignment questions: 15.25

OptionalOptional

This presentation is an augmented version of the presentation accompanying Chapter 15 of J. Hull’s textbook “Option’s Futures and other Derivatives”, 6e, Prentice Hall, 2006, downloaded from http://www.rotman.utoronto.ca/~hull


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