1 The Greek Letters Chapter 17. 2 Goals OTC risk management by option market makers may be problematic due to unique features of the options that are.

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The Greek LettersThe Greek Letters

Chapter 17Chapter 17

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GoalsGoals OTC risk management by option OTC risk management by option

market makers may be problematic market makers may be problematic due to unique features of the options due to unique features of the options that are not available on exchanges.that are not available on exchanges.

Many dimensions of risk (greeks) Many dimensions of risk (greeks) must be managed so that all risks must be managed so that all risks are acceptable.are acceptable.

Synthetic options and portfolio Synthetic options and portfolio insurance insurance

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DeltaDelta Delta (Delta () is the rate of change of the option ) is the rate of change of the option

price with respect to the underlying:price with respect to the underlying:

==f/f/SS Remember, this creates a hedge Remember, this creates a hedge

(replication of option payoff):(replication of option payoff):

**S - S - f = 0f = 0 Two ways to think about it:Two ways to think about it:

If If S =$1 => S =$1 => f = $f = $ If short 1 option, then need to buy If short 1 option, then need to buy

shares to hedge itshares to hedge it

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ExampleExample Let Let cc=0.7. What does this mean?=0.7. What does this mean?

--c + 0.7c + 0.7S = 0S = 0 Short call can be hedged by Short call can be hedged by

buying 0.7 shares orbuying 0.7 shares or For $1 change in S, C changes For $1 change in S, C changes

by 70 cents.by 70 cents.

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Example: Cost of hedgingExample: Cost of hedging Assume the following conditions: Assume the following conditions:

r = 10%, r = 10%, T = 6 months; T = 6 months; SS00 = 20; u=1.1; = 20; u=1.1; d = 0.9; d = 0.9; K = 20. K = 20.

Compute the cost of hedging a short Compute the cost of hedging a short call position on a two-period binomial call position on a two-period binomial tree (each period = 3 months).tree (each period = 3 months).

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Example: Cost of hedgingExample: Cost of hedging SSuuuu=24.2; =24.2; SSudud=S=Sdudu=19.8; =19.8; SSdddd=16.2=16.2 CCuuuu=4.2; C=4.2; Cudud=C=Cdudu=C=Cdddd=0=0 Risk-neutral probability: Risk-neutral probability:

p = (ep = (e0.1/40.1/4 – 0.9)/(1.1-0.9)=0.62658 – 0.9)/(1.1-0.9)=0.62658

CCuu=e=e-0.1/4-0.1/4*p*4.2 = 2.56667; C*p*4.2 = 2.56667; C00 = 1.5685 = 1.5685

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Example (cont’d)Example (cont’d) The delta is changing over time:The delta is changing over time:

uu = (4.2-0)/(24.2-19.8) = 0.95455 = (4.2-0)/(24.2-19.8) = 0.95455 00 = (2.56667-0)/(22-18) = 0.64167 = (2.56667-0)/(22-18) = 0.64167 dd = 0 = 0

Assume that the stock ends up going Assume that the stock ends up going up twice in a row: uu moveup twice in a row: uu move

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Example (cont’d)Example (cont’d) Hedging cost can be computed as follows:Hedging cost can be computed as follows:

t=0 t=0 : the risk-free portfolio is - C + 0.64167*S: the risk-free portfolio is - C + 0.64167*S Cost(t=0) = 0.64267*20 = 12.8334 (borrow it)Cost(t=0) = 0.64267*20 = 12.8334 (borrow it) t=1t=1: : uu = 0.95455 => buy ( = 0.95455 => buy (uu - - 00) = 0.31288 ) = 0.31288

more sharesmore shares Cost(t=1) = 0.31288*22 = 6.88336Cost(t=1) = 0.31288*22 = 6.88336 t=2t=2 : option is exercised by the long position : option is exercised by the long position

holderholder You get:-C + You get:-C + uu*S = 0.95455*24.2 – 4.2 *S = 0.95455*24.2 – 4.2

=18.90011 (inflow)=18.90011 (inflow)

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Example (cont’d)Example (cont’d)

Present Value of the costs:Present Value of the costs: PVPVt=0t=0 = 12.8334 + 6.88336e = 12.8334 + 6.88336e-0.1/4-0.1/4

– – 18.90011e18.90011e-0.1/2-0.1/2 = 1.5685 = call = 1.5685 = call premiumpremium

In practice, transaction costs and In practice, transaction costs and discreteness of hedging lead to discreteness of hedging lead to imperfect hedgingimperfect hedging

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Example (cont’d)Example (cont’d) What if the path is What if the path is dudu??

dd = 0 = 0

No shares needed to hedge the short call No shares needed to hedge the short call as there is no uncertainty about its value as there is no uncertainty about its value any more (it is OTM).any more (it is OTM).

At home confirm that in this case, i.e., At home confirm that in this case, i.e., dudu path, the cost of hedging is also equal to path, the cost of hedging is also equal to call premium and that it does not depend call premium and that it does not depend on the path of the underlying asset.on the path of the underlying asset.

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Example (cont’d)Example (cont’d) What if the stock pays a dividend What if the stock pays a dividend

yield q = 4%? Tree is the same. yield q = 4%? Tree is the same. What are the What are the ’s now?’s now?

Need to take into account both Need to take into account both the cap. gains and the dividends the cap. gains and the dividends now:now:

- C- Cuu + + uu S Suu

-Cuu + u *Suueqt

Cud + u *Sudeqt

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Example (cont’d)Example (cont’d)

Thus,Thus,

Delta is smaller. Is option price Delta is smaller. Is option price bigger or lower in this case? bigger or lower in this case? What about costs of hedging?What about costs of hedging?

94505.0

95455.04/04.0

eS

Ce tq

u

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Deltas for Currency and Deltas for Currency and Futures OptionsFutures Options

Let Let ff be the price of an option. be the price of an option. Currency options:Currency options:

Futures options (F is the futures Futures options (F is the futures price):price):

S

fe tr

uf

F

fe tr

u

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Two Special CasesTwo Special Cases Forwards:Forwards:

SS00

For For S – fS – f to be a hedge, to be a hedge, = = 1(perfect short hedge)1(perfect short hedge)

It is all cap gains here, no income.It is all cap gains here, no income.

Su

Su – Ke-r(T-t)=fu

Sd

Sd – Ke-r(T-t)=fd

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Two Special Cases cont’dTwo Special Cases cont’d Futures:Futures:

SS00

For For S – f to be a hedge, S – f to be a hedge, = e = er(T-r(T-t)t) (perfect short hedge); can ignore (perfect short hedge); can ignore t, if t, if small.small.

It is all income here, no cap gains.It is all income here, no cap gains.

Su

Fu – F0

Sd

Fd – F0

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Using Futures for Delta Using Futures for Delta HedgingHedging

Futures can be used to make an option position Futures can be used to make an option position --neutral.neutral.

The delta of an option contract on a dividend-paying The delta of an option contract on a dividend-paying asset is asset is ee-(-(r-qr-q))TT times the delta of a spot contract: times the delta of a spot contract:

The position required in futures for delta hedging is The position required in futures for delta hedging is therefore therefore ee--((r-qr-q))TT times the position required in the times the position required in the corresponding spot contractcorresponding spot contract

TqreS

c

f

S

S

c

f

c )(

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Using Futures for Delta Using Futures for Delta HedgingHedging

Example: hedging a currency option on Example: hedging a currency option on £ £ requires a short position in £458,000. If r=10% requires a short position in £458,000. If r=10% and rf = 13%, how many 9-month futures and rf = 13%, how many 9-month futures contracts on £ would achieve the same contracts on £ would achieve the same objective?objective?

One contract is for One contract is for £62,500 => need to sell £62,500 => need to sell about 7 contractsabout 7 contracts

442,468000,458 12/9)13.1(. ef

c

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ThetaTheta Theta (Theta () of a derivative (or ) of a derivative (or

portfolio of derivatives) is the rate portfolio of derivatives) is the rate of change of the value with respect of change of the value with respect to the passage of timeto the passage of time

0t

c

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GammaGamma Gamma (Gamma () is the rate of change ) is the rate of change

of delta (of delta () with respect to the ) with respect to the price of the underlying assetprice of the underlying asset

In our previous example:In our previous example:

Assets linear in S (futures, stock) Assets linear in S (futures, stock) do not affect Gamma.do not affect Gamma.

23864.01822

095455.0

du

du

SSS

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Gamma Addresses Delta Gamma Addresses Delta Hedging Errors Caused By Hedging Errors Caused By

Curvature Curvature

S

CStock price

S’

Callprice

C’C’’

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Interpretation of GammaInterpretation of Gamma For a delta neutral portfolio, For a delta neutral portfolio, tt + ½ + ½SS 22

S

Negative Gamma

S

Positive Gamma

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VegaVega Vega (Vega () is the rate of change of ) is the rate of change of

the value of a derivatives portfolio the value of a derivatives portfolio with respect to volatilitywith respect to volatility

See Figure 17.11 for the variation See Figure 17.11 for the variation

of of with respect to the stock price with respect to the stock price for a call or put optionfor a call or put option

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Managing Delta, Gamma, Managing Delta, Gamma, & Vega& Vega

can be changed by taking a position in the can be changed by taking a position in the underlying or the futures on itunderlying or the futures on it

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

The Greeks of a portfolio are computed as follows:The Greeks of a portfolio are computed as follows:

N

iiin

1

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Delta and Gamma of a Delta and Gamma of a Protective PutProtective Put

Protective put = long in the put and Protective put = long in the put and the stock. The put has a delta of -0.4. the stock. The put has a delta of -0.4. Let a call on the stock have a delta of Let a call on the stock have a delta of 0.6 and a gamma of 2.0.6 and a gamma of 2.

Delta of the protective put:Delta of the protective put: = = pp + + SS = -0.4+1 = 0.6 (why is it equal = -0.4+1 = 0.6 (why is it equal

to call delta?)to call delta?) Gamma: Gamma: = 2+0=2 = 2+0=2 You cannot change gamma buy trading You cannot change gamma buy trading

sharesshares

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RhoRho

Rho is the rate of change of Rho is the rate of change of the value of a derivative with the value of a derivative with respect to the interest raterespect to the interest rate

For currency options there are For currency options there are 2 rho’s2 rho’s

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Hedging in PracticeHedging in Practice

Traders usually ensure that their Traders usually ensure that their portfolios are delta-neutral at least portfolios are delta-neutral at least once a dayonce a day

Whenever the opportunity arises, Whenever the opportunity arises, they improve gamma and vegathey improve gamma and vega

As portfolio becomes larger hedging As portfolio becomes larger hedging becomes less expensivebecomes less expensive

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Hedging vs Creation of an Hedging vs Creation of an Option SyntheticallyOption Synthetically

When we are hedging we take When we are hedging we take

positions that offset positions that offset , , , , , etc., etc. When we create an option When we create an option

synthetically we take positions synthetically we take positions

that match that match &&

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Portfolio InsurancePortfolio Insurance In October of 1987 many portfolio In October of 1987 many portfolio

managers attempted to create a put managers attempted to create a put option on a portfolio syntheticallyoption on a portfolio synthetically

This involves initially selling enough of This involves initially selling enough of the portfolio (or of index futures) to the portfolio (or of index futures) to match the match the of the put option of the put option

In our earlier example: at t=0 we need In our earlier example: at t=0 we need to sell -to sell -pp = 1 – 0.64167 = 0.35833 = = 1 – 0.64167 = 0.35833 = 35.83% of the portfolio and invest the 35.83% of the portfolio and invest the proceeds into riskless assets.proceeds into riskless assets.

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Portfolio InsurancePortfolio Insurance Can achieve the same with index Can achieve the same with index

futures. One benefit is lower futures. One benefit is lower transactions costs.transactions costs.

RecallRecall

Remember, T is the maturity of Remember, T is the maturity of the futures.the futures.

S

pe

f

p Tqr

)(

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Portfolio InsurancePortfolio Insurancecontinuedcontinued

As the value of the portfolio As the value of the portfolio increases, the increases, the of the put becomes of the put becomes less negative and some of the less negative and some of the original portfolio is repurchasedoriginal portfolio is repurchased

As the value of the portfolio As the value of the portfolio decreases, the decreases, the of the put becomes of the put becomes more negative and more of the more negative and more of the portfolio must be soldportfolio must be sold

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Portfolio InsurancePortfolio Insurancecontinuedcontinued

The strategy did not work well on The strategy did not work well on October 19, 1987...October 19, 1987...