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14

Chapter Fourteen

The Greek Letters

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Executive Summary

This chapter covers the way in which traders working for financial institutions and market makers on the floor of an exchange hedge a portfolio of derivatives.

The software, DerivaGem for Excel, can be used to chart the relationships between any of the Greek letters and variables such as S0, K, r, and T.

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Chapter Outline14.1 Illustration14.2 Naked and Covered Positions14.3 A Stop-Loss Strategy14.4 Delta Hedging14.5 Theta14.6 Gamma14.7 Relationship Between Delta, Theta, and Gamma14.8 Vega14.9 Rho14.10 Hedging in Practice14.11 Scenario Analysis14.12 Portfolio Insurance14.13 Stock Market Volatility

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14.1 Illustration

• A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend paying stock

• S0 = $49, K = $50, r = 5%, = 20%, T = 20 weeks, = 13%

• The Black-Scholes value of the option is around $240,000• How does the bank hedge its risk to lock in a $60,000

profit?

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14.2 Naked and Covered Positions

Naked position

Take no action: wait for expiry and “hope for the best”

Covered position

Buy 100,000 shares today

this amounts to a covered call position

Both strategies leave the bank exposed to significant risk.

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14.2 Naked and Covered Positions

• Neither a naked position nor a covered position provides a satisfactory hedge.

Put-call parity shows that the exposure from writing a covered call is the same as the exposure from writing a naked put.

• For a perfect hedge the standard deviation of the cost of writing and hedging the option is zero.

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14.3 A Stop-Loss Strategy

This involves:• Buying 100,000 shares as soon as price reaches $50• Selling 100,000 shares as soon as price falls below

$50This deceptively simple hedging strategy does not work well in practice:

Purchases and subsequent sales cannot be made at K.Transactions costs could easily eat the option premium

and then some.

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14.4 Delta Hedging

• Most traders use more sophisticated hedging schemes.

• These involve calculating measures such as delta, gamma, and vega.

• Delta was introduced in Chapter 10• Delta is very closely related to the idea of the

replicating portfolio intuition.

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Delta (See Figure 14.2, page 302)

• Delta () is the rate of change of the option price with respect to the underlying security

• If you have a pricing equation, just take a derivative with respect to S

Optionprice

A

BSlope =

Stock price

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Delta Hedging

• This involves maintaining a delta neutral portfolio

• The delta of a European call on a stock paying dividends at rate q is N(d 1)e– qT

• The delta of a European put is

e– qT [N (d 1) – 1]

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Variation of Delta with Stock Price

Stock price

of call

Stock price

of put+1

-1

K K

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Delta Hedging

• The hedge position must be frequently rebalanced• Delta hedging a written option involves a “buy

high, sell low” trading rule• See Tables 14.2 (page 307) and 14.3 (page 308) for

examples of delta hedging

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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 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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Transactions costs

• Maintaining a delta-neutral position in a single option and the underlying assets is likely to be prohibitively expensive due to transactions costs.

• However, for a large portfolio of options, delta hedging is much more feasible: – Only 1 trade in the underlying assets is necessary to zero

out delta for the whole portfolio.

– The transactions costs of delta hedging could then be spread over many different trades.

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14.5 Theta

• Theta () of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time

• See Figure 14.5 for the variation of with respect to the stock price for a European call

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14.6 Gamma

• Gamma () is the rate of change of delta () with respect to the price of the underlying asset

• See Figure 14.9 for the variation of with respect to the stock price for a call or put option

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Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 14.7, page 312)

S

CStock price

S’

Callprice

C’C’’

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Interpretation of Gamma• For a delta neutral portfolio, t + ½S 2

S

Negative Gamma

S

Positive Gamma

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14.7 Relationship Between , and

• Delta () is the rate of change of the option price with respect to the underlying security

• Gamma () is the rate of change of delta () with respect to the price of the underlying asset

• Theta () of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time

S

f

2

2

S

f

S

t

f

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14.7 Relationship Between , and

• The price of a single derivative dependent on a non-dividend paying stock must satisfy the Black-Scholes-Merton differential equation:

rfS

fS

S

frS

t

f

2

222

2

1

S

f

2

2

S

f

S

t

f

rfSrS 22

2

1

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14.7 Relationship Between , and

For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q

( )r q S S r1

22 2

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14.8 Vega

• Vega () is the rate of change of the value of a derivatives portfolio with respect to volatility

• See Figure 14.11 for the variation of with respect to the stock price for a call or put option

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Managing Delta, Gamma, & Vega

can be changed by taking a position in the underlying

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

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14.9 Rho

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

• For currency options there are 2 rhos

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14.10 Hedging in Practice

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

• Whenever the opportunity arises, they improve gamma and vega

• As a portfolio becomes larger hedging becomes less expensive

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Hedging vs. Creation of a Synthetic Option

• When we are hedging we take positions that offset , , , etc.

• When we create an option synthetically we take positions that match &

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14.11 Scenario Analysis

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

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14.12 Portfolio Insurance

• It is essential to understand what portfolio insurance is:

• Portfolio insurance involves creating a long position in an option synthetically.

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Portfolio Insurance

• In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically

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

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Portfolio Insurance

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

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

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Portfolio Insurance

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

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14.13 Stock Market Volatility

Trading itself is a cause of volatility.Portfolio insurance schemes such as those just described have the potential

to increase volatility.When the market declines, they cause portfolio managers to either sell stock

or to sell index futures contracts. Either action may accentuate the decline.Selling obviously carries the potential to drive down pricesThe sale of index futures contracts is liable to drive down futures prices, this creates selling pressure on stocks via the index arbitrage mechanism.

Whether the portfolio insurance schemes affect volatility depends on how easily the market can absorb the trades that are generated by portfolio insurance.

Widespread use of portfolio insurance could have a destabilizing effect on the market—which would of course increase the necessity of portfolio insurance.

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Summary

• Delta () is the rate of change of the option price with respect to the underlying security

• Gamma () is the rate of change of delta () with respect to the price of the underlying asset

• Theta () of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time

S

f

2

2

S

f

S

t

f

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Summary

• Vega () is the rate of change of the option price with respect to the volatility of the underlying security

• Rho () of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the interest rate

f

r

f