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22 Pages

### CH15HullOFOD6thEd

Course: FIR 7721, Fall 2008
School: U. Memphis
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Word Count: 902

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Greek The Letters Chapter 15 Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.1 Example A bank has sold for \$300,000 a European call option on 100,000 shares of a nondividend paying stock S = 49, K = 50, r = 5%, = 20%, 0 T = 20 weeks, = 13% The Black-Scholes value of the option is \$240,000 How does the bank hedge its risk to lock in a \$60,000 profit? Options, Futures,...

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Greek The Letters Chapter 15 Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.1 Example A bank has sold for \$300,000 a European call option on 100,000 shares of a nondividend paying stock S = 49, K = 50, r = 5%, = 20%, 0 T = 20 weeks, = 13% The Black-Scholes value of the option is \$240,000 How does the bank hedge its risk to lock in a \$60,000 profit? Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.2 Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.3 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 \$50 This deceptively simple hedging strategy does not work well Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.4 Delta (See Figure 15.2, page 345) Delta () is the rate of change of the option price with respect to the underlying Option price Slope = A Stock price 15.5 B Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 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] Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.6 Delta Hedging continued The hedge position must be frequently rebalanced Delta hedging a written option involves a "buy high, sell low" trading rule See Tables 15.2 (page 350) and 15.3 (page 351) for examples of delta hedging Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.7 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 Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.8 Theta Theta () of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time The 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 remaining the same, the value of the option declines Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.9 Gamma Gamma () is the rate of change of delta () with respect to the price of the underlying asset Gamma is greatest for options that are close to the money (see Figure 15.9, page 358) Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.10 Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 15.7, page 355) Call price C'' C' C Stock price S S' 15.11 Options, and Futures, Other Derivatives, 6th Edition, Copyright John C. Hull 2005 Interpretation of Gamma For a delta neutral portfolio, t + S 2 S S Positive Gamma Negative Gamma 15.12 Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 Relationship Between Delta, Gamma, and Theta For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q 1 2 2 + (r - q ) S + S = r 2 Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.13 Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility Vega tends to be greatest for options that are close to the money (See Figure 15.11, page 361) Vega Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.14 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 Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.15 Rho Rho is the rate of change of the value of a derivative with respect to the interest rate currency options there are 2 rhos For Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.16 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 portfolio becomes larger hedging becomes less expensive Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005 15.17 Scenario Analysis A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities Options, Futures, and Othe...

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