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6.1 The Greek Letters Lecture 6 6.2 Example • A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock • S0 = 49, X = 50, r = 5%, s = 20%, T = 20 weeks, • The Black-Scholes value of the option is $240,000 • How does the bank hedge its risk? 6.3 Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk 6.4 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 6.5 Delta • Delta (D) is the rate of change of the option price with respect to the underlying Option price Slope = D B A Stock price 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 – long in a call is delta-hedged by a short position in the stock • The delta of a European put is e– qT [N (d 1) – 1] - long in a put is delta-hedged by a long position in the stock 6.6 6.7 1 X 6.8 Delta Hedging continued • The hedge position must be frequently rebalanced • Delta hedging a written option involves a “buy high, sell low” trading rule • see delta_hedging.xls 6.9 Delta Hedging continued • delta_hedging is then equivalent to create a long position in the option synthetically • this means that you can create a new way of managing money that replicates the non-linear pay-off of an option 6.10 Theta • Theta (Q) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time • (almost) always negative 6.11 x theta for a european call 6.12 Gamma • Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset • See Figure for the variation of G with respect to the stock price for a call or put option 6.13 x gamma for a european call/put Gamma Addresses Delta Hedging 6.14 Errors Caused By Curvature Call price C’’ C’ C Stock price S S ’ 6.15 Relationship Among Delta, Gamma, and Theta For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q. From Black-Scholes equation: 2 ƒ ƒ ƒ 2 2 rS ½s S rƒ 2 t S S 1 2 2 Q (r q) SD s S G r 2 Interpretation of Gamma 6.16 • For a delta neutral portfolio, D Q Dt + ½GDS 2 D D DS DS Positive Gamma long call+short stock long put + long stock Negative Gamma viceversa 6.17 Vega • Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility • See Figure for the variation of n with respect to the stock price for a call or put option 6.18 x vega for a european call/put 6.19 Managing Delta, Gamma, & Vega D can be changed by taking a position in the underlying • To adjust G & n it is necessary to take a position in an option or other derivative 6.20 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