《金融期货与期权》（英文版） Chapter 14 The greek letters

The greek letters Chapter 14 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.1 The Greek Letters Chapter 14

14.2 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%,o=20%, T=20 Weeks, u=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, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.2 Example • A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock • S0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks, m = 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?

14.3 Naked covered positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.3 Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk

14.4 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 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.4 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

Delta(See Figure 14.2, page 302) Delta(A)is the rate of change of the option price with respect to the underlying Option price S|ope=△ a Stock price Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.5 Delta (See Figure 14.2, page 302) • Delta (D) is the rate of change of the option price with respect to the underlying Option price A B Slope = D Stock price

Delta Hedging 14.6 This involves maintaining a delta neutral portfolio The delta of a european call on a stock paying dividends at rate q is n(d de-q4 The delta of a european put is eq[N(d1)-1] Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.6 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]

14.7 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 14.2(page 307) and 14.3 (page 308 )for examples of delta hedging Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.7 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 14.2 (page 307) and 14.3 (page 308) for examples of delta hedging

14.8 Using Futures for Delta Hedging The delta of a futures contract is elra)r times the delta of a spot contract The position required in futures for delta hedging is therefore e-(r-q) times the position required in the corresponding spot contract Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.8 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

14.9 Theta Theta(o)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 o with respect to the stock price for a European call Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.9 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 • See Figure 14.5 for the variation of Q with respect to the stock price for a European call

14.10 G amma Gamma r)is the rate of change of delta (a)with respect to the price of the underlying asset See Figure 14.9 for the variation of d with respect to the stock price for a call or put option Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 14.10 Gamma • Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset • See Figure 14.9 for the variation of G with respect to the stock price for a call or put option