1、Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,1,The Greek Letters,Chapter 17,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,2,Example,A bank has sold for $300,000 a European call option on 100,000 shar

2、es of a non-dividend 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?,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,3,

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, 7th International Edition, Copyright John C. Hull 2008,4,Stop-Loss Strategy,This involves: Buying 100,

4、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, 7th International Edition, Copyright John C. Hull 2008,5,Delta (See Figure 17.2, page 353),Delta (D) is

5、 the rate of change of the option price with respect to the underlying,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,6,Delta Hedging,This involves maintaining a delta neutral portfolio The delta of a European call on a non-dividend paying stock is N

6、(d 1) The delta of a European put on the stock is N (d 1) 1,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,7,Delta Hedgingcontinued,The hedge position must be frequently rebalanced Delta hedging a written option involves a “buy high, sell low” trading

7、 rule See Tables 17.2 (page 356) and 17.3 (page 357) for examples of delta hedging,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,8,Theta,Theta (Q) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the pa

8、ssage 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 a long option declines See Figure 17.5 for the variation of Q with respect to the stock price for a European call,O

9、ptions, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,9,Gamma,Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset Gamma is greatest for options that are close to the money (see Figure 17.9, page 364),Options, Futures

10、, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,10,Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 17.7, page 361),S,C,Stock price,S,Call price,C,C,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,11,Inte

11、rpretation of Gamma,For a delta neutral portfolio, DP Q Dt + GDS 2,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,12,Relationship Between Delta, Gamma, and Theta (page 365),For a portfolio of derivatives on a stock paying a continuous dividend yield a

12、t rate q,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,13,Vega,Vega (n) 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 17.11, page

13、 366),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,14,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,Options, Futures, and Oth

14、er Derivatives, 7th International Edition, Copyright John C. Hull 2008,15,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,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008

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, 7th International Edition, Copyr

16、ight John C. Hull 2008,17,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 Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,18,Greek Let

17、ters for Options on an Asset that Provides a Dividend Yield at Rate q,See Table 17.6 on page 370,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,19,Futures Contract Can Be Used for Hedging,The delta of a futures contract on an asset paying a yield at r

18、ate q 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, 7th International Edition, Copyright John C. Hull 2008,20,Hedging vs Cr

19、eation of an Option Synthetically,When we are hedging we take positions that offset D, G, n, etc. When we create an option synthetically we take positions that match D, G, & n,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,21,Portfolio Insurance,In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically This involves initiall