期权期货及其衍生品第18弹.ppt

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1、Chapter 18The Greek Letters,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,1,Example,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%,s=20%,T=20 weeks,m=13%The Black-Scholes-Merton value of the option is

2、$240,000How does the bank hedge its risk to lock in a$60,000 profit?,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,2,Naked&Covered Positions,Naked positionTake no actionCovered positionBuy 100,000 shares todayWhat are the risks associated with these strategies?,Options

3、,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,3,Stop-Loss Strategy,This involves:Buying 100,000 shares as soon as price reaches$50Selling 100,000 shares as soon as price falls below$50,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,4,Stop-Loss St

4、rategy continued,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,5,Ignoring discounting,the cost of writing and hedging the option appears to be max(S0K,0).What are we overlooking?,Delta(See Figure 18.2,page 381),Delta(D)is the rate of change of the option price with res

5、pect to the underlying,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,6,Hedge,Trader would be hedged with the position:short 1000 optionsbuy 600 sharesGain/loss on the option position is offset by loss/gain on stock positionDelta changes as stock price changes and time

6、passesHedge position must therefore be rebalanced,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,7,Delta Hedging,This involves maintaining a delta neutral portfolioThe delta of a European call on a non-dividend paying stock is N(d 1)The delta of a European put on the st

7、ock is N(d 1)1,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,8,The Costs in Delta Hedgingcontinued,Delta hedging a written option involves a“buy high,sell low”trading rule,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,9,First Scenario for

8、 the Example:Table 18.2 page 384,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,10,Second Scenario for the Example Table 18.3,page 385,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,11,Theta,Theta(Q)of a derivative(or portfolio of derivativ

9、es)is the rate of change of the value with respect to the passage of timeThe 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 call or put option declines,Options,Futures,and Othe

10、r Derivatives,8th Edition,Copyright John C.Hull 2012,12,Theta for Call Option:K=50,s=25%,r=5%T=1,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,13,Gamma,Gamma(G)is the rate of change of delta(D)with respect to the price of the underlying assetGamma is greatest for optio

11、ns that are close to the money,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,14,Gamma for Call or Put Option:K=50,s=25%,r=5%T=1,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,15,Gamma Addresses Delta Hedging Errors Caused By Curvature(Figu

12、re 18.7,page 389),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,16,S,C,Stock price,S,Callprice,C,C,Interpretation of Gamma,For a delta neutral portfolio,DP Q Dt+GDS 2,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,17,Relationship Between D

13、elta,Gamma,and Theta(page 393),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,18,For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q it follows from the Black-Scholes-Merton differential equation that,Vega,Vega(n)is the rate of change

14、of the value of a derivatives portfolio with respect to volatility,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,19,Vega for Call or Put Option:K=50,s=25%,r=5%T=1,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,20,Taylor Series Expansion(Ap

15、pendix to Chapter 18),The value of a portfolio of derivatives dependent on an asset is a function of of the asset price S,its volatility s,and time t,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,21,Managing Delta,Gamma,&Vega,Delta can be changed by taking a position i

16、n the underlying assetTo adjust gamma and vega it is necessary to take a position in an option or other derivative,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,22,Example,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,23,What position in

17、option 1 and the underlying asset will make the portfolio delta and gamma neutral?Answer:Long 10,000 options,short 6000 of the assetWhat position in option 1 and the underlying asset will make the portfolio delta and vega neutral?Answer:Long 4000 options,short 2400 of the asset,Example continued,Opt

18、ions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,24,What position in option 1,option 2,and the asset will make the portfolio delta,gamma,and vega neutral?We solve5000+0.5w1+0.8w2=08000+2.0w1+1.2w2=0to get w1=400 and w2=6000.We require long positions of 400 and 6000 in option

19、 1 and option 2.A short position of 3240 in the asset is then required to make the portfolio delta neutral,Rho,Rho is the rate of change of the value of a derivative with respect to the interest rate,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,25,Hedging in Practice,

20、Traders usually ensure that their portfolios are delta-neutral at least once a dayWhenever the opportunity arises,they improve gamma and vegaThere are economies of scaleAs portfolio becomes larger hedging becomes less expensive per option in the portfolio,Options,Futures,and Other Derivatives,8th Ed

21、ition,Copyright John C.Hull 2012,26,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,8th Edition,Copyright John C.Hull 2012,27,Greek Letters for

22、European Options on an Asset that Provides a Yield at Rate q,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,28,Futures Contract Can Be Used for Hedging,The delta of a futures contract on an asset paying a yield at rate q is e(rq)T times the delta of a spot contractThe p

23、osition required in futures for delta hedging is therefore e(rq)T times the position required in the corresponding spot contract,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,29,Hedging vs Creation of an Option Synthetically,When we are hedging we take positions that o

24、ffset delta,gamma,vega,etcWhen we create an option synthetically we take positions that matchdelta,gamma,vega,etc,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,30,Portfolio Insurance,In October of 1987 many portfolio managers attempted to create a put option on a portf

25、olio syntheticallyThis involves initially selling enough of the portfolio(or of index futures)to match the D of the put option,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,31,Portfolio Insurancecontinued,As the value of the portfolio increases,the D of the put becomes

26、 less negative and some of the original portfolio is repurchasedAs the value of the portfolio decreases,the D of the put becomes more negative and more of the portfolio must be sold,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,32,Portfolio Insurancecontinued,The strategy did not work well on October 19,1987.,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,33,