1、Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.1The Greek LettersChapter 13Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.2ExampleA FI has SOLD for $300

2、,000 a European call on100,000 shares of a non-dividend paying stock:S0 = 49 X = 50r = 5% = 20% = 13% T = 20 weeksThe Black-Scholes value of the option is $240,000How does the FI hedge its risk?Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Norma

3、l University13.3Naked & Covered PositionsNaked position (裸期權頭寸策略)Take NO actionCovered position(抵補期權頭寸策略)Buy 100,000 shares todayBoth strategies leave the FI exposedto significant riskOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal Universi

4、ty13.4Stop-Loss StrategyThis involvesFully covering the option as soon as it movesin-the-moneyStaying naked the rest of the time This deceptively simple hedging strategydoes NOT work well !Transactions costs, discontinuity of prices, andthe bid-ask bounce kills itOptions, Futures, and Other Derivati

5、ves, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.5DeltaDelta () is the rate of change of the option price with respect to the underlyingFigure 13.2 (p. 311)Option PriceABStock PriceSlope = Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTa

6、ng Yincai, 2003, Shanghai Normal University13.6Delta HedgingThis involves maintaining a delta neutral portfolioThe delta of a European call on a stock paying dividends at a rate q is The delta of a European put is The hedge position must be frequently rebalancedDelta hedging a written option involve

7、s a“BUY high, SELL low” trading ruleOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.7Delta Neutral Portfolio Example(in-the-money) Cum. Cost of Cost Stock Shares Shares Incl. Int.Week Price Delta Purch. Purch. Interest Cost 049

8、.000 0.522 52,2002,557.8 2,557.8 2.5 148.120 0.458 (6,400) (308.0) 2,252.3 2.2 247.370 0.400 (5,800) (274.7) 1,979.8 1.9 1854.620 0.990 1,200 65.5 5,197.3 5.0 1955.870 1.000 1,000 55.9 5,258.2 5.1 2057.250 1.000 0 0.0 5,263.3Table 13.2 (p. 314)Options, Futures, and Other Derivatives, 4th edition 200

9、0 by John C. HullTang Yincai, 2003, Shanghai Normal University13.8Delta Neutral Portfolio Example(out-of-the-money) Cum. Cost of Cost Stock Shares Shares Incl. Int.Week Price Delta Purch. Purch. Interest Cost 049.000 0.522 52,2002,557.8 2,557.8 2.5 149.750 0.568 4,600 228.0 2,789.2 2.7 252.000 0.705

10、 13,700 712.4 3,504.3 3.4 1848.130 0.183 12,100 582.4 1,109.6 1.1 1946.630 0.007 (17,600) (820.7) 290.0 0.3 2048.120 0.000 (700) (33.7) 256.6Table 13.3 (p. 315)Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.9Delta for FuturesF

11、rom Chapter 3, we havewhere T* is the maturity of futures contractThus, the delta of a futures contract isSo, if HA is the required position in the asset for delta hedging and HF is the required position in futures for the same delta hedging,Options, Futures, and Other Derivatives, 4th edition 2000

12、by John C. HullTang Yincai, 2003, Shanghai Normal University13.10Delta for other FuturesFor a stock or stock index paying a continuous dividend,For a currency,Speculative Markets, Finance 665 Spring 2003Brian BalyeatOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai

13、, 2003, Shanghai Normal University13.11GammaGamma () is the rate of change of delta () with respect to the price of the underlyingFigure 13.9 (p. 325) for a call or putGammaStock PriceXOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal Univers

14、ity13.12Equation for GammaThe Gamma () for a European call or put paying a continuous dividend q iswhereOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.13Gamma Addresses Delta Hedging Errors Caused By CurvatureFigure 13.7 (p. 3

15、22)Call PriceSCStock PriceSCCOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.14ThetaTheta () of a derivative (or a portfolio ofderivatives) is the rate of change of the value with respect to the passage of timeFigure 13.6 (p. 3

16、21)0ThetaTime to MaturityAt-the-MoneyIn-the-MoneyOut-of-the-MoneyOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.15Equations for ThetaThe Theta () of an European call option paying a dividend at rate q isThe Theta () of an Euro

17、pean put option paying a dividend at rate q isOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.16Relationship Among Delta, Gamma, and, ThetaFor a non-dividend paying stockThis follows from the Black-Scholes differential equation

18、Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.17VegaVega () is the rate of change of a derivatives portfolio with respect to volatilityFigure 14.11 (p. 317) for a call or putVegaStock PriceXOptions, Futures, and Other Derivat

19、ives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.18Equation for VegaThe Vega () for a European call or put paying a continuous dividend q isOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.19

20、Managing Delta, Gamma, and Vega can be changed by taking a position in theunderlyingTo adjust and it is necessary to take a position in an option or other derivativeOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.20Hedging Exam

21、ple(ref. p.324,p327)Assume that a company has a portfolio of the following S&P100 stock options Type Position Delta Gamma Vega Call 20000.62.21.8 Call-5000.10.60.2 Put1000-0.21.30.7 Put-1500-0.71.81.4An option is available which has a delta of 0.6, a gamma of 1.8, and a vega of 0.1.What position in

22、the traded option and the S&P100 would make the portfolio both gamma and delta neutral?Both vega and delta neutral?Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.21Hedging Example(continued)First, calculate the delta, gamma, a

23、nd vega of the portfolio.deltap= 2000*0.6 - 500*0.1 +1000*(-0.2) -1500*(-0.7) = +2000gammap = 2000*2.2 - 500*0.6 +1000* 1.3 -1500* 1.8= +2700vegap = 2000*1.8 - 500*0.2 +1000* 0.7 -1500* 1.4 = +2100To be gamma neutral, we need to add -2700/1.8 = -1500traded options ( )This changes the delta of the ne

24、w portfolio to be -1500*0.6 + 2000 = 1100In addition to selling 1500 traded options, we would need a short position of 1100 shares in the indexOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.22Hedging Example(continued)To be ve

25、ga neutral, we need to add -2100/0.1 = -21000traded options (i.e. short 21000 options)( )This changes the delta of the new portfolio to be -21000*0.6 + 2000 = -10600In addition to shorting the 21000 traded options, we would need a long position of 10600 shares in the indexTo be delta, gamma, and veg

26、a neutral we would need a second (independent) option. We would then solve a system of two equations in 2 unknowns to determine how many of each type of option needs to be purchased to be both gamma and vega neutral. Then, we take a position in the underlying to assure delta neutrality.Options, Futu

27、res, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.23Hedging Example(continued)Assume that a second option is available which has a delta of 0.2, a gamma of 0.9, and a vega of 0.8.Solving 2 equations with 2 unknowns, we haveThe solution to thi

28、s system is OPT1 = -200 and OPT2 = - 2600This gives a new ofThus, 1,360 shares must be shorted to become delta neutralOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.24RhoRho is the rate of change of the value of aderivative wi

29、th respect to the interest rateFor currency options there are 2 rhosOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.25Equations for RhoThe Rho () of an European call option paying a dividend at rate q isThe Rho () of an Europea

30、n put option paying a dividend at rate q isThe same formulas apply to European call and put options on non-dividend stock Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.26Equations for Rho in Currency Options In addition to th

31、e two previous formulas, which correspond to the domestic interest rate r, we have those rhos correspond to rfThe Rho (f) of an European call currency option isThe Rho (f) of an European put currency option isOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003,

32、 Shanghai Normal University13.27Hedging in PracticeTraders usually ensure that their portfolios are delta-neutral at least once a dayWhenever the opportunity arises, they improve gamma and vegaAs portfolio becomes larger hedging becomes less expensiveOptions, Futures, and Other Derivatives, 4th edit

33、ion 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.28Scenario AnalysisScenario analysis and the calculation of value at risk (VaR) is an alternative to relying exclusively on , , , etc.Typical VaR question: What loss level are we 99% certain will not beexceeded over the next 10

34、days?Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.29Hedging vs. Creation of an Option SyntheticallyWhen we are hedging,we take positions that offset , , , etc.When we create an option synthetically,we take positions that mat

35、ch , , and Thus, the procedure for creating anoption position synthetically is the reverse of the procedure for hedging the option position.Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.30Portfolio InsuranceIn October 1987, m

36、any portfolio managers attempted to create put options on their portfolios by matching This involves initially SELLING enough of the portfolio (or of index futures) to match the of the put optionAs the value of the portfolio increases, the of the put becomes less negative and the position in the por

37、tfolio is increasedAs the value of the portfolio decreases, the of the put becomes more negative and more of the portfolio must be SOLDThis strategy did NOT work well on October 19, 1987 Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal Unive

38、rsity13.31Portfolio Insurance ExampleA fund manager has a well-diversified portfolio that mirrors the performance of the S&P500 and is worth $90 million. The value of the S&P500 is 300 and the portfolio manager would like to insure against a reduction of more than 5% in the value of the portfolio ov

39、er the next six months. The risk-free rate is 6% per annum. The dividend yield on both the portfolio and the S&P500 is 3% and the volatility of the index is 30% per annum.Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.32Portfo

40、lio Insurance Example(continued)If the fund manager buys traded European options, how much would the insurance cost?If the value of the portfolio falls by 5%, so does the index asReturn from Change in Portfolio-5.0%in 6 mthsDividends from Portfolio1.5% per 6 mthsTotal Portfolio Return-3.5%per 6 mths

41、Risk-free rate3.0%per 6 mthsExcess Portfolio Return-6.5%per 6 mthsExcess Index Return-6.5% per 6 mthsTotal Index Return-3.5%per 6 mthsDividends from Index1.5%per 6 mthsIncrease in Value of Index -5.0%in 6 mthsThus, we need to evaluate a put option on the S&P500 with a strike of 300*(1.0-0.05) = 300*

42、0.95 = 285Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.33Portfolio Insurance Example(continued)UsingSo, we have the total cost of the hedge beingOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Y

43、incai, 2003, Shanghai Normal University13.34Portfolio Insurance Example(continued)Explain carefully alternative strategies open to the fund manager involving traded European call options, and show that they lead to the same resultFrom the put-call parityThis shows that a put option can be created by

44、 buying a call option, selling (or shorting) e-qT of the index, and lending the net present value of the strike at the risk-free rate of interest.Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.35Portfolio Insurance Example(continued)Applying this to this situation, the fund manager could,1. Sell 90e-0.03*6/12 = $88.66 million of stock2. Buy 300,000 call options on the S&P500 with exerciseprice = 285 and 6 months to maturity3. Invest remaining cash at the risk-free rate of 6%Thus, $1.34 million of stock