Chapter 18The Greek Letters1期权期货及其衍生品第18弹ExampleA 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 $240,000How does the bank hedge its risk to lock in a $60,000 profit?2期权期货及其衍生品第18弹Naked & Covered PositionsNaked positionTake no actionCovered positionBuy 100,000 shares todayWhat are the risks associated with these strategies?3期权期货及其衍生品第18弹Stop-Loss StrategyThis involves:Buying 100,000 shares as soon as price reaches $50Selling 100,000 shares as soon as price falls below $504期权期货及其衍生品第18弹Stop-Loss Strategy continuedIgnoring discounting, the cost of writing and hedging the option appears to be max(S0K, 0). What are we overlooking?5期权期货及其衍生品第18弹Delta (See Figure 18.2, page 381)Delta (D) is the rate of change of the option price with respect to the underlying Call optionpriceABSlope = D = 0.6Stock price6期权期货及其衍生品第18弹HedgeTrader 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 passesHedge position must therefore be rebalanced 7期权期货及其衍生品第18弹Delta HedgingThis 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 stock is N (d 1) 18期权期货及其衍生品第18弹The Costs in Delta HedgingcontinuedDelta hedging a written option involves a “buy high, sell low” trading rule9期权期货及其衍生品第18弹First Scenario for the Example: Table 18.2 page 384WeekStock priceDeltaShares purchasedCost ($000)CumulativeCost ($000)Interest049.000.52252,2002,557.82,557.82.5148.120.458(6,400)(308.0)2,252.32.2247.370.400(5,800)(274.7)1,979.81.9.1955.871.0001,00055.95,258.25.12057.251.000005263.310期权期货及其衍生品第18弹Second Scenario for the Example Table 18.3, page 385WeekStock priceDeltaShares purchasedCost ($000)CumulativeCost ($000)Interest049.000.52252,2002,557.82,557.82.5149.750.568 4,600228.92,789.22.7252.000.70513,700712.43,504.33.4.1946.630.007(17,600)(820.7)290.00.32048.120.000(700)(33.7)256.611期权期货及其衍生品第18弹ThetaTheta (Q) of a derivative (or portfolio of derivatives) 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 declines12期权期货及其衍生品第18弹Theta for Call Option: K=50, s s = 25%, r = 5% T = 113期权期货及其衍生品第18弹GammaGamma (G) is the rate of change of delta (D) with respect to the price of the underlying assetGamma is greatest for options that are close to the money14期权期货及其衍生品第18弹Gamma for Call or Put Option: K=50, s s = 25%, r = 5% T = 115期权期货及其衍生品第18弹Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 18.7, page 389) SCStock priceSCallpriceCC16期权期货及其衍生品第18弹Interpretation of GammaFor a delta neutral portfolio, DP Q Dt + GDS 2 DPDS Negative GammaDPDS Positive Gamma17期权期货及其衍生品第18弹Relationship Between Delta, Gamma, and Theta (page 393)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 18期权期货及其衍生品第18弹VegaVega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility19期权期货及其衍生品第18弹Vega for Call or Put Option: K=50, s s = 25%, r = 5% T = 120期权期货及其衍生品第18弹Taylor Series Expansion (Appendix 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 t21期权期货及其衍生品第18弹Managing Delta, Gamma, & VegaDelta can be changed by taking a position in the underlying assetTo adjust gamma and vega it is necessary to take a position in an option or other derivative22期权期货及其衍生品第18弹ExampleDeltaGammaVegaPortfolio050008000Option 10.60.52.0Option 20.50.81.2What position in 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 asset23期权期货及其衍生品第18弹Example continuedDeltaGammaVegaPortfolio050008000Option 10.60.52.0Option 20.50.81.2What 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 1 and option 2. A short position of 3240 in the asset is then required to make the portfolio delta neutral24期权期货及其衍生品第18弹RhoRho is the rate of change of the value of a derivative with respect to the interest rate25期权期货及其衍生品第18弹Hedging in PracticeTraders 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 portfolio26期权期货及其衍生品第18弹Scenario AnalysisA scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities27期权期货及其衍生品第18弹Greek Letters for European Options on an Asset that Provides a Yield at Rate q Greek LetterCall OptionPut OptionDeltaGammaThetaVegaRho28期权期货及其衍生品第18弹Futures Contract Can Be Used for HedgingThe 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 position required in futures for delta hedging is therefore e(rq)T times the position required in the corresponding spot contract29期权期货及其衍生品第18弹Hedging vs Creation of an Option SyntheticallyWhen we are hedging we take positions that offset delta, gamma, vega, etcWhen we create an option synthetically we take positions that matchdelta, gamma, vega, etc30期权期货及其衍生品第18弹Portfolio InsuranceIn October of 1987 many portfolio managers attempted to create a put option on a portfolio syntheticallyThis involves initially selling enough of the portfolio (or of index futures) to match the D of the put option31期权期货及其衍生品第18弹Portfolio InsurancecontinuedAs the value of the portfolio increases, the D of the put becomes 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 sold32期权期货及其衍生品第18弹Portfolio InsurancecontinuedThe strategy did not work well on October 19, 1987.33期权期货及其衍生品第18弹