Ch13_The Greek Letters(金融工程華東師范大學 湯銀才)

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 University13.2Example A FI has SOLD for $30

2、0,000 a European call on100,000 shares of a non-dividend paying stock:S0 = 49 X = 50r = 5% = 20% = 13% T = 20 weeks The Black-Scholes value of the option is $240,000 How does the FI hedge its risk?Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai No

3、rmal University13.3Naked & Covered Positions Naked position (裸期權(quán)頭寸策略)Take NO action Covered position(抵補期權(quán)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

4、 University13.4Stop-Loss StrategyThis involves Fully covering the option as soon as it movesin-the-money Staying 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 Ot

5、her Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.5Delta Delta () is the rate of change of the option price with respect to the underlying Figure 13.2 (p. 311)fSOption PriceABStock PriceSlope = Options, Futures, and Other Derivatives, 4th edition 2000 b

6、y John C. HullTang Yincai, 2003, Shanghai Normal University13.6Delta Hedging This involves maintaining a delta neutral portfolio The 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 rebalanced Delta hedging a wri

7、tten option involves a“BUY high, SELL low” trading ruleqTdNe)(1qTdN e 1)(1Options, 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

8、 Delta Purch. Purch. Interest Cost 049.000 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,

9、and Other Derivatives, 4th edition 2000 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

10、 4,600 228.0 2,789.2 2.7 252.000 0.705 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 N

11、ormal University13.9Delta for Futures From Chapter 3, we havewhere T* is the maturity of futures contract Thus, the delta of a futures contract is So, 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,*00erTSF *e)e(rTr

12、TSSSFArTArTFHHH*ee1Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.10Delta for other Futures For a stock or stock index paying a continuous dividend, For a currency,Speculative Markets, Finance 665 Spring 2003Brian BalyeatATqrF

13、HH*)(eATrrFHHf*)(eOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.11Gamma Gamma () is the rate of change of delta () with respect to the price of the underlying Figure 13.9 (p. 325) for a call or put22SfSGammaStock PriceXOption

14、s, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.12Equation for Gamma The Gamma () for a European call or put paying a continuous dividend q iswhereTSdNqT01e )( 2/1121e21)()( ddndNOptions, Futures, and Other Derivatives, 4th edition 2

15、000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.13Gamma Addresses Delta Hedging Errors Caused By Curvature Figure 13.7 (p. 322)Call PriceSCStock PriceSCCOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.14Theta

16、 Theta () of a derivative (or a portfolio ofderivatives) is the rate of change of the value with respect to the passage of time Figure 13.6 (p. 321)ft0ThetaTime to MaturityAt-the-MoneyIn-the-MoneyOut-of-the-MoneyOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 20

17、03, Shanghai Normal University13.15Equations for Theta The Theta () of an European call option paying a dividend at rate q is The Theta () of an European put option paying a dividend at rate q is)(ee)(2e)( 21010dNrKdNqSTdNSrTqTqTc)(ee )(2e)( 21010dNrKdNqSTdNSrTqTqTpOptions, Futures, and Other Deriva

18、tives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.16Relationship Among Delta, Gamma, and, Theta For a non-dividend paying stock This follows from the Black-Scholes differential equation2212(13.7)rSSrf222122(11.15)fffrSSrftSSOptions, Futures, and Other Derivatives

19、, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.17Vega Vega () is the rate of change of a derivatives portfolio with respect to volatility Figure 14.11 (p. 317) for a call or putVegaStock PriceXfOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. Hu

20、llTang Yincai, 2003, Shanghai Normal University13.18Equation for Vega The Vega () for a European call or put paying a continuous dividend q isqTdNTSe)( 10Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.19Managing Delta, Gamma,

21、and Vega can be changed by taking a position in theunderlying To 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 Example(ref. p.324,p327) A

22、ssume 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 the traded optio

23、n 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, and vega of

24、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 new portfolio

25、 to be -1500*0.6 + 2000 = 1100In addition to selling 1500 traded options, we would need a short position of 1100 shares in the index*0pTOPT Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.22Hedging Example(continued) To be vega

26、 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 index To be delta, gamma, and veg

27、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.*0pTOPTOption

28、s, Futures, 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 have The solut

29、ion to this system is OPT1 = -200 and OPT2 = - 2600 This gives a new of Thus, 1,360 shares must be shorted to become delta neutral02*8 . 01*1 . 0210002*9 . 01*8 . 12700OPTOPTOPTOPT13602 . 0*)2600(6 . 0*)200(20002 . 0*26 . 0*12000OPTOPTOptions, Futures, and Other Derivatives, 4th edition 2000 by John

30、 C. HullTang Yincai, 2003, Shanghai Normal University13.24Rho Rho is the rate of change of the value of aderivative with respect to the interest rate For currency options there are 2 rhosrho frOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal

31、 University13.25Equations for Rho The Rho () of an European call option paying a dividend at rate q is The Rho () of an European put option paying a dividend at rate q is The same formulas apply to European call and put options on non-dividend stock 2rhoe()rTXTN d2rhoe()rTXTN dOptions, Futures, and

32、Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.26Equations for Rho in Currency Options In addition to the two previous formulas, which correspond to the domestic interest rate r, we have those rhos correspond to rf The Rho (f) of an European call c

33、urrency option is The Rho (f) of an European put currency option is)(eohr10dNTSTrff)(eohr10dNTSTrffOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.27Hedging in Practice Traders usually ensure that their portfolios are delta-neu

34、tral at least once a day Whenever the opportunity arises, they improve gamma and vega As portfolio becomes larger hedging becomes less expensiveOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.28Scenario Analysis Scenario analys

35、is 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 days?Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai

36、 Normal University13.29Hedging vs. Creation of an Option Synthetically When we are hedging,we take positions that offset , , , etc. When we create an option synthetically,we take positions that match , , and Thus, the procedure for creating anoption position synthetically is the reverse of the proce

37、dure for hedging the option position.Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 2003, Shanghai Normal University13.30Portfolio Insurance In October 1987, many portfolio managers attempted to create put options on their portfolios by matching This involves i

38、nitially SELLING enough of the portfolio (or of index futures) to match the of the put option As the value of the portfolio increases, the of the put becomes less negative and the position in the portfolio is increased As the value of the portfolio decreases, the of the put becomes more negative and

39、 more of the portfolio must be SOLD This 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 University13.31Portfolio Insurance Example A fund manager has a well-diversified portfolio that mirrors

40、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 over the next six months. The risk-free rate is 6% per annum. The dividend yield on both the

41、 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 University13.32Portfolio Insurance Example(continued) If the fund manager buys traded European options, how

42、 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 mthsRisk-free rate3.0%per 6 mthsExcess Portfolio Return-6.5%per 6 mthsExcess Index Retur

43、n-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*0.95 = 285Options, Futures, and Other Derivatives, 4th edition 2000 by John C. H

44、ullTang Yincai, 2003, Shanghai Normal University13.33Portfolio Insurance Example(continued) UsingSo, we have the total cost of the hedge being4182. 0)(2064. 03378. 0)(4186. 02211dNddNd000,755,4$300000,000,90*85.15201e()e()rTqTpXNdSNd2010.06*(6/12)0.03*(6/12)e()e()285*0.4182300*0.337815.85rTqTpXNdSNd

45、eeOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, 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

46、 same resultFrom the put-call parityThis shows that a put option can be created by 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.00eeeeqTrTqTrTSpcXpcSXOptions, Futures, and Other Derivatives, 4th editi

47、on 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 t

48、o maturity3. Invest remaining cash at the risk-free rate of 6% Thus, $1.34 million of stock is retained The value of one call is The total cost of the call options is 300,000*34.80 = $10.44 mill80.345818.0*e2856622.0*e300)(e)(e12/6*06.012/6*03.0210dNKdNScrTqTOptions, Futures, and Other Derivatives, 4th edition