Ch13_TheGreekLetters(金融工程学,华东师大).ppt

Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.1,The Greek LettersChapter 13,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.2,Example,A FI has SOLD for$300,000 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 Normal University,13.3,Naked&Covered Positions,Naked position(裸期权头寸策略)Take NO actionCovered position(抵补期权头寸策略)Buy 100,000 shares todayBoth strategies leave the FI exposedto significant risk,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.4,Stop-Loss Strategy,This 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 it,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.5,Delta,Delta()is the rate of change of the option price with respect to the underlyingFigure 13.2(p.311),Option Price,A,B,Stock Price,Slope=,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.6,Delta Hedging,This 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 involves a“BUY high,SELL low”trading rule,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.7,Delta Neutral Portfolio Example(in-the-money),Table 13.2(p.314),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.8,Delta Neutral Portfolio Example(out-of-the-money),Table 13.3(p.315),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.9,Delta for Futures,From 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 by John C.HullTang Yincai,2003,Shanghai Normal University,13.10,Delta for other Futures,For a stock or stock index paying a continuous dividend,For a currency,Speculative Markets,Finance 665 Spring 2003Brian Balyeat,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.11,Gamma,Gamma()is the rate of change of delta()with respect to the price of the underlyingFigure 13.9(p.325)for a call or put,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.12,Equation for Gamma,The Gamma()for a European call or put paying a continuous dividend q iswhere,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.13,Gamma Addresses Delta Hedging Errors Caused By Curvature,Figure 13.7(p.322),Call Price,S,C,Stock Price,S,C,C,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.14,Theta,Theta()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.321),0,Theta,Time to Maturity,At-the-Money,In-the-Money,Out-of-the-Money,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.15,Equations for Theta,The Theta()of an European call option paying a dividend at rate q isThe Theta()of an European put option paying a dividend at rate q is,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.16,Relationship Among Delta,Gamma,and,Theta,For a non-dividend paying stockThis follows from the Black-Scholes differential equation,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.17,Vega,Vega()is the rate of change of a derivatives portfolio with respect to volatilityFigure 14.11(p.317)for a call or put,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.18,Equation for Vega,The Vega()for a European call or put paying a continuous dividend q is,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.19,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 derivative,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.20,Hedging Example(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 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 University,13.21,Hedging Example(continued),First,calculate the delta,gamma,and 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 new 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 index,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.22,Hedging Example(continued),To be vega 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 vega 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,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.23,Hedging 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 this system is OPT1=-200 and OPT2=-2600This gives a new ofThus,1,360 shares must be shorted to become delta neutral,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.24,Rho,Rho is the rate of change of the value of aderivative with respect to the interest rateFor currency options there are 2 rhos,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.25,Equations for Rho,The Rho()of an European call option paying a dividend at rate q isThe Rho()of an European 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 University,13.26,Equations 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 rfThe Rho(f)of an European call currency option isThe Rho(f)of an European put currency option is,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.27,Hedging in Practice,Traders 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 expensive,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.28,Scenario Analysis,Scenario 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 days?,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.29,Hedging 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 procedure for hedging the option position.,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.30,Portfolio Insurance,In October 1987,many 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 portfolio 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 University,13.31,Portfolio Insurance Example,A 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 over 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 University,13.32,Portfolio 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 mthsRisk-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*0.95=285,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.33,Portfolio Insurance Example(continued),UsingSo,we have the total cost of the hedge being,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.34,Portfolio 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 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 University,13.35,Portfolio 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 is retainedThe value of one call isThe total cost of the call options is 300,000*34.80=$10.44 mill,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.36,Portfolio Insurance Example(continued),The value of the portfolio at the end of the six months is pay-off of putdividends future value of price of putNote that,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.37,Portfolio Insurance Example(continued),If the fund manager decides to provide insurance by keeping part of the portfolio in risk-free securities,what should the initial position be?The delta of one put option isThis indicates that 33.27%of the portfolio(i.e.$29.94 million)should be initially sold and invested in risk-free securities,3327,.,0,1,6622,.,0,*,e,1,),(,*,e,12,/,6,*,03,.,0,1,-,=,-,=,-,=,-,-,d,N,qT,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.38,Portfolio Insurance Example(continued),If the fund manager decides to provide insurance by using nine-month index futures,what should the initial position be?The delta of a nine-month index futures contract isFrom before,the spot position is 29,940,000,so we need contracts.,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,13.39,Assignments,13.2,13.8,13.10,13.12,13.14,13.15,13.18,13.19,13.22Assignment Questions,