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1、Chapter 12Binomial Trees,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,1,A Simple Binomial Model,A stock price is currently$20In 3 months it will be either$22 or$18,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,2,A Call Option(Figure 12.1
2、,page 254),A 3-month call option on the stock has a strike price of 21.,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,3,Stock Price=$18Option Price=$0,Setting Up a Riskless Portfolio,For a portfolio that is long D shares and a short 1 call option values arePortfolio is
3、 riskless when 22D 1=18D or D=0.25,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,4,Valuing the Portfolio(Risk-Free Rate is 12%),The riskless portfolio is:long 0.25 sharesshort 1 call optionThe value of the portfolio in 3 months is 22 0.25 1=4.50The value of the portfol
4、io today is 4.5e0.120.25=4.3670,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,5,Valuing the Option,The portfolio that is long 0.25 sharesshort 1 option is worth 4.367The value of the shares is 5.000(=0.25 20)The value of the option is therefore 0.633(5.000 0.633=4.367)
5、,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,6,Generalization(Figure 12.2,page 255),A derivative lasts for time T and is dependent on a stock,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,7,Generalization(continued),Value of a portfolio
6、 that is long D shares and short 1 derivative:The portfolio is riskless when S0uD u=S0dD d or,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,8,S0uD u,S0dD d,Generalization(continued),Value of the portfolio at time T is S0uD uValue of the portfolio today is(S0uD u)erTAno
7、ther expression for the portfolio value today is S0D fHence=S0D(S0uD u)erT,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,9,Generalization(continued),Substituting for D we obtain=pu+(1 p)d erT where,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull
8、 2012,10,p as a Probability,It is natural to interpret p and 1-p as probabilities of up and down movementsThe value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,11,Ris
9、k-Neutral Valuation,When the probability of an up and down movements are p and 1-p the expected stock price at time T is S0erTThis shows that the stock price earns the risk-free rateBinomial trees illustrate the general result that to value a derivative we can assume that the expected return on the
10、underlying asset is the risk-free rate and discount at the risk-free rateThis is known as using risk-neutral valuation,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,12,Original Example Revisited,p is the probability that gives a return on the stock equal to the risk-fr
11、ee rate:20e 0.12 0.25=22p+18(1 p)so that p=0.6523Alternatively:,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,13,Valuing the Option Using Risk-Neutral Valuation,The value of the option is e0.120.25(0.65231+0.34770)=0.633,Options,Futures,and Other Derivatives,8th Editio
12、n,Copyright John C.Hull 2012,14,Irrelevance of Stocks Expected Return,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,15,When we are valuing an option in terms of the price of the underlying asset,the probability of up and down movements in the real world are irrelevantT
13、his is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant,A Two-Step ExampleFigure 12.3,page 260,K=21,r=12%Each time step is 3 months,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,16,Valuing a Ca
14、ll OptionFigure 12.4,page 260,Value at node B=e0.120.25(0.65233.2+0.34770)=2.0257Value at node A=e0.120.25(0.65232.0257+0.34770)=1.2823,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,17,A Put Option ExampleFigure 12.7,page 263,K=52,time step=1yrr=5%,u=1.32,d=0.8,p=0.628
15、2,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,18,What Happens When the Put Option is American(Figure 12.8,page 264),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,19,The American feature increases the value at node C from 9.4636 to 12.00
16、00.This increases the value of the option from 4.1923 to 5.0894.,Delta,Delta(D)is the ratio of the change in the price of a stock option to the change in the price of the underlying stockThe value of D varies from node to node,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2
17、012,20,Choosing u and d,One way of matching the volatility is to setwhere s is the volatility and Dt is the length of the time step.This is the approach used by Cox,Ross,and Rubinstein,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,21,Girsanovs Theorem,Volatility is the
18、 same in the real world and the risk-neutral worldWe can therefore measure volatility in the real world and use it to build a tree for the an asset in the risk-neutral world,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,22,Assets Other than Non-Dividend Paying Stocks,F
19、or options on stock indices,currencies and futures the basic procedure for constructing the tree is the same except for the calculation of p,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,23,The Probability of an Up Move,Options,Futures,and Other Derivatives,8th Edition
20、,Copyright John C.Hull 2012,24,Proving Black-Scholes-Merton from Binomial Trees(Appendix to Chapter 12),Option is in the money when j a where so that,25,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,Proving Black-Scholes-Merton from Binomial Trees continued,The express
21、ion for U1 can be writtenwhere Both U1 and U2 can now be evaluated in terms of the cumulative binomial distributionWe now let the number of time steps tend to infinity and use the result that a binomial distribution tends to a normal distribution,26,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,
