Ch09-IntroductiontoBinomialTrees金融工程學(xué),華東

1、2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.1Introduction toBinomial TreesChapter 92021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.2A Simple Binom

2、ial Modelof Stock Price MovementsIn a binomial model, the stock priceat the BEGINNING of a periodcan lead to only 2 stock pricesat the END of that period2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.3Option Pricing Based on

3、 the Assumption of No Arbitrage Opportunities Procedures: Establish a portfolio of stock and option Value the Portfolio no arbitrage opportunities no uncertainty at maturity no risk with the portfolio risk-free interest earnedValue the optionRisk-free interest = value of portfolio today2021/9/11Opti

4、ons, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.4A Simple Binomial Model:ExampleA stock price is currently $20In three months it will be either $22 or $18Stock Price = $22Stock Price = $18Stock price = $202021/9/11Options, Futures, and Ot

5、her Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.5Stock Price = $22Option Price = $1Stock Price = $18Option Price = $0Stock price = $20Option Price=?A Call Option A 3-month call option on the stock has a strike price of $21. Figure 9.1 (P.202)2021/9/11Options

6、, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.6Consider the Portfolio:LONG D sharesSHORT 1 call optionFigure 9.1 becomesPortfolio is riskless when 22D 1 = 18D or D = 0.2522D 118DSetting Up a Riskless PortfolioS0 = 202021/9/11Options, Futur

7、es, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.7Valuing the Portfolio( with Risk-Free Rate 12% )The riskless portfolio is: LONG 0.25 shares SHORT 1 call optionThe value of the portfolio in 3 months is22 * 0.25 - 1 = 4.50 = 18 * 0.25The value of th

8、e portfolio today is 4.50e-0.12*0.25=4.36702021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.8Valuing the OptionThe portfolio that is:LONG 0.25 sharesSHORT 1 call optionis worth 4.367The value of the shares is5.000 = 0.25 * 20T

9、he value of the option is therefore0.633 = 5.000 - 4.3672021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.9Generalization Consider a derivativethat lasts for time T andthat is dependent on a stockFigure 9.2 (P.203)S0u uS0d dS02

10、021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.10Generalization (continued)Consider the portfolio that is:LONG sharesSHORT 1 derivative Figure 9.2 becomesThe portfolio is riskless when S0uD u = S0d D d or whenS0uD uS0 dD dS0

11、- f2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.11Generalization (continued)Value of the portfolio at time T is S0u D uValue of the portfolio today is (S0u D u )erTAnother expression for the portfolio value today is S0 D f

12、Hence, = S0 D (S0u D u )erT 2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.12Generalization(continued)Substituting for D we obtain = p u + (1 p )d erTwhere 2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 b

13、y John C. HullTang Yincai, Shanghai Normal University9.13Generalization (continued) : Proof with an ExampleThis is known as the No Arbitrage methodologyIn our earlier example f=0.633 and =0.25If f S0-f=0.25*20-0.6=4.44.367 t = 0 ST=18 ST=22Buy call-0.600 0 1 Sell Shares5.000 -18*0.25=-4.50 -22*0.25=

14、-5.50 Lend 4.367 at r-4.367 4.50 4.50 Net Flows0.033 0 02021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.14Generalization (continued) : Proof with an ExampleIf f 0.633, e.g. f=0.65 = S0-f=0.25*20-0.65=4.35 9.46376What Happens

15、When anOption is American?72 048 43220601.414740 12505.0894ABCDFE6282.08.02.18.0ee1.0*0.05=-=-=DdudpTrRule:The value of the option at the final nodes is the same for the European optionAt earlier nodes it is the greater of - The value given by (9.2) - The payoff from early exercise2021/9/11Options,

16、Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.29DeltaDelta () is the ratio ofthe change in the price of a stock option tothe change in the price of the underlying stockThe value of varies from node to node 2021/9/11Options, Futures, and Othe

17、r Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.30Using Binomial Trees in PracticeRealistically, only 1 or 2 time steps is not nearlyenough. Practitioners usually use 30 or more.The values for u and d are usually determined from the stocks volatilityIf stock p

18、rices are assumed to be lognormal (then geometric returns are normal), then2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University9.31Importance of a Stocks VolatilityLets look at two examples, both as 3 month callswith X=21 and where r = 0Case I: S0u = 22 Case II: S0u = 26 fu = 1 fu = 5 S0=20 S0=20 f =0.5 f =2.5 S0d = 18 S0d = 14 fd = 0 fd = 0 In both cases, p=0.5 5.06.03.07.03.17.017.03.17.0ee5.02.01.09.01.19.019.01.19.0ee12/3*0212/3*01=-=-=-=-=-=-=DDdudpdudptrtrImportance of a Stocks VolatilityImportanc