toBinomialTrees(金融工程-华东师范大学汤银才)

1、Introduction toBinomial TreesChapter 91Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullA Simple Binomial Modelof Stock Price Movements In a binomial model, the stock priceat the BEGINNING of a periodcan lead to only 2 stock pricesat the END of that period2Options, Futures, a

2、nd Other Derivatives, 4th edition 2000 by John C. HullOption Pricing Based on 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 ear

3、ned Value the option Risk-free interest = value of portfolio today3Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullA Simple Binomial Model:Example A stock price is currently $20 In three months it will be either $22 or $18Stock Price = $22Stock Price = $18Stock price = $204O

4、ptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullStock 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)5Options, Futures, and Other Deriva

5、tives, 4th edition 2000 by John C. Hull Consider the Portfolio:LONG D sharesSHORT 1 call option Figure 9.1 becomes Portfolio is riskless when 22D 1 = 18D or D = 0.2522D 118DSetting Up a Riskless PortfolioS0 = 206Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullValuing the Por

6、tfolio( with Risk-Free Rate 12% ) The riskless portfolio is: LONG 0.25 shares SHORT 1 call option The value of the portfolio in 3 months is22 * 0.25 - 1 = 4.50 = 18 * 0.25 The value of the portfolio today is 4.50e-0.12*0.25=4.36707Options, Futures, and Other Derivatives, 4th edition 2000 by John C.

7、HullValuing the Option The portfolio that is:LONG 0.25 sharesSHORT 1 call optionis worth 4.367 The value of the shares is5.000 = 0.25 * 20 The value of the option is therefore0.633 = 5.000 - 4.3678Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullGeneralization Consider a deri

8、vativethat lasts for time T andthat is dependent on a stock Figure 9.2 (P.203)S0u uS0d dS09Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullGeneralization (continued) Consider the portfolio that is:LONG D sharesSHORT 1 derivative Figure 9.2 becomes The portfolio is riskless w

9、hen S0uD u = S0d D d or whendSuSffdu00 DS0uD uS0 dD dDS0 - f10Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullGeneralization (continued) Value of the portfolio at time T is S0u D u Value of the portfolio today is (S0u D u )erT Another expression for the portfolio value today

10、 is S0 D f Hence, = S0 D (S0u D u )erT 11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullGeneralization(continued) Substituting for D we obtain = p u + (1 p )d erTwhere pedudrT12Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullGeneralization (continue

11、d) : Proof with an Example This is known as the No Arbitrage methodology In our earlier example f=0.633 and D=0.25 If f D S0-f=0.25*20-0.6=4.44.367 t = 0 ST=18 ST=22Buy call-0.600 0 1 Sell D Shares5.000 -18*0.25=-4.50 -22*0.25=-5.50 Lend 4.367 at r-4.367 4.50 4.50 Net Flows0.033 0 013Options, Future

12、s, and Other Derivatives, 4th edition 2000 by John C. HullGeneralization (continued) : Proof with an Example If f 0.633, e.g. f=0.65 = D S0-f=0.25*20-0.65=4.35 9.46376What Happens When anOption is American?72 048 43220601.414740 12505.0894ABCDFE6282.08 .02.18 .0ee1.0*0.05DdudpTrRule:The value of the

13、 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 exercise28Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullDeltaDelta (D) is the ratio ofthe change in the price of a stock o

14、ption tothe change in the price of the underlying stockThe value of D varies from node to node dSuSffdu00D29Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullUsing Binomial Trees in Practice Realistically, only 1 or 2 time steps is not nearlyenough. Practitioners usually use 3

15、0 or more. The values for u and d are usually determined from the stocks volatility If stock prices are assumed to be lognormal (then geometric returns are normal), thenudupdudtrteeDD130Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullImportance of a Stocks Volatility Lets lo

16、ok 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*01DDdudpdudptrtrImportance of a Stocks VolatilityImportance of a Stocks VolatilityImportance of a Stocks VolatilityImportance