单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Tang Yincai,Shanghai Normal University,9.,*,Introduction toBinomial Trees,Chapter 9,1,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,A Simple,Binomial Model,of,Stock Price,Movements,In a,binomial model,the,stock price,at the,BEGINNING,of a periodcan lead to only 2 stock pricesat the,END,of that period,2,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Option 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 earned,Value the option,Risk-free interest=value of portfolio today,3,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,A Simple Binomial Model:Example,A stock price is currently$20,In three months it will be either$22 or$18,Stock Price=$22,Stock Price=$18,Stock price=$20,4,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Stock Price=$22,Option Price=$1,Stock Price=$18,Option Price=$0,Stock price=$20,Option Price=?,A Call Option,A 3-month call option on the stock has a strike price of$21.,Figure 9.1(P.202),5,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Consider the Portfolio:LONG,D,sharesSHORT 1 call option,Figure 9.1 becomes,Portfolio is,riskless,when 22,D,1=18,D,or,D,=0.25,22,D,1,18,D,Setting Up a Riskless Portfolio,S,0,=20,6,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Valuing the Portfolio,(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,=4.3670,7,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Valuing the Option,The portfolio that is:,LONG,0.25,shares,SHORT,1,call,option,is worth 4.367,The,value,of the,shares,is5.000=0.25*20,The,value,of the,option,is therefore0.633=5.000-4.367,8,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Generalization,Consider a,derivative,that lasts for time,T,andthat is,dependent,on a stock,Figure 9.2(P.203),S,0,u,u,S,0,d,d,S,0,9,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Generalization,(continued),Consider the portfolio that is:,LONG,shares,SHORT,1 derivative,Figure 9.2 becomes,The portfolio is riskless when,S,0,u,D,u,=,S,0,d,D,d,or when,S,0,u,D,u,S,0,d,D,d,S,0,-,f,10,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Generalization,(continued),Value of the portfolio at time,T,is,S,0,u,D,u,Value of the portfolio,today,is,(S,0,u,D,u,)e,rT,Another expression for the portfolio value,today,is,S,0,D,f,Hence,=S,0,D,(,S,0,u,D,u,)e,rT,11,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Generalization,(continued),Substituting for,D,we obtain,=,p,u,+(1,p,),d,e,rT,where,12,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Generalization,(continued),:Proof with an Example,This is known as the,No Arbitrage,methodology,In our earlier example,f,=0.633 and,=0.25,If,f,S,0,-f,=0.25*20-0.6=4.44.367,t,=0,S,T,=,18,S,T,=22,Buy,call-0.600 0 1,Sell,Shares5.000 -18*0.25=-4.50 -22*0.25=-5.50,Lend,4.367 at,r-,4.367 4.50,4.50,Net Flows,0.033,0 0,13,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Generalization,(continued),:Proof with an Example,If,f,0.633,e.g.,f,=0.65=,S,0,-f,=0.25*20-0.65=4.35 9.46376,What Happens When anOption is,American,?,72,0,48,4,32,20,60,1.4147,40,12,50,5.0894,A,B,C,D,F,E,6282,.,0,8,.,0,2,.,1,8,.,0,e,e,1.0,*,0.05,=,-,-,=,-,-,=,D,d,u,d,p,T,r,Rule:,The value of the option at the final nodes is the same for the European option,At earlier nodes it is the greater of,-The value given by(9.2),-The payoff from early exercise,28,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Delta,Delta(,)is the,ratio,ofthe,change,in the price of a stock,option,tothe,change,in the price of the,underlying,stock,The,value,of,varies,from node to node,29,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Using,Binomial Trees,in Practice,Realistically,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 volatility,If stock prices are assumed to be lognormal(then geometric returns are normal),then,30,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull,Importance,of a Stocks,Volatility,Lets look at two examples,both as 3 month callswith X=21 and where,r,=0,Case I:,S,0,u=22,Case II:,S,0,u=26,f,u,=1,f,u,=5 S,0,=20 S,0,=20,f=0.5,f=2.5,S,0,d=18 S,0,d=14 f,d,=0,f,d,=0,In both cases,p,=0.5,5,.,0,6,.,0,3,.,0,7,.,0,3,.,1,7,.,0,1,7,.,0,3,.,1,7,.,0,e,e,5,.,0,2,.,0,1,.,0,9,.,0,1,.,1,9,.,0,1,9,.,0,1,.,1,9,.,0,e,e,12,/,3,*,0,2,12,/,3,*,0,1,=,