第10章　二叉树模型介绍.pptVIP

Introduction toBinomial TreesChapter 10 A Simple Binomial Model 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 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option (Figure 10.1, page 200) A 3-month call option on the stock has a strike price of 21. Consider the Portfolio: long D shares short 1 call option Portfolio is riskless when 22D ?1 = 18D or D = 0.25 22D ?1 18D Setting Up a Riskless Portfolio Valuing the Portfolio(Risk-Free Rate is 12%) The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22?.25 ?1 = 4.50 The value of the portfolio today is 4.5e ?0.12?.25 = 4.3670 Valuing the Option The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.25?0 ) The value of the option is therefore 0.633 (= 5.000 ?4.367 ) Generalization (Figure 10.2, page 202) A derivative lasts for time T and is dependent on a stock S0 僽 S0d 僤 S0 ? Generalization(continued) Consider the portfolio that is long D shares and short 1 derivative The portfolio is riskless when S0uD ?僽 = S0d D ?僤 or S0 uD ?僽 S0dD ?僤 S0?f Generalization(continued) Value of the portfolio at time T is S0u D ?僽 Value of the portfolio today is (S0u D ?僽 )e杛T Another expression for the portfolio value today is S0D ?f Hence ?= S0D ?(S0u D ?僽 )e杛T Generalization(continued) Substituting for D we obtain ?= [ p 僽 + (1 ?p )僤 ]e杛T where Risk-Neutral Valuation ?= [ p 僽 + (1 ?p )僤 ]e-rT The variables p and (1 ?p ) can be interpreted as the risk-neutral probabilities of up and down movements The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate