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1、Introduction toBinomial TreesChapter 9,1,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,A 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 period,2,Options, Futur
2、es, 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 intere
3、st 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 p
4、rice = $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, Fu
5、tures, 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 22D 1 = 18D or D = 0.25,Setting Up a Riskless Portfolio,S0 = 20,6,Options, Futures, and Other Derivatives, 4th edition 2000 by John C.
6、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 ed
7、ition 2000 by John C. Hull,Valuing 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 - 4.367,8,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Gene
8、ralization,Consider a derivativethat lasts for time T andthat is dependent on a stock Figure 9.2 (P.203),9,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Generalization (continued),Consider the portfolio that is:LONG sharesSHORT 1 derivative Figure 9.2 becomes The portfoli
9、o is riskless when S0uD u = S0d D d or when,S0uD u,S0 dD d,S0 - f,10,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Generalization (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 val
10、ue today is S0 D f Hence, = S0 D (S0u D u )erT,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 erT where,12,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Generalization (co
11、ntinued) : Proof with an Example,This is known as the No Arbitrage methodology In our earlier example f=0.633 and =0.25 If 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=-5.50 Lend 4.367 at r-4.367 4.50 4.50 Net Flows0.033 0 0,13,Options, Fut
12、ures, 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 = S0-f=0.25*20-0.65=4.354.367 t = 0 ST=18ST=22Buy Shares-5.000 18*0.25=4.50 22*0.25=5.50 Borrow 4.367 at r4.367 -4.50 -4.50 Sell call0.650 0 -1 Net Flows0.017 0 0,
13、14,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Irrelevance of a Stocks Expected Return,When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant This is because in our formula f = S0 - (S0u-fu) e-rTf does not involve t
14、he probability of the stock moving up or down It does not matter if we say the probability of an increase is 50% or 80% we get the same result,15,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Irrelevance of a Stocks E(R)Proof: (continued),Lets call pu the probability of a
15、n increase in the stock price and pd=1- pu the probability of a stock decrease S0 - f = pu(S0u-fu)+ pd(S0d-fd) e-kT where k is the appropriate rate for the risk involved However, is chosen such thatS0u-fu= S0d-fd and we know that pd=1- pu Substituting, S0 - f = pu(S0u-fu)+ (1- pu)(S0u-fu) e-kT = (S0
16、u-fu)e-rT as since this is risk-free, k = r No pus or pds left, thus probability of stock increase is irrelevant,16,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Irrelevance of a Stocks E(R)(continued),The probability of an increase in the stock price is irrelevant because options are redundant securities In our two-step models, we form a risk-less portfolio with stock a
