期货理论与实务英文版课件：Chapter 8 Option Pricing

1、Chapter 8Option Pricingmain content8.1 European Options in the Binomial Tree Model8.1.1 One Step8.1.2 Two Steps8.1.3 General N-Step Model8.1.4 CoxRossRubinstein Formula8.2 American Options in the Binomial Tree Model8.3 BlackScholes FormulaBy a European derivative security or contingent claim with st

2、ock S as the underlying asset we mean a random variable of the form D(T) = f(S(T), where f is a given function, called the payoff . TheoremSuppose that for any contingent claim D(T) there exists a replication strategy, that is, an admissible strategy x(t), y(t) with final value V (T) = D(T). Then th

3、e price D(0) of the contingent claim at time 0 must be equal to that of the replicating strategy, V (0) = D(0).8.1 European Options in the Binomial TreeModelOne StepWe assume that the random stock price S(1) at time 1 may take two values denoted by Su = S(0)(1 + u), Sd = S(0)(1 + d), with probabilit

4、ies p and 1 p, respectively.To replicate a general derivativesecurity with payoff f we need to solve the system of equationsx(1)Su + y(1)(1 + r) = f(Su),x(1)Sd + y(1)(1 + r) = f(Sd),The initial value of the replicating portfolio is x(1)S(0)+y(1). By Theorem beforeExerciseFind a formula for the price

5、 CE(0) of a call option if r = 0 and S(0) = X = 1 dollar. Compute the price for u = 0.05 and d = 0.05, and also for u = 0.01 and d = 0.19. Draw a conclusion about the relationship between the variance of the return on stock and that on the optionTheoremThe expectation of the discounted payoff comput

6、ed with respect to the riskneutral probability is equal to the present value of the contingent claim,8.1.2 Two StepsThe expectation of the discounted payoff computed with respect to the riskneutral probability is equal to the present value of the derivative security,8.1.3 General N-Step ModelThe val

7、ue of a European derivative security with payoff f(S(N) in the Nstep binomial model is the expectation of the discounted payoff under the risk-neutral probability:8.1.4 CoxRossRubinstein FormulaIn the binomial model the price of a European call and put option with strike price X to be exercised afte

8、r N time steps is given by(m,N, p) denotes the cumulative binomial distribution with N trials and probability p of success in each trial,ExerciseLet S(0) = 50 dollars, r = 5%, u = 0.3 and d = .0.1. Find the price of a European call and put with strike price X = 60 dollars to be exercised after N = 3

9、 time steps.8.2 American Options in the Binomial TreeModelThe option can be exercised at any time step n such that 0 n N, with payoff f(S(n). Of course, it can be exercised only once. The price of an American option at time n will be denoted by DA(n).To begin with, we shall analyse an American optio

10、n expiring after 2 time steps. Unless the option has already been exercised, at expiry it will be worthThe value of waiting can be computed by treating f(S(2) as a one-step European contingent claim to be priced at time 1, which gives the valueThe American option at time 1 will, therefore, be worth

11、the higher of the two,A similar argument gives the American option value at time 0,ExampleTo illustrate the above procedure we consider an American put option with strike price X = 80 dollars expiring at time 2 on a stock with initial price S(0) = 80 dollars in a binomial model with u = 0.1, d = 0.0

12、5 and r = 0.05.At time 1 the option writer can choose between exercising the option immediately or waiting until time 2. In the up state at time 1 the immediate payoff and the value of waiting are both zero. In the down state the immediate payoff is 4 dollars, while the value of waiting is 2.48 doll

13、ars. At time 0 the choice is, once again, between the payoff, which is zero, or the value of waiting, which is 1.27 dollars.DefinitionAn American derivative security or contingent claim with payoff function f expiring at time N is a sequence of random variables defined by backward induction:8.3 Blac

14、kScholes FormulaThe time t price of a European call with strike price X and exercise time T, where t T, is given byTarget:Have a Test:Transformation:Martingale:Pricing BlackScholes formula and CoxRossRubinstein formulaThe precise relationship comes from a version of the Central Limit Theorem: It can

15、 be shown that the option price given by the CoxRossRubinstein formula tends to that in the BlackScholes formula in the continuous time limit described in Chapter 3.European call with strike X = 100 on a stock with S(0) = 100, = 0.3 and m = 0.2. The continuous compounding interest rate is taken to be r = 0.2. The option price is computed in two ways:a) (solid lin