-期货与股票P41.pptVIP

Lecture #9: Black-Scholes option pricing formula ·??????? Brownian Motion The first formal mathematical model of financial asset prices, developed by Bachelier (1900), was the continuous-time random walk, or Brownian motion. This continuous-time process is closely related to the discrete-time versions of the random walk. ·??????? The discrete-time random walk Pk = Pk-1 + ?k, ?k = ? (-?) with probability ? (1-?), P0 is fixed. Consider the following continuous time process Pn(t), t ? [0, T], which is constructed from the discrete time process Pk, k=1,..n as follows: Let h=T/n and define the process Pn(t) = P[t/h] = P[nt/T] , t ? [0, T], where [x] denotes the greatest integer less than or equal to x. Pn(t) is a left continuous step function. We need to adjust ?, ? such that Pn(t) will converge when n goes to infinity. Consider the mean and variance of Pn(T): E(Pn(T)) = n(2?-1) ? Var (Pn(T)) = 4n?(?-1) ?2 We wish to obtain a continuous time version of the random walk, we should expect the mean and variance of the limiting process P(T) to be linear in T. Therefore, we must have n(2?-1) ? ? ?T 4n?(?-1) ?2 ??T This can be accomplished by setting ? = ?*(1+??h /?), ?=??h ·??????? The continuous time limit It cab be shown that the process P(t) has the following three properties: 1. For any t1 and t2 such that 0 ? t1 t2 ? T: P(t1)-P(t2) ??(?(t2-t1), ?2(t2-t1)) 2. For any t1, t 2 , t3, and t4 such that 0 ? t1 t2 t1 t2 ? t3 t4? T, the increment P(t2)- P(t1) is statistically independent of the increment P(t4)- P(t3). 3. The sample paths of P(t) are continuous. P(t) is called arithmetic Brownian motion or Winner process. If we set ?=0, ?=1, we obtain standard Brownian Motion which is denoted as B(t). Accordingly, P(t) = ?t + ?B(t) Consider the following moments: E[P(t) | P(t0)] = P(t0) +?(t-t0) Var[P(t) | P(t0)] = ?2(t-t0) Cov(P(t1),P(t2) = ?2 min(t1,t2) Since Var[ (B(t+h)-B(t))/h ] = ?2/h, therefore, the derivative of Brownian moti