期货与股票ppt.ppt

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 plished by setting
= ½*(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 motion, B’(t) does not exist in the ord