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Black-Scholes option pricing formula 英文版
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Lecture #9: Black-Scholes option pricing formula
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
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