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【正文】 partial differential equation: ?C/?t + r S?C/?S + 189。 ?2C/?S2 ?2(S,t) ?C/?S ?(S,t)dt]dt [?C/?S ?(S,t) ?C/?S ?(S,t)]dz or dP = [?C/?t + 189。 ?2C/?S2 ?2(S,t)]dt + [?C/?S ?(S,t)]dz Consider a portfolio P, bination of S and C to eliminate uncertainty: P = C + ?C/?S S , the dynamics of P is dP = dC + ?C/?S dS, dP = [?C/?S ?(S,t) + ?C/?t + 189。 ?2f/?P2 (dP)2 ? Applications in Finance A lognormal distribution for stock price returns is the standard model used in financial economics. Given some reasonable assumptions about the random behavior of stock returns, a lognormal distribution is implied. These assumptions will characterize lognomal distribution in a very intuitive manner. Let S(t) be the stock39。Lecture 9: BlackScholes option pricing formula ? Brownian Motion The first formal mathematical model of financial asset prices, developed by Bachelier (1900), was the continuoustime random walk, or Brownian motion. This continuoustime process is closely related to the discretetime versions of the random walk. ? The discretetime random walk Pk = Pk1 + ?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 acplished by setting ? = 189。*(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) ??(?(t2t1), ?2(t2t1)) 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) |
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