定價策略：black-scholesoptionpricingformula-wenkub
2023-02-13 22:39:38 本頁面
【正文】 enerate to a fixed point. Assumption 13 implies that for any infinitesimal time subintervals, the distribution for the continuously pounded return z(t) has a normal distribution with mean ?h, and variance ?2h. This implies that S(t) is lognormally distributed. ? Lognormal distribution At time t t+h lnSt+h ~ ?[lnSt+(??2/2)h,?] where ?(m,s) denotes a normal distribution with mean m and standard deviation s. ? Continuously pounded return ln(St+h/St) ~ ?[(??2/2)h,?] ? Expected returns Et[ln(St+h/St)] = (??2/2)h Et[St+h/St] = exp(?h) ? Variance of returns Vart[ln(St+h/St)] = ?2h Vart[St+h/St] = exp(2?h)(exp(?2h)1) ? Estimation of ? n+1: number of stock observations Sj: stock price at the end of jth interval, j=1,…n h: length of time intervals in years Let uj = ln[Sj+Dj)/Sj1] u = (u1+…+u n)/n is an estimator for (??2/2)h, s={ [(u1u)2+…+(u nu)2]/(n1)}1/2 is an estimator for ?h1/2. Example: Daily returns Day Closing price Dividend Daily Return 07/04 08/04 09/04 10/04 11/04 14/04 15/04 16/04 17/04 18/04 21/04 22/04 0 0 0 0 0 0 0 0 0 0 0 0 Day Closing price Dividend Daily Return 23/04 24/04 25/04 28/04 29/04 30/04 01/05 02/05 05/05 Mean . Annualized Annualized Mean(250 d) . (250 d) 0 0 0 0 0 0 0 0 0 % % ? Fundamental equation for derivative securities Stock price follows Ito process: dS = ?(S,t)dt + ?(S,t)dz At this point, we assume ?(S,t) =?S, and ?(S,t)= ?S Let C(S,t) be a derivative security, according to Ito’s lemma, the process followed by C is dC = [?C/?S ?(S,t) + ?C/?t + 189。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
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