定價(jià)策略black-scholesoptionpricingformula(1)(完整版)
【正文】 Example: Synthetic put option Day Closing price Daily Return Maturity Delta Stock Position Overall Cash 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 23/04 24/04 25/04 28/04 29/04 30/04 01/05 02/05 05/05 0 0 0 0 0 0 0 0 0 Mean . a. m. 250 d a. . 250 d % % X= T= r= P=9 ? Duration of an option An option39。 ?2C/?S2 ?2(S,t)]dt The portfolio is a riskfree portfolio, hence it should earn risk free return, . dP/P = [?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 have n(2?1) ? ? ?T 4n?(?1) ?2 ??T This can be acplished by setting ? = 189。 ?2C/?S2 ?2(S,t)]dt / [ C + ?C/?S S] = r dt, rearranging terms leads to the well known BS partial differential equation: ?C/?t + r S?C/?S + 189。s ? is its partial derivative with respect to a change in the continuously pounded interest rate. Specifically, the call option pricing formula (Black and Scholes) is c=SN(d1)XerTN(d2) where d1=[ln(S/X)+(r+?2/2)T]/(?T1/2), d2=d1?T1/2 It follows that ?c/?r = XTerTN(d2) and (?c/?r)/c = (X/c)TerTN(d2)0 The total differential of the call option can be written as dc = ?c/?r dr + ?c/?S dS Dc=(dc/dr)/c =(?c/c)/?r ?c/?S (dS/dr)= = (X/c)TerTN(d2)(S/c)N(d1)Ds Con
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