定價競爭策略英文-在線瀏覽
2025-02-25 02:40本頁面
【正文】 pounded return on the stock over the period [0 T] is the sum of the continuously pounded returns over the n subintervals. Assumption A1. The returns {z(j)} are i... Assumption A2. E[z(t)]=?h, where ? is the expected continuously pounded return per unit time. Assumption A3. var[z(t)]=?2h. Technically, these assumptions ensure that as the time decrease proportionally, the behavior of the distribution for S(t) dose not explode nor degenerate 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。 ?2C/?S2 ?2(S,t)]dt [?C/?S ?(S,t)]dz + ?C/?S[?(S,t)dt + ?(S,t)dz] Collecting terms involving dt and dz together we get dP = [?C/?S ?(S,t) + ?C/?t + 189。 ?2C/?S2 ?2(S,t)]dt The portfolio is a riskfree portfolio, hence it should earn risk free return, . dP/P = [?C/?t + 189。 ?2S2?2C/?S2 – rC = 0
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