定價策略black-scholesoptionpricingformula(1)-文庫吧在線文庫
【正文】 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。s change with respect to an increase in sock price ?c=?p=N39。 :58:1400:58Feb2312Feb23 1故人江海別，幾度隔山川。 00:58:1400:58:1400:582/12/2023 12:58:14 AM 1成功就是日復一日那一點點小小努力的積累。 , February 12, 2023 閱讀一切好書如同和過去最杰出的人談話。 2023年 2月 12日星期日 12時 58分 14秒 00:58:1412 February 2023 1一個人即使已登上頂峰，也仍要自強不息。勝人者有力，自勝者強。 2023年 2月 12日星期日 上午 12時 58分 14秒 00:58: 1楚塞三湘接，荊門九派通。 2023年 2月 上午 12時 58分 :58February 12, 2023 1行動出成果，工作出財富。(d1)eqT Rho: with respect to an increase in interest rate ?c=XTerTN(d2) ?p=XTerTN(d2) Example1: X=$70, T= S= X= T= r= . = q = European Option Prices d1= N(d1)= d2= N(d2)= Call= Put= Delta= Gamma= Theta = Vega= Rho= ? Synthetic option Set aside cash in the amount equal to the model value. Maintain the stock position equal to the delta of the target option. Cash balance is invested in riskfree assets to earn interests. Close the position at the desired matuirity. If the model is good, the terminal payoff of this dynamic strategy should be close to the payoff of the target option at the maturity.
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