定價策略black-scholesoptionpricingformula(1)(存儲版)
【正文】 above rule we can calculate (dP)2 = ?2dt. It is not a random variable! ? Geometric Brownian motion If the arithmetic Brownian motion P(t) is taken to be the price of some asset, the price may be negative. The price process p(t)= exp(P(t)), where P(t) is the arithmetic Brownian motion, is called geometric Brownian motion or lognormal diffusion. ? Ito’s Lemma Although the first plete mathematical theory of Brownian motion is due to Wiener(1923), it is the seminal contribution of Ito (1951) that is largely responsible for the enormous number of applications of Brownian motion to problems in mathematics, statistics, physics, chemistry, biology, engineering, and of course, financial economics. In particular, Ito constructs a broad class of continuous time stochastic process based on Brownian motion – now known as Ito process or Ito stochastic differential equations – which is closed under general nonlinear transformation. Ito (1951) provides a formula – Ito’s lemma for calculating explicitly the stochastic differential equation that governs the dynamics of f(P,t): df(P,t) = ?f/?P dP + ?f/?t dt + 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。(d1)?eqT/(2T1/2) qSN(d1)eqT+rXerTN(d2) Vega: with respect to an increase in volatility ?c=?p=ST1/2N39。 2023年 2月 12日星期日 上午 12時 58分 14秒 00:58: 1比不了得就不比，得不到的就不要。 :58:1400:58:14February 12, 2023 1意志堅強的人能把世界放在手中像泥塊一樣任意揉捏。 00:58:1400:58:1400:58Sunday, February 12, 2023 1知人者智，自知者明。 上午 12時 58分 14秒 上午 12時 58分 00:58: MOMODA POWERPOINT Lorem ipsum dolor sit, eleifend nulla ac, fringilla purus. Nulla iaculis tempor felis amet, consectetur adipiscing elit. Fusce id urna blanditut cursus. 感謝您的下載觀看 專家告訴 。 00:58:1400:58:1400:582/12/2023 12:58:14 AM 1越是沒有本領(lǐng)的就越加自命不凡。 :58:1400:58Feb2312Feb23 1世間成事，不求其絕對圓滿，留一份不足，可得無限完美。 00:58:1400:58:1400:58Sunday, February 12, 2023 1乍見翻疑夢，相悲各問年。(d1)eqT/(S?T1/2) Theta with respect to a decrease in maturity ?c=SN39。 ?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。s price at date t. We subdivided the time horizon [0 T] into n equally spaced subintervals of length h. We write S(ih) as S(i), i=0,1,…,n. Let z(i) be the continuous pounded rate of return over [ (i1)h ih], ie S(i)=S(i1)exp(z(i)), i=1,2,..,n. It is clear that S(i)=S(0)exp[z(1)+z(2)+…+z(i)]. The continuous 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
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