定價(jià)策略：black-scholes option pricing formula-文庫(kù)吧
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【正文】 | 0 0 Using th 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。 ?2f/?P2 (dP)2 ? Applications in Finance A lognormal distribution for stock price returns is the standard model used in financial economics. Given some reasonable assumptions about the random behavior of stock returns, a lognormal distribution is implied. These assumptions will characterize lognomal distribution in a very intuitive manner. Let S(t) be the stock39。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 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(?2
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