金融随机分析 III It.ppt

* COROLLARY3.3.5 (The Ito isometry) for all EXAMPLE 3.3.6 Assume B0=0. Then EX Assume B0=0. Prove that * Some properties of the Ito integral Let f,g ? V(0,T] and let 0≤SUT. Then is FT-measurable. * (7) there exists a t-continuous version of (5) is a martingale w. r. t. Ft * 3.4 Ito process and the Ito formula DEFINTION 3.4.1 Let Bt be 1-dimensional Brownian motion on (?,F, P) , A (1-dimensional) Ito process (or stochastic integral) is a stochastic process Xt on (?,F, P) of the form Where , u is Ft–adapted and * THEOREM3.4.2 (The 1-dimensional Ito formula) Let Xt be an Ito process give by Is again an Ito process, and Then Where is computed according to the rules * So Ito formula can also be write as EXAMPLE 3.4.3 let Xt=Bt,， p42 EXAMPLE3.4.4 What is THEOREM3.4.5 (INTEGRATION BY PARTS). Suppose f(s, ? )=f(s) only depends on s and that f is continuous and of bounded variation in [0,t] . Then Proof * EXAMPLE 3.4.6 Suppose Where Bs is Brownian motion in R, Then Define Zt is a martingale for t?T, provided that Corollary * Example 3.4.7 Mean and variance of the Cox-Ingersoll-Ross process The Cox-Ingersoll-Ross model for interest rates is In integral form, this equation is Then the mean and variance of r(t) are * 3.5 Black-Scholes-Mertion formula Wealth of an investor. An investor begins with initial wealth X0 and at each time t, holds Δ(t) shares of stock: Let X(t) denote the wealth of the investor at time t. Then Value of European option v(t; x) be the solution to the Black- Scholes partial differential equation * Black-Scholes-Mertion formula: Consider a European call option that pays at time T. c(t,x) -- value of the call at time t and the the stock price x. Then where (解法参见姜礼尚期权定价的数学模型和方法 p80 §5.3) Put-Call parity (参阅习题2.11) * The derivatives of the function c(t,x) w.r.t various variables are called the Greeks: delta: theta: gamma: vega: * Option value Stock price delta-neutral and long gamma