同济大学金融衍生物定价理论课件 第四章.pptVIP

Chapter 4 Brownian Motion It? Formula Stochastic Process The price movement of an underlying asset is a stochastic process. The French mathematician Louis Bachelier was the first one to describe the stock share price movement as a Brownian motion in his 1900 doctoral thesis. introduction to the Brownian motion derive the continuous model of option pricing giving the definition and relevant properties Brownian motion derive stochastic calculus based on the Brownian motion including the Ito integral Ito formula. All of the description and discussion emphasize clarity rather than mathematical rigor. Coin-tossing Problem Define a random variable It is easy to show that it has the following properties: are independent Random Variable With the random variable, define a random variable and a random sequence Random Walk Consider a time period [0,T], which can be divided into N equal intervals. Let Δ=T\ N, t_n=nΔ ,(n=0,1,\cdots,N), then A random walk is defined in [0,T]: is called the path of the random walk. Distribution of the Path Let T=1,N=4,Δ=1/4, Form of Path the path formed by linear interpolation between the above random points. For Δ=1/4 case, there are 2^4=16 paths. Properties of the Path Central Limit Theorem For any random sequence where the random variable X~ N(0,1), i.e. the random variable X obeys the standard normal distribution: E(X)=0,Var(X)=1. Application of Central Limit Them. Consider limit as Δ→ 0. Definition of Winner Process(Brownian Motion) 1) Continuity of path: W(0)=0,W(t) is a continuous function of t. 2) Normal increments: For any t0,W(t)~ N(0,t), and for 0 s t, W(t)-W(s) is normally distributed with mean 0 and variance t-s, i.e., 3) Independence of increments: for any choice of in [0,T] with the increments are independent. Cont