文档介绍：Chapter 4
Brownian Motion
& Itô Formula
布朗运动与伊藤公式
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.
布朗运动与伊藤公式
Coin-tossing Problem
Define a random variable
It is easy to show that it has the following properties:
& are independent
布朗运动与伊藤公式
Random Variable
With the random variable, define a random variable and a random sequence
布朗运动与伊藤公式
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.
布朗运动与伊藤公式
Distribution of the Path
Let T=1,N=4,Δ=1/4,
布朗运动与伊藤公式
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.
t
S
1
布朗运动与伊藤公式
Properties of the Path
布朗运动与伊藤公式
Central Limit Theorem
For any random sequence
where the random variable X~ N(0,1), . the random variable X obeys the standard normal distribution:
E(X)=0,Var(X)=1.
布朗运动与伊藤公式
Application of Central Limit Them.
Consider limit as Δ→ 0.
布朗运动与伊藤公式