布朗运动与伊藤公式.ppt

Chapter 4
Brownian Motion
& Itô Formula
1
Stochastic Process
The price movement of an underlying asset is a stochhe increments

are independent.
11
Continuous Models of Asset Price Movement
Introduce the discounted value
of an underlying asset as follows:
in time interval [t,t+Δt], the BTM can be written as
12
Lemma
If ud=1, σis the volatility, letting

then under the martingale measure Q,
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Proof of the Lemma
According to the definition of martingale measure Q, on [t,t+Δt],

thus by straightforward computation,
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Proof of the Lemma
Moreover, since
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Proof of the Lemma cont.
by the assumption of the lemma,

input these values to the ori. equation.
This completes the proof of the lemma.
16
Geometric Brownian Motion
By Taylor expansion
neglecting the higher order terms of Δt, we have
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Geometric Brownian Motion cont.
By definition
therefore after partitioning [0,T], at each instant ,
.
18
Geometric Brownian Motion cont.-
19
Geometric Brownian Motion cont.--
This means the underlying asset price movement as a continuous stochastic process, its logarithmic function is described by the Brownian motion. The underlying asset price S(t) is said to fit geometric Brownian motion.
This means: Corresponding to the discrete BTM of the underlying asset price in a risk-neutral world (. under the martingale measure), its continuous model obeys the geometric Brownian motion .
20
Definition of Quadratic Variation
Let function f(t) be given in [0,T], and Π be a partition of the interval [0,T]:
the quadratic variation of f(t) is defined by
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Quadratic Variation for classical function
22
Theorem
Let Π be any partition of the interval [0,T], then the quadratic variation of a Brownian motion has a limit as follows:
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Path of a Brownian motion
For any let be an
arbitrary partition of the interval and be the q