1. Motivation
Definition: Let be Brownian motion with drift coefficient
and variance parameter
. Let
. Then
is geometric Brownian motion.
Motivation: Let be the price of a stock at time
(where n is distance); Let
be the fractional increase/decrease in the price of the stock from time n-1 to time n.
We suppose that are i.i.d. Then
The process looks like a random walk.
2. Property of Geometric Brownian Motion
Proof: ![Rendered by QuickLaTeX.com E[Y(t) | Y(s) = y_s] = E[e^{X(t)} | X(s) = ln y_s]](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-701e5d1e797fb903bd2750dec8a1eed7_l3.png)
![Rendered by QuickLaTeX.com E[Y(t) | Y(s) = y_s] = E[e^{X(t)} | X(s) = ln y_s]](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-701e5d1e797fb903bd2750dec8a1eed7_l3.png)
= ![Rendered by QuickLaTeX.com E[e^{X(t) - X(s) + X(s)} | X(s) = ln y_s]](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-cbaa4bc6913e4710b685d37a0c9ed632_l3.png)
![Rendered by QuickLaTeX.com E[e^{X(t) - X(s) + X(s)} | X(s) = ln y_s]](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-cbaa4bc6913e4710b685d37a0c9ed632_l3.png)
= ![Rendered by QuickLaTeX.com y_s E[e^{X(t) - X(s)}]](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-8719921d7edc09270eaaf88041d3ce07_l3.png)
![Rendered by QuickLaTeX.com y_s E[e^{X(t) - X(s)}]](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-8719921d7edc09270eaaf88041d3ce07_l3.png)
- E[Y(t) | Y(s) = y_s] = y_s e^{mu(t-s) + sigma^2(t-s)/2}
Proof: If W ~
, then
is lognormal with mean ![Rendered by QuickLaTeX.com E[e^W] = e^{mu + sigma^2/2}](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-8f034b090dfb3a6326d07789f19c1cc4_l3.png)
Thus, {X(t) – X(s)} ~
, since
is Brownian motion with drift
![Rendered by QuickLaTeX.com N(mu, sigma^2)](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-d2835efa95cd63c054fc29b112fcffb2_l3.png)
![Rendered by QuickLaTeX.com e^W](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-980f68af1d16129b9b17aa25e9314edf_l3.png)
![Rendered by QuickLaTeX.com E[e^W] = e^{mu + sigma^2/2}](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-8f034b090dfb3a6326d07789f19c1cc4_l3.png)
Thus, {X(t) – X(s)} ~
![Rendered by QuickLaTeX.com N[mu(t-s), sigma^2(t-s)]](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-fe97cb2671709fb45d594be3e856395c_l3.png)
![Rendered by QuickLaTeX.com X(t)](http://mytechroad.com/wp-content/ql-cache/quicklatex.com-1a6a816769366d6fc41b67d7eab4ae2b_l3.png)
- Note: if
, then
. Thus
is increasing even though the jump process with
is symmetric
3. Example
Question: You invest 1000 dollars in the stock market. Suppose that the stock market can be modeled using geometric Brownian motion with an average daily return of 0.03% and a standard deviation of 1.02%. what is the probability that your money increases after 1 year (260 business days?) 10 years? 30 years?
Question: You invest 1000 dollars in the stock market. Suppose that the stock market can be modeled using geometric Brownian motion with an average daily return of 0.03% and a standard deviation of 1.02%. what is the probability that your money increases after 1 year (260 business days?) 10 years? 30 years?
Answer: Let , where
is standard Brownian Motion.
Let where
is standard Brownian motion,
is Brownian motion with drift, and
is geometric Brownian motion.
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