[Stochastic Process] Geometric Brownian Motion

1. Motivation

Definition: Let X(t) be Brownian motion with drift coefficient mu and variance parameter sigma^2. Let Y(t) = e^{-X(t)}. Then Y(t) is geometric Brownian motion.

Motivation: Let Y(_n) be the price of a stock at time n (where n is distance); Let X(n) = frac{Y_n}{Y_+{n-1}} be the fractional increase/decrease in the price of the stock from time n-1 to time n.
We suppose that X_n are i.i.d. Then
Y_n = X_n Y_{n-1} = X_n X_{n-1} X_{n-1} cdots X_1 Y_0
ln Y_n = ln X_n X_{n-1} X_{n-1} cdots X_1 Y_0 = ln Y_0 + sum^n_{i=1} ln X_i

The process ln(Y_n) looks like a random walk.

2. Property of Geometric Brownian Motion

  • E[Y(t) | Y(s) = y_s] = y_s E[e^{{X(t) - X(s)}}]
Proof: E[Y(t) | Y(s) = y_s] = E[e^{X(t)} | X(s) = ln y_s]
                                             = E[e^{X(t) - X(s) + X(s)} | X(s) = ln y_s]
                                             = y_s E[e^{X(t) - X(s)}]
  • E[Y(t) | Y(s) = y_s] = y_s e^{mu(t-s) + sigma^2(t-s)/2}
Proof: If W ~ N(mu, sigma^2), then e^W is lognormal with mean E[e^W] = e^{mu + sigma^2/2}
Thus, {X(t) – X(s)} ~ N[mu(t-s), sigma^2(t-s)], since X(t) is Brownian motion with drift

  • Note: if mu = 0, then E[Y(t)|Y(0) = y_0 ] = y_0  e^{sigma^2 t/2}. Thus E[Y(t)] is increasing even though the jump process with X(t) 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?

Answer: Let X(t) = mu t + sigma B(t), where B(t) is standard Brownian Motion.
Let Y(t) = Y_0 e^{X(t)} where B(t) is standard Brownian motion, X(t) is Brownian motion with drift, and Y(t) is geometric Brownian motion.

Pr(Y(t) > Y_0) = Pr(Y_0 e^{X(t)} > Y_0)
                           = Pr(e^{X(t)} > 1)
                           = Pr(X(t) > ln 1 = 0)
                           = Pr(N(mu t, sigma^2 t) > 0)
                           = 1 - Phi(frac{-mu t}{sigma sqrt{t}})
                           = Phi(frac{mu sqrt{t}}{sigma})

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