Analysis of Financial Time Series, Third Edition

6.4 Distributions of Stock Prices and Log Returns

The result of the previous section shows that if one assumes that price of a stock follows the geometric Brownian motion

then the logarithm of the price follows a generalized Wiener process

where Pt is the price of the stock at time t and wt is a Wiener process. Therefore, the change in log price from time t to T is normally distributed as

(6.9)

Consequently, conditional on the price Pt at time t, the log price at time T > t is normally distributed as

(6.10)

Using the result of lognormal distribution discussed in Chapter 1, we obtain the (conditional) mean and variance of PT as

Note that the expectation confirms that μ is the expected rate of return of the stock.

The prior distribution of stock price can be used to make inferences. For example, suppose that the current price of stock A is $50, the expected return of the stock is 15% per annum, and the volatility is 40% per annum. Then the expected price of stock A in 6 months (0.5 year) ...