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
Consequently, conditional on the price Pt at time t, the log price at time T > t is normally distributed as
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) and the associated variance are given by
The standard deviation of the price 6 months from now is 
.
Next, let r be the continuously compounded rate of return per annum from time t to T. Then we have
where T and t are measured in years. Therefore,
By Eq. (6.9), we have
Consequently, the distribution of the continuously compounded rate of return per annum is
The continuously compounded rate of return is, therefore, normally distributed with mean μ − σ2/2 and standard deviation 
.
Consider a stock with an expected rate of return of 15% per annum and a volatility of 10% per annum. The distribution of the continuously compounded rate of return of the stock over 2 years is normal with mean 0.15 − 0.01/2 = 0.145 or 14.5% per annum and standard deviation 
or 7.1% per annum. These results allow us to construct confidence intervals (CI) for r. For instance, a 95% CI for r is 0.145 ± 1.96 × 0.071 per annum (i.e., 0.6%, 28.4%).