Options, Futures, and Other Derivatives, 10th Edition

15.2 THE DISTRIBUTION OF THE RATE OF RETURN

The lognormal property of stock prices can be used to provide information on the probability distribution of the continuously compounded rate of return earned on a stock between times 0 and T. If we define the continuously compounded rate of return per annum realized between times 0 and T as x, then

$$S_T=S_0{e}^{xT}$$

so that

$$x=\frac{1}{T}\ln \frac{S_T}{S_0}$$(15.6)

From equation (15.2), it follows that

$$x\sim \varphi \left(\mu -\frac{{\sigma }^2}{2},\frac{{\sigma }^2}{T}\right)$$(15.7)

Thus, the continuously compounded rate of return per annum is normally distributed with mean $\mu -{\sigma }^2/2$ and standard deviation $\sigma /\sqrt{T}$. As T increases, the standard deviation of x declines. To understand the reason for this, consider two cases: $T=1\text{and}T\text{=}2\text{0}$. We are more certain about the average return per year ...