Options, Futures, and Other Derivatives, 10th Edition

14.7 THE LOGNORMAL PROPERTY

We now use Itô’s lemma to derive the process followed by ln S when S follows the process in equation (14.13). We define

$$G=1\text{n}S$$

Since

$$\frac{\partial G}{\partial S}=\frac{1}{S},\frac{{\partial }^{2}G}{\partial S^2}=-\frac{1}{S^2},\frac{\partial G}{\partial t}=0$$

it follows from equation (14.14) that the process followed by G is

$$dG=\left(\mu -\frac{{\sigma }^2}{2}\right)dt+\sigma dz$$(14.17)

Since μ and σ are constant, this equation indicates that $G=\text{ln}S$ follows a generalized Wiener process. It has constant drift rate $\mu -{\sigma }^2/2$ and constant variance rate σ². The change in ln S between time 0 and some future time T is therefore normally distributed, with mean $(\mu -{\sigma }^2/2)T$ and variance σ²T. This means that

$$\text{ln}{S}_T-\text{ln}{S}_0\sim \varphi \left[\left(\mu -\frac{{\sigma }^2}{2}\right)T,{\sigma }^{2}T\right]$$(14.18)

or

$$\text{ln}{S}_T\sim \varphi \left[\text{ln}{S}_0+\left(\mu -\frac{{\sigma }^2}{2}\right)T,{\sigma }^{2}T\right]$$(14.19)

where S_T is the stock price ...