HardStochastic Processes

Expectation of GBM

Goldman SachsMorgan Stanley

A stock follows geometric Brownian motion $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$ with $S_0$ given. What is $\mathbb{E}[S_T]$ ?

Approach

Apply Itô to $\ln S_t$ .

Use $\mathbb{E}[e^{aZ}] = e^{a^2/2}$ for standard normal $Z$ .