Exercise: The Exponential Martingale of Brownian Motion

Prerequisites: Brownian Motion, Conditional Expectation, Log-Normal Distribution

Problem

1. Show that $\mathbb{E}[M_t] = 1$ for every $t \ge 0$. (Use the MGF of a normal distribution.)
2. Prove that $(M_t)_{t \ge 0}$ is a martingale with respect to the natural filtration of $W$. That is, for $0 \le s \le t$:

1. Financial interpretation: the quantity $\sigma^2 / 2$ is the "drift correction" that appears in the log-return of a geometric Brownian motion. Explain in one paragraph why $M_t$ being a martingale is the mathematical content of the statement "discounted asset prices are martingales under the risk-neutral measure" in its simplest possible form.
2. Extension: What goes wrong if you drop the $-\tfrac{1}{2}\sigma^2 t$ term? Compute $\mathbb{E}[\exp(\sigma W_t)]$ without the correction, and observe how it depends on $t$.

Hint

For parts 1–2, use $W_t - W_s \sim \mathcal{N}(0, t-s)$ is independent of $\mathcal{F}_s$, and the MGF identity $\mathbb{E}[e^{\lambda Z}] = e^{\lambda^2/2}$ for $Z \sim \mathcal{N}(0, 1)$. Factor $M_t$ into a $\mathcal{F}_s$-measurable piece times a piece involving $W_t - W_s$.