Solution: The Exponential Martingale of Brownian Motion

Exercise: The Exponential Martingale of Brownian Motion

Part 1 — Expected value at a single time

Since $W_t \sim \mathcal{N}(0, t)$ , the MGF of a Gaussian gives:

\mathbb{E}[e^{\sigma W_t}] = e^{\sigma^2 t / 2}

Therefore:

\mathbb{E}[M_t] = \mathbb{E}[e^{\sigma W_t - \sigma^2 t / 2}] = e^{-\sigma^2 t / 2}\,\mathbb{E}[e^{\sigma W_t}] = e^{-\sigma^2 t / 2} \cdot e^{\sigma^2 t / 2} = 1

The $-\tfrac{1}{2}\sigma^2 t$ is exactly the correction needed to cancel the $+\tfrac{1}{2}\sigma^2 t$ Jensen bump contributed by exponentiating a mean-zero Gaussian.

Part 2 — Martingale property

Split the exponent using $W_t = W_s + (W_t - W_s)$ :

M_t = \exp\!\left(\sigma W_s - \tfrac{1}{2}\sigma^2 s\right) \cdot \exp\!\left(\sigma(W_t - W_s) - \tfrac{1}{2}\sigma^2(t - s)\right) = M_s \cdot R_{s, t}

where $R_{s, t} := \exp\!\left(\sigma(W_t - W_s) - \tfrac{1}{2}\sigma^2(t - s)\right)$ . Now:

$M_s$ is $\mathcal{F}_s$ -measurable (it depends only on $W_u$ for $u \le s$ ).
$R_{s, t}$ depends only on the increment $W_t - W_s$ , which is independent of $\mathcal{F}_s$ by (BM2).
$W_t - W_s \sim \mathcal{N}(0, t - s)$ , so applying part 1 with parameter $t - s$ :

\mathbb{E}[R_{s, t}] = \exp\!\left(-\tfrac{1}{2}\sigma^2(t - s) + \tfrac{1}{2}\sigma^2(t - s)\right) = 1

Using measurability of $M_s$ and independence of $R_{s, t}$ from $\mathcal{F}_s$ (standard take-out-what-is-known and independence tricks for conditional expectation):

\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s \cdot \mathbb{E}[R_{s, t} \mid \mathcal{F}_s] = M_s \cdot \mathbb{E}[R_{s, t}] = M_s \cdot 1 = M_s

Hence $(M_t)$ is a martingale. $\square$

Part 3 — Financial interpretation

Under the risk-neutral measure $\mathbb{Q}$ , a non-dividend-paying stock following geometric Brownian motion has dynamics $dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}$ , with solution:

S_t = S_0 \exp\!\left((r - \tfrac{1}{2}\sigma^2)t + \sigma W_t^{\mathbb{Q}}\right)

The discounted price $e^{-rt}S_t$ is:

e^{-rt}S_t = S_0 \exp\!\left(\sigma W_t^{\mathbb{Q}} - \tfrac{1}{2}\sigma^2 t\right) = S_0 \cdot M_t

That is, the discounted price is exactly

S_0

times the exponential martingale we just analysed. The martingale property

\mathbb{E}^{\mathbb{Q}}[M_t \mid \mathcal{F}_s] = M_s

is therefore identical to the no-arbitrage condition "discounted asset prices are

\mathbb{Q}

-martingales." The

-\tfrac{1}{2}\sigma^2 t

correction is not cosmetic — it is the term that makes the martingale property hold, and through it, the fair-pricing formula

V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[\text{payoff}]

is internally consistent.

Part 4 — Without the correction

\mathbb{E}[\exp(\sigma W_t)] = e^{\sigma^2 t / 2}

This grows exponentially in

t

. If we used

\tilde{M}_t := \exp(\sigma W_t)

(no correction), then

\mathbb{E}[\tilde{M}_t]

is not constant, so

\tilde{M}_t

is a submartingale, not a martingale — the process has an upward drift in expectation. Financially, this would correspond to a mispricing: an asset whose discounted-price expectation drifts upward admits an arbitrage (buy and hold; it grows faster than the risk-free rate on average). The

-\tfrac{1}{2}\sigma^2 t

subtracts exactly enough drift to cancel the Jensen bump and restore the fair-pricing martingale.

Takeaways

$M_t = \exp(\sigma W_t - \tfrac{1}{2}\sigma^2 t)$ is the simplest nontrivial martingale driven by Brownian motion. All the machinery of risk-neutral pricing is built on exponential martingales of this form (with $\sigma$ possibly replaced by a process or a vector).
The $-\tfrac{1}{2}\sigma^2 t$ is a Jensen correction. It is exactly the $\sigma^2/2$ drift term that appears in the log-return of GBM, and it reflects the same phenomenon every time: the expected value of an exponentiated random variable is larger than the exponential of its expectation.
Martingale = no-arbitrage. The martingale property of the discounted price is the mathematical expression of absence of arbitrage, not a coincidence.
Girsanov preview. The exponential martingale $M_t$ is also the Radon-Nikodym density that changes the measure from $\mathbb{P}$ to $\mathbb{Q}$ in Girsanov's theorem. A later lesson formalises this.