Mean of e to a zero-mean normal

Question

Suppose $X$ is normally distributed with mean $0$ and variance $\sigma^2$ . Find $\mathbb{E}[e^X]$ .

Short answer

$e^{\sigma^2/2}$ . This is the normal moment generating function $\mathbb{E}[e^{tX}]=e^{t^2\sigma^2/2}$ at $t=1$ ; complete the square in the exponent of the Gaussian integral. It is the lognormal mean that drives the $-\tfrac12\sigma^2$ drift correction in finance.

Adapted from Crack, Heard on the Street, Q4.26, p.42; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

The Normal Distribution
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free