Exp Value of exp(Normal)

This question focuses on finding the expectation of an exponential function of a normally distributed random variable. The setup is deliberately simple: a single normal variable with specified mean and variance, and an exponential applied directly to it. Despite looking like a basic probability exercise, the calculation connects to how exponentials of Gaussian variables behave, which is fundamental in models where log-returns are normal and asset prices become lognormal. Candidates are expected to recognize that this is not a generic integral, but a structured one that exploits the special form of the normal distribution.

To answer it cleanly, you need familiarity with the moment-generating function of a normal variable and how to evaluate expectations of the form E[exp(tX)]. The problem leans on completing the square in the exponent or, equivalently, recalling the closed-form expression for the normal MGF. Interviewers look for comfort moving between density integrals and MGFs, correct handling of parameters, and an understanding of why exponentials of normals are analytically convenient in probability and quantitative finance.

What it tests

Whenever you are asked for the expected value of an exponential function of a normal random variable, you are dealing with the moment-generating function (MGF) of the normal distribution. The MGF for a normal variable $X \sim N(\mu, \sigma^2)$ is $M_X(t) = e^{\mu t + \frac{1}{2}\sigma^2 t^2}$, which encodes all moments and, crucially, allows you to evaluate $E[e^{tX}]$ directly by plugging in $t$. This works because the exponential function and the normal density combine in such a way that the resulting integral can always be completed to a square, making the calculation tractable. The underlying reason is that the normal distribution is 'exponential quadratic', so exponentials of linear functions of $X$ preserve the form of the density and allow closed-form integration. This property is central to many results in quantitative finance and probability theory, especially when modeling log-normal processes or pricing derivatives.