Exercise: Reading Moments from the MGF — Normal Distribution

Prerequisites: Moment Generating Functions, Normal Distribution

Problem

Let $X \sim \mathcal{N}(0, \sigma^2)$.

1. State $M_X(s)$. Write it as a Taylor series in $s$ up to order $s^8$.

2. By reading off Taylor coefficients, derive the formula $\mathbb{E}[X^{2k}] = (2k - 1)!!\cdot \sigma^{2k}, \qquad \mathbb{E}[X^{2k+1}] = 0,$ for $k = 0, 1, 2, 3, 4$. Verify $(2k - 1)!! = 1, 1, 3, 15, 105, \ldots$ (the double factorial of odd numbers).

3. Kurtosis of the normal. Recall excess kurtosis is $\mathbb{E}[X^4]/\sigma^4 - 3$. Verify this is $0$ for the normal — the definitional fact that "normal has zero excess kurtosis."
4. Black-Scholes application. For $W_T \sim \mathcal{N}(0, T)$, compute $\mathbb{E}[W_T^4]$ as a function of $T$. Explain why this appears in the variance of integrated Brownian motion $\int_0^T W_t^2\,dt$ (see the quadratic variation discussion in the Itô's lemma lesson).

Hint

For part 1, $M_X(s) = \exp(\sigma^2 s^2/2)$, and $\exp(u) = 1 + u + u^2/2! + u^3/3! + u^4/4! + \ldots$ with $u = \sigma^2 s^2/2$. Collect powers of $s$.