Solution: Reading Moments from the MGF — Normal Distribution

Exercise: Reading Moments from the MGF: Normal Distribution

Part 1

$M_X(s) = \exp(\sigma^2 s^2/2)$ . With $u = \sigma^2 s^2/2$ :

M_X(s) = 1 + u + \tfrac{u^2}{2} + \tfrac{u^3}{6} + \tfrac{u^4}{24} + \cdots = 1 + \tfrac{\sigma^2 s^2}{2} + \tfrac{\sigma^4 s^4}{8} + \tfrac{\sigma^6 s^6}{48} + \tfrac{\sigma^8 s^8}{384} + \cdots

Only even powers of $s$ appear.

Part 2

Compare to the general series $M_X(s) = \sum_k \mathbb{E}[X^k] s^k/k!$ . Matching coefficient of $s^{2k}$ :

\frac{\mathbb{E}[X^{2k}]}{(2k)!} = \frac{1}{k!}\cdot \frac{\sigma^{2k}}{2^k}.

So $\mathbb{E}[X^{2k}] = (2k)!/(2^k\,k!)\cdot \sigma^{2k}$ . The combinatorial factor $(2k)!/(2^k k!)$ equals $(2k - 1)!! = 1\cdot 3\cdot 5 \cdots (2k - 1)$ . Values:

$k$	$\mathbb{E}[X^{2k}]/\sigma^{2k}$	$(2k - 1)!!$
$0$	$1$	$1$
$1$	$1$	$1$
$2$	$3$	$1\cdot 3 = 3$
$3$	$15$	$1\cdot 3\cdot 5 = 15$
$4$	$105$	$1\cdot 3\cdot 5\cdot 7 = 105$

Odd moments are $0$ because only even powers of $s$ appear in the MGF Taylor series — and by symmetry of the normal distribution around $0$ , the odd central moments must vanish.

Part 3

Excess kurtosis:

\mathbb{E}[X^4]/\sigma^4 - 3 = 3 - 3 = 0

. ✓ This is the defining property: zero excess kurtosis characterises the normal among light-tailed distributions (at the level of the fourth moment). Distributions with positive excess kurtosis ("leptokurtic") have heavier tails than normal — essentially every empirical financial return distribution.

Part 4

For $W_T \sim \mathcal{N}(0, T)$ : $\sigma^2 = T$ , so $\mathbb{E}[W_T^4] = 3T^2$ .

Application. The quadratic variation of Brownian motion is

\langle W, W\rangle_T = T

— deterministic. But we can also compute

\mathbb{E}[W_T^2] = T

directly. More subtle:

\mathbb{E}[(W_T^2 - T)^2] = \mathbb{E}[W_T^4] - 2T\mathbb{E}[W_T^2] + T^2 = 3T^2 - 2T^2 + T^2 = 2T^2

. So

\text{Var}(W_T^2) = 2T^2

, with standard deviation

T\sqrt{2}

. This is a strong form of concentration around the mean: the

L^2

fluctuations of

W_T^2

grow proportionally with

T

, which is what makes the quadratic-variation limit rigorous.

Integrated-Brownian-motion variance computations use these fourth-moment identities routinely (via Itô isometry and Fubini arguments).

Takeaways

Normal moments are $(2k - 1)!!\,\sigma^{2k}$ . The double factorial appears because of the $1/2^k$ and $1/k!$ in the Taylor expansion of $\exp$ .
Zero excess kurtosis is a normal-distribution fingerprint. Any departure flags heavier-than-normal tails, a universal feature of real returns.
$\mathbb{E}[W_T^4] = 3T^2$ is the most commonly-cited Brownian-motion fourth moment; it appears in quadratic-variation estimates, Itô-isometry variance bounds, and virtually every variance-of-a-square-of-normal calculation.
MGF Taylor coefficients are the cleanest way to derive all moments of a family at once.