Normal Distribution

Motivation: why this matters in quant finance

The normal distribution is the benchmark law for additive uncertainty. It is not important because markets are exactly Gaussian; it is important because sums of many small shocks often look Gaussian, and because the Gaussian gives a clean baseline against which fat tails, skew, jumps, and volatility clustering can be seen.

In quant finance, the normal appears in three different jobs. Brownian increments are normal. Log-returns in the Black-Scholes model are normal. Regression errors and coefficient estimates are often analysed through Gaussian or near-Gaussian approximations. These are not the same application, so the lesson should not reduce the normal to one generic bell-curve story.

Definition

A random variable

X

is normally distributed with mean

\mu

and variance

\sigma^2

, written

X \sim \mathcal{N}(\mu,\sigma^2)

, if

f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right), \qquad x\in\mathbb{R}.

The standard normal is $Z\sim\mathcal{N}(0,1)$ , with density $\phi$ and CDF $\Phi$ :

\phi(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2}, \qquad \Phi(z)=\int_{-\infty}^{z}\phi(u)\,du.

Any normal variable can be standardised:

Z=\frac{X-\mu}{\sigma}, \qquad \mathbb{P}(X\leq x)=\Phi\left(\frac{x-\mu}{\sigma}\right).

This is why probability tables, regression test statistics, and Black-Scholes $d_1,d_2$ terms all reduce to the same standard normal CDF.

Key Properties

The normal is symmetric, so $\phi(-z)=\phi(z)$ and $\Phi(-z)=1-\Phi(z)$ . Its MGF is

M_X(t)=\exp\left(\mu t+\frac{\sigma^2t^2}{2}\right),

which gives the important identity

\mathbb{E}[e^X]=e^{\mu+\sigma^2/2}.

Independent linear combinations stay normal:

\sum_i a_iX_i\sim \mathcal{N}\left(\sum_i a_i\mu_i,\sum_i a_i^2\sigma_i^2\right).

That closure is why Gaussian portfolio models are so tractable. It is also why they can become dangerously comfortable: tractability is not evidence that the tails are right.

In Quant Finance

A standard Brownian motion satisfies

W_t-W_s\sim\mathcal{N}(0,t-s).

Under geometric Brownian motion,

\ln\frac{S_T}{S_0}\sim\mathcal{N}\left(\left(\mu-\frac{\sigma^2}{2}\right)T,\sigma^2T\right),

so prices are log-normal, not normal.

ISL's regression examples use normal errors as a model for random noise around a systematic relationship. In finance this is the same diagnostic idea: explain the signal, then inspect whether the residual distribution is close enough to Gaussian for the intended use.

Worked Example: Gaussian VaR

If daily portfolio return is $R\sim\mathcal{N}(0,0.01^2)$ , the 99% one-day loss quantile is

-\Phi^{-1}(0.01)\cdot0.01\approx2.326\%.

A five-sigma daily loss has probability about $2.87\times10^{-7}$ under this model. If such events occur every few years, the problem is not arithmetic; it is the Gaussian tail assumption.

Common Confusions and Pitfalls

Stock prices are normal. In the standard model, log-returns are normal and prices are log-normal.

A Gaussian first pass is harmless. It can be harmless for central estimates and harmful for tail capital.

The normal is only useful when the data are normal. The Central Limit Theorem makes normal approximations useful for sums and averages even when the original variables are not normal.

Where This Goes Next

Log-Normal Distribution: exponentiating a normal gives the positive price distribution in Black-Scholes.
Student's t-Distribution: keeps the bell-shaped centre but thickens the tails.
Chi-Squared and Related Distributions: sums of squared normals drive variance estimation and the $t$ statistic.
Central Limit Theorem: explains why Gaussian limits appear so often.

References

Bertsekas and Tsitsiklis, Introduction to Probability, 2nd ed., Ch. 3 Sec. 3.3 and Ch. 4 Sec. 4.4.
Mosteller, Fifty Challenging Problems in Probability, Problems 31-33, for approximation discipline and the habit of checking what the model is counting.
James, Witten, Hastie, and Tibshirani, An Introduction to Statistical Learning, 2nd ed., Ch. 3, for Gaussian errors, standard errors, confidence intervals, and $t$ -statistics.