Random Variables and Distributions

Quantitative finance models uncertain quantities as random variables: tomorrow's return, the number of defaults in a portfolio, the time until the next trade. The modelling work is not deriving the distribution, it is choosing one, and the choice is usually forced by what the quantity physically cannot do rather than by what the data looks like.

Let the constraint pick the distribution

Returns: normal, approximately. Symmetric, defined on the whole real line, and justified by the central limit theorem if a return aggregates many small independent effects. Convenient because sums of normals are normal, so a portfolio of jointly normal assets is itself normal and needs no new machinery.

Prices: lognormal. A price cannot go negative, which rules out a normal immediately. If log returns are normal then the price is lognormal, which respects positivity and captures the multiplicative nature of compounding. This is the assumption behind Black-Scholes.

Up-down moves: binomial. Discrete steps, which is the basis of the binomial tree.

Defaults and rare events: Poisson. Counting infrequent events over a fixed period.

Waiting times: exponential. Time until the next trade, or the next default.

The normal-to-lognormal step, in one line

The pairing of "normal returns, lognormal prices" is stated so often that its content gets lost. It is just a change of variable.

If the log return over a period is $R = \ln(S_T/S_0)$ and $R \sim \mathcal{N}(\mu, \sigma^2)$, then

and an exponential of a normal is lognormal by definition. Positivity is automatic, since $e^R > 0$ whatever $R$ does.

One consequence catches people out. Because the exponential is convex, Jensen's inequality puts the mean above the median:

That $\frac{1}{2}\sigma^2$ is not a rounding term. It is the reason the drift in geometric Brownian motion has a variance correction in it, and the reason a typical path ends below the expected value. Most outcomes are below average, and the average is pulled up by a thin right tail.

Where the normal assumption fails

Financial returns have fat tails and negative skew. Extreme moves occur far more often than a normal predicts, and downside moves are sharper than upside ones.

This is why a model assuming normality prices tail risk far too cheaply, and why the volatility smile exists at all: it is the options market stating, in prices, that it does not believe the assumption.

The usual repairs, in increasing order of cost: the t-distribution buys fatter tails for one extra parameter, jump diffusion adds discrete shocks and generates steep short-dated skew, and stochastic volatility makes $\sigma$ itself random, which produces smiles naturally and fits their dynamics better.

The practical stance

Use the normal for intuition, for the central limit theorem, and for anything depending on the centre of a distribution. Do not use it for tails, and say so explicitly whenever a result rests on one.

That distinction, between results driven by the body of a distribution and results driven by its tail, is the most useful single habit in applied quantitative finance, and it is the one that separates a model being wrong from a model being dangerous.