Log-Normal Distribution

Motivation: why this matters in quant finance

The log-normal distribution is the distribution of a positive quantity whose logarithm is normal. That is the right shape for a price model built from multiplicative compounding. Returns over short intervals multiply wealth; logs convert that product into a sum.

This makes the log-normal the distribution of prices in the Black-Scholes framework. It also teaches a broader modelling lesson: a transformation can change the meaning of parameters. The mean of $\ln X$ is not the mean of $X$ .

Definition

A positive random variable

X

is log-normal with parameters

\mu

and

\sigma^2

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

Equivalently, $X=e^Y$ with $Y\sim\mathcal{N}(\mu,\sigma^2)$ . Its density is

f(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{(\ln x-\mu)^2}{2\sigma^2}\right), \qquad x>0.

The CDF is a normal CDF after taking logs:

F(x)=\Phi\left(\frac{\ln x-\mu}{\sigma}\right), \qquad x>0.

Key Properties

The central tendency measures separate:

\operatorname{Mode}(X)=e^{\mu-\sigma^2}, \qquad \operatorname{Median}(X)=e^\mu, \qquad \mathbb{E}[X]=e^{\mu+\sigma^2/2}.

Thus mode < median < mean. A few very large upside outcomes pull the mean above the typical path.

The variance is

\operatorname{Var}(X)=e^{2\mu+\sigma^2}(e^{\sigma^2}-1).

Products are closed: if independent log-normal variables have log-parameters $(\mu_i,\sigma_i^2)$ , their product is log-normal with log-mean $\sum_i\mu_i$ and log-variance $\sum_i\sigma_i^2$ . Sums are not closed, which is why portfolios of log-normal asset values often require approximation or simulation.

In Quant Finance

Under geometric Brownian motion,

dS_t=\mu S_t\,dt+\sigma S_t\,dW_t.

Applying Itô's Lemma to

\ln S_t

gives

\ln S_T=\ln S_0+\left(\mu-\frac{\sigma^2}{2}\right)T+\sigma W_T.

Therefore $S_T$ is log-normal. Under the risk-neutral measure, replace $\mu$ by $r$ :

S_T=S_0\exp\left(\left(r-\frac{\sigma^2}{2}\right)T+\sigma\sqrt{T}Z\right).

The Black-Scholes formula is a discounted expectation under this distribution. The $\Phi(d_1)$ and $\Phi(d_2)$ terms are normal-CDF calculations created by the log-normal terminal price.

ISL's additive Gaussian error setup is a useful contrast. Regression noise is often modelled as additive; asset prices are often modelled through multiplicative shocks. Choosing levels or logs is a modelling decision, not a formatting choice.

Worked Example: Mean versus Median Price

Let $S_0=100$ , $\mu=8\%$ , $\sigma=30\%$ , and $T=5$ . Then

\mathbb{E}[S_T]=100e^{0.08\cdot5}=149.18,

while the median is

100e^{(0.08-0.045)5}=119.12.

The expected price is much higher than the median because the distribution is right-skewed.

Common Confusions and Pitfalls

$\mu$ and $\sigma$ are the mean and standard deviation of $X$ . They belong to

\ln X

A log-normal tail fixes market crashes. It gives positive, right-skewed prices, but it does not capture jumps, volatility clustering, or negative return skew.

A basket of log-normal prices is log-normal. Products preserve log-normality; sums generally do not.

Where This Goes Next

Normal Distribution: all log-normal calculations reduce to normal calculations in log-space.
The Black-Scholes formula: prices options by integrating a payoff against a log-normal terminal stock price.
Binomial Tree Model: discrete multiplicative trees converge toward log-normal terminal prices.

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 40-43, for transformed positive quantities and expectation after random construction.
James, Witten, Hastie, and Tibshirani, An Introduction to Statistical Learning, 2nd ed., Ch. 3, for additive Gaussian modelling as a contrast to multiplicative log-normal modelling.