The Lognormal Distribution
A positive random variable X is lognormal when its
logarithm is normal. That is the whole definition: if
\ln X \sim N(\mu, \sigma^2), \qquad\text{equivalently}\qquad X = e^{Y}, \quad Y \sim N(\mu, \sigma^2),
then X is lognormal with parameters
\mu and \sigma^2. Because
e^{Y} is always positive, X lives on
(0, \infty) — it can never be zero or negative. Note the parameters
\mu, \sigma describe the normal on the log scale, not the
mean and standard deviation of X itself.
The density, mean, and median
Changing variables from Y = \ln X back to
X introduces a factor of 1/x, giving the
density (for x > 0):
f(x) = \frac{1}{x\,\sigma\sqrt{2\pi}}\,\exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \qquad x > 0.
This curve is right-skewed: it rises from zero, peaks, then trails off with a
long tail to the right. The mean and median are no longer equal — the long right tail drags the
mean above the median:
\mathbb{E}[X] = \exp\!\left(\mu + \tfrac12\sigma^2\right), \qquad \text{median}(X) = e^{\mu}.
The extra \tfrac12\sigma^2 in the mean is the fingerprint of the
skew: more spread on the log scale pushes the average further above the typical (median) value.
Watch the skew
Slide the log-scale parameters. The whole curve stays on the positive side
x > 0, hugging zero on the left and stretching into a long tail on
the right. Raising \sigma exaggerates the skew dramatically; raising
\mu slides the bulk to larger values.
Why prices are lognormal
Two facts make this the natural model for a stock price S_t. First, a
price cannot go negative — and the lognormal lives exactly on
(0, \infty). Second, returns compound
multiplicatively: a price is its starting value times a product of many small growth
factors, and the logarithm turns that product into a sum of log-returns. If
each log-return is
normal,
their sum \ln(S_t/S_0) is normal — so
S_t itself is lognormal. This is precisely the distribution that
falls out of geometric Brownian motion, the engine of the Black–Scholes model.