The Gamma Distribution

Trade fills on a thin order book don't arrive on a fixed schedule — they arrive at random, one at a time. The exponential distribution models the wait for a single fill. But what if you care about the wait for the fifth fill, or you want a flexible, always-positive shape for a quantity like realized variance that can't be normal (since variance can't go negative)? Both needs point to the same distribution: the gamma.

The analogy: waiting for the k-th bus

If buses arrive at random, independent intervals (a Poisson process) with an average wait of $1/\lambda$ between each, the exponential distribution tells you how long until the next bus. The gamma distribution tells you how long until the $k$-th bus arrives — the sum of $k$ independent exponential waits stacked end to end. As $k$ grows, that sum smooths out (by a CLT-like effect) from a sharply skewed shape into something closer to a bell curve, but for small $k$ it stays distinctly right-skewed — exactly the shape you'd expect for "total time until several things have happened."

Writing it down

The gamma distribution with shape $k$ (or $\alpha$) and rate $\lambda$ (or scale $\theta = 1/\lambda$) has density

$$f(x; k, \lambda) = \frac{\lambda^k x^{k-1} e^{-\lambda x}}{\Gamma(k)}, \qquad x > 0,$$

where $\Gamma(k)$ is the gamma function, a continuous generalization of factorial ($\Gamma(k) = (k-1)!$ for integer $k$). In words: the density rises like $x^{k-1}$ (governed by shape) and decays like $e^{-\lambda x}$ (governed by rate), with $\Gamma(k)$ just a normalizing constant so the total probability integrates to 1. Mean is $k/\lambda$, variance is $k/\lambda^2$. Two special cases matter: $k=1$ recovers the exponential distribution exactly, and setting $k = \nu/2$, $\lambda = 1/2$ recovers the chi-squared distribution with $\nu$ degrees of freedom.

The rising-then-falling shape here mirrors gamma's density for small shape $k$: a fast rise from zero (no mass exactly at zero once $k>1$), a peak, then an exponential-style decay — try different exponents mentally as different values of $k$ and watch the peak shift right and the tail thin as $k$ grows.

Worked example 1: waiting for the fifth trade

A stock's trades on a thin book arrive as a Poisson process averaging $\lambda = 2$ per minute. The time until the 5th trade is $\text{Gamma}(k=5, \lambda=2)$, with mean $5/2 = 2.5$ minutes and variance $5/4=1.25$ minutes², so a standard deviation of about $1.12$ minutes. A trader waiting for enough liquidity to accumulate (5 trades' worth) should plan around 2.5 minutes, but be prepared for real waits ranging roughly 1.4 to 3.6 minutes (mean ± 1 sd) fairly often, since the distribution is still visibly right-skewed at $k=5$.

Worked example 2: modeling realized variance

Daily realized variance estimates from intraday returns are strictly positive and right-skewed — never normal. Suppose historical data fits a $\text{Gamma}(k=4, \lambda=800)$ to daily variance (in return² units), giving mean $4/800 = 0.005$ (matching an average daily vol near $\sqrt{0.005}\approx 7.1\%$, unrealistically high here for illustration — treat as stylized). What's the probability realized variance exceeds double its mean, i.e. $x > 0.01$? Using the gamma CDF (from tables or software) for $k=4,\lambda=800$ at $x=0.01$: $P(X>0.01) \approx 0.091$ — about a 9% chance on any given day, a number a normal-based model applied to variance (which would nonsensically allow negative variance) simply cannot produce honestly.

This is the discrete cousin of the story: a Poisson count with rate $\lambda$ counts how many events land in a fixed window, while the gamma distribution with the same rate measures the waiting time until a fixed number of events land — two sides of the same arrival process, one counting, one timing.

What this means in practice

• Modeling any strictly-positive, skewed quantity. Trade durations, time-to-fill, drawdown recovery times, realized variance — anywhere a normal distribution's negative tail is nonsensical, gamma is a natural default.
• Bayesian priors. The gamma distribution is the conjugate prior for the rate of a Poisson process (like arrival rate of trades or defaults), making posterior updates closed-form and fast.
• Interest rate models. The Cox-Ingersoll-Ross (CIR) short-rate model's stationary distribution is a gamma distribution, which is exactly why CIR rates stay positive while mean-reverting.