Variance Gamma Model

Black-Scholes assumes a stock's randomness arrives at a perfectly steady rate: the variance of a return over any interval is exactly proportional to its length, no faster, no slower. Real markets don't trade at a steady rate at all — some days are frantic, some are dead quiet, and news doesn't arrive on a metronome. The Variance Gamma (VG) model builds that unevenness directly into the process: instead of running a Brownian motion on ordinary clock time, it runs the same Brownian motion on a random, jittery clock — and that one change is enough to generate jumps, skew, and fat tails from only three extra parameters.

A stopwatch that speeds up and slows down

Picture a stopwatch that doesn't tick at a fixed rate — it randomly speeds up during frantic bursts of activity and randomly slows down during quiet lulls, though on average it still reads the same total elapsed time as a normal clock. Now run an ordinary, steady random walk, but let it evolve according to that jittery stopwatch instead of the wall clock. Bursts of "fast time" produce clusters of big moves that look like jumps; stretches of "slow time" produce the calm periods in between. That's exactly what Variance Gamma does mathematically.

$$X_t = \theta\,G_t + \sigma\,W_{G_t}, \qquad G_t \sim \text{Gamma}\left(\frac{t}{\nu}, \nu\right)$$

In plain English: $W$ is ordinary Brownian motion, but instead of running it for $t$ units of calendar time, it's run for $G_t$ units of random trading time, where $G_t$ is drawn from a Gamma distribution with mean $t$ — so on average the clock keeps pace with the calendar, but any particular stretch can run fast or slow. $\theta$ adds a drift proportional to the random clock itself (controlling skew — more downside jump risk if $\theta<0$), and $\nu$ controls how jittery the clock is (the "variance of the gamma clock" — larger $\nu$ means more erratic trading time, and therefore fatter tails).

Worked example 1: what the clock looks like. Set $t=1$ (one year) and $\nu=0.2$. The gamma clock has mean $t=1$ and variance $\nu t = 0.2$, so a typical draw of $G_1$ might land anywhere roughly between $0.4$ and $1.9$ about 90% of the time (a gamma distribution with these parameters is right-skewed — most draws cluster a bit below 1, with an occasional much larger draw). A draw of $G_1=1.9$ means that "year" experienced nearly twice as much trading activity as an average year — the Brownian motion inside gets to unfold for almost double the usual amount of randomness, producing an outsized move with no change to the calendar at all.

Worked example 2: recovering the ordinary case. As $\nu \to 0$, the Gamma distribution for $G_t$ collapses onto a spike exactly at $t$ — the clock stops being random and just reads calendar time. Concretely, a Gamma$(t/\nu,\nu)$ distribution has variance $\nu t$, which shrinks to zero as $\nu\to0$: with $\nu=0.01$, the clock's standard deviation is only $\sqrt{0.01\times1}=0.1$ around a mean of 1, barely different from a fixed clock, and $X_t$ converges to ordinary Brownian motion with drift $\theta t$ and volatility $\sigma\sqrt t$ — plain geometric Brownian motion, recovered as the special case $\nu=0$.

Compare a smooth GBM path to what a VG path looks like: bursts of rapid moves separated by flatter stretches, driven entirely by the random speed of the underlying clock rather than by any change in the calendar-time volatility parameter.

What this means in practice

VG needs only three parameters ($\sigma$, $\nu$, $\theta$) yet produces a genuine implied volatility skew and excess kurtosis without adding a second stochastic-volatility factor the way Heston does, which makes it popular for markets where a Heston-style calibration is overkill but Black-Scholes clearly misprices the tails. Like other tractable jump-and-skew models, VG is priced through its known closed-form characteristic function (see Characteristic Function Pricing) rather than a closed-form density, which doesn't exist for this process.