The Variance Gamma Distribution

Stock returns are famously not normal: they have fatter tails and often noticeable skew, but the normal distribution has neither dial to turn. The variance gamma (VG) distribution fixes this by building returns from a small twist on Brownian motion: instead of letting time run at a constant, deterministic rate, it lets a Brownian motion with drift run on a random clock — specifically, the clock itself follows a gamma distribution. On days when the random clock ticks fast, more randomness accumulates and you get a big move; on slow-clock days, very little happens. Averaging over all the possible clock speeds produces a distribution with heavier tails than the normal (because some days genuinely get a lot more "time") and, if the underlying drift is nonzero, skew as well.

The distribution has four parameters: $\sigma$ (volatility of the underlying Brownian motion), $\theta$ (drift, controlling skew), $\nu$ (the variance of the random clock, controlling excess kurtosis — how fat the tails are), and a location/mean parameter. Setting $\nu \to 0$ removes the randomness in the clock entirely and the VG distribution collapses back to an ordinary normal distribution — so VG is literally "normal plus one extra tail/skew knob," which is why it's a popular first step up from Black-Scholes-style modelling: it reprices the option-pricing formula with a distribution that can actually match an observed volatility smile, using a characteristic-function approach rather than a closed-form density.