the variance-gamma process

The variance-gamma process is Brownian motion with drift run on a gamma random clock — the textbook example of subordination and one of the most popular pure-jump models for asset log-returns. It captures the empirical facts the Black-Scholes Gaussian misses: heavier tails, excess kurtosis, and skewness, while remaining analytically tractable with a closed-form characteristic function.

Construct it as X_t = theta G_t + sigma B_{G_t}, where G_t is a gamma subordinator with unit mean rate (so E[G_t] = t) and variance rate kappa, B is an independent Brownian motion, theta is a drift and sigma a volatility. Time-changing Brownian motion by the gamma clock produces a pure-jump Lévy process of finite variation with characteristic exponent psi(theta) given by subordination, psi_X(u) = -(1/kappa) log(1 - i theta kappa u + sigma^2 kappa u^2 / 2). Its three economically meaningful parameters are sigma (volatility), theta (skew: theta < 0 gives a left-heavy crash-prone return), and kappa (kurtosis: larger kappa fattens the tails). The process has NO Brownian (continuous) component despite being built from Brownian motion — the random time change converts the diffusion into infinitely many small jumps; it has finite variation and an infinite Lévy measure with a known density.

VG is used to price options consistently across strikes (it fits the volatility smile far better than Black-Scholes) and to model returns with fat tails and skew. The caveat worth stating: although VG is born from Brownian motion, the result is a finite-variation PURE-JUMP process with no continuous martingale part — so VG markets behave differently from diffusion markets (e.g. hedging and the local behavior of quadratic variation differ). And the gamma clock must be independent of the driving Brownian motion for the clean subordination formula; correlated versions are different models.