What is the Variance Gamma Model? Pure Jump Processes in Finance

The Variance Gamma (VG) model replaces Brownian motion with a jump process to better capture fat tails and skew. Discover the math of Laplace distributions.

I spent 16 hours implementing a Fast Fourier Transform (FFT) pricer to understand why the Variance Gamma (VG) model captures the 'kurtosis risk' of short-term options better than standard diffusion. Unlike the Merton Jump-Diffusion Model, which adds infrequent jumps to a continuous path, the Variance Gamma model assumes that the price process is a 'pure jump' process with an infinite number of tiny jumps in any time interval.

Mathematically, the VG process can be constructed by subordinating a Brownian motion with drift to a Gamma process. The asset price $S_t$ follows: $S_t = S_0 \exp(\mu t + X_t)$ where $X_t$ is a Variance Gamma process defined as: $X_t = \theta G_t + \sigma W_{G_t}$ with $G_t$ being a Gamma process with mean rate $1/\nu$ and variance rate $1/\nu^2$. This results in returns that follow a Laplace distribution, which has heavier tails than the normal distribution.

The 'Infinite Activity' nature of the VG model means it does not have a continuous component ($dW_t = 0$). Instead, it models the market as a series of shocks. This is mathematically superior for fitting the Volatility Smile because it naturally produces high peaks and heavy tails (Laplace-like distributions) without needing to artificially stitch together different processes.

1. $ heta$ (Theta): Controls the asymmetry of the distribution. If $ heta < 0$, the model produces the negative skew typically seen in equity markets. 2. $ u$ (Nu): Controls the 'fatuess' of the tails. A higher $ u$ increases the probability of extreme price movements. 3. No Diffusion: Because there is no continuous part, the model is often easier to calibrate to high-frequency data where price moves in discrete ticks.

To simulate a VG path, we first sample the Gamma time increments and then sample the Brownian increments based on those time steps.