Variance Gamma (VG) Option Pricing Lab

1. Pure Jump Lévy Processes & Brownian Subordination

Traditional option pricing models, starting from Black, Scholes, and Merton (1973), postulate that asset prices follow continuous geometric Brownian motion with Gaussian normal log-returns. However, empirical market returns exhibit two ubiquitous characteristics: leptokurtosis (peaked center and fat tails) and negative skewness (asymmetric downside crashes).

Dilip B. Madan, Peter P. Carr, and Eric C. Chang (1998), developing earlier work by Madan and Seneta (1990), introduced the Variance Gamma (VG) model. VG is a pure jump Lévy process with infinite activity and finite variation. Unlike jump-diffusion models (such as Merton 1976) which blend continuous diffusion with rare Poisson jumps, the VG model has no continuous diffusion component at all ($W_t \equiv 0$). The asset price moves exclusively via an infinite sequence of price jumps in any finite time interval.

2. The Stochastic Operational Time Clock

The core economic intuition of Variance Gamma is stochastic subordination. In financial markets, information does not arrive at a constant calendar pace; trading volume and market stress cause economic 'business time' to speed up during active trading periods and slow down during quiet hours.

Mathematically, the VG process $X(t; \sigma, \nu, \theta)$ is defined as a standard Brownian motion with drift $\theta$ and diffusion $\sigma$, evaluated at a random operational time $\gamma(t; 1, \nu)$ governed by an independent Gamma process:

where $\gamma(t; 1, \nu)$ has independent Gamma-distributed increments with mean $\mathbb{E}[\Delta \gamma] = \Delta t$ and variance $\text{Var}(\Delta \gamma) = \nu \Delta t$. The parameter $\nu$ represents the variance rate of the economic clock.

3. Analytical Moments and Parameter Roles

The Variance Gamma process yields closed-form analytical moments for log-returns over horizon $T$:

• Mean: $\mathbb{E}[X_T] = \theta T$
• Total Variance: $\text{Var}(X_T) = (\sigma^2 + \nu \theta^2) T \implies \sigma_{\text{total}} = \sqrt{\sigma^2 + \nu \theta^2}$
• Skewness: $\text{Skew}(X_T) = \frac{\theta \nu (3\sigma^2 + 2\nu \theta^2)}{(\sigma^2 + \nu \theta^2)^{3/2} \sqrt{T}}$
• Excess Kurtosis: $\text{Kurt}(X_T) - 3 = 3 \nu \left( \frac{\sigma^4 + 2\nu \theta^4 + 4\sigma^2 \theta^2 \nu}{(\sigma^2 + \nu \theta^2)^2} \right) \frac{1}{T}$

Notice that when $\theta < 0$, the skewness is negative, directly capturing the downside equity volatility skew. When $\nu \to 0$, the excess kurtosis vanishes, and the VG process collapses smoothly to standard Black-Scholes Gaussian diffusion.

4. Difference of Two Gamma Processes

Another remarkable property of the VG process is that it can be expressed as the difference of two independent, monotonic Gamma processes representing upward price runs $G_p(t)$ (buying pressure) and downward price drops $G_n(t)$ (selling pressure):

with parameters $\mu_p = \frac{1}{2}\sqrt{\theta^2 + 2\sigma^2/\nu} + \frac{1}{2}\theta$ and $\mu_n = \frac{1}{2}\sqrt{\theta^2 + 2\sigma^2/\nu} - \frac{1}{2}\theta$.

5. Option Pricing via Characteristic Function Fourier Inversion

Under the risk-neutral pricing measure $\mathbb{Q}$, the asset price evolves as $S_T = S_0 \exp((r - q + \omega)T + X_T)$, where the drift compensator $\omega$ guarantees that $\mathbb{E}^{\mathbb{Q}}[S_T] = S_0 e^{(r-q)T}$:

The closed-form characteristic function of $\ln(S_T)$ is:

European call option prices are then evaluated via Gil-Pelaez inversion: