# Risk Neutral Variance Gamma

In the risk neutral version of the Variance Gamma model the stock dynamics are

$$S_T=S_0 e^{ (r-q+\omega)t + X(t;\sigma,\nu,\theta)}$$

with

$$\omega=\frac{1}{\nu}\ln\left(1-\theta \nu - \frac{\sigma^2 \nu }{2}\right)$$

and where

$$X(t;\sigma,\nu,\theta) = \theta G(t; \nu)+\sigma G(t;\nu)W_t$$

is a Variance Gamma process where $G(t;\nu)$ is a Gamma distribution with mean $t$ and variance $\nu t$, and $W$ is $N(0,1)$.

For the VG process $X(t;\sigma,\nu,\theta)$ $\sigma$ controls the volatility, while $\nu$ and $\theta$ jointly control the asymmetry and kurtosis. Specifically, if $\theta = 0$, the VG process is symmetric and $\nu$ determines the excess kurtosis, but in other cases both $\nu$ and $\theta$ are joinly needed to obtain the higher moments.

However, I wonder how these interpretations translate to the risk neutral process for $S_T$. In particular:

1. Is there any simple interpretation of how $\sigma$, $\nu$ and $\theta$ affect the risk neutral process?
2. How does $\sigma$, $\nu$ and $\theta$ relate to the four first moments of $S_T$ (or $\frac{ S_T} { S_0}$)? This thesis (pages 31/32) obtains these four moments for $X(t;\sigma,\nu,\theta)$, but I have not seen a similar result for $S_T$ (or $\frac{ S_T} { S_0}$).
• The link appears broken. – Gordon Mar 10 '16 at 16:50
• Updated with a new link. – sets Sep 7 '16 at 9:50