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This paper seems to outline what you are looking for. You want to be careful about mean/variance/kurtosis to make sure you are working in the correct measure.


VG belongs in the family of variance-mean mixture models. Given a horizon $T$ the distribution of log-returns $f$ is a mixture of Gaussians $f_G$ with randomised mean and variance. The randomisation density is $g$ and its mean and variance increase with $T$. For the VG process this randomised factor is Gamma-distributed. More concretely, denote with ...


The parameters θ, ν and r need to be estimated from the sample with some technique, but unfortunately there is no easy way to do that for a VG process. There is, for example, "maximum likelihood estimation" that gives you the parameters that are "most likely" to have generated your sample, assuming your sample comes from a VG process. But MLE involves ...


If you say stock prices are following GBM then you can say $dS_t = \mu S_tdt + \sigma S_t dW_t$ solving which it brings where $\sigma$ is volatility and $r$ is risk free rate . **EDITED For a Variance Gamma process theta is the deterministic drift in subordinated Brownian motion and sigma standard deviation in subordinated Brownian motion. I ...

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