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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 ...


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.


If the tail index is $\alpha$ then moments beyond that do not exist. The fact that all moments exist for NIG and VG indicates that the tail index is infinite (like the Gaussian).


In general, there cannot be a closed-form solution of a random coefficients VG model. The reason is the drift-restriction that needs to be imposed to ensure that the discounted price process is a martingale under the risk-neutral measure. Using the bank account as numeraire, the restriction is $$ \frac{1}{\beta} > \theta + \frac{\sigma^2}{2} $$ where ...


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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