# How to simulate stock prices using variance gamma process?

I want to simulate stock prices with the variance gamma process. The model is given by:

$S_T=S_0 e^{ {[}(r-1)T + \omega + z{]}}$

where

$S_0=$ starting value

$T=$ Time

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

$r=$ interest rate

$z=$ normally distributed variable with mean $\theta g$ and standard deviation $\sigma \sqrt{g}$

I know, that I have to simulate first the g values by a random generator (using gamma function with parameters), then generate random numbers z using the g's. But my problem is, how does I specify the three parameters $\nu$ and $\theta$ and r? The T means years, so if I have e.g. 10 trading days, this would be 10 divided by 365. I had a another simulation with the geometric brownian motion before, there I used the sample mean, sample standard deviation, 22 trading days, and starting value 20. So I thought to make it comparable:

$T=22/365$

$S_0=20$

Nut what about $\theta$, $\nu$ and r? Is r just the sample mean?

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Hi, welcome to Quant.SE. Please don't hesitate to register in order to help the site grow and make it out of beta. –  SRKX Nov 22 '12 at 14:42

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.

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Any chance you could summarize the contents of that paper here? –  chrisaycock Dec 26 '12 at 17:12
It's a basic survey/study of a variance gamma model vs black-scholes, using calibrations to both historical as well as implied data. –  experquisite Dec 27 '12 at 4:12

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 choose mu in (0.1,0.3) and volatility estimate by GARCH or around 15% lower to 30% upper for a typical simulation

HTH

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@ Ashwani Roy no, this is not an appropriate answer to my question, I did a GBM simulation already, now I want to do it with VG –  user1690846 Nov 22 '12 at 9:54
Sorry , totally misunderstood . Edited based on what i know about Levy's process models –  ash Nov 22 '12 at 10:32
Why are these equations presented as images? You can enter $\LaTeX$ code and it will be converted into text appropriate for a browser. –  chrisaycock Nov 22 '12 at 13:44
@chrisaycock Sorry I am new to answering here. Will try this in future. Apologies if it caused any problems –  ash Nov 22 '12 at 13:55
@AshwaniRoy I edited the answer and convereted the first equation to $\LaTeX$ for you so you can use it as an example to convert the other and future. Thanks for taking the time to answer! –  Louis Marascio Nov 22 '12 at 15:54

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 computing the likelihood function of a VG process which is extremely complex by itself (check its pdf on the VG process wikipedia page).

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