Option Prices in the Variance Gamma Model
The demonstration compares the values of the vanilla European Call and Put options in the Black-Scholes model with the values of the same options in the Variance Gamma model. The strike price is fixed at 100. The control parameters volatility, risk-free interest rate and time to expiry are shared by both models while the parameters "drift" and "gamma variance" affect only the Variance Gamma model.
For small values of the "gamma variance" parameter options prices in the Black-Scholes and Variance Gamma models almost coincide but in general the relationship between them is complex, with Variance Gamma both overpricing and underpricing Black-Scholes, depending on whether the options are in, at or out of the money and the values of the other parameters.
Several methods of computing European options prices in the Variance Gamma model are known. Madan, Carr and Chang  give an "explicit" formula involving special functions, which however is difficult to implement in practice due to the presence of a singularity in one of the special functions involved. Since the state price density (i.e. "risk neutral" PDF) of the risk adjusted price process used in  is known in explicit form, one can use numerical integration to compute options prices, but again this seems to suffer from numerical difficulties. Better results are obtained by using the Fourier transform approach described in , which is what we use in this demonstration.
. D. B. Madan, P. P. Carr, and E. C. Chang, "The Variance Gamma Model and Option Pricing," European Finance Review, 2(1), 1998 pp. 79–105.
. P.P.Carr and D.B.Madan, "Option valuation using the Fast Fourier Transform", Journal of Computational Finance, 1999 vol. 2 pp. 61-73