Variance gamma process
subjected to a random time change which follows a gamma process
This last characterization gives an understanding of the structure of the sample path with location and sizes of jumps.
and is given by The variance is given as The 3rd central moment is The 4th central moment is The VG process can be advantageous to use when pricing options since it allows for a wider modeling of skewness and kurtosis than the Brownian motion does.
As such the variance gamma model allows to consistently price options with different strikes and maturities using a single set of parameters.
Madan and Seneta present a symmetric version of the variance gamma process.
[4] Madan, Carr and Chang [1] extend the model to allow for an asymmetric form and present a formula to price European options under the variance gamma process.
Hirsa and Madan show how to price American options under variance gamma.
[5] Fiorani presents numerical solutions for European and American barrier options under variance gamma process.
[6] He also provides computer code to price vanilla and barrier European and American barrier options under variance gamma process.
Lemmens et al.[7] construct bounds for arithmetic Asian options for several Lévy models including the variance gamma model.
The pure jump nature of the process and the possibility to control skewness and kurtosis of the distribution allow the model to price correctly the risk of default of securities having a short maturity, something that is generally not possible with structural models in which the underlying assets follow a Brownian motion.
Fiorani, Luciano and Semeraro[8] model credit default swaps under variance gamma.
In an extensive empirical test they show the overperformance of the pricing under variance gamma, compared to alternative models presented in literature.
Monte Carlo methods for the variance gamma process are described by Fu (2000).
is integer the variance gamma distribution can be represented as a 2-EPT probability density function.
Under this assumption it is possible to derive closed form vanilla option prices and their associated Greeks.
