The CGMY model is definitely an attractive one, for its flexibility in mod- elling processes of both finite and infinite activity and of both finite and infi- nite variation. Moreover it allows to work with L´evy density not completely monotone. Clearly, being the variance gamma obtainable as a particular parametric case of the CGMY, from a theoretical point of view the CGMY is more desirable. However as shown in section (3.7) and more in particular in section (3.7.3), the additional properties that the CGMY exhibits do not seem to make noticeable improvement over the variance gamma, at least in the examples analyzed in the current literature. Price processes seem to be completely monotone and also from an intuitive point of view, it is more in- teresting to think about a stochastic process which exhibits large jumps less often than small jumps. Hence the additional CGMY feature represented by non monotonicity does not appear of much practical use. As for finite and infinite variation and finite and infinite activity, Carr, Geman, Madan and Yor observe that the large majority of the statistical processes are consis- tent with finite variation and infinite activity. Moreover where the processes are of finite activity, the variance gamma process null hypothesis cannot be rejected. Furthermore, substantially all the risk neutral processes are consis- tent with the variance gamma model. This is reasonable because the pricing process focuses on large moves while small moves do not get too much atten- tion, while the infinite variation comes from a high degree of activity near zero which does not seem to be justified. For all these reasons, it appears
that the variance gamma model allows to obtain results comparable to the CGMY in terms of quality of modelling of the price processes at a lower cost represented by an easier model having one less variable. It would be inter- esting to investigate which stocks may benefit from the additional properties of the CGMY.
On the other side the stochastic volatility extension for both the variance gamma and the CGMY model, in the form of VGSA and CGMYSA, seems very promising. Of particular appeal from both a theoretical and a practical point of view is the fact that the models fit well options of different maturities and of different strikes. This result is generally not reached by option pric- ing model as reflected by the practice of comparing the models partitioning the data by term and moneyness in order to get adequate pricing quality27. The fact that the Variance Gamma can be extended to include a stochas- tic volatility is very positive, because it shows the quality of the model and opens perspective of future research. Moreover the fact that the stochastic volatility is obtained as a time changed process with the same approach used to define variance gamma as a time changed Brownian motion, shows that the subordination technique can be applied in a consistent and successful way to obtained the desired results. However we believe that a good strategy to treat these models is to investigate deeply the variance gamma first before working on further extensions, because there are still areas of interest inside the variance gamma model which can be better understood and whose study may facilitate the analysis of any further extension. For these reasons, we decide to concentrate our research effort on the variance gamma process in this work, leaving the stochastic volatility extensions for future research.
27This is for example the approach followed by Gurdip Bakshi, Charles Cao and Zhiwu
Chen, “Empirical Performance of Alternative Option Pricing Models”,The Journal of Fi- nance, Vol. 52, No. 5, December 1997, pages 2003-2049.
Chapter 4
Numerical Solution for the
European and American Plain
Vanilla Option Price Under the
Variance Gamma Process
4.1
Introduction
As we have seen, an analytical solution is known for the pricing of European options, when the underlying follows a variance gamma process. However in the case of American options, we cannot solve the problem in an analytical way and we have therefore to use numerical methods. Numerical solution of the option price under variance gamma via Monte Carlo has been presented by Ribeiro and Webber1. They use a gamma bridge in conjunction with a stratified sampling to price both vanilla and some exotic options, including barrier options. K¨ellezi and Webber2present a lattice method to solve option pricing under a L´evy process and have variance gamma as an example. In particular, they relate the transition density function of a L´evy process to its representation as time changed Brownian motion and to its time-copula,
1Nick Webber and Claudia Ribeiro, “Valuing Path-Dependent Options in the Variance-
Gamma Model by Monte Carlo with a Gamma Bridge”, Journal of Computational Fi- nance, Vol. 7, No. 2, Winter 2003/2004, pages 81-100.
2Evis K¨ellezi and Nick Webber, “Numerical Methods for L´evy Processes: Lattice Meth-
leading to an alternative derivation of the lattice.A finite difference method which can be applied to solve only the case of European options when the underlying asset is driven by a L´evy process, and hence also by a variance gamma process as a particular case, has also been recently presented by Cont and Voltchkova3.
The scheme presented here is instead a slight modification of the numer- ical solution of the differential equation in terms of finite difference scheme as presented by Hirsa and Madan4for American vanilla options.
Part of the developments presented in this chapter has been the subject of a project for the Master in Mathematics of Finance at Columbia University in New York in the Spring 2001.