Numerical Methods for Lévy Processes: Lattice Methods and the Density, the Subordinator and the Time Copula

TLDR: In this paper, the authors apply the lattice to models based on the variance-gamma, NIG and Meixner processes, contrasting the numerical difficulties in each case, concluding that current methods based, directly or indirectly, on low order branching, are unlikely to be capable of calibrating to market prices.

Abstract: Evidence from the financial markets suggests that empirical returns distributions, both historical and implied, do not arise from diffusion processes. A growing literature models the returns process as a Lévy process, finding a number of explicit formulae for the values of some derivatives in special cases. Practical use of these models has been hindered by a relative paucity of numerical methods that can be used when explicit solutions are not present. This paper investigates lattice methods that can be used when the returns process is Lévy. We relate the transition density function of a Lévy process to its representation as a time-changed Brownian motion and to its time-copula, leading to alternative derivations of the lattice. We apply the lattice to models based on the variance-gamma, NIG and Meixner processes, contrasting the numerical difficulties in each case. We discuss implications for implied pricing, concluding that current methods based, directly or indirectly, on low order branching, are unlikely to be capable of calibrating to market prices. ∗We gratefully acknowledge the help and support of Manfred Gilli and the hospitality of the Department of Econometrics, University of Geneva. We would like to thank Grace Kuan and Stewart Hodges for their comments and advice. The paper has benefited from comments by Lynda McCarthy, Peter Carr, Philip Schönbucher, Steve Heston, Mark Broadie, Chris Rogers, Rupert Brotherton-Ratcliffe and Alessio Sancetta, and from participants at the 8th CAP workshop, New York, and QMF 2002, Sydney..