The author would like to thank Dimitrina Dimitrova for her invaluable help in developing the Mathematica implementation and the numerical validation of the LG and VG models of Section 4 and for carefully reading the draft of the paper which lead to numerous corrections and notational improvements. The author is also indebted to Zvetan Ignatov for his constructive comments and suggestions, which helped improve the paper.
LÉVY PROCESSES INDUCED BY DIRICHLET (B-)SPLINES: MODELING MULTIVARIATE ASSET PRICE DYNAMICS
Article first published online: 13 FEB 2012
© 2012 Wiley Periodicals, Inc.
Volume 23, Issue 2, pages 217–247, April 2013
How to Cite
Kaishev, . V. K. (2013), LÉVY PROCESSES INDUCED BY DIRICHLET (B-)SPLINES: MODELING MULTIVARIATE ASSET PRICE DYNAMICS. Mathematical Finance, 23: 217–247. doi: 10.1111/j.1467-9965.2011.00504.x
Earlier versions of this paper have been presented at the QMF 2006 conference in Sydney (see Kaishev 2006a) and also at the King’s College Financial Mathematics and Applied Probability Seminars, December 2009. The author extends his thanks to the participants of these events, for their useful feedback.
- Issue published online: 5 MAR 2013
- Article first published online: 13 FEB 2012
- Manuscript received May 2010; final revision received April 2011.
- LG (Lévy) process;
- (multivariate) variance gamma process;
- bilateral gamma process;
- Dirichlet spline;
- simplex spline;
- Dirichlet bridge sampling;
- FX modeling
We consider a new class of processes, called LG processes, defined as linear combinations of independent gamma processes. Their distributional and path-wise properties are explored by following their relation to polynomial and Dirichlet (B-)splines. In particular, it is shown that the density of an LG process can be expressed in terms of Dirichlet (B-)splines, introduced independently by Ignatov and Kaishev and Karlin, Micchelli, and Rinott. We further show that the well-known variance gamma (VG) process, introduced by Madan and Seneta, and the bilateral gamma (BG) process, recently considered by Küchler and Tappe are special cases of an LG process. Following this LG interpretation, we derive new (alternative) expressions for the VG and BG densities and consider their numerical properties. The LG process has two sets of parameters, the B-spline knots and their multiplicities, and offers further flexibility in controlling the shape of the Levy density, compared to the VG and the BG processes. Such flexibility is often desirable in practice, which makes LG processes interesting for financial and insurance applications. Multivariate LG processes are also introduced and their relation to multivariate Dirichlet and simplex splines is established. Expressions for their joint density, the underlying LG-copula, the characteristic, moment and cumulant generating functions are given. A method for simulating LG sample paths is also proposed, based on the Dirichlet bridge sampling of gamma processes, due to Kaishev and Dimitriva. A method of moments for estimation of the LG parameters is also developed. Multivariate LG processes are shown to provide a competitive alternative in modeling dependence, compared to the various multivariate generalizations of the VG process, proposed in the literature. Application of multivariate LG processes in modeling the joint dynamics of multiple exchange rates is also considered.