A Family of Density Expansions for Lévy-Type Processes

Pagliarani, S., A. Pascucci, and C. Riga (2013). Adjoint expansions in local lévy models. SIAM J. Finan. Math. 4(1), 265–296.

30 Pages Posted: 7 Apr 2013 Last revised: 28 Dec 2013

See all articles by Matthew Lorig

Matthew Lorig

University of Washington - Applied Mathematics

Stefano Pagliarani

DEAMS, Università di Trieste

Andrea Pascucci

University of Bologna - Department of Mathematics

Date Written: December 27, 2013

Abstract

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Lévy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Lévy measure. Generalizing and extending the novel adjoint expansion technique of Pagliarani, Pascucci, and Riga (2013), we derive a family of asymptotic expansions for the transition density of the underlying as well as for European-style option prices and defaultable bond prices. For the density expansion, we also provide error bounds for the truncated asymptotic series. Additionally, for pure diffusion processes, we derive an asymptotic expansion for the implied volatility induced by European calls/puts. Our method is numerically efficient; approximate transition densities and European option prices are computed via Fourier transforms; approximate bond prices are computed as finite series. Additionally, as in Pagliarani et al. (2013), for models with Gaussian-type jumps, approximate option prices can be computed in closed form. Numerical examples confirming the effectiveness and versatility of our method are provided, as is sample Mathematica code.

Keywords: Local volatility, Lévy-type process, Asymptotic expansion, Pseudo-differential calculus, Defaultable asset

JEL Classification: C00

Suggested Citation

Lorig, Matthew and Pagliarani, Stefano and Pascucci, Andrea, A Family of Density Expansions for Lévy-Type Processes (December 27, 2013). Pagliarani, S., A. Pascucci, and C. Riga (2013). Adjoint expansions in local lévy models. SIAM J. Finan. Math. 4(1), 265–296., Available at SSRN: https://ssrn.com/abstract=2245118 or http://dx.doi.org/10.2139/ssrn.2245118

Matthew Lorig (Contact Author)

University of Washington - Applied Mathematics ( email )

Seattle, WA
United States

Stefano Pagliarani

DEAMS, Università di Trieste ( email )

Via Valerio n. 4/1
Trieste
Italy

HOME PAGE: http://www.cmap.polytechnique.fr/~pagliarani/

Andrea Pascucci

University of Bologna - Department of Mathematics ( email )

Piazzadi Porta San Donato, 5
Bologna, 40126
Italy

HOME PAGE: http://www.dm.unibo.it/~pascucci

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