The Annals of Applied Probability

A family of density expansions for Lévy-type processes

Matthew Lorig, Stefano Pagliarani, and Andrea Pascucci

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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 [SIAM J. Financial Math. 4 (2013) 265–296], 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. 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, Pascucci and Riga (2013), for models with Gaussian-type jumps, approximate option prices can be computed in closed form. Sample Mathematica code is provided.

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Ann. Appl. Probab., Volume 25, Number 1 (2015), 235-267.

First available in Project Euclid: 16 December 2014

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Primary: 35R09: Integro-partial differential equations [See also 45Kxx] 60G99: None of the above, but in this section 91G20: Derivative securities 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Local volatility Lévy-type process asymptotic expansion pseudo-differential calculus defaultable asset


Lorig, Matthew; Pagliarani, Stefano; Pascucci, Andrea. A family of density expansions for Lévy-type processes. Ann. Appl. Probab. 25 (2015), no. 1, 235--267. doi:10.1214/13-AAP994.

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