A family of density expansions for Lévy-type processes

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 [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.