Asymptotics for d-dimensional lévy-type processes

Abstract

We consider a general d-dimensional Lévy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator A(t) of the Lévy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stochastic volatility models with jumps are provided in order to illustrate the numerical accuracy of our approach. The methods described in this paper extend the results from Corielli et al. (SIAM J Financ Math 1:833–867, 2010, [4]), Pagliarani and Pascucci (Int. J. Theor. Appl. Financ. 16(8):1–35, 2013, [20]) to Lorig et al. (Analytical expansions for parabolic equations, 2013, [13]) forMarkov diffusions to Markov processes with jumps.