Lévy models amenable to efficient calculations

Abstract

In our previous publications (IJTAF 2019, Math. Finance 2020), we introduced a general class of SINH-regular processes and demonstrated that efficient numerical methods for the evaluation of the Wiener-Hopf factors and various probability distributions (prices of options of several types) in Lévy models can be developed using only a few general properties of the characteristic exponent ψ. Essentially all popular Lévy processes enjoy these properties. In the present paper, we define classes of Stieltjes-Lévy processes (SL-processes) as processes with completely monotone Lévy densities of positive and negative jumps, and signed Stieltjes-Lévy processes (sSL-processes) as processes with densities representable as differences of completely monotone densities. We demonstrate that 1) all crucial properties of ψ are consequences of the representationwhere ST (G) is the Stieltjes transform of the (signed) Stieltjes measure G and a ± j ≥ 0; 2) essentially all popular processes other than Merton's model and Meixner processes are SLprocesses; 3) Meixner processes are sSL-processes; 4) under a natural symmetry condition, essentially all popular classes of Lévy processes are SL-or sSL-subordinated Brownian motion.