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 $\psi$. 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 $\psi$ are consequences of a certain representation of the characteristic exponent in terms of a pair of Stieltjes measures or a pair of differences of two Stieltjes measures (SL- and sSL-processes); (2) essentially all popular processes other than Merton’s model and Meixner processes are SL-processes; (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. We use the properties of (s)SL-processes to derive new formulas for the Wiener–Hopf factors ${\varphi }_q^{\pm }$ for small $q$ in terms of the absolute continuous components of SL-measures and their densities, and calculate the leading terms of the survival probability also in terms of the absolute continuous components of SL-measures and their densities. The lower tail probability is calculated for more general classes of SINH-regular processes.