L\'evy models amenable to efficient calculations

TL;DR

This paper introduces classes of Stieltjes-Lévy processes with specific monotonicity properties, demonstrating their broad applicability and how they enable efficient numerical calculations for Lévy models used in financial mathematics.

Contribution

It defines SL- and sSL-processes based on monotone Lévy densities, showing their relevance to popular Lévy models and facilitating efficient computation of Wiener-Hopf factors.

Findings

SL-processes encompass most popular Lévy models.

Meixner processes are classified as sSL-processes.

Efficient numerical methods are applicable to these classes.

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 L\'evy models can be developed using only a few general properties of the characteristic exponent $\psi$. Essentially all popular L\'evy processes enjoy these properties. In the present paper, we define classes of Stieltjes-L\'evy processes (SL-processes) as processes with completely monotone L\'evy densities of positive and negative jumps, and signed Stieltjes-L\'evy 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 the representation $\psi(\xi)=(a^+_2\xi^2-ia^+_1\xi)ST(\cG_+)(-i\xi)+(a^-_2\xi^2+ia^-_1\xi)ST(\cG_-)(i\xi)+(\sg^2/2)\xi^2-i\mu\xi$, where $ST(\cG)$ is the Stieltjes transform of the (signed) Stieltjes measure $\cG$ and $a^\pm_j\ge 0$; 2) essentially all popular processes other than Merton's model and Meixner processes areSL-processes; 3) Meixner processes are sSL-processes; 4) under a natural symmetry condition, essentially all popular classes of L\'evy processes are SL- or sSL-subordinated Brownian motion.