Quasi self-dual exponential L\'evy processes

TL;DR

This paper characterizes quasi self-dual exponential Lévy processes, providing new conditions and inversion formulas that facilitate arbitrage-free modeling in financial markets with semi-static hedging.

Contribution

It offers novel characterizations of quasi self-duality for exponential Lévy processes and derives explicit inversion formulas for key model parameters.

Findings

Derived equivalent conditions for quasi self-duality.

Provided closed-form inversion formulas for well-known models.

Established relations between model parameters and market costs.

Abstract

The important application of semi-static hedging in financial markets naturally leads to the notion of quasi self-dual processes. The focus of our study is to give new characterizations of quasi self-duality for exponential L\'evy processes such that the resulting market does not admit arbitrage opportunities. We derive a set of equivalent conditions for the stochastic logarithm of quasi self-dual martingale models and derive a further characterization of these models not depending on the L\'evy-Khintchine parametrization. Since for non-vanishing order parameter two martingale properties have to be satisfied simultaneously, there is a non-trivial relation between the order and shift parameter representing carrying costs in financial applications. This leads to an equation containing an integral term which has to be inverted in applications. We first discuss several important properties of this equation and, for some well-known models, we derive a family of closed-form inversion formulae leading to parameterizations of sets of possible combinations in the corresponding parameter spaces of well-known L\'evy driven models.