Levy preservation and associated properties for $f$-divergence minimal equivalent martingale measures

TL;DR

This paper investigates the properties of $f$-divergence minimal martingale measures in exponential Levy models, focusing on their preservation, invariance, and scaling, and provides conditions for their existence based on the form of $f$.

Contribution

It characterizes the form of $f$ that ensures key properties of $f$-divergence minimal martingale measures and establishes conditions for their existence in exponential Levy models.

Findings

Identifies conditions on $f$ for Levy preservation and invariance properties.

Provides a decomposition of $f$-divergence minimal martingale measures.

Establishes necessary and sufficient conditions for the existence of $f$-minimal martingale measures for common $f$-divergences.

Abstract

We study such important properties of $f$-divergence minimal martingale measure as Levy preservation property, scaling property, invariance in time property for exponential Levy models. We give some useful decomposition for $f$-divergence minimal martingale measures and we answer on the question which form should have $f$ to ensure mentioned properties. We show that $f$ is not necessarily common $f$-divergence. For common $f$-divergences, i.e. functions verifying $f"(x) = ax^ {\gamma},\, a>0,\, \gamma \in \mathbb R$, we give necessary and sufficient conditions for existence of $f$-minimal martingale measure.