Minimal $f^q$-martingale measures for exponential L\'evy processes

TL;DR

This paper characterizes the minimal $f^q$-martingale measures for exponential Lévy processes, providing explicit formulas, existence conditions, and convergence results, thereby advancing the understanding of alternative risk-neutral measures in financial modeling.

Contribution

It offers necessary and sufficient conditions for the existence of $f^q$-minimal martingale measures and derives explicit formulas for their densities, extending the theory of equivalent martingale measures.

Findings

Explicit formula for the density of $Q_q$

Necessary and sufficient conditions for existence of $Q_q$

Convergence of $Q_q$ to minimal entropy measure as $q o 1$

Abstract

Let $L$ be a multidimensional L\'evy process under $P$ in its own filtration. The $f^q$-minimal martingale measure $Q_q$ is defined as that equivalent local martingale measure for $\mathcal {E}(L)$ which minimizes the $f^q$-divergence $E[(dQ/dP)^q]$ for fixed $q\in(-\infty,0)\cup(1,\infty)$. We give necessary and sufficient conditions for the existence of $Q_q$ and an explicit formula for its density. For $q=2$, we relate the sufficient conditions to the structure condition and discuss when the former are also necessary. Moreover, we show that $Q_q$ converges for $q\searrow1$ in entropy to the minimal entropy martingale measure.