Local risk-minimization for exponential additive processes

TL;DR

This paper develops explicit formulas and numerical methods for local risk-minimization in exponential additive models, extending the theory beyond Lévy processes to more complex additive processes with time-dependent measures.

Contribution

It provides necessary integrability conditions for deriving local risk-minimization strategies in exponential additive models, a significant extension from Lévy process frameworks.

Findings

Derived explicit expressions for LRM strategies in additive models.

Established integrability conditions for the Lévy measure.

Numerical experiments with variance-gamma scaled Sato process.

Abstract

We explore local risk-minimization, a quadratic hedging method for incomplete markets, in exponential additive models. The objectives are to derive explicit mathematical expressions and to conduct numerical experiments. While local risk-minimization is well studied for L\'evy processes, little is known for the additive process case because, unlike L\'evy processes, the L\'evy measure for an additive process depends on time, which significantly complicates the mathematical framework. This paper shall provide a set of necessary conditions for deriving expressions for LRM strategies in exponential additive models, as integrability conditions on the L\'evy measure, which allow us to confirm whether these conditions are satisfied for given concrete models. In the final section, we introduce the variance-gamma scaled self-decomposable process, a Sato process that generalizes the variance-gamma process, as a primary example, and perform numerical experiments.