Local risk-minimization for exponential additive processes

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.