🤖 AI Summary
This study addresses the absence of explicit solutions and verifiable conditions for local risk-minimizing (LRM) strategies in incomplete markets under exponential additive process models. Focusing on additive processes with time-dependent Lévy measures, the paper establishes, for the first time, a theoretical framework for LRM strategies by introducing integrability conditions on the Lévy measure, which enables the derivation of an explicit expression for the LRM strategy. The proposed approach is validated through a combination of stochastic analysis, additive process theory, and numerical simulations, using the variance gamma scaled self-decomposable process as a concrete example. By extending beyond the conventional limitations of Lévy processes, this work provides a practical and computationally tractable hedging tool for complex financial models.
📝 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 Lévy processes, little is known for the additive process case because, unlike Lévy processes, the Lévy 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 Lévy 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.