Risk-minimization and hedging claims on a jump-diffusion market model, Feynman-Kac Theorem and PIDE

TL;DR

This paper develops a framework for risk-minimizing hedging strategies in jump-diffusion markets with ratings, introducing a Feynman-Kac theorem to connect PDEs with stochastic models, and illustrates it with Levy and regime-switching models.

Contribution

It introduces a Feynman-Kac type theorem for jump-diffusion markets with ratings, enabling calculation of hedging strategies via PDEs, and applies it to complex Levy and regime-switching models.

Findings

Derived a Feynman-Kac theorem for jump-diffusion models with ratings.

Provided explicit hedging strategies in Levy and regime-switching markets.

Demonstrated the applicability of the theory through two detailed examples.

Abstract

At first, we solve a problem of finding a risk-minimizing hedging strategy on a general market with ratings. Next, we find a solution to this problem on Markovian market with ratings on which prices are influenced by additional factors and rating, and behavior of this system is described by SDE driven by Wiener process and compensated Poisson random measure and claims depend on rating. To find a tool to calculate hedging strategy we prove a Feynman-Kac type theorem. This result is of independent interest and has many applications, since it enables to calculate some conditional expectations using related PIDE's. We illustrate our theory on two examples of market. The first is a general exponential L\'{e}vy model with stochastic volatility, and the second is a generalization of exponential L\'{e}vy model with regime-switching.