Thiele's PIDE for unit-linked policies in the Heston-Hawkes stochastic volatility model

TL;DR

This paper derives Thiele's differential equation for unit-linked policies within an extended Heston-Hawkes stochastic volatility model, enabling practical reserve computation in life insurance with complex volatility features.

Contribution

It extends the Heston model by incorporating a Hawkes process and derives Thiele's equation for this new framework, facilitating reserve calculations.

Findings

Derived Thiele's differential equation for the model

Identified risk-neutral measures for pricing

Provided a practical method for reserve computation

Abstract

The main purpose of the paper is to derive Thiele's differential equation for unit-linked policies in the Heston-Hawkes stochastic volatility model introduced in arXiv:2210.15343. This model is an extension of the well-known Heston model that incorporates the volatility clustering feature by adding a compound Hawkes process in the volatility. Since the model is arbitrage-free, pricing unit-linked policies via the equivalence principle under a risk neutral probability measure is possible. Studying the moments of the variance and certain stochastic exponentials, a suitable family of risk neutral probability measures is found. The established and practical method to compute reserves in life insurance is by solving Thiele's equation, which is crucial to guarantee the solvency of the insurance company.