Life insurance pricing in a Heston market with CIR interests

In finance, the Heston model describes a stochastic process with a constant drift and a sto-chastic diffusion coefficient which is modeled by a Cox-Ingersoll-Ross process. In this publication we extend this model by allowing for stochastic behavior of the drift as well using an affine Cox-Ingersoll-Ross model for stochastic interest rates. This modified version of the Heston model allows for more degrees of freedom in comparison to standard financial models as the Black-Scholes model or the standard Heston model and is perfectly suitable to model portfolios consisting of a stock and a bond. We show that this modified version of the Heston model is an affine process whose density and characteristic function can by analytically derived using Fourier transformations and solving certain Riccati equations. Such a closed form solution is a fundamental tool to study the impact of the parameters on the process and its characteristics (e. g. expectation, variance or density function). Application in Finance/Insurance: The Heston model as introduced above is currently used as the “standard” model for deriving risk-return profiles – i. e. the probability distribution of returns from a client’s perspective – for different (financial) old age provision products in the German market. In this setting (often complex) old age provision products are analyzed applying Monte-Carlo simulation assuming the modified Heston model. Our work now provides an analytical treatment of the model itself which hence facilitates the explanation of results and allows for quickly assessing the impact of e. g. (isolated) changes on the assumed parameters such as interest rates, risk premium or average volatilities. Terms: Heston model, Cox-Ingersoll-Ross model, affine process, risk-return-profile