A flexible lattice model for fair policy valuations under multiple risk factors

We propose a flexible lattice model for pricing insurance contracts by considering both financial and actuarial risk factors. The simultaneous consideration of all such sources of risk is suggested by the fact that long-term policy benefits are usually linked to the performance of a reference fund invested in financial assets. Consequently, the consideration of a stochastic process to model not only the reference fund but also the other sources of risk is more appropriate for policy evaluations (see, for instance, Hilliard and Schwartz \cite{jk1} and Van-Haastrecht et al. \cite{vlps}). In this direction, we work in a stochastic interest rate framework, taking properly into account mortality risk and embedding additional guarantees, like a surrender option, into the policy contract. The latter aspect is challenging on a computational point of view since the surrender option has the feature of an American-style option. The considered contracts may not be evaluated by explicit formulae due to the unknown distribution of the optimal exercise time, and efficient numerical methods need. Under this perspective, our contribution provides a flexible model since it allows to discretize the most common dynamics used in financial and actuarial literature. Hence, an additional advantage of the model is that the insurer may choose the most appropriate dynamics to capture interest rate and mortality risks to obtain a fair policy valuation. Mortality and interest rate dynamics are discretized by two different binomial recombining trees. The reference fund dynamic is discretized through a bivariate tree to capture the influence of the stochastic interest rate on the fund value. Finally, embedding mortality risk for policy valuation, we obtain a trivariate binomial lattice presenting eight branches for each node, where the joint probabilities of the possible jumps are computed to capture the proper pairwise process correlation.