Valuation of Annuity Guarantees Under a Self-Exciting Switching Jump Model

Abstract

This article investigates the valuation of annuity guarantees under a regime-switching model when the dynamics of the underlying stock price follow a self-exciting switching jump-diffusion process. In this framework, we add a jump component to a regime-switching geometric Brownian for large shocks on the stock price. The intensity of shock arrivals is a Hawkes process modulated by a continuous time hidden Markov chain with a finite number of states. The interest rate used for discounting is stochastic and correlated to the stock market. In an incomplete market, we define an equivalent martingale measure to price a variable annuity contract that guarantees a minimum living or death benefit. Under this equivalent martingale measure, we propose closed-form approximation formulas using the inverse Fourier transform technique. A numerical implementation highlights the impact of self-exciting jumps and economic regimes on the valuation of guarantees.

Affine Heston model style with self-exciting jumps and long memory

“…Motivated by the empirical results, authors truncate this intensity to the second exponent and ignore the first exponent term. However, recent work of , Hainaut and Moraux [2018 b] and Njike and Hainaut [2022] have shown the role of linear HP when it comes to replicate features of asset's return time series. In combination with the self-excitation property of the linear HP, Hainaut [2021] has designed a non-markov linear HP for claims process that has memory.…”