Valuation of a Guaranteed Minimum Maturity Benefit in a Self-exciting Stochastic Mortality Model

The Guaranteed Minimum Maturity Benefit is quite a popular feature embedded in several unit-linked policies offered by insurance companies. The value of this benefit depends on several processes assumed to describe both the mortality and the financial dynamics, typically represented by interest rates and by the fund associated to the unit-linked policy. A large literature is devoted to the valuation of GMMB for different mortality models, in particular when the mortality dynamics is described by affine models of diffusion type. In the present paper we assume for the mortality dynamics a self-exciting behaviour, described by a Hawkes-type process with exponential kernel, which allows to keep both the Markov and the affine features, but introduces jumps with a stochastic intensity. This type of dynamics, exhibiting a jump clustering property, is quite convenient in order to describe mortality in some critical situations, like epidemics, when contagions phenomena make the probability of jumps arrivals higher when a jump is already occurred. By assuming a diffusion type dynamics for both the fund and the interest rates and introducing all the possible correlations among the diffusion processes necessary in order to get a realistic dynamics, we take advantage of the affine features of the model proposed and compute in a semi-explicit form the GMMB