Delta and Gamma Hedging of Mortality and Interest Rate Risk

Abstract

This paper studies the hedging problem of life insurance policies, when the mortality and interest rates are stochastic. We focus primarily on stochastic mortality. We represent death arrival as the first jump time of a doubly stochastic process, i.e. a jump process with stochastic intensity. We propose a Delta-Gamma Hedging technique for mortality risk in this context. The risk factor against which to hedge is the difference between the actual mortality intensity in the future and its forecast today, the instantaneous forward intensity. We specialize the hedging technique first to the case in which survival intensities are affine, then to Ornstein-Uhlenbeck and Feller processes, providing actuarial justifications for this restriction. We show that, without imposing no arbitrage, we can get equivalent probability measures under which the HJM condition for no arbitrage is satisfied. Last, we extend our results to the presence of both interest rate and mortality risk, when the forward interest rate follows a constant-parameter Hull and White process. We provide a UK calibrated example of Delta and Gamma Hedging of both mortality and interest rate risk.