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 primar- ily on stochastic mortality. We represent death arrival as the rst jump time of a doubly stochastic process, i.e. a jump process with stochastic intensity. We propose a Delta-Gamma Hedging technique for mortal- ity risk in this context. The risk factor against which to hedge is the dierence between the actual mortality intensity in the future and its "forecast" today, the instantaneous forward intensity. We specialize the hedging technique rst to the case in which survival intensities are ane, then to Ornstein-Uhlenbeck and Feller processes, providing actuarial justications for this restriction. We show that, without im- posing no arbitrage, we can get equivalent probability measures under which the HJM condition for no arbitrage is satised. Last, we ex- tend 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.