Optimal Consumption, Investment and Life Insurance With Surrender Option Guarantee

Abstract

We consider an investor, with an uncertain life time, endowed with deterministic labor income, who has the possibility to continuously invest in a Black-Scholes market and to buy life insurance or annuities. We solve the optimal consumption, investment and life insurance problem when the investor is restricted to fulfill an American capital guarantee. By allowing the guarantee to depend, in a very general way, on the past we include, among other possibilities, the interesting case of a minimum rate of return guarantee, commonly offered by pension companies. The optimal strategies turn out to be on option based portfolio insurance form, but since the capital guarantee is valid at every intermediate point in time, re-calibration is needed whenever the constraint is active.

Keywords: Stochastic Control; Martingale Method; Life Insurance; Rate Guarantee; Option Based Portfolio Insurance; CRRA Utility