Analytic valuation of GMDB options with utility based asset allocation

A number of analytic solutions have been found for Variable Annuity Guaranteed Minimum Death Benefit (GMDB) option values under a variety of mortality laws. To date, the solutions are for Risk-Neutral valuation only. Where policyholder decisions are allowed, it is assumed that they act to maximize the risk-neutral value of the GMDB. We examine situations where the asset allocation decisions are made to maximize expected utility rather than option value. We find analytic solutions for both return of premium and ratchet options at small values of bequest motive for a number of mortality laws.