Valuing guaranteed minimum death benefits in variable annuities and the option to lapse∗

Many variable annuities provide money-back guarantees and market guarantees on invested principal. Embedded in some of these guarantees are stochastic maturity put options with adjustable strike prices. These variable annuities can be surrendered or lapsed at any time. The lapse option when exercised rationally represents an American style sell-back option that is exercised by the policyholder when the embedded put option is out-of-the-money. The death benefits we consider are only exercised involuntary, that is, upon the death of the policyholder. Critical to the valuation analysis is that the embedded put options have stochastic maturity and that the policyholder can exercise the lapse, or early-exercise, option feature to increase the value of the contract and thereby exposing the insurance company to loss of fees. We analyze specific variable annuity products by focusing on the lapse option when exercised either rationally or irrationally, taking into account the mortality risk and surrender charges.