Motivated by the Guaranteed Minimum Death Benefits (GMDB) in variable annuities, we are interested in valuing equity-linked options whose expiry date is the time of the death of the policyholder. Because the time-until-death distribution can be approximated by linear combinations of exponential distributions or mixtures of Erlang distributions, the analysis can be reduced to the case where the time-until-death distribution is exponential or Erlang.
We present two probability methods to price American options with an exponential expiry date. Both methods give the same results. An American option with Erlang expiry date can be seen as an extension of the exponential expiry date case. We calculate its price as the sum of the price of the corresponding European option and the early exercise premium. Because the optimal exercise boundary takes the form of a staircase, the pricing formula is a triple sum. We determine the optimal exercise boundary recursively by imposing the “smooth pasting” condition. The examples of the put option, the exchange option, and the maximum option are provided to illustrate how the methods work.
Another issue related to variable annuities is the surrender behavior of the policyholders. To model this behavior, we suggest using barrier options. We generalize the reflection principle and use it to derive explicit formulas for outside barrier options, double barrier options with constant barriers, and double barrier options with time varying exponential barriers.
Finally, we provide a method to approximate the distribution of the time-until-death random variable by combinations of exponential distributions or mixtures of Erlang distributions. Compared to directly fitting the distributions, my method has two advantages: 1) It is more robust to the initial guess. 2) It is more likely to obtain the global minimizer.