Randomization and the valuation of guaranteed minimum death benefits

In this article, we focus on death-linked contingent claims (GMDBs) paying a random financial return at a random time of death in the general case where financial returns follow a regime switching model with two-sided phase-type jumps. We approximate the distribution of the remaining lifetime by either a series of Erlang distributions or a Laguerre series expansion, whose capability to fit the tail of the observed mortality data turns out to be much better than the commonly used series of exponential distributions. More precisely, we develop a Laurent series expansion of the discounted Laplace transform of the subordinated process at an Erlang distributed time, which leads to explicit formulae for European-type GMDB as well as related risk measures such as the Value-at-Risk (VaR) and the Conditional-Tail-Expectation (CTE). We further concentrate upon path-dependent GMDBs with lookback features like dynamic fund protection or dynamic withdrawal benefits, by relying on a Sylvester equation approach. The advantage of our approaches is that our results are of semi-closed form, avoiding numerical Fourier inversion or Monte-Carlo simulation, leading to fast evaluation. This is necessary in risk-management, in particularly for nested simulation in the framework of Solvency II. Several numerical experiments are included. Our results have implications beyond life-insurance and GMDBs, namely in all situations where randomization or Erlangization replaces known quantities, like, for example, model parameters, by random variables. In Finance, it is for example well-known that a random maturity time leads to much more convenient valuation formulas that well approximate its non-random counterpart.