A subordinated Markov model for stochastic mortality

Abstract

In this paper we propose a subordinated Markov model for modeling stochastic mortality. The aging process of a life is assumed to follow a finite-state Markov process with a single absorbing state and the stochasticity of mortality is governed by a subordinating gamma process. We focus on the theoretical development of the model and have shown that the model exhibits many desirable properties of a mortality model and meets many model selection criteria laid out in Cairns et al. (ASTIN Bull 36:79–120, 2006; Scand Actuar J 2:79–113, 2008). The model is flexible and fits either historical mortality data or projected mortality data well. We also explore applications of the proposed model to the valuation of mortality-linked securities. A general valuation framework for valuing mortality-linked products is presented for this model. With a proposed risk loading mechanism, we can make an easy transition from the physical measure to a risk-neutral measure and hence is able to calibrate the model to market information. The phase-type structure of the model allows us to apply the matrix-analytic methods that have been extensively used in ruin theory in actuarial science and queuing theory in operations research (see Asmussen in Applied probability and queues. Wiley, New York, 1987; Asmussen and Albrecher in Ruin probabilities, 2nd edn. World Scientific Publishing, Singapore, 2010; Neuts in Matrix-geometrix solutions in stochastic models. Johns Hopkins University Press, Baltimore, 1981). As a result, many quantities of interest such as the distribution of future survival rates, prediction intervals, the term structure of mortality as well as the value of caps and floors on the survival index can be obtained analytically.