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A subordinated Markov model for stochastic mortality. (English) Zbl 1273.91239

Summary: 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 [A. J. G. Cairns et al., Astin Bull. 36, No. 1, 79–120 (2006; Zbl 1162.91403); Scand. Actuar. J. 2008, No. 2–3, 79–113 (2008; Zbl 1224.91048)]. 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 [S. Asmussen, Applied probability and queues. 2nd revised and extended ed. New York, NY: Springer (2003; Zbl 1029.60001); S. Asmussen and H. Albrecher, Ruin probabilities. 2nd ed. Hackensack, NJ: World Scientific (2010; Zbl 1247.91080); M. F. Neuts, Matrix-geometric solutions in stochastic models. An algorithmic approach. London: The Johns Hopkins University Press (1981; Zbl 0469.60002)]. 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.

MSC:

91B30 Risk theory, insurance (MSC2010)
91B70 Stochastic models in economics
91G20 Derivative securities (option pricing, hedging, etc.)
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