Acknowledgments: The research reported in this paper is partially supported by MUR - Ministero Università e Ricerca (Roma, Italy), 40%, 2007, under grant: “The impact of population ageing on financial markets, intermediaries and financial stability”. The support and sponsorship of MUR are gratefully acknowledged.

Abstract

In this paper we consider the problem of pricing life insurance contracts using ideas and methods taken from mathematical finance, more specifically taken from the solution given in mathematical finance to the option pricing problem. The methodology developed is rather general, however, to fix the ideas, we focus our attention on the problem of pricing a pure endowment policy also known as Guaranteed Minimum Income Benefit Contract. In particular we consider a pure endowment policy that has its life contingent payout linked to the performance of a risky asset (such as, for example, a stock, a market index, or an interest rate). The behaviour of the value of the risky asset is modeled using a multiscale stochastic volatility model given by a system of three stochastic differential equations. This model has been introduced in [1] in the context of mathematical finance. The mortality risk is modeled via a stochastic differential equation suggested in [2], [3] that describes the time evolution of the mortality rate. That is the time dynamics of the state variables of the model used to price a pure endowment policy is defined by the system of four stochastic differential equations obtained considering simultaneously the two models mentioned above. Under the assumption that the financial and the mortality risks are independent a formula to price the simplest pure endowment policy is derived as the product of the discounted expected value of the money that must be paid at maturity of the contract to the policy holder if the policy holder is alive at maturity time and of the relevant survival probability. The relevant survival probability is the probability that an individual belonging to the ``cohort" of the policy holder is alive at maturity time given the fact that he is alive at the time when the contract is subscribed. A ``cohort" is defined as the subset of the population made by the individuals born in a given year. A computational method to evaluate the pricing formula derived is developed. This computational method largely overcomes in accuracy and computational efficiency the performance of the Monte Carlo method used in a straightforward way to evaluate the same formula. The computational method proposed is based on two formulae: a semi-explicit formula to price in the multiscale stochastic volatility model adopted a European call option [1] and an explicit formula for a family of conditioned transition probability densities associated to the hazard rate in the mortality risk model considered. This last formula leads to an ad hoc Monte Carlo method to evaluate the survival probability. Furthermore, in the limit when the volatility parameter of the mortality risk model goes to zero an asymptotic formula for the survival probability is derived. In its region of validity the numerical evaluation of this asymptotic formula leads to a method to price the pure endowment policy that greatly outperforms the computational methods mentioned above based on Monte Carlo. Some numerical experiments on test problems are presented. Elementary test problems that can be solved explicitly are used to show the performance of the numerical methods proposed and the accuracy of the results obtained. More complex test problems are considered. On these last test problems a comparison between the results obtained with the numerical methods presented is carried out. Finally the range of validity of the asymptotic formula for the survival probability is studied. Auxiliary material that assists an understanding of this work is on this website. A more general reference to the work of some of the authors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance/ .

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References

[1]

Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F. (2009). An explicitly solvable multi-scale stochastic volatility model: option pricing and calibration problems . Journal of Futures Markets, 29(9), 862-893.

[2]

Giacometti, R., Ortobelli, S., Bertocchi, M.I. (2009). A stochastic model for mortality rate on Italian data. Submitted to Journal of Optimization Theory and Applications.

[3]

Milevsky, M. A., Promislow, S.D. (2001). Mortality derivatives and the option to annuitise. Insurance: Mathematics and Economics, 29, 299-318.

 

 

 

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