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On the Risk-Neutral Valuation of Life Insurance Contracts with Numerical Methods in View

Published online by Cambridge University Press:  09 August 2013

Daniel Bauer ,
Daniela Bergmann  and
Rüdiger Kiesel
Daniela Bergmann
Affiliation:
Institute of Insurance, Ulm University, Helmholtzstraβe 18, 89069 Ulm, Germany, Tel.: +49-731-50-31187, Fax: +49-731-50-31239, [email protected]
Rüdiger Kiesel
Affiliation:
Chair for Energy Trading and Financial Services, University of Duisburg-Essen, and, Centre of Mathematics for Applications, University of Oslo, Universitätsstraβe 12, 45141 Essen, Germany, Tel.: +49-201-183-4963, Fax: +49-201-183-4974, [email protected]
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Abstract

In recent years, market-consistent valuation approaches have gained an increasing importance for insurance companies. This has triggered an increasing interest among practitioners and academics, and a number of specific studies on such valuation approaches have been published.

In this paper, we present a generic model for the valuation of life insurance contracts and embedded options. Furthermore, we describe various numerical valuation approaches within our generic setup. We particularly focus on contracts containing early exercise features since these present (numerically) challenging valuation problems.

Based on an example of participating life insurance contracts, we illustrate the different approaches and compare their efficiency in a simple and a generalized Black-Scholes setup, respectively. Moreover, we study the impact of the considered early exercise feature on our example contract and analyze the influence of model risk by additionally introducing an exponential Lévy model.

Keywords

Life insuranceRisk-neutral valuationEmbedded optionsBermudan optionsNested simulationsPDE methodsLeast-squares Monte Carlo
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 40 , Issue 1 , May 2010 , pp. 65 - 95
Copyright
Copyright © International Actuarial Association 2010

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References

Aase, K. and Persson, S.A. (1994) Pricing of unit-linked life insurance policies. Scandinavian Actuarial Journal 1, 26–52.CrossRefGoogle Scholar
Lebensversicherungs-AG, Allianz (2006) Geschäftsbericht. http://www.allianz.com/images/pdf/azlgb.pdf. Google Scholar
Andersen, L. and Andreasen, J. (2000) Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Review of Derivatives Research 4, 231–262.Google Scholar
Andreatta, G. and Corradin, S. (2003) Valuing the surrender options embedded in a portfolio of Italian life guar anteed participating policies: a least squares Monte Carlo approach. Working Paper, University of California, Berkeley.Google Scholar
Ballotta, L. (2006) A Lévy process-based framework for the fair valuation of participating life insurance contracts. Insurance: Mathematics and Economics 37, 173–196.Google Scholar
Barndorff-Nielsen, O.E. (1998) Processes of normal inverse Gaussian type. Finance and Stochastics 2, 41–68.Google Scholar
Bauer, D., Kiesel, R., Kling, A. and Ruβ, J. (2006) Risk-neutral valuation of participating life insurance contracts. Insurance: Mathematics and Economics 39, 171–183.Google Scholar
Bauer, D., Kling, A. and Ruβ, J. (2008) A universal pricing framework for guaranteed minimum benefits in variable annuities. ASTIN Bulletin 38, 621–651.Google Scholar
Bayraktar, E. and Ludkovski, M. (2009) Relative hedging of systematic mortality risk. North American Actuarial Journal 13, 106–140.Google Scholar
Becherer, D. (2003) Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insurance: Mathematics and Economics 33, 1–28.Google Scholar
Bertsekas, D. (1995) Dynamic Programming and Optimal Control, Vols. I and II. Athena Scientific.Google Scholar
Biffis, E. (2005) Affine processes for dynamic mortality and actuarial valuations. Insurance: Mathematics and Economics 37, 443–468.Google Scholar
Blake, D., Cairns, A. and Dowd, K. (2006) Living with mortality: longevity bonds and other mortality-linked securities. British Actuarial Journal 12, 153–197.CrossRefGoogle Scholar
Brennan, M. and Schwartz, E. (1976) The pricing of equity-linked life insurance policies with an asset value guarantee. Journal of Financal Economics 3, 195–213.Google Scholar
Briys, E. and de Varenne, F. (1997) On the risk of insurance liabilities: Debunking some common pitfalls. The Journal of Risk and Insurance 64, 673–694.CrossRefGoogle Scholar
Carpenter, J. (1998) The exercise and valuation of executive stock options. Journal of Financial Economics 48, 127–158.Google Scholar
Clement, E., Lamberton, D. and Protter, P. (2002) An analysis of a least squares regression method for American option pricing. Finance and Stochastics 6, 449–471.CrossRefGoogle Scholar
Cont, R. and Tankov, P. (2003) Financial Modelling with Jump Processes. Chapman & Hall/CRC Press, Boca Raton, FL.Google Scholar
Cont, R. and Voltchkova, E. (2005) A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM Journal on Numerical Analysis 43, 1596–1626.Google Scholar
Dahl, M. (2004) Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insurance: Mathematics and Economics 35, 113–136.Google Scholar
Dahl, M., Melchior, M., and Møller, T. (2008) On systematic mortality risk and risk-minimization with survivor swaps. Scandinavian Actuarial Journal 2008, 114–146.Google Scholar
Delbaen, F. and Schachermayer, W. (1994) A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300, 463–520.CrossRefGoogle Scholar
Gatzert, N. and Schmeiser, H. (2008) Assessing the risk potential of premium payment options in participating life insurance contracts. The Journal of Risk and Insurance 75, 691–712.CrossRefGoogle Scholar
Glasserman, P. (2003) Monte Carlo Methods in Financial Engineering. Series: Stochastic Modelling and Applied Probability, volume 53. Springer Verlag, Berlin.Google Scholar
Grosen, A. and Jørgensen, P. (2000) Fair valuation of life insurance liabilities: The impact of interest rate guarantees, surrender options and bonus policies. Insurance: Mathematics and Economics 26, 37–57.Google Scholar
Hull, J. (2000) Options, Futures & other Derivatives. Prentice-Hall, Upper Saddle River, NJ, 4th edition.Google Scholar
Ingersoll, J. (2006) The subjective and objective evaluation of incentive stock options. Journal of Business 79, 453–487.Google Scholar
International Accounting Standards Board (2007) Press release (03.05.2007). http://www.iasb.org/News/Press+Releases/Press+Releases.htm. Google Scholar
Jackson, K., Jaimungal, S. and Surkov, V. (2008) Fourier space time-stepping for option pricing with Lévy models. Journal of Computational Finance 12, 1–29.Google Scholar
Kassberger, S., Kiesel, R. and Liebmann, T. (2008) Fair valuation of insurance contracts under Lévy process specifications. Insurance: Mathematics and Economics 42, 419–433.Google Scholar
Kling, A., Richter, A. and Ruß, J. (2007) The interaction of guarantees, surplus distribution, and asset allocation in with profit life insurance policies. Insurance: Mathematics and Economics 40, 164–178.Google Scholar
Kling, A., Ruβ, J. and Schmeiser, H. (2006) Analysis of embedded options in individual pension schemes in Germany. The Geneva Risk and Insurance Review 31, 43–60.Google Scholar
Lando, D. (1998) On Cox processes and credit risky securities. Review of Derivatives Research 2, 99–120.Google Scholar
Longstaff, F. and Schwartz, E. (2001) Valuing american options by simulation: A simple least-squares approach. The Review of Financial Studies 14, 113–147.Google Scholar
Lord, R., Fang, F., Bervoets, F. and Oosterlee, C. (2008) A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes. SIAM Journal on Scientific Computing 30, 1678–1705.Google Scholar
Matache, A.M., von Petersdorff, T. and Schwab, C. (2004) Fast deterministic pricing of options on Levy driven assets. European Series in Applied and Industrial Mathematics: Mathematical Modelling and Numerical Analysis 38, 37–71.Google Scholar
Miltersen, K. and Persson, S.A. (2003) Guaranteed investment contracts: Distributed and undistributed excess return. Scandinavian Actuarial Journal 4, 257–279.CrossRefGoogle Scholar
Møller, T. (2001) Risk-minimizing hedging strategies for insurance payment processes. Finance and Stochastics 5, 419–446.Google Scholar
Møller, T. (2003) Indifference pricing of insurance contracts in a product space model. Finance and Stochastics 7, 197–217.Google Scholar
Nordahl, H. (2008) Valuation of life insurance surrender and exchange options. Insurance: Mathematics and Economics 42, 909–919.Google Scholar
Riesner, M. (2006) Hedging life insurance contracts in a Lévy process financial market. Insurance: Mathematics and Economics 38, 599–608.Google Scholar
Schoutens, W. (2003) Lévy Processes in Finance, Pricing Financial Derivatives. Wiley.Google Scholar
Schweizer, M. (1995) On the minimal martingale measure and the Föllmer-Schweizer decomposition. Stochastic Analysis and Applications 13, 573–599.CrossRefGoogle Scholar
Steffensen, M. (2000) A no arbitrage approach to Thiele's differential equation. Insurance: Mathematics and Economics 27, 201–214.Google Scholar
Steffensen, M. (2002) Intervention options in life insurance. Insurance: Mathematics and Economics 31, 71–85.Google Scholar
Tanskanen, A. and Lukkarinen, J. (2003) Fair valuation of path-dependent participating life insurance contracts. Insurance: Mathematics and Economics 33, 595–609.Google Scholar
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–188.Google Scholar
Zaglauer, K. and Bauer, D. (2008) Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment. Insurance: Mathematics and Economics 43, 29–40.Google Scholar