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A recursive approach to mortality-linked derivative pricing. (English) Zbl 1218.91156

Summary: We develop a recursive method to derive an exact numerical and nearly analytical representation of the Laplace transform of the transition density function with respect to the time variable for time-homogeneous diffusion processes. We further apply this recursion algorithm to the pricing of mortality-linked derivatives. Given an arbitrary stochastic future lifetime \(\mathbb T\), the probability distribution function of the present value of a cash flow depending on \(\mathbb T\) can be approximated by a mixture of exponentials, based on Jacobi polynomial expansions. In case of mortality-linked derivative pricing, the required Laplace inversion can be avoided by introducing this mixture of exponentials as an approximation of the distribution of the survival time \(\mathbb T\) in the recursion scheme. This approximation significantly improves the efficiency of the algorithm.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
65R10 Numerical methods for integral transforms
44A10 Laplace transform
91B30 Risk theory, insurance (MSC2010)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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[1] Aït-Sahalia, Yacine, Transition densities for interest rate and other non-linear diffusions, Journal of finance, 54, 1361-1395, (1999)
[2] Biffis, Enrico, Affine processes for dynamic mortality and actuarial valuations, Insurance: mathematics and economics, 37, 443-468, (2005) · Zbl 1129.91024
[3] Cairns, Andrew J.G.; Blake, David; Dowd, Kevin, A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration, Journal of risk and insurance, 73, 687-718, (2006)
[4] Cairns, Andrew J.G.; Blake, David; Dowd, Kevin; Coughlan, Guy D.; Epstein, David; Ong, Alen; Balevich, Igor, A quantitative comparison of stochastic mortality models using data from england and wales and the united states, North American actuarial journal, 13, 1, 1-35, (2009)
[5] Dahl, M., Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: mathematics and economics, 35, 113-136, (2004) · Zbl 1075.62095
[6] Denuit, Michel; Devolder, Pierre; Goderniaux, Anne-Cécile, Securitization of longevity risk: pricing survivor bonds with Wang transform in the lee – carter framework, Journal of risk and insurance, 74, 1, 87-113, (2007)
[7] Duffie, Darrell; Pan, Jun; Singleton, Kenneth, Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68, 1343-1376, (2000) · Zbl 1055.91524
[8] Dufresne, Daniel, Fitting combinations of exponentials to probability distributions, Applied stochastic models in business and industry, 23, 23-48, (2007) · Zbl 1142.60321
[9] Dufresne, Daniel, Stochastic life annuities, North American actuarial journal, 11, 136-157, (2007)
[10] Embrechts, Paul, Actuarial versus financial pricing of insurance, Journal of risk finance, 1, 17-26, (2000)
[11] Goovaerts, Marc J., 1976. Bijdrage tot het pad-integraalformalisme. In: RUG faculteit wetenschappen. Gent.266 p.
[12] Goovaerts, Marc J.; Broeckx, F., Analytical treatment of a periodic \(\delta\)-function potential in the path-integral formalism, SIAM journal on applied mathematics, 45, 479-490, (1985)
[13] Goovaerts, Marc J.; Schepper, Ann De; Decamps, Marc, Closed-form approximations for diffusion densities: a path integral approach, Journal of computational and applied mathematics, 164-165, 337-364, (2004) · Zbl 1039.60070
[14] Goovaerts, Marc J.; Laeven, Roger, Actuarial risk measures for financial derivative pricing, Insurance: mathematics and economics, 42, 540-547, (2008) · Zbl 1152.91444
[15] Goovaerts, Marc J., Laeven, Roger J.A., Shang, Zhaoning, 2010. Transform analysis and asset pricing for diffusion processes: a recursive approach. Working Paper.
[16] Grosche, Christian, Path integral for potential problems with \(\delta\)-function perturbation, Journal of physics A: mathematical and general, 23, 5205-5234, (1990) · Zbl 0715.58048
[17] Jensen, Bjarke; Poulsen, Rolf, Transition densities of diffusion processes: numerical comparison of approximation techniques, The journal of derivatives, 9, 18-32, (2002)
[18] Lauschagne, C.; Offwood, T., A note on the connection between the esscher – girsanov transform and the Wang transform, Insurance: mathematics and economics, 47, 3, 385-390, (2010) · Zbl 1231.60062
[19] Lee, Ronald, The lee – carter method of forecasting mortality, with various extensions and applications, North American actuarial journal, 4, 80-93, (2000) · Zbl 1083.62535
[20] Lee, Ronald; Carter, Lawrence R., Modeling and forecasting the time series of US mortality, Journal of American statistical association, 87, 659-671, (1992) · Zbl 1351.62186
[21] Lin, Yijia; Cox, Samuel H., Securitization of mortality risk in life annuities, Journal of risk and insurance, 72, 2, 227-252, (2005)
[22] Milevsky, M.; Promislow, D., Mortality derivatives and the option to annuitise, Insurance: mathematics and economics, 29, 299-318, (2001) · Zbl 1074.62530
[23] Møller, Thomas, On valuation and risk management at the interface of insurance and finance, British actuarial journal, 8, 787-827, (2002)
[24] Pelsser, Antoon, On the applicability of the Wang transform for pricing financial risks, ASTIN bulletin, 38, 171-181, (2008) · Zbl 1169.91343
[25] Renshaw, A.E.; Haberman, S., Lee – carter mortality forecasting with age-specific enhancement, Insurance: mathematics and economics, 33, 255-272, (2003) · Zbl 1103.91371
[26] Wang, S., A class of distortion operators for pricing financial and insurance risks, Journal of risk and insurance, 67, 1, 15-36, (2000)
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