×

Pricing participating products under a generalized jump-diffusion model. (English) Zbl 1141.91386

Summary: We propose a model for valuing participating life insurance products under a generalized jump-diffusion model with a Markov-switching compensator. It also nests a number of important and popular models in finance, including the classes of jump-diffusion models and Markovian regime-switching models. The Esscher transform is employed to determine an equivalent martingale measure. Simulation experiments are conducted to illustrate the practical implementation of the model and to highlight some features that can be obtained from our model.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60J60 Diffusion processes
60J75 Jump processes (MSC2010)
91B30 Risk theory, insurance (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] A. Grosen and P. L. Jørgensen, “Fair valuation of life insurance liabilities: the impact of interest rate guarantees, surrender options, and bonus policies,” Insurance: Mathematics & Economics, vol. 26, no. 1, pp. 37-57, 2000. · Zbl 0977.62108 · doi:10.1016/S0167-6687(99)00041-4
[2] M. J. Brennan and E. S. Schwartz, “The pricing of equity-linked life insurance policies with an asset value guarantee,” Journal of Financial Economics, vol. 3, no. 3, pp. 195-213, 1976.
[3] M. J. Brennan and E. S. Schwartz, “Alternative investment strategies for the issuers of equity-linked life insurance with an asset value guarantee,” Journal of Business, vol. 52, no. 1, pp. 63-93, 1979.
[4] P. P. Boyle and E. S. Schwartz, “Equilibrium prices of guarantees under equity-linked contracts,” The Journal of Risk and Insurance, vol. 44, no. 4, pp. 639-660, 1977.
[5] D. Prieul, V. Putyatin, and T. Nassar, “On pricing and reserving with-profits life insurance contracts,” Applied Mathematical Finance, vol. 8, no. 3, pp. 145-166, 2001. · Zbl 1026.91060 · doi:10.1080/13504860110111279
[6] A. R. Bacinello, “Fair pricing of life insurance participating policies with a minimum interest rate guaranteed,” Astin Bulletin, vol. 31, no. 2, pp. 275-297, 2001. · Zbl 1098.91537 · doi:10.2143/AST.31.2.1006
[7] A. R. Bacinello, “Fair valuation of a guaranteed life insurance participating contract embedding a surrender option,” The Journal of Risk and Insurance, vol. 70, no. 3, pp. 461-487, 2003. · doi:10.1111/1539-6975.t01-1-00060
[8] A. Grosen and P. L. Jørgensen, “Life insurance liabilities at market value: an analysis of insolvency risk, bonus policy, and regulatory intervention rules in a barrier option framework,” The Journal of Risk and Insurance, vol. 69, no. 1, pp. 63-91, 2002. · doi:10.1111/1539-6975.00005
[9] C. C. Chu and Y. K. Kwok, “Pricing participating policies with rate guarantees,” International Journal of Theoretical and Applied Finance, vol. 9, no. 4, pp. 517-532, 2006. · Zbl 1184.91110 · doi:10.1142/S0219024906003688
[10] T. K. Siu, “Fair valuation of participating policies with surrender options and regime switching,” Insurance: Mathematics & Economics, vol. 37, no. 3, pp. 533-552, 2005. · Zbl 1129.60062 · doi:10.1016/j.insmatheco.2005.05.007
[11] D. B. Madan, P. P. Carr, and E. C. Chang, “The variance gamma process and option pricing,” European Finance Review, vol. 2, no. 1, pp. 79-105, 1998. · Zbl 0937.91052 · doi:10.1023/A:1009703431535
[12] P. Carr, H. German, D. B. Madan, and M. Yor, “The fine structure of asset returns: an empirical investigation,” Journal of Business, vol. 75, no. 2, pp. 305-332, 2002.
[13] X. Guo, “Information and option pricings,” Quantitative Finance, vol. 1, no. 1, pp. 38-44, 2001. · doi:10.1080/713665550
[14] J. Buffington and R. J. Elliott, “Regime switching and European options,” in Stochastic Theory and Control (Lawrence, KS, 2001), K. S. Lawrence, Ed., vol. 280 of Lecture Notes in Control and Information Sciences, pp. 73-82, Springer, Berlin, Germany, 2002. · Zbl 1073.91027 · doi:10.1007/3-540-48022-6_5
[15] J. Buffington and R. J. Elliott, “American options with regime switching,” International Journal of Theoretical and Applied Finance, vol. 5, no. 5, pp. 497-514, 2002. · Zbl 1107.91325 · doi:10.1142/S0219024902001523
[16] R. J. Elliott, L. Chan, and T. K. Siu, “Option pricing and Esscher transform under regime switching,” Annals of Finance, vol. 1, no. 4, pp. 423-432, 2005. · Zbl 1233.91270 · doi:10.1007/s10436-005-0013-z
[17] R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models: Estimation and Control, vol. 29 of Applications of Mathematics, Springer, New York, NY, USA, 1995. · Zbl 0819.60045
[18] L. F. James, “Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics,” 2002, http://arxiv.org/abs/math.PR/0205093.
[19] L. F. James, “Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages,” The Annals of Statistics, vol. 33, no. 4, pp. 1771-1799, 2005. · Zbl 1078.62106 · doi:10.1214/009053605000000336
[20] M. Perman, J. Pitman, and M. Yor, “Size-biased sampling of Poisson point processes and excursions,” Probability Theory and Related Fields, vol. 92, no. 1, pp. 21-39, 1992. · Zbl 0741.60037 · doi:10.1007/BF01205234
[21] H. Föllmer and D. Sondermann, “Hedging of contingent claims under incomplete information,” in Contributions to Mathematical Economics, W. Hildenbrand and A. Mas-Colell, Eds., pp. 205-223, North Holland, Amsterdam, The Netherlands, 1986. · Zbl 0663.90006
[22] H. Föllmer and M. Schweizer, “Hedging of contingent claims under incomplete information,” in Applied Stochastic Analysis (London, 1989), M. H. A. Davis and R. J. Elliott, Eds., vol. 5 of Stochastics Monographs, pp. 389-414, Gordon and Breach, New York, NY, USA, 1991. · Zbl 0738.90007
[23] D. Duffie and H. R. Richardson, “Mean-variance hedging in continuous time,” The Annals of Applied Probability, vol. 1, no. 1, pp. 1-15, 1991. · Zbl 0735.90021 · doi:10.1214/aoap/1177005978
[24] M. Schweizer, “Approximation pricing and the variance-optimal martingale measure,” The Annals of Probability, vol. 24, no. 1, pp. 206-236, 1996. · Zbl 0854.60045 · doi:10.1214/aop/1042644714
[25] M. H. A. Davis, “Option pricing in incomplete markets,” in Mathematics of Derivative Securities (Cambridge, 1995), M. A. H. Dempster and S. R. Pliska, Eds., vol. 15 of Publications of the Newton Institute, pp. 216-226, Cambridge University Press, Cambridge, UK, 1997. · Zbl 0914.90017
[26] H. U. Gerber and E. S. W. Shiu, “Option pricing by Esscher transforms,” Transactions of the Society of Actuaries, vol. 46, pp. 99-191, 1994.
[27] F. Esscher, “On the probability function in the collective theory of risk,” Skandinavisk Aktuarietidskrift, vol. 15, pp. 175-195, 1932. · Zbl 0004.36101
[28] R. J. Elliott, Stochastic Calculus and Applications, vol. 18 of Applications of Mathematics, Springer, New York, NY, USA, 1982. · Zbl 0503.60062
[29] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 2nd edition, 2003. · Zbl 1018.60002
[30] J. M. Harrison and D. M. Kreps, “Martingales and arbitrage in multiperiod securities markets,” Journal of Economic Theory, vol. 20, no. 3, pp. 381-408, 1979. · Zbl 0431.90019 · doi:10.1016/0022-0531(79)90043-7
[31] J. M. Harrison and S. R. Pliska, “Martingales and stochastic integrals in the theory of continuous trading,” Stochastic Processes and Their Applications, vol. 11, no. 3, pp. 215-260, 1981. · Zbl 0482.60097 · doi:10.1016/0304-4149(81)90026-0
[32] J. M. Harrison and S. R. Pliska, “A stochastic calculus model of continuous trading: complete markets,” Stochastic Processes and Their Applications, vol. 15, no. 3, pp. 313-316, 1983. · Zbl 0511.60094 · doi:10.1016/0304-4149(83)90038-8
[33] F. Delbaen and W. Schachermayer, “A general version of the fundamental theorem of asset pricing,” Mathematische Annalen, vol. 300, no. 1, pp. 463-520, 1994. · Zbl 0865.90014 · doi:10.1007/BF01450498
[34] P. W. Glynn, “Optimization of stochastic systems via simulation,” in Proceedings of the 21st Conference on Winter Simulation (WSC ’89), pp. 90-105, Society for Computer Simulation, Washington, DC, USA, December 1989. · doi:10.1145/76738.76750
[35] M. Broadie and P. Glasserman, “Estimating security price derivatives using simulation,” Management Science, vol. 42, no. 2, pp. 269-285, 1996. · Zbl 0881.90018 · doi:10.1287/mnsc.42.2.269
[36] E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, and N. Touzi, “Applications of Malliavin calculus to Monte Carlo methods in finance,” Finance and Stochastics, vol. 3, no. 4, pp. 391-412, 1999. · Zbl 0947.60066 · doi:10.1007/s007800050068
[37] E. Fournié, J.-M. Lasry, J. Lebuchoux, and P.-L. Lions, “Applications of Malliavin calculus to Monte-Carlo methods in finance-II,” Finance and Stochastics, vol. 5, no. 2, pp. 201-236, 2001. · Zbl 0973.60061 · doi:10.1007/PL00013529
[38] N. Chen and P. Glasserman, “Malliavin Greeks without Malliavin calculus,” Stochastic Processes and Their Applications, vol. 117, no. 11, pp. 1689-1723, 2007. · Zbl 1133.60030 · doi:10.1016/j.spa.2007.03.012
[39] J. A. León, J. L. Solé, F. Utzet, and J. Vives, “On Lévy processes, Malliavin calculus and market models with jumps,” Finance and Stochastics, vol. 6, no. 2, pp. 197-225, 2002. · Zbl 1005.60067 · doi:10.1007/s007800100055
[40] Y. El-Khatib and N. Privault, “Computations of Greeks in a market with jumps via the Malliavin calculus,” Finance and Stochastics, vol. 8, no. 2, pp. 161-179, 2004. · Zbl 1098.91050 · doi:10.1007/s00780-003-0111-6
[41] M. H. A. Davis and M. P. Johansson, “Malliavin Monte Carlo Greeks for jump diffusions,” Stochastic Processes and Their Applications, vol. 116, no. 1, pp. 101-129, 2006. · Zbl 1081.60040 · doi:10.1016/j.spa.2005.08.002
[42] K. Aase, B. Øksendal, N. Privault, and J. Ubøe, “White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance,” Finance and Stochastics, vol. 4, no. 4, pp. 465-496, 2000. · Zbl 0963.60065 · doi:10.1007/PL00013528
[43] R. J. Elliott and A. J. Royal, “Asset prices with regime switching variance gamma dynamics,” in Handbook on Mathematical Finance, A. Bensoussan and Q. Zhang, Eds., Elsevier, Amsterdam, The Netherlands, 2007. · Zbl 1180.91141
[44] J. Lee and Y. Kim, “Sampling methods for NTR processes,” in Proceedings of International Workshop for Statistics (SRCCS ’04), Lecture Note, Seoul, Korea, June 2004.
[45] P. Glasserman, Monte Carlo Methods in Financial Engineering, vol. 53 of Applications of Mathematics, Springer, New York, NY, USA, 2004. · Zbl 1038.91045
[46] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics, Springer, Berlin, Germany, 1992. · Zbl 0752.60043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.