Document Type : Research Article
Authors
1 Shahid Beheshti University (SBU)
2 Department of Actuarial Science, Shahid Beheshti University, Tehran, Iran
Abstract
In this paper, we design a pure-endowment insurance contract and obtain the optimal strategy and consumption for a policyholder with CRRA utility function. In this contract, premiums are received from the policyholder at certain times. The
insurer undertakes to pay the premiums by a certain guarantee rate, in addition, by investing in a portfolio of risky and risk free assets share invest pro ts. We used Variance Gamma process as a representative of in nite activity jump models
and sensitivity of jump parameters in an uncertainty nancial market has been studied. Also we compared results using by two forces of mortality.
Keywords
[1] At-Sahalia, Y., Cacho-Diaz, J., and Hurd, T. R. (2009). Portfolio choice with jumps: A closed-form solution. Annals of Applied Probability, 19(2), 556{584.
[2] Applebaum, D. (2009). Levy processes and stochastic calculus. Cambridge university press.
[3] Barigou, K., and Delong, L. (2020). Pricing equity-linked life insurance contracts with multiple risk factors by neural networks. arXiv preprint arXiv:2007.08804.
[4] Ba~nos, D., Lagunas-Merino, M., and Ortiz-Latorre, S. (2020). Variance and interest rate risk in unit-linked insurance policies. Risks, 8(3), 84.
[5] Bosserhoff, F., and Stadje, M. (2021). Time-consistent mean-variance investment with unit linked life insurance contracts in a jump-diffusion setting. Insurance: Mathematics and Economics, (100), 130-146.
[6] Campbell, J. Y., Champbell, J. J., Campbell, J. W., Lo, A. W., Lo, A. W., and MacKinlay, A. C. (1997). The econometrics of nancial markets. princeton University press.
[2] Applebaum, D. (2009). Levy processes and stochastic calculus. Cambridge university press.
[3] Barigou, K., and Delong, L. (2020). Pricing equity-linked life insurance contracts with multiple risk factors by neural networks. arXiv preprint arXiv:2007.08804.
[4] Ba~nos, D., Lagunas-Merino, M., and Ortiz-Latorre, S. (2020). Variance and interest rate risk in unit-linked insurance policies. Risks, 8(3), 84.
[5] Bosserhoff, F., and Stadje, M. (2021). Time-consistent mean-variance investment with unit linked life insurance contracts in a jump-diffusion setting. Insurance: Mathematics and Economics, (100), 130-146.
[6] Campbell, J. Y., Champbell, J. J., Campbell, J. W., Lo, A. W., Lo, A. W., and MacKinlay, A. C. (1997). The econometrics of nancial markets. princeton University press.
[7] Ceci, C., Colaneri, K., and Cretarola, A. (2020). Indifference pricing of pure endowments via BSDEs under partial information. Scandinavian Actuarial Journal, 2020(10), 904-933.
[8] Ceci, C., Colaneri, K., and Cretarola, A. (2017). Unit-linked life insurance policies: Optimal hedging in partially observable market models. Insurance: Mathematics and Economics, (76), 149-163.
[9] Cont, R., and Tankov, P. (2004). Financial Modelling with jump processes. Chapman and amp.
[10] Cont, R., and Tankov, P. (2009). Constant proportion portfolio insurance in the presence of jumps in asset prices. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 19(3), 379{401.
[11] Deelstra, G., Grasselli, M., and Koehl, P. F. (2004). Optimal design of the guarantee for de ned contribution funds. Journal of economic dynamics and control, 28(11), 2239{2260.
[12] Devroye, L. (2006). Nonuniform random variate generation. Handbooks in operations research and management science, 13, 83{121.
[13] Dong, Y., and Zheng, H. (2019). Optimal investment of DC pension plan under short-selling constraints and portfolio insurance. Insurance: Mathematics and Economics, 85, 47{59.
[14] Dong, Y., and Zheng, H. (2020). Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to de ned contribution pension plan. European Journal of Operational Research, 281(2), 341{356.
[15] Gradshteyn, I. S., and Ryzhik, I. M. (2014). Table of integrals, series, and products. Academic press.
[16] Huertas, K. E. A. (2021). Mathematical reserves of unit-linked insurance policies with fractional volatility (Master's thesis).
[17] Iscanoglu-Cekic, A. (2016). An optimal Turkish private pension plan with a guarantee feature.Risks, 4(3), 19.
[18] Kirkby, J. L., and Nguyen, D. (2021). Equity-Linked Guaranteed Minimum Death Bene ts with Dollar Cost Averaging. Insurance: Mathematics and Economics.
[19] Kou, S. G. (2002). A jump-diffusion model for option pricing. Management science, 48(8),1086{1101.
[20] Kou, S. G., and Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management science, 50(9), 1178{1192.
[21] Kung, K. L., and Yang, S. Y. (2020). Optimal Consumption and Investment Problem Incorporating Housing and Life Insurance Decisions: The Continuous Time Case. Journal of Risk and Insurance, 87(1), 143-171.
[22] Lamberton, D., and Lapeyre, B. (2007). Introduction to stochastic calculus applied to nance. CRC press.
[23] Li, D., Rong, X., and Zhao, H. (2016). Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model. Computational and Applied Mathematics, 35(2), 533-557.
[24] Liang, X., and Lu, Y. (2017). Indifference pricing of a life insurance portfolio with risky asset driven by a shot-noise process. Insurance: Mathematics and Economics, 77, 119-132.
[25] Madan, D. B., and Seneta, E. (1990). The variance gamma (VG) model for share market returns. Journal of business, 511-524.
[26] Madan, D. (2001). Financial modeling with discontinuous price processes. Levy processes-theory and applications. Birkhauser, Boston.
[27] Markowitz, H. (1952). Portfolio selection. the Journal of Finance, 7 (1), 77{91.
[28] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The review of Economics and Statistics, 247{257.
[29] Merton, R. C. (1975). Optimum consumption and portfolio rules in a continuous-time model. In Stochastic Optimization Models in Finance. Academic Press.
[30] Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of nancial economics, 3(1-2), 125{144.
[31] ksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science and Business Media.
[32] Richard, S. F. (1975). Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics, 2(2), 187-203.
[8] Ceci, C., Colaneri, K., and Cretarola, A. (2017). Unit-linked life insurance policies: Optimal hedging in partially observable market models. Insurance: Mathematics and Economics, (76), 149-163.
[9] Cont, R., and Tankov, P. (2004). Financial Modelling with jump processes. Chapman and amp.
[10] Cont, R., and Tankov, P. (2009). Constant proportion portfolio insurance in the presence of jumps in asset prices. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 19(3), 379{401.
[11] Deelstra, G., Grasselli, M., and Koehl, P. F. (2004). Optimal design of the guarantee for de ned contribution funds. Journal of economic dynamics and control, 28(11), 2239{2260.
[12] Devroye, L. (2006). Nonuniform random variate generation. Handbooks in operations research and management science, 13, 83{121.
[13] Dong, Y., and Zheng, H. (2019). Optimal investment of DC pension plan under short-selling constraints and portfolio insurance. Insurance: Mathematics and Economics, 85, 47{59.
[14] Dong, Y., and Zheng, H. (2020). Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to de ned contribution pension plan. European Journal of Operational Research, 281(2), 341{356.
[15] Gradshteyn, I. S., and Ryzhik, I. M. (2014). Table of integrals, series, and products. Academic press.
[16] Huertas, K. E. A. (2021). Mathematical reserves of unit-linked insurance policies with fractional volatility (Master's thesis).
[17] Iscanoglu-Cekic, A. (2016). An optimal Turkish private pension plan with a guarantee feature.Risks, 4(3), 19.
[18] Kirkby, J. L., and Nguyen, D. (2021). Equity-Linked Guaranteed Minimum Death Bene ts with Dollar Cost Averaging. Insurance: Mathematics and Economics.
[19] Kou, S. G. (2002). A jump-diffusion model for option pricing. Management science, 48(8),1086{1101.
[20] Kou, S. G., and Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management science, 50(9), 1178{1192.
[21] Kung, K. L., and Yang, S. Y. (2020). Optimal Consumption and Investment Problem Incorporating Housing and Life Insurance Decisions: The Continuous Time Case. Journal of Risk and Insurance, 87(1), 143-171.
[22] Lamberton, D., and Lapeyre, B. (2007). Introduction to stochastic calculus applied to nance. CRC press.
[23] Li, D., Rong, X., and Zhao, H. (2016). Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model. Computational and Applied Mathematics, 35(2), 533-557.
[24] Liang, X., and Lu, Y. (2017). Indifference pricing of a life insurance portfolio with risky asset driven by a shot-noise process. Insurance: Mathematics and Economics, 77, 119-132.
[25] Madan, D. B., and Seneta, E. (1990). The variance gamma (VG) model for share market returns. Journal of business, 511-524.
[26] Madan, D. (2001). Financial modeling with discontinuous price processes. Levy processes-theory and applications. Birkhauser, Boston.
[27] Markowitz, H. (1952). Portfolio selection. the Journal of Finance, 7 (1), 77{91.
[28] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The review of Economics and Statistics, 247{257.
[29] Merton, R. C. (1975). Optimum consumption and portfolio rules in a continuous-time model. In Stochastic Optimization Models in Finance. Academic Press.
[30] Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of nancial economics, 3(1-2), 125{144.
[31] ksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science and Business Media.
[32] Richard, S. F. (1975). Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics, 2(2), 187-203.
[33] Vahabi, S., and Najafabadi, A. T. P. (2022). Optimal investment strategy for a DC pension fund plan in a nite horizon time: an optimal stochastic control approach. Annals of Actuarial Science, 1-17.
[34] Wang, Y., Zhang, Z., and Yu, W. (2021). Pricing equity-linked death bene ts by complex Fourier series expansion in a regime-switching jump diffusion model. Applied Mathematics and Computation, (399), 126031.
[35] Wang, W., Shen, Y., Qian, L., and Yang, Z. (2021). Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering. Journal of Industrial and Management Optimization.
[36] Xu, W., and Gao, J. (2020). An Optimal Portfolio Problem of DC Pension with Input-Delay and Jump-Diffusion Process. Mathematical Problems in Engineering, 2020, 1{9.
[37] Yao, H., Chen, P., Zhang, M., and Li, X. (2020). Dynamic discrete-time portfolio selection for de ned contribution pension funds with in ation risk. Journal of Industrial and Management Optimization, 13(5), 1{30.
[38] Yaari, M. E. (1965). Uncertain lifetime, life insurance, and the theory of the consumer. The Review of Economic Studies, 32(2), 137-150.
[34] Wang, Y., Zhang, Z., and Yu, W. (2021). Pricing equity-linked death bene ts by complex Fourier series expansion in a regime-switching jump diffusion model. Applied Mathematics and Computation, (399), 126031.
[35] Wang, W., Shen, Y., Qian, L., and Yang, Z. (2021). Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering. Journal of Industrial and Management Optimization.
[36] Xu, W., and Gao, J. (2020). An Optimal Portfolio Problem of DC Pension with Input-Delay and Jump-Diffusion Process. Mathematical Problems in Engineering, 2020, 1{9.
[37] Yao, H., Chen, P., Zhang, M., and Li, X. (2020). Dynamic discrete-time portfolio selection for de ned contribution pension funds with in ation risk. Journal of Industrial and Management Optimization, 13(5), 1{30.
[38] Yaari, M. E. (1965). Uncertain lifetime, life insurance, and the theory of the consumer. The Review of Economic Studies, 32(2), 137-150.
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