April  2013, 9(2): 411-429. doi: 10.3934/jimo.2013.9.411

Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model

1. 

Research Center of International Finance and Risk Management, East China Normal University, Shanghai, 200241, China

2. 

Department of Mathematics, Ningbo University, Ningbo, 315211, China

3. 

School of Finance and Statistics, East China Normal University, Shanghai, 200241

Received  November 2011 Revised  January 2013 Published  February 2013

This paper extends the model in Riesner (2007) to a Markov modulated Lévy process. The parameters of the Lévy process switch over time according to the different states of an economy, which is described by a finite-state continuous time Markov chain. Employing the local risk minimization method, we find an optimal hedging strategy for a general payment process. Finally, we give an example for single unit-linked insurance contracts with guarantee to display the specific locally risk-minimizing hedging strategy.
Citation: Linyi Qian, Wei Wang, Rongming Wang. Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model. Journal of Industrial and Management Optimization, 2013, 9 (2) : 411-429. doi: 10.3934/jimo.2013.9.411
References:
[1]

K. Aase and S.-A. Persson, Pricing of unit-linked life insurance policies, Scandinavian Actuarial Journal, 1994, 26-52. doi: 10.1080/03461238.1994.10413928.

[2]

J. P. Ansel and C. Stricker, Décomposition de Kunita-Watanabe, in "Séminaire de Probabilités," XXVII, Lecture Notes in Mathematics, 1557, Springer, Berlin, (1993), 30-32. doi: 10.1007/BFb0087960.

[3]

J. Bi and J. Guo, Hedging unit-linked life insurance contracts in a financial market driven by shot-noise processes, Applied Stochastic Models In Business And Industry, 26 (2010), 609-623. doi: 10.1002/asmb.807.

[4]

T. Chan, Pricing contingent claims on stocks driven by Lévy processes, The Annals of Applied Probability, 9 (1999), 504-528. doi: 10.1214/aoap/1029962753.

[5]

A. Deshpande and M. K. Ghosh, Risk minimizing option pricing in a regime switching market, Stochastic Analysis and Applications, 26 (2008), 313-324. doi: 10.1080/07362990701857194.

[6]

R. J. Elliott, L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432.

[7]

H. Föllmer and M. Schweizer, Hedging of contingent claims under incomplete information, in "Applied Stochastic Analysis" (eds. M. Davis and R. Elliot) (London, 1989), Stochastic Monographs, 5, Gordon and Breach, New York, (1991), 389-414.

[8]

H. Föllmer and D. Sondermann, Hedging of non-redundant contingent claims, in "Contributions to Mathematical Economics" (eds. W. Hildenbrand and A. Mas-Colell), North-Holland, Elsevier, (1986), 205-223.

[9]

M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions, SIAM Journal of Contral and Optimization, 35 (1997), 1952-1988. doi: 10.1137/S0363012996299302.

[10]

J. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Ecomometrica, 57 (1989), 357-384. doi: 10.2307/1912559.

[11]

J. Hoem, Markov chain models in life insurance, Blätter der Deut. Gesell. Versicherungsmath, 9 (1969), 91-107.

[12]

S. Lin, K. Tan and H. Yang, Pricing annuity guarantees under a regime-switching model, North American Actuarial Journal, 13 (2009), 316-332. doi: 10.1080/10920277.2009.10597557.

[13]

T. Møller, Risk-minimizing hedging strategies for unit-linked life insurance contracts, ASTIN Bulletin, 28 (1998), 17-47.

[14]

T. Møller, Risk-mimizing hedging strategies for insurance payment processes, Finance and Stochastics, 5 (2001), 419-446. doi: 10.1007/s007800100041.

[15]

R. Norberg, Hattendorff's theorem and Thiele's differential equation generalized, Scandinavian Actuarial Journal, 1992, 2-14. doi: 10.1080/03461238.1992.10413894.

[16]

M. Riesner, Hedging life insurance contracts in a Lévy process financial market, Insurance: Mathematics and Economics, 38 (2006), 599-608. doi: 10.1016/j.insmatheco.2005.12.004.

[17]

M. Riesner, Locally risk-minimizing hedging of insurance payment streams, Astin Bulletin, 37 (2007), 67-91. doi: 10.2143/AST.37.1.2020799.

[18]

M. Schweizer, Option hedging for semimartingales, Stochastic Processes and Their Applications, 37 (1991), 339-363. doi: 10.1016/0304-4149(91)90053-F.

[19]

M. Schweizer, Risk-minimizing hedging strategies under restricted information, Mathematical Finance, 4 (1994), 327-342. doi: 10.1111/j.1467-9965.1994.tb00062.x.

[20]

M. Schweizer, A guided tour through quadratic hedging approaches, in "Option Pricing, Interest Rates and Risk Management," Handbooks in Mathematical Finance, Cambridge University Press, (2001), 538-574. doi: 10.1017/CBO9780511569708.016.

[21]

M. Schweizer, Local risk-minimization for multidimensional assets and payment streams, in "Advances in Mathematics of Finance," Banach Center Publications, 83, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 213-229. doi: 10.4064/bc83-0-13.

[22]

L. Qian, H. Yang and R. Wang, Locally risk-minimizing hedging strategies for unit-linked life insurance contracts under a regime switching Lévy model, Frontiers of Mathematics in China, 6 (2011), 1185-1202. doi: 10.1007/s11464-011-0100-6.

[23]

N. Vandaele and M. Vanmaele, A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market, Insurance: Mathematics and Economics, 42 (2008), 1128-1137. doi: 10.1016/j.insmatheco.2008.03.001.

[24]

T. Choulli, N. Vandaele and M. Vanmaele, The Föllmer-Schweizer decomposition: Comparison and description, Stochastic Processes and their Applications, 120 (2010), 853-872. doi: 10.1016/j.spa.2010.02.004.

[25]

L. Xu and R. Wang, Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate, Journal of Industrial and Management Optimization, 2 (2006), 165-175. doi: 10.3934/jimo.2006.2.165.

show all references

References:
[1]

K. Aase and S.-A. Persson, Pricing of unit-linked life insurance policies, Scandinavian Actuarial Journal, 1994, 26-52. doi: 10.1080/03461238.1994.10413928.

[2]

J. P. Ansel and C. Stricker, Décomposition de Kunita-Watanabe, in "Séminaire de Probabilités," XXVII, Lecture Notes in Mathematics, 1557, Springer, Berlin, (1993), 30-32. doi: 10.1007/BFb0087960.

[3]

J. Bi and J. Guo, Hedging unit-linked life insurance contracts in a financial market driven by shot-noise processes, Applied Stochastic Models In Business And Industry, 26 (2010), 609-623. doi: 10.1002/asmb.807.

[4]

T. Chan, Pricing contingent claims on stocks driven by Lévy processes, The Annals of Applied Probability, 9 (1999), 504-528. doi: 10.1214/aoap/1029962753.

[5]

A. Deshpande and M. K. Ghosh, Risk minimizing option pricing in a regime switching market, Stochastic Analysis and Applications, 26 (2008), 313-324. doi: 10.1080/07362990701857194.

[6]

R. J. Elliott, L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432.

[7]

H. Föllmer and M. Schweizer, Hedging of contingent claims under incomplete information, in "Applied Stochastic Analysis" (eds. M. Davis and R. Elliot) (London, 1989), Stochastic Monographs, 5, Gordon and Breach, New York, (1991), 389-414.

[8]

H. Föllmer and D. Sondermann, Hedging of non-redundant contingent claims, in "Contributions to Mathematical Economics" (eds. W. Hildenbrand and A. Mas-Colell), North-Holland, Elsevier, (1986), 205-223.

[9]

M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions, SIAM Journal of Contral and Optimization, 35 (1997), 1952-1988. doi: 10.1137/S0363012996299302.

[10]

J. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Ecomometrica, 57 (1989), 357-384. doi: 10.2307/1912559.

[11]

J. Hoem, Markov chain models in life insurance, Blätter der Deut. Gesell. Versicherungsmath, 9 (1969), 91-107.

[12]

S. Lin, K. Tan and H. Yang, Pricing annuity guarantees under a regime-switching model, North American Actuarial Journal, 13 (2009), 316-332. doi: 10.1080/10920277.2009.10597557.

[13]

T. Møller, Risk-minimizing hedging strategies for unit-linked life insurance contracts, ASTIN Bulletin, 28 (1998), 17-47.

[14]

T. Møller, Risk-mimizing hedging strategies for insurance payment processes, Finance and Stochastics, 5 (2001), 419-446. doi: 10.1007/s007800100041.

[15]

R. Norberg, Hattendorff's theorem and Thiele's differential equation generalized, Scandinavian Actuarial Journal, 1992, 2-14. doi: 10.1080/03461238.1992.10413894.

[16]

M. Riesner, Hedging life insurance contracts in a Lévy process financial market, Insurance: Mathematics and Economics, 38 (2006), 599-608. doi: 10.1016/j.insmatheco.2005.12.004.

[17]

M. Riesner, Locally risk-minimizing hedging of insurance payment streams, Astin Bulletin, 37 (2007), 67-91. doi: 10.2143/AST.37.1.2020799.

[18]

M. Schweizer, Option hedging for semimartingales, Stochastic Processes and Their Applications, 37 (1991), 339-363. doi: 10.1016/0304-4149(91)90053-F.

[19]

M. Schweizer, Risk-minimizing hedging strategies under restricted information, Mathematical Finance, 4 (1994), 327-342. doi: 10.1111/j.1467-9965.1994.tb00062.x.

[20]

M. Schweizer, A guided tour through quadratic hedging approaches, in "Option Pricing, Interest Rates and Risk Management," Handbooks in Mathematical Finance, Cambridge University Press, (2001), 538-574. doi: 10.1017/CBO9780511569708.016.

[21]

M. Schweizer, Local risk-minimization for multidimensional assets and payment streams, in "Advances in Mathematics of Finance," Banach Center Publications, 83, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 213-229. doi: 10.4064/bc83-0-13.

[22]

L. Qian, H. Yang and R. Wang, Locally risk-minimizing hedging strategies for unit-linked life insurance contracts under a regime switching Lévy model, Frontiers of Mathematics in China, 6 (2011), 1185-1202. doi: 10.1007/s11464-011-0100-6.

[23]

N. Vandaele and M. Vanmaele, A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market, Insurance: Mathematics and Economics, 42 (2008), 1128-1137. doi: 10.1016/j.insmatheco.2008.03.001.

[24]

T. Choulli, N. Vandaele and M. Vanmaele, The Föllmer-Schweizer decomposition: Comparison and description, Stochastic Processes and their Applications, 120 (2010), 853-872. doi: 10.1016/j.spa.2010.02.004.

[25]

L. Xu and R. Wang, Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate, Journal of Industrial and Management Optimization, 2 (2006), 165-175. doi: 10.3934/jimo.2006.2.165.

[1]

Wei Wang, Yang Shen, Linyi Qian, Zhixin Yang. Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2369-2399. doi: 10.3934/jimo.2021072

[2]

Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100

[3]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022007

[4]

Zhimin Zhang, Eric C. K. Cheung. A note on a Lévy insurance risk model under periodic dividend decisions. Journal of Industrial and Management Optimization, 2018, 14 (1) : 35-63. doi: 10.3934/jimo.2017036

[5]

Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5101-5134. doi: 10.3934/dcdsb.2020335

[6]

Alexander Melnikov, Hongxi Wan. CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 343-368. doi: 10.3934/puqr.2021017

[7]

Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial and Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010

[8]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic and Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53

[9]

Engel John C Dela Vega, Robert J Elliott. Conditional coherent risk measures and regime-switching conic pricing. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 267-300. doi: 10.3934/puqr.2021014

[10]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027

[11]

Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial and Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001

[12]

A. Settati, A. Lahrouz, Mohamed El Fatini, A. El Haitami, M. El Jarroudi, M. Erriani. A Markovian switching diffusion for an SIS model incorporating Lévy processes. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 209-229. doi: 10.3934/dcdsb.2022072

[13]

Yang Yang, Kaiyong Wang, Jiajun Liu, Zhimin Zhang. Asymptotics for a bidimensional risk model with two geometric Lévy price processes. Journal of Industrial and Management Optimization, 2019, 15 (2) : 481-505. doi: 10.3934/jimo.2018053

[14]

Hui Gao, Chuancun Yin. A Lévy risk model with ratcheting and barrier dividend strategies. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022025

[15]

Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298

[16]

Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315

[17]

Ming Yan, Hongtao Yang, Lei Zhang, Shuhua Zhang. Optimal investment-reinsurance policy with regime switching and value-at-risk constraint. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2195-2211. doi: 10.3934/jimo.2019050

[18]

W.C. Ip, H. Wong, Jiazhu Pan, Keke Yuan. Estimating value-at-risk for chinese stock market by switching regime ARCH model. Journal of Industrial and Management Optimization, 2006, 2 (2) : 145-163. doi: 10.3934/jimo.2006.2.145

[19]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[20]

Karel Kadlec, Bohdan Maslowski. Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4039-4055. doi: 10.3934/dcdsb.2020137

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (91)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]