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 & 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.  doi: 10.1080/03461238.1994.10413928.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

R. Norberg, Hattendorff's theorem and Thiele's differential equation generalized,, Scandinavian Actuarial Journal, 1992 (): 2.  doi: 10.1080/03461238.1992.10413894.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

R. Norberg, Hattendorff's theorem and Thiele's differential equation generalized,, Scandinavian Actuarial Journal, 1992 (): 2.  doi: 10.1080/03461238.1992.10413894.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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