American Institute of Mathematical Sciences

• Previous Article
Approach to the consistency and consensus of Pythagorean fuzzy preference relations based on their partial orders in group decision making
• JIMO Home
• This Issue
• Next Article
Lookback option pricing problem of mean-reverting stock model in uncertain environment
doi: 10.3934/jimo.2020062

Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform

 1 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 2 Department of General Education, Army Engineering University of PLA, Nanjing 211101, China

* Corresponding author: Peibiao Zhao

Received  March 2019 Revised  December 2019 Published  March 2020

Fund Project: This work was supported by NNSF of China (No.11871275; No.11371194)

The present paper investigates an optimal reinsurance-investment problem with Hyperbolic Absolute Risk Aversion (HARA) utility. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer. The insurer is allowed to purchase reinsurance from the reinsurer. Both the insurer and the reinsurer are assumed to invest in one risk-free asset and one risky asset whose price follows Heston's SV model. Our aim is to seek optimal investment-reinsurance strategies to maximize the expected HARA utility of the insurer's and the reinsurer's terminal wealth. In the utility theory, HARA utility consists of power utility, exponential utility and logarithmic utility as special cases. In addition, HARA utility is seldom studied in the optimal investment and reinsurance problem due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the insurer. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solutions of optimal investment-reinsurance strategies. Moreover, some special cases are also discussed in detail. Finally, some numerical examples are presented to illustrate the impacts of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategies.

Citation: Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020062
References:
 [1] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005.  Google Scholar [2] L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.  Google Scholar [3] K. Borch, The optimal reinsurance treaty, ASTIN Bulletin, 5 (1969), 293-297.   Google Scholar [4] G. Chacko and L. M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, The Review of Financial Studies, 18 (2005), 1369-1402.   Google Scholar [5] H. Chang and K. Chang, Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform, Insurance Math. Econom., 72 (2017), 215-227.  doi: 10.1016/j.insmatheco.2016.10.014.  Google Scholar [6] J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance Math. Econom., 45 (2009), 9-18.  doi: 10.1016/j.insmatheco.2009.02.006.  Google Scholar [7] H. U. Gerber, An Introduction to Mathematical Risk Theory, in S. S. Huebner Foundation Monograph Series, 8, Richard D. Irwin, Inc., Homewood, Ⅲ., 1979.  Google Scholar [8] J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9058-9.  Google Scholar [9] M. Grasselli, A stability result for the HARA class with stochastic interest rates, Insurance Math. Econom., 33 (2003), 611-627.  doi: 10.1016/j.insmatheco.2003.09.003.  Google Scholar [10] M. Gu, Y. Yang, S. Li and J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46 (2010), 580-587.  doi: 10.1016/j.insmatheco.2010.03.001.  Google Scholar [11] E. J. Jung and J. H. Kim, Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance Math. Econom., 51 (2012), 667-673.  doi: 10.1016/j.insmatheco.2012.09.009.  Google Scholar [12] V. Henderson, Explicit solutions to an optimal portfolio choice problem with stochastic income, J Econ. Dyn. Control, 29 (2005), 1237-1266.  doi: 10.1016/j.jedc.2004.07.004.  Google Scholar [13] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar [14] Y. Huang, X. Yang and J. Zhou, Robust optimal investment and reinsurance problem for a general insurance company under Heston model, Math. Methods Oper. Res., 85 (2017), 305-326.  doi: 10.1007/s00186-017-0570-8.  Google Scholar [15] Y. Huang, Y. Ouyang, L. Tang and J. Zhou, Robust optimal investment and reinsurance problem for the product of the insurer's and the reinsurer's utilities, J. Comput. Appl. Math., 344 (2018), 532-552.  doi: 10.1016/j.cam.2018.05.060.  Google Scholar [16] D. Li, X. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, J. Comput. Appl. Math., 283 (2015), 142-162.  doi: 10.1016/j.cam.2015.01.038.  Google Scholar [17] D. Li, X. Rong and H. Zhao, Optimal reinsurance-investment problem for maximizing the product of the insurer's and the reinsurer's utilities under a CEV model, J. Comput. Appl. Math., 255 (2014), 671-683.  doi: 10.1016/j.cam.2013.06.033.  Google Scholar [18] D. Li, X. Rong and H. Zhao, Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model, Comp. Appl. Math., 35 (2016), 533-557.  doi: 10.1007/s40314-014-0204-1.  Google Scholar [19] Z. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.  Google Scholar [20] Z. Liang and K. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899.  Google Scholar [21] X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the CEV mode, N. Am. Actuar. J., 15 (2011), 417-431.  doi: 10.1080/10920277.2011.10597628.  Google Scholar [22] J. Liu, Portfolio selection in stochastic environments, The Review of Financial Studies, 20 (2007), 1-39.  doi: 10.1093/rfs/hhl001.  Google Scholar [23] S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.  Google Scholar [24] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.  Google Scholar [25] D.-L. Sheng, Explicit solution of reinsurance-investment problem for an insurer with dynamic income under Vasicek model, Adv. Math. Phys., 2016 (2016), Art. ID 1967872, 13 pp. doi: 10.1155/2016/1967872.  Google Scholar [26] Z. Sun, X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, J Math. Anal. Appl., 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053.  Google Scholar [27] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 328 (2018), 414-431.  doi: 10.1016/j.cam.2017.08.001.  Google Scholar [28] J. Xiao, Z. Hong and C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302-310.  doi: 10.1016/j.insmatheco.2006.04.007.  Google Scholar [29] B. Yi, Z. Li, F. G. Viens and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model, Insurance Math. Econom., 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011.  Google Scholar [30] H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance Math. Econom., 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004.  Google Scholar [31] X. Zheng, J. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model, Insurance Math. Econom., 67 (2016), 77-87.  doi: 10.1016/j.insmatheco.2015.12.008.  Google Scholar [32] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance Math. Econom., 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006.  Google Scholar [33] Y. Zhang and P. Zhao, Optimal reinsurance-investment problem with dependent risks based on Legendre transform, Journal of Industrial & Management Optimization, (2019). doi: 10.3934/jimo.2019011.  Google Scholar [34] H. Zhao, C. Weng, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, Sci. China Math., 60 (2017), 317-344.  doi: 10.1007/s11425-015-0542-7.  Google Scholar

show all references

References:
 [1] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005.  Google Scholar [2] L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.  Google Scholar [3] K. Borch, The optimal reinsurance treaty, ASTIN Bulletin, 5 (1969), 293-297.   Google Scholar [4] G. Chacko and L. M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, The Review of Financial Studies, 18 (2005), 1369-1402.   Google Scholar [5] H. Chang and K. Chang, Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform, Insurance Math. Econom., 72 (2017), 215-227.  doi: 10.1016/j.insmatheco.2016.10.014.  Google Scholar [6] J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance Math. Econom., 45 (2009), 9-18.  doi: 10.1016/j.insmatheco.2009.02.006.  Google Scholar [7] H. U. Gerber, An Introduction to Mathematical Risk Theory, in S. S. Huebner Foundation Monograph Series, 8, Richard D. Irwin, Inc., Homewood, Ⅲ., 1979.  Google Scholar [8] J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9058-9.  Google Scholar [9] M. Grasselli, A stability result for the HARA class with stochastic interest rates, Insurance Math. Econom., 33 (2003), 611-627.  doi: 10.1016/j.insmatheco.2003.09.003.  Google Scholar [10] M. Gu, Y. Yang, S. Li and J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46 (2010), 580-587.  doi: 10.1016/j.insmatheco.2010.03.001.  Google Scholar [11] E. J. Jung and J. H. Kim, Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance Math. Econom., 51 (2012), 667-673.  doi: 10.1016/j.insmatheco.2012.09.009.  Google Scholar [12] V. Henderson, Explicit solutions to an optimal portfolio choice problem with stochastic income, J Econ. Dyn. Control, 29 (2005), 1237-1266.  doi: 10.1016/j.jedc.2004.07.004.  Google Scholar [13] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar [14] Y. Huang, X. Yang and J. Zhou, Robust optimal investment and reinsurance problem for a general insurance company under Heston model, Math. Methods Oper. Res., 85 (2017), 305-326.  doi: 10.1007/s00186-017-0570-8.  Google Scholar [15] Y. Huang, Y. Ouyang, L. Tang and J. Zhou, Robust optimal investment and reinsurance problem for the product of the insurer's and the reinsurer's utilities, J. Comput. Appl. Math., 344 (2018), 532-552.  doi: 10.1016/j.cam.2018.05.060.  Google Scholar [16] D. Li, X. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, J. Comput. Appl. Math., 283 (2015), 142-162.  doi: 10.1016/j.cam.2015.01.038.  Google Scholar [17] D. Li, X. Rong and H. Zhao, Optimal reinsurance-investment problem for maximizing the product of the insurer's and the reinsurer's utilities under a CEV model, J. Comput. Appl. Math., 255 (2014), 671-683.  doi: 10.1016/j.cam.2013.06.033.  Google Scholar [18] D. Li, X. Rong and H. Zhao, Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model, Comp. Appl. Math., 35 (2016), 533-557.  doi: 10.1007/s40314-014-0204-1.  Google Scholar [19] Z. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.  Google Scholar [20] Z. Liang and K. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899.  Google Scholar [21] X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the CEV mode, N. Am. Actuar. J., 15 (2011), 417-431.  doi: 10.1080/10920277.2011.10597628.  Google Scholar [22] J. Liu, Portfolio selection in stochastic environments, The Review of Financial Studies, 20 (2007), 1-39.  doi: 10.1093/rfs/hhl001.  Google Scholar [23] S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.  Google Scholar [24] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.  Google Scholar [25] D.-L. Sheng, Explicit solution of reinsurance-investment problem for an insurer with dynamic income under Vasicek model, Adv. Math. Phys., 2016 (2016), Art. ID 1967872, 13 pp. doi: 10.1155/2016/1967872.  Google Scholar [26] Z. Sun, X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, J Math. Anal. Appl., 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053.  Google Scholar [27] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 328 (2018), 414-431.  doi: 10.1016/j.cam.2017.08.001.  Google Scholar [28] J. Xiao, Z. Hong and C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302-310.  doi: 10.1016/j.insmatheco.2006.04.007.  Google Scholar [29] B. Yi, Z. Li, F. G. Viens and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model, Insurance Math. Econom., 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011.  Google Scholar [30] H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance Math. Econom., 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004.  Google Scholar [31] X. Zheng, J. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model, Insurance Math. Econom., 67 (2016), 77-87.  doi: 10.1016/j.insmatheco.2015.12.008.  Google Scholar [32] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance Math. Econom., 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006.  Google Scholar [33] Y. Zhang and P. Zhao, Optimal reinsurance-investment problem with dependent risks based on Legendre transform, Journal of Industrial & Management Optimization, (2019). doi: 10.3934/jimo.2019011.  Google Scholar [34] H. Zhao, C. Weng, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, Sci. China Math., 60 (2017), 317-344.  doi: 10.1007/s11425-015-0542-7.  Google Scholar
Effect of $x$ on $q^*_{HARA}$
Effects of $y$ on pH ARA*
Effect of $r$ on $q^*_{\exp}$ and $p^*_{\exp}$
Effect of v on qexp* and pexp*
Effect of r on qexp* and pexp*
Effect of x on π1 HARA*
Effect of y on π2HARA*
Effect of v on π1HARA* and π2HARA*
Effect of β on π1 exp*
Effect of α on π1 exp*
Effect of σ on π1 exp*
 [1] Yan Zhang, Peibiao Zhao. Optimal reinsurance-investment problem with dependent risks based on Legendre transform. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1457-1479. doi: 10.3934/jimo.2019011 [2] Lin Xu, Rongming Wang, Dingjun Yao. On maximizing the expected terminal utility by investment and reinsurance. Journal of Industrial & Management Optimization, 2008, 4 (4) : 801-815. doi: 10.3934/jimo.2008.4.801 [3] Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial & Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044 [4] Nan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Markowitz's mean-variance optimization with investment and constrained reinsurance. Journal of Industrial & Management Optimization, 2017, 13 (1) : 375-397. doi: 10.3934/jimo.2016022 [5] Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080 [6] Nan Zhang, Linyi Qian, Zhuo Jin, Wei Wang. Optimal stop-loss reinsurance with joint utility constraints. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020001 [7] Yan Zhang, Yonghong Wu, Benchawan Wiwatanapataphee, Francisca Angkola. Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. Journal of Industrial & Management Optimization, 2020, 16 (1) : 71-101. doi: 10.3934/jimo.2018141 [8] Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2020008 [9] Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009 [10] Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control & Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003 [11] Tao Chen, Wei Liu, Tao Tan, Lijun Wu, Yijun Hu. Optimal reinsurance with default risk: A reinsurer's perspective. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020103 [12] Pavol Kútik, Karol Mikula. Diamond--cell finite volume scheme for the Heston model. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 913-931. doi: 10.3934/dcdss.2015.8.913 [13] Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020226 [14] Xiaoshan Chen, Xun Li, Fahuai Yi. Optimal stopping investment with non-smooth utility over an infinite time horizon. Journal of Industrial & Management Optimization, 2019, 15 (1) : 81-96. doi: 10.3934/jimo.2018033 [15] Jiapeng Liu, Ruihua Liu, Dan Ren. Investment and consumption in regime-switching models with proportional transaction costs and log utility. Mathematical Control & Related Fields, 2017, 7 (3) : 465-491. doi: 10.3934/mcrf.2017017 [16] Zhongbao Zhou, Yanfei Bai, Helu Xiao, Xu Chen. A non-zero-sum reinsurance-investment game with delay and asymmetric information. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020004 [17] Yan Zeng, Zhongfei Li. Optimal reinsurance-investment strategies for insurers under mean-CaR criteria. Journal of Industrial & Management Optimization, 2012, 8 (3) : 673-690. doi: 10.3934/jimo.2012.8.673 [18] Ming Yan, Hongtao Yang, Lei Zhang, Shuhua Zhang. Optimal investment-reinsurance policy with regime switching and value-at-risk constraint. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2195-2211. doi: 10.3934/jimo.2019050 [19] Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020143 [20] Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019127

2019 Impact Factor: 1.366

Metrics

• PDF downloads (132)
• HTML views (266)
• Cited by (0)

Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]