随机金融市场环境下的最优再保险–投资策略
|
摘要
为了对冲保险风险,保险公司可以向再保险公司购买比例再保险;同时,为了保值增值,保险公司将其财富投资于金融市场.假设盈余过程由带漂移的布朗运动所驱动,利率满足仿射利率模型,股票波动率满足Heston随机波动率模型.应用随机最优控制和HJB方程方法得到了指数效用下最优再保险–投资策略的显式解.给出数值算例并分析了模型参数对最优再保险策略和最优投资策略的影响.研究结果表明:最优再保险策略不仅依赖于保险市场参数,而且依赖于金融市场参数;随机利率与随机波动率模型下的最优再保险–投资策略与利率动态密切相关,而与波动率动态无关;再保险行为对投资于股票的数量没有影响,而对投资于零息票债券的数量产生较大的影响.
Insurance company can purchase proportional reinsurance to hedge the risk of insurance. Meantime, insurance company can invest its wealth into the financial market to preserve or increase the value. In this paper, surplus process is supposed to be driven by Brownian motion with drift, short rate is described by stochastic affine interest rate model and the volatility of stock price is governed by Heston’s stochastic volatility model. By using the technique of stochastic dynamic programming and Hamilton-Jocabi-Bellman(HJB) equation, optimal reinsurance-investment strategy with exponential utility is obtained in explicit form. A numerical example is given to analyze the sensitivity of optimal reinsurance-investment strategy to model parameters. Research results display that optimal reinsurance strategy does not only depend on the parameters of insurance market, but also depends on the parameters of financial market; optimal reinsurance-investment strategies with stochastic interest rate and stochastic volatility are closely related to the dynamics of interest rate and have nothing to do with the dynamics of volatility; the reinsurance behavior has no effect on the amount in the stock, yet has a considerable influence on the amount in the zero-coupon bond.
Insurance company can purchase proportional reinsurance to hedge the risk of insurance. Meantime, insurance company can invest its wealth into the financial market to preserve or increase the value. In this paper, surplus process is supposed to be driven by Brownian motion with drift, short rate is described by stochastic affine interest rate model and the volatility of stock price is governed by Heston’s stochastic volatility model. By using the technique of stochastic dynamic programming and Hamilton-Jocabi-Bellman(HJB) equation, optimal reinsurance-investment strategy with exponential utility is obtained in explicit form. A numerical example is given to analyze the sensitivity of optimal reinsurance-investment strategy to model parameters. Research results display that optimal reinsurance strategy does not only depend on the parameters of insurance market, but also depends on the parameters of financial market; optimal reinsurance-investment strategies with stochastic interest rate and stochastic volatility are closely related to the dynamics of interest rate and have nothing to do with the dynamics of volatility; the reinsurance behavior has no effect on the amount in the stock, yet has a considerable influence on the amount in the zero-coupon bond.
引文
[1] BROWNE S. Optimal investment policies for a firm with a random risk process:exponential utility and minimizing probablity of ruin.Mathematics of Operations Research, 1995, 20(4):937–957.
[2] WANG N. Optimal investment for an insurer with exponential utility preference. Insurance:Mathematics and Economics, 2007, 40(1):77–84.
[3] RONG Ximin, FAN Lixin. Insurer’s optimal investment strategy under constant elasticity of variance model. Systems Engineering–Theory&Practice, 2012, 32(12):2619–2628.(荣喜民,范立鑫.常弹性方差模型下保险人的最优投资策略.系统工程理论与实践, 2012, 32(12):2619–2628.)
[4] PROMISLOW S, YOUNG V R. Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal, 2005, 9(3):110–128.
[5] SCHMIDLI H. On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability, 2002, 12(3):890–907.
[6] BI J N, GUO J Y. Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer. Journal of Optimization Theory&Applications, 2013, 157(1):252–275.
[7] LIANG Z B, BI J N, YUEN K C, et al. Optimal mean–variance reinsurance and investment in a jump-diffusion financial market with common shock dependence. Mathematical Methods of Operations Research, 2016, 84(1):155–181.
[8] GU A L, GUO X P, LI Z F, et al. Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model. Insurance:Mathematics and Economics, 2012, 51(3):674–684.
[9] ZHAO H, RONG X M, ZHAO Y G. Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model. Insurance:Mathematics and Economics, 2013, 53(3):504–514.
[10] GU Ailing, CHEN Shumin. Optimal excess-of-loss reinsurance and investment strategy under state-dependent utility function. Operations Research Transactions, 2016, 20(1):91–104.(谷爱玲,陈树敏.状态相依效用下的超额损失再保险–投资策略.运筹学学报, 2016, 20(1):91–104.)
[11] LI D P, RONG X M, ZHAO H. Optimal reinsurance-investment problem for maximizing the product of the insurer’s and the reinsurer’s utilities under a CEV model. Journal of Computational and Applied Mathematics, 2014, 255(2):671–683.
[12] YI B, LI Z F, VIENS F G, et al. Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model. Insurance Mathematics&Economics, 2013, 53(3):601–614.
[13] LI Z F, ZENG Y, LAI Y Z. Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model. Insurance:Mathematics and Economics, 2012, 50(1):191–203.
[14] ZHAO H, WENG C G, SHEN Y, et al. Time-consistent investmentreinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models. Science China Mathematics, 2017,60(2):1–28.
[15] GUAN G H, LIANG Z X. Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks. Insurance Mathematics&Economics, 2014, 55(1):105–115.
[16] LI D P, RONG X M, ZHAO H. Time-consistent reinsurance–investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk. Insurance Mathematics&Economics, 2015, 64(1):28–44.
[17] LIU J. Portfolio selection in stochastic environments. The Review of Financial Studies, 2007, 20(1):1–39.
[18] LI J Z, WU R. Optimal investment problem with stochastic interest rate and stochastic volatility:maximizing a power utility. Applied Stochastic Models in Business and Industry, 2009, 25(3):407–420.
[19] NOH E J, KIM J H. An optimal portfolio model with stochastic volatility and stochastic interest rate. Journal of Mathematical Analysis and Applications, 2011, 375(2):510–522.
[20] CHANG H, RONG X M. An investment and consumption problem with CIR interest rate and stochastic volatility. Abstract and Applied Analysis, 2013, 2013(1):1–12, http://dx.doi.org/10.1155/2013/219397.
[21] WEI W, FAN S H, CHANG H. Optimal reinsurance and investment problem with stochastic interest rate and stochastic volatility in the mean-variance framework. WSEAS Transactions on Mathematics, 2016, 15(2):226–240.
[22] DEELSTRA G, GRASSELLI M, KOEHL P F. Optimal investment strategies in the presence of a minimum guarantee. Insurance:Mathematics and Economics, 2003, 33(1):189–207.
[23] GAO J W. Stochastic optimal control of DC pension funds. Insurance:Mathematics and Economics, 2008, 42(3):1159–1164.
[24] MA Wei, GU Mengdi. Optimal proportional reinsurance and investment under stochastic interest rate. Journal of Systems&Management, 2013, 22(2):274–277.(马威,顾孟迪.随机利率下的再保险与投资策略.系统管理学报,2013, 22(2):274–277.)
[25] GUAN G H, LIANG Z X. Optimal management of DC pension plan in a stochastic interest rates and stochastic volatility framework. Insurance:Mathematics and Economics, 2014, 57(1):58–66.
[26] YAO H X, YANG Z, CHEN P. Markowitz’s mean–variance defined contribution pension fund management under inflation:a continuoustime model. Insurance:Mathematics and Economics, 2013, 53(3):851–863.
[2] WANG N. Optimal investment for an insurer with exponential utility preference. Insurance:Mathematics and Economics, 2007, 40(1):77–84.
[3] RONG Ximin, FAN Lixin. Insurer’s optimal investment strategy under constant elasticity of variance model. Systems Engineering–Theory&Practice, 2012, 32(12):2619–2628.(荣喜民,范立鑫.常弹性方差模型下保险人的最优投资策略.系统工程理论与实践, 2012, 32(12):2619–2628.)
[4] PROMISLOW S, YOUNG V R. Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal, 2005, 9(3):110–128.
[5] SCHMIDLI H. On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability, 2002, 12(3):890–907.
[6] BI J N, GUO J Y. Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer. Journal of Optimization Theory&Applications, 2013, 157(1):252–275.
[7] LIANG Z B, BI J N, YUEN K C, et al. Optimal mean–variance reinsurance and investment in a jump-diffusion financial market with common shock dependence. Mathematical Methods of Operations Research, 2016, 84(1):155–181.
[8] GU A L, GUO X P, LI Z F, et al. Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model. Insurance:Mathematics and Economics, 2012, 51(3):674–684.
[9] ZHAO H, RONG X M, ZHAO Y G. Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model. Insurance:Mathematics and Economics, 2013, 53(3):504–514.
[10] GU Ailing, CHEN Shumin. Optimal excess-of-loss reinsurance and investment strategy under state-dependent utility function. Operations Research Transactions, 2016, 20(1):91–104.(谷爱玲,陈树敏.状态相依效用下的超额损失再保险–投资策略.运筹学学报, 2016, 20(1):91–104.)
[11] LI D P, RONG X M, ZHAO H. Optimal reinsurance-investment problem for maximizing the product of the insurer’s and the reinsurer’s utilities under a CEV model. Journal of Computational and Applied Mathematics, 2014, 255(2):671–683.
[12] YI B, LI Z F, VIENS F G, et al. Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model. Insurance Mathematics&Economics, 2013, 53(3):601–614.
[13] LI Z F, ZENG Y, LAI Y Z. Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model. Insurance:Mathematics and Economics, 2012, 50(1):191–203.
[14] ZHAO H, WENG C G, SHEN Y, et al. Time-consistent investmentreinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models. Science China Mathematics, 2017,60(2):1–28.
[15] GUAN G H, LIANG Z X. Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks. Insurance Mathematics&Economics, 2014, 55(1):105–115.
[16] LI D P, RONG X M, ZHAO H. Time-consistent reinsurance–investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk. Insurance Mathematics&Economics, 2015, 64(1):28–44.
[17] LIU J. Portfolio selection in stochastic environments. The Review of Financial Studies, 2007, 20(1):1–39.
[18] LI J Z, WU R. Optimal investment problem with stochastic interest rate and stochastic volatility:maximizing a power utility. Applied Stochastic Models in Business and Industry, 2009, 25(3):407–420.
[19] NOH E J, KIM J H. An optimal portfolio model with stochastic volatility and stochastic interest rate. Journal of Mathematical Analysis and Applications, 2011, 375(2):510–522.
[20] CHANG H, RONG X M. An investment and consumption problem with CIR interest rate and stochastic volatility. Abstract and Applied Analysis, 2013, 2013(1):1–12, http://dx.doi.org/10.1155/2013/219397.
[21] WEI W, FAN S H, CHANG H. Optimal reinsurance and investment problem with stochastic interest rate and stochastic volatility in the mean-variance framework. WSEAS Transactions on Mathematics, 2016, 15(2):226–240.
[22] DEELSTRA G, GRASSELLI M, KOEHL P F. Optimal investment strategies in the presence of a minimum guarantee. Insurance:Mathematics and Economics, 2003, 33(1):189–207.
[23] GAO J W. Stochastic optimal control of DC pension funds. Insurance:Mathematics and Economics, 2008, 42(3):1159–1164.
[24] MA Wei, GU Mengdi. Optimal proportional reinsurance and investment under stochastic interest rate. Journal of Systems&Management, 2013, 22(2):274–277.(马威,顾孟迪.随机利率下的再保险与投资策略.系统管理学报,2013, 22(2):274–277.)
[25] GUAN G H, LIANG Z X. Optimal management of DC pension plan in a stochastic interest rates and stochastic volatility framework. Insurance:Mathematics and Economics, 2014, 57(1):58–66.
[26] YAO H X, YANG Z, CHEN P. Markowitz’s mean–variance defined contribution pension fund management under inflation:a continuoustime model. Insurance:Mathematics and Economics, 2013, 53(3):851–863.