# American Institute of Mathematical Sciences

July  2022, 18(4): 2369-2399. doi: 10.3934/jimo.2021072

## Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering

 1 School of Mathematics and Statistics, Ningbo University, 818 Fenghua Road, Ningbo 315211, China 2 School of Risk and Actuarial Studies and CEPAR, University of New South Wales, Sydney, NSW 2052, Australia 3 Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai 200241, China 4 Department of Mathematical Sciences, Ball State University, Muncie 47304, USA

* Corresponding author: Linyi Qian

Received  July 2020 Revised  February 2021 Published  July 2022 Early access  April 2021

Fund Project: This work was supported by the Humanity and Social Science Youth Foundation of the Ministry of Education of China (18YJC910012), the National Natural Science Foundation of China (11771147, 12071147), "Shuguang Program" supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission(18SG25), the State Key Program of National Natural Science Foundation of China (71931004), the Discovery Early Career Researcher Award (DE200101266) of the Australian Research Council, the Zhejiang Provincial Natural Science Foundation of China (LY17G010003), the 111 Project(B14019), Ningbo City Natural Science Foundation (202003N4144) and the Humanity and Social Science Foundation of Ningbo University (XPYB19002)

This paper studies the hedging problem of unit-linked life insurance contracts in an incomplete market presence of self-exciting (clustering) effect, which is described by a Hawkes process. Applying the local risk-minimization method, we manage to obtain closed-form expressions of the locally risk-minimizing hedging strategies for both pure endowment and term insurance contracts. Besides, we demonstrate the existence of the minimal martingale measure and perform numerical analyses. Our numerical results indicate that jump clustering has a significant impact on the optimal hedging strategies.

Citation: 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
##### References:
 [1] Y. Aït-Sahalia, J. Cacho-Diaz and R. J. A. Laeven, Modeling financial contagion using mutually exciting jump processes, Journal of Financial Economics, 117 (2015), 585-606.  doi: 10.1016/j.jfineco.2015.03.002. [2] Y. Aït-Sahalia and T. R. Hurd, Portfolio choice in markets with contagion, Journal of Financial Economics, 14 (2016), 1-28.  doi: 10.1093/jjfinec/nbv024. [3] T. Arai, Y. Imai and R. Suzuki, Numerical analysis on local risk-minimization for exponential L$\acute{e}$vy models, International Journal of Theoretical and Applied Finance, 19 (2016), 1650008, 27 pp. doi: 10.1142/S0219024916500084. [4] T. Arai, Y. Imai and R. Suzuki, Local risk-minimization for Barndorff-Nielsen and Shephard models, Finance and Stochastic, 21 (2017), 551-592.  doi: 10.1007/s00780-017-0324-8. [5] C. G. Bowsher, Modelling security market events in continuous time: Intensity based, multivariate point process models, Journal of Econometrics, 141 (2007), 876-912.  doi: 10.1016/j.jeconom.2006.11.007. [6] C. Ceci, K. Colaneri and A. Cretarola, Hedging of unit-linked life insurance contracts with unobservable mortality hazard rate via local risk-minimization, Insurance: Mathematics and Economics, 60 (2015), 47-60.  doi: 10.1016/j.insmatheco.2014.10.013. [7] C. Ceci, K. Colaneri and A. Cretarola, Unit-linked life insurance policies: Optimal hedging in partially observable market models, Insurance: Mathematics and Economics, 76 (2017), 149-163.  doi: 10.1016/j.insmatheco.2017.07.005. [8] T. Chan, Pricing contingent claims on stocks driven by L$\acute{e}$vy processes, The Annals of Applied Probability, 9 (1999), 504-528.  doi: 10.1214/aoap/1029962753. [9] T. Choulli, L. Krawczyk and C. Stricker, $\mathscr{E}$-martingales and their applications in mathematical finance, The Annals of Applied Probability, 26 (1998), 853-876.  doi: 10.1214/aop/1022855653. [10] T. Choulli, N. Vandaele and M. Vanmaele, The F$\ddot{o}$llmer-Schweizer decomposition: comparison and description, Stochastic Processes and their Applications, 120 (2010), 853-872.  doi: 10.1016/j.spa.2010.02.004. [11] S. N. Cohen and R. J. Elliott, Stochastic Calculus and Applications, Probability and its Applications. Springer, Cham, 2015. doi: 10.1007/978-1-4939-2867-5. [12] N. Dacev, The necessity of legal arrangement of unit-linked life insurance products, UTMS Journal of Economics, 8 (2017), 259-269. [13] A. Dassios and H. Zhao, Exact simulation of Hawkes process with exponentially decaying intensity, Electronic Communications in Probability, 18 (2013), 1-13.  doi: 10.1214/ECP.v18-2717. [14] E. Errais, K. Giesecke and L. R. Goldberg, Affine point processes and portfolio credit risk, SIAM Journal on Financial Mathematics, 1 (2010), 642-665.  doi: 10.1137/090771272. [15] H. F$\ddot{o}$llmer and D. Sondermann, Hedging of non-redundant contingent claims, Contributions to Mathematical Economics. In honor of G. Debreu (Eds. W. Hildenbrand and A. Mas-Colell), Elsevier Science Publ., North-Holland, (1986), 205–223. [16] D. Hainaut, A bivariate Hawkes process for interest rate modeling, Economic Modelling, 57 (2016), 180-196.  doi: 10.1016/j.econmod.2016.04.016. [17] D. Hainaut and F. Moraux, Hedging of options in presence of jump clustering, Journal of Computational Finance, 28 (2018), 1-35.  doi: 10.21314/JCF.2018.354. [18] A. G. Hawkes, Point spectra of some mutually exciting point processes, Journal of the Royal Statistical Society. Series B, 33 (1971), 438-443.  doi: 10.1111/j.2517-6161.1971.tb01530.x. [19] A. G. Hawkes, Spectra of some self exciting and mutually exciting point processes, Biometrika, 58 (1971), 83-90.  doi: 10.1093/biomet/58.1.83. [20] A. G. Hawkes, Hawkes processes and their applications to finance: A review, Quantitative Finance, 17 (2018), 193-198.  doi: 10.1080/14697688.2017.1403131. [21] L. F. B. Henriksen and T. Møller, Local risk-minimization with longevity bonds, Applied Stochastic Models in Business and Industry, 31 (2015), 241-263.  doi: 10.1002/asmb.2028. [22] T. Kokholm, Pricing and hedging of derivatives in contagious markets, Journal of Banking and Finance, 66 (2016), 19-34.  doi: 10.1016/j.jbankfin.2016.01.012. [23] K. Lee and S. Song, Insiders' hedging in a jump diffusion model, Quantitative Finance, 7 (2007), 537-545.  doi: 10.1080/14697680601043191. [24] K. Lee and P. Protter, Hedging claims with feedback jumps in the price process, Communications on Stochastic Analysis, 2 (2008), 125-143.  doi: 10.31390/cosa.2.1.09. [25] J. Li and A. Szimayer, The uncertain mortality intensity framework: Pricing and hedging unit-linked life insurance contracts, Insurance: Mathematics and Economics, 49 (2011), 471-486.  doi: 10.1016/j.insmatheco.2011.08.001. [26] Y. Ma, K. Shrestha and W. Xu, Pricing vulnerable options with jump clustering, The Journal of Futures Markets, 37 (2017), 1155-1178.  doi: 10.1002/fut.21843. [27] T. Møller, Risk minimizing hedging strategies for unit-linked life insurance contracts, Astin Bulletin, 28 (1998), 17-47.  doi: 10.2143/AST.28.1.519077. [28] O. Nteukam T., F. Planchet and P.-E. Thérond, Optimal strategies for hedging portfolios of unit-linked life insurance contracts with minimum death guarantee, Insurance: Mathematics and Economics, 48 (2011), 161-175.  doi: 10.1016/j.insmatheco.2010.10.011. [29] J. Pansera, Discrete-time local risk-minimization of payment processes and applications to equity-linked life-insurance contracts, Insurance: Mathematics and Economics, 50 (2012), 1-11.  doi: 10.1016/j.insmatheco.2011.10.001. [30] S.-H. Park and K. Lee, Insiders' hedging in a stochastic volatility model, IMA Journal of Management and Mathematics, 27 (2016), 281-2951.  doi: 10.1093/imaman/dpu023. [31] E. Platen and N. Bruti-Liberati, Numerical solution of stochastic differential equations with jumps in finance, Springer, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-13694-8. [32] M. Riesner, Hedging life insurance contracts in a L$\acute{e}$vy process financial market, Insurance: Mathematics and Economics, 38 (2006), 599-608.  doi: 10.1016/j.insmatheco.2005.12.004. [33] M. Schweizer, Hedging of Options in a General Semimartingale Model, Ph.D thesis, Zurich University, Switzerland, 1988. [34] M. Schweizer, A guided tour through quadratic hedging approaches, in Handbooks in Mathematical Finance: Option Pricing, Interest Rates and Risk Management, Cambridge University Press, Cambridge, (2001), 538–574. doi: 10.1017/CBO9780511569708.016. [35] Y. Shen and B. Zou, Mean-variance portfolio selection in contagious markets, preprint. doi: 10.13140/RG.2.2.36243.02088. [36] G. Stabile and G. L. Torrisi, Risk processes with non-stationary Hawkes claims arrivals, Methodology and Computing in Applied Probability, 12 (2010), 415-429.  doi: 10.1007/s11009-008-9110-6. [37] N. Vandaele and M. Vanmaele, A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a L$\acute{e}$vy process financial market, Insurance: Mathematics and Economics, 42 (2008), 1128-1137.  doi: 10.1016/j.insmatheco.2008.03.001. [38] X. Zhang, J. Xiong and Y. Shen, Bond and option pricing for interest rate model with clustering effects, Quantitative Finance, 18 (2018), 969-981.  doi: 10.1080/14697688.2017.1388534. [39] L. Zhu, Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps, Journal of Applied Probability, 51 (2014), 699-712.  doi: 10.1239/jap/1409932668.

show all references

##### References:
 [1] Y. Aït-Sahalia, J. Cacho-Diaz and R. J. A. Laeven, Modeling financial contagion using mutually exciting jump processes, Journal of Financial Economics, 117 (2015), 585-606.  doi: 10.1016/j.jfineco.2015.03.002. [2] Y. Aït-Sahalia and T. R. Hurd, Portfolio choice in markets with contagion, Journal of Financial Economics, 14 (2016), 1-28.  doi: 10.1093/jjfinec/nbv024. [3] T. Arai, Y. Imai and R. Suzuki, Numerical analysis on local risk-minimization for exponential L$\acute{e}$vy models, International Journal of Theoretical and Applied Finance, 19 (2016), 1650008, 27 pp. doi: 10.1142/S0219024916500084. [4] T. Arai, Y. Imai and R. Suzuki, Local risk-minimization for Barndorff-Nielsen and Shephard models, Finance and Stochastic, 21 (2017), 551-592.  doi: 10.1007/s00780-017-0324-8. [5] C. G. Bowsher, Modelling security market events in continuous time: Intensity based, multivariate point process models, Journal of Econometrics, 141 (2007), 876-912.  doi: 10.1016/j.jeconom.2006.11.007. [6] C. Ceci, K. Colaneri and A. Cretarola, Hedging of unit-linked life insurance contracts with unobservable mortality hazard rate via local risk-minimization, Insurance: Mathematics and Economics, 60 (2015), 47-60.  doi: 10.1016/j.insmatheco.2014.10.013. [7] C. Ceci, K. Colaneri and A. Cretarola, Unit-linked life insurance policies: Optimal hedging in partially observable market models, Insurance: Mathematics and Economics, 76 (2017), 149-163.  doi: 10.1016/j.insmatheco.2017.07.005. [8] T. Chan, Pricing contingent claims on stocks driven by L$\acute{e}$vy processes, The Annals of Applied Probability, 9 (1999), 504-528.  doi: 10.1214/aoap/1029962753. [9] T. Choulli, L. Krawczyk and C. Stricker, $\mathscr{E}$-martingales and their applications in mathematical finance, The Annals of Applied Probability, 26 (1998), 853-876.  doi: 10.1214/aop/1022855653. [10] T. Choulli, N. Vandaele and M. Vanmaele, The F$\ddot{o}$llmer-Schweizer decomposition: comparison and description, Stochastic Processes and their Applications, 120 (2010), 853-872.  doi: 10.1016/j.spa.2010.02.004. [11] S. N. Cohen and R. J. Elliott, Stochastic Calculus and Applications, Probability and its Applications. Springer, Cham, 2015. doi: 10.1007/978-1-4939-2867-5. [12] N. Dacev, The necessity of legal arrangement of unit-linked life insurance products, UTMS Journal of Economics, 8 (2017), 259-269. [13] A. Dassios and H. Zhao, Exact simulation of Hawkes process with exponentially decaying intensity, Electronic Communications in Probability, 18 (2013), 1-13.  doi: 10.1214/ECP.v18-2717. [14] E. Errais, K. Giesecke and L. R. Goldberg, Affine point processes and portfolio credit risk, SIAM Journal on Financial Mathematics, 1 (2010), 642-665.  doi: 10.1137/090771272. [15] H. F$\ddot{o}$llmer and D. Sondermann, Hedging of non-redundant contingent claims, Contributions to Mathematical Economics. In honor of G. Debreu (Eds. W. Hildenbrand and A. Mas-Colell), Elsevier Science Publ., North-Holland, (1986), 205–223. [16] D. Hainaut, A bivariate Hawkes process for interest rate modeling, Economic Modelling, 57 (2016), 180-196.  doi: 10.1016/j.econmod.2016.04.016. [17] D. Hainaut and F. Moraux, Hedging of options in presence of jump clustering, Journal of Computational Finance, 28 (2018), 1-35.  doi: 10.21314/JCF.2018.354. [18] A. G. Hawkes, Point spectra of some mutually exciting point processes, Journal of the Royal Statistical Society. Series B, 33 (1971), 438-443.  doi: 10.1111/j.2517-6161.1971.tb01530.x. [19] A. G. Hawkes, Spectra of some self exciting and mutually exciting point processes, Biometrika, 58 (1971), 83-90.  doi: 10.1093/biomet/58.1.83. [20] A. G. Hawkes, Hawkes processes and their applications to finance: A review, Quantitative Finance, 17 (2018), 193-198.  doi: 10.1080/14697688.2017.1403131. [21] L. F. B. Henriksen and T. Møller, Local risk-minimization with longevity bonds, Applied Stochastic Models in Business and Industry, 31 (2015), 241-263.  doi: 10.1002/asmb.2028. [22] T. Kokholm, Pricing and hedging of derivatives in contagious markets, Journal of Banking and Finance, 66 (2016), 19-34.  doi: 10.1016/j.jbankfin.2016.01.012. [23] K. Lee and S. Song, Insiders' hedging in a jump diffusion model, Quantitative Finance, 7 (2007), 537-545.  doi: 10.1080/14697680601043191. [24] K. Lee and P. Protter, Hedging claims with feedback jumps in the price process, Communications on Stochastic Analysis, 2 (2008), 125-143.  doi: 10.31390/cosa.2.1.09. [25] J. Li and A. Szimayer, The uncertain mortality intensity framework: Pricing and hedging unit-linked life insurance contracts, Insurance: Mathematics and Economics, 49 (2011), 471-486.  doi: 10.1016/j.insmatheco.2011.08.001. [26] Y. Ma, K. Shrestha and W. Xu, Pricing vulnerable options with jump clustering, The Journal of Futures Markets, 37 (2017), 1155-1178.  doi: 10.1002/fut.21843. [27] T. Møller, Risk minimizing hedging strategies for unit-linked life insurance contracts, Astin Bulletin, 28 (1998), 17-47.  doi: 10.2143/AST.28.1.519077. [28] O. Nteukam T., F. Planchet and P.-E. Thérond, Optimal strategies for hedging portfolios of unit-linked life insurance contracts with minimum death guarantee, Insurance: Mathematics and Economics, 48 (2011), 161-175.  doi: 10.1016/j.insmatheco.2010.10.011. [29] J. Pansera, Discrete-time local risk-minimization of payment processes and applications to equity-linked life-insurance contracts, Insurance: Mathematics and Economics, 50 (2012), 1-11.  doi: 10.1016/j.insmatheco.2011.10.001. [30] S.-H. Park and K. Lee, Insiders' hedging in a stochastic volatility model, IMA Journal of Management and Mathematics, 27 (2016), 281-2951.  doi: 10.1093/imaman/dpu023. [31] E. Platen and N. Bruti-Liberati, Numerical solution of stochastic differential equations with jumps in finance, Springer, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-13694-8. [32] M. Riesner, Hedging life insurance contracts in a L$\acute{e}$vy process financial market, Insurance: Mathematics and Economics, 38 (2006), 599-608.  doi: 10.1016/j.insmatheco.2005.12.004. [33] M. Schweizer, Hedging of Options in a General Semimartingale Model, Ph.D thesis, Zurich University, Switzerland, 1988. [34] M. Schweizer, A guided tour through quadratic hedging approaches, in Handbooks in Mathematical Finance: Option Pricing, Interest Rates and Risk Management, Cambridge University Press, Cambridge, (2001), 538–574. doi: 10.1017/CBO9780511569708.016. [35] Y. Shen and B. Zou, Mean-variance portfolio selection in contagious markets, preprint. doi: 10.13140/RG.2.2.36243.02088. [36] G. Stabile and G. L. Torrisi, Risk processes with non-stationary Hawkes claims arrivals, Methodology and Computing in Applied Probability, 12 (2010), 415-429.  doi: 10.1007/s11009-008-9110-6. [37] N. Vandaele and M. Vanmaele, A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a L$\acute{e}$vy process financial market, Insurance: Mathematics and Economics, 42 (2008), 1128-1137.  doi: 10.1016/j.insmatheco.2008.03.001. [38] X. Zhang, J. Xiong and Y. Shen, Bond and option pricing for interest rate model with clustering effects, Quantitative Finance, 18 (2018), 969-981.  doi: 10.1080/14697688.2017.1388534. [39] L. Zhu, Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps, Journal of Applied Probability, 51 (2014), 699-712.  doi: 10.1239/jap/1409932668.
Paths of intensity process $\lambda_t$ and stock price process $S_t$
Number of stock $\xi_{t}^{PE\ast}$ with different strike prices $K$
Effects of the jump size $Z$ on $\xi_{0}^{PE\ast}$ and $\Delta_{0}$. We assume the jump size $Z_j\in U(-0.1, 0.1)$ and $Z_j\in U(-0.5, 0.5)$ in the left panel and the right panel of Figure 3, respectively
Effects of the intensity $\lambda$ on $\xi_{0}^{PE\ast}$
 [1] 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 [2] 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 [3] Xingchun Wang. Pricing path-dependent options under the Hawkes jump diffusion process. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022024 [4] Jie Sun, Honglei Xu, Min Zhang. A new interpretation of the progressive hedging algorithm for multistage stochastic minimization problems. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1655-1662. doi: 10.3934/jimo.2019022 [5] Jingzhen Liu, Shiqi Yan, Shan Jiang, Jiaqin Wei. Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022040 [6] Ailing Shi, Xingyi Li, Zhongfei Li. Optimal portfolio selection with life insurance under subjective survival belief and habit formation. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022051 [7] Xiaoyu Xing, Caixia Geng. Optimal investment-reinsurance strategy in the correlated insurance and financial markets. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3445-3459. doi: 10.3934/jimo.2021120 [8] 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 [9] Xi Chen, Zongrun Wang, Songhai Deng, Yong Fang. Risk measure optimization: Perceived risk and overconfidence of structured product investors. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1473-1492. doi: 10.3934/jimo.2018105 [10] Yufei Sun, Grace Aw, Kok Lay Teo, Guanglu Zhou. Portfolio optimization using a new probabilistic risk measure. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1275-1283. doi: 10.3934/jimo.2015.11.1275 [11] Burak Ordin. The modified cutting angle method for global minimization of increasing positively homogeneous functions over the unit simplex. Journal of Industrial and Management Optimization, 2009, 5 (4) : 825-834. doi: 10.3934/jimo.2009.5.825 [12] 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 [13] Steve Drekic, Jae-Kyung Woo, Ran Xu. A threshold-based risk process with a waiting period to pay dividends. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1179-1201. doi: 10.3934/jimo.2018005 [14] Qiuli Liu, Xiaolong Zou. A risk minimization problem for finite horizon semi-Markov decision processes with loss rates. Journal of Dynamics and Games, 2018, 5 (2) : 143-163. doi: 10.3934/jdg.2018009 [15] Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial and Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191 [16] Nana Wan, Li Li, Xiaozhi Wu, Jianchang Fan. Risk minimization inventory model with a profit target and option contracts under spot price uncertainty. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2827-2845. doi: 10.3934/jimo.2021093 [17] 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 and Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143 [18] Bernard Host, Alejandro Maass, Servet Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1423-1446. doi: 10.3934/dcds.2003.9.1423 [19] Xuanhua Peng, Wen Su, Zhimin Zhang. On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1967-1986. doi: 10.3934/jimo.2019038 [20] Kai Kang, Taotao Lu, Jing Zhang. Financing strategy selection and coordination considering risk aversion in a capital-constrained supply chain. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1737-1768. doi: 10.3934/jimo.2021042

2021 Impact Factor: 1.411

## Tools

Article outline

Figures and Tables