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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  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 & Management Optimization, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] N. Dacev, The necessity of legal arrangement of unit-linked life insurance products, UTMS Journal of Economics, 8 (2017), 259-269.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] K. Lee and S. Song, Insiders' hedging in a jump diffusion model, Quantitative Finance, 7 (2007), 537-545.  doi: 10.1080/14697680601043191.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] M. Schweizer, Hedging of Options in a General Semimartingale Model, Ph.D thesis, Zurich University, Switzerland, 1988. Google Scholar [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.  Google Scholar [35] Y. Shen and B. Zou, Mean-variance portfolio selection in contagious markets, preprint. doi: 10.13140/RG.2.2.36243.02088.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] N. Dacev, The necessity of legal arrangement of unit-linked life insurance products, UTMS Journal of Economics, 8 (2017), 259-269.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] K. Lee and S. Song, Insiders' hedging in a jump diffusion model, Quantitative Finance, 7 (2007), 537-545.  doi: 10.1080/14697680601043191.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] M. Schweizer, Hedging of Options in a General Semimartingale Model, Ph.D thesis, Zurich University, Switzerland, 1988. Google Scholar [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.  Google Scholar [35] Y. Shen and B. Zou, Mean-variance portfolio selection in contagious markets, preprint. doi: 10.13140/RG.2.2.36243.02088.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Paths of intensity process $\lambda_t$ and stock price process $S_t$
Number of stock $\xi_{t}^{PE\ast}$ with different strike prices $K$
, respectively">Figure 3.  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 $\xi_{0}^{PE\ast}$
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