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May  2022, 18(3): 1689-1707. doi: 10.3934/jimo.2021039

First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits

Meiqiao Ai 1, , Zhimin Zhang 1,, and  Wenguang Yu 2,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2. 

School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China

* Corresponding author: Zhimin Zhang

Received  August 2020 Revised  November 2020 Published  May 2022 Early access  March 2021

Fund Project: Zhimin Zhang is supported by the National Natural Science Foundation of China [grant number 11871121], Natural Science Foundation Project of CQ CSTC [grant number cstc2019jcyj-msxmX0004] and the Fundamental Research Funds for the Central Universities (project number 2020CDJSK02ZH03). Wenguang Yu is supported by the National Social Science Foundation of China (No. 15BJY007), the Taishan Scholars Program of Shandong Province (No. tsqn20161041), the Humanities and Social Sciences Project of the Ministry Education of China (No. 19YJA910002), the Natural Science Foundation of Shandong Province (No. ZR2018MG002), the Fostering Project of Dominant Discipline and Talent Team of Shandong Province Higher Education Institutions (No. 1716009), Shandong Provincial Social Science Project Planning Research Project (No. 19CQXJ08), the Risk Management and Insurance Research Team of Shandong University of Finance and Economics, Excellent Talents Project of Shandong University of Finance and Economics, the Collaborative Innovation Center Project of the Transformation of New and old Kinetic Energy and Government Financial Allocation

This paper studies some first passage time problems in a refracted jump diffusion process with hyper-exponential jumps. Closed-form expressions for four functions associated with the first passage time are obtained by solving some ordinary integro-differential equations. In addition, the obtained results are used to value equity-linked death benefit products with state-dependent fees. Specifically, we obtain the closed-form Laplace transform of the fair value of barrier option, which is further recovered by the bilateral Abate-Whitt algorithm. Numerical results confirm that the proposed approach is efficient.

Keywords: Refracted jump diffusion processes, First passage time, Laplace transform, Equity-linked death benefits.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1689-1707. doi: 10.3934/jimo.2021039
References:
[1]

J. Abate and W. Whitt, The Fourier-series method for inverting transforms of probability distributions, Queueing Systems, 10 (1992), 5-87.  doi: 10.1007/BF01158520.

[2]

S. Asmussen, F. Avram and M. R. Pistorius, Russian and American put options under exponential phase-type Lévy models, Stochastic Processes and their Applications, 109 (2004), 79-111.  doi: 10.1016/j.spa.2003.07.005.

[3]

D. Bauer, A. Kling and J. Russ, A universal pricing framework for guaranteed minimum benefits in variable annuities, Astin Bulletin, 38 (2008), 621-651.  doi: 10.1017/S0515036100015312.

[4]

C. Bernard, M. Hardy and A. Mackay, State-dependent fees for variable annuity guarantees, Astin Bulletin, 44 (2014), 559-585.  doi: 10.1017/asb.2014.13.

[5]

R. F. Botta and C. M. Harris, Approximation with generalized hyperexponential distributions: Weak convergence results, Queueing Systems, 1 (1986), 169-190.  doi: 10.1007/BF01536187.

[6]

N. Cai, On first passage times of a hyper-exponential jump diffusion process, Operations Research Letters, 37 (2009), 127-134.  doi: 10.1016/j.orl.2009.01.002.

[7]

N. Cai, N. Chen and X. Wan, Pricing double-barrier options under a flexible jump diffusion model, Operations Research Letters, 37 (2009), 163-167.  doi: 10.1016/j.orl.2009.02.006.

[8]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081. 

[9]

D. Dufresne, Fitting combinations of exponentials to probability distributions, Applied Stochastic Models in Business and Industry, 23 (2007), 23-48.  doi: 10.1002/asmb.635.

[10]

H. U. Gerber, E. S. W. Shiu and N. Smith, Maximizing Dividends without Bankruptcy, Astin Bulletin, 36 (2006), 5-23.  doi: 10.1017/S0515036100014392.

[11]

H. U. Gerber, E. S. W. Shiu and H. Yang, Valuing equity-linked death benefits and other contingent options: A discounted density approach, Insurance: Mathematics and Economics, 51 (2012), 73-92.  doi: 10.1016/j.insmatheco.2012.03.001.

[12]

H. U. Gerber, E. S. W. Shiu and H. Yang, Valuing equity-linked death benefits in jump diffusion models, Insurance: Mathematics and Economics, 53 (2013), 615-623.  doi: 10.1016/j.insmatheco.2013.08.010.

[13]

O. Kella and W. Whitt, Useful martingales for stochastic storage processes with Lévy-type input, Journal of Applied Probability, 29 (1992), 396-403.  doi: 10.2307/3214576.

[14]

S. G. Kou and H. Wang, First passage times of a jump diffusion process, Advances in Applied Probability, 35 (2003), 504-531.  doi: 10.1239/aap/1051201658.

[15]

A. Kyprianou and R. Loeffeng, Refracted Lévy processes, Annales de l'Institut Henri Poincaré Probabilités et Statistiques, 46 (2010), 24-44.  doi: 10.1214/08-AIHP307.

[16]

A. Kyprianou, J. C. Pardo and J. L. Pérez, Occupation times of refracted Lévy processes, Journal of Theoretical Probability, 27 (2014), 1292-1315.  doi: 10.1007/s10959-013-0501-4.

[17]

B. Li and X. Zhou, On weighted occupation times for refracted spectrally negative Lévy processes, Journal of Mathematical Analysis and Applications, 466 (2018), 215-237.  doi: 10.1016/j.jmaa.2018.05.077.

[18]

C. C. Siu, S. C. P. Yam and H. Yang, Valuing equity-linked death benefits in a regime-switching framework, Astin Bulletin, 45 (2015), 355-395.  doi: 10.1017/asb.2014.32.

[19]

E. R. Ulm, The effect of the real option to transfer on the value of guaranteed minimum death benefits, The Journal of Risk and Insurance, 73 (2006), 43-69.  doi: 10.1111/j.1539-6975.2006.00165.x.

[20]

E. R. Ulm, Analytic solution for return of premium and rollup guaranteed minimum death benefit options under some simple mortality laws, Astin Bulletin, 38 (2008), 543-563.  doi: 10.1017/S0515036100015282.

[21]

C. Yin, Y. Shen and Y. Wen, Exit problems for jump processes with applications to dividend problems, Journal of Computational and Applied Mathematics, 245 (2013), 30-52.  doi: 10.1016/j.cam.2012.12.004.

[22]

Z. Zhang and Y. Yong, Valuing guaranteed equity-linked contracts by Laguerre series expansion, Journal of Computational and Applied Mathematics, 357 (2019), 329-348.  doi: 10.1016/j.cam.2019.02.032.

[23]

Z. Zhang, Y. Yong and W. Yu, Valuing equity-linked death benefits in general exponential Lévy models, Journal of Computational and Applied Mathematics, 365 (2020), 112377, 18pp. doi: 10.1016/j.cam.2019.112377.

[24]

J. Zhou and L. Wu, Occupation times of refracted double exponential jump diffusion processes, Statistics and Probability Letters, 106 (2015), 218-227.  doi: 10.1016/j.spl.2015.07.023.

[25]

J. Zhou and L. Wu, The time of deducting fees for variable annuities under the state-dependent fee structure, Insurance: Mathematics and Economics, 61 (2015), 125-134.  doi: 10.1016/j.insmatheco.2014.12.008.

[26]

J. Zhou and L. Wu, Valuing equity-linked death benefits with a threshold expense strategy, Insurance: Mathematics and Economics, 62 (2015), 79-90.  doi: 10.1016/j.insmatheco.2015.03.002.

[27]

J. Zhou and L. Wu, The distribution of refracted Lévy processes with jumps having rational Laplace transforms, Journal of Applied Probability, 54 (2017), 1167-1192.  doi: 10.1017/jpr.2017.58.

show all references

References:
[1]

J. Abate and W. Whitt, The Fourier-series method for inverting transforms of probability distributions, Queueing Systems, 10 (1992), 5-87.  doi: 10.1007/BF01158520.

[2]

S. Asmussen, F. Avram and M. R. Pistorius, Russian and American put options under exponential phase-type Lévy models, Stochastic Processes and their Applications, 109 (2004), 79-111.  doi: 10.1016/j.spa.2003.07.005.

[3]

D. Bauer, A. Kling and J. Russ, A universal pricing framework for guaranteed minimum benefits in variable annuities, Astin Bulletin, 38 (2008), 621-651.  doi: 10.1017/S0515036100015312.

[4]

C. Bernard, M. Hardy and A. Mackay, State-dependent fees for variable annuity guarantees, Astin Bulletin, 44 (2014), 559-585.  doi: 10.1017/asb.2014.13.

[5]

R. F. Botta and C. M. Harris, Approximation with generalized hyperexponential distributions: Weak convergence results, Queueing Systems, 1 (1986), 169-190.  doi: 10.1007/BF01536187.

[6]

N. Cai, On first passage times of a hyper-exponential jump diffusion process, Operations Research Letters, 37 (2009), 127-134.  doi: 10.1016/j.orl.2009.01.002.

[7]

N. Cai, N. Chen and X. Wan, Pricing double-barrier options under a flexible jump diffusion model, Operations Research Letters, 37 (2009), 163-167.  doi: 10.1016/j.orl.2009.02.006.

[8]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081. 

[9]

D. Dufresne, Fitting combinations of exponentials to probability distributions, Applied Stochastic Models in Business and Industry, 23 (2007), 23-48.  doi: 10.1002/asmb.635.

[10]

H. U. Gerber, E. S. W. Shiu and N. Smith, Maximizing Dividends without Bankruptcy, Astin Bulletin, 36 (2006), 5-23.  doi: 10.1017/S0515036100014392.

[11]

H. U. Gerber, E. S. W. Shiu and H. Yang, Valuing equity-linked death benefits and other contingent options: A discounted density approach, Insurance: Mathematics and Economics, 51 (2012), 73-92.  doi: 10.1016/j.insmatheco.2012.03.001.

[12]

H. U. Gerber, E. S. W. Shiu and H. Yang, Valuing equity-linked death benefits in jump diffusion models, Insurance: Mathematics and Economics, 53 (2013), 615-623.  doi: 10.1016/j.insmatheco.2013.08.010.

[13]

O. Kella and W. Whitt, Useful martingales for stochastic storage processes with Lévy-type input, Journal of Applied Probability, 29 (1992), 396-403.  doi: 10.2307/3214576.

[14]

S. G. Kou and H. Wang, First passage times of a jump diffusion process, Advances in Applied Probability, 35 (2003), 504-531.  doi: 10.1239/aap/1051201658.

[15]

A. Kyprianou and R. Loeffeng, Refracted Lévy processes, Annales de l'Institut Henri Poincaré Probabilités et Statistiques, 46 (2010), 24-44.  doi: 10.1214/08-AIHP307.

[16]

A. Kyprianou, J. C. Pardo and J. L. Pérez, Occupation times of refracted Lévy processes, Journal of Theoretical Probability, 27 (2014), 1292-1315.  doi: 10.1007/s10959-013-0501-4.

[17]

B. Li and X. Zhou, On weighted occupation times for refracted spectrally negative Lévy processes, Journal of Mathematical Analysis and Applications, 466 (2018), 215-237.  doi: 10.1016/j.jmaa.2018.05.077.

[18]

C. C. Siu, S. C. P. Yam and H. Yang, Valuing equity-linked death benefits in a regime-switching framework, Astin Bulletin, 45 (2015), 355-395.  doi: 10.1017/asb.2014.32.

[19]

E. R. Ulm, The effect of the real option to transfer on the value of guaranteed minimum death benefits, The Journal of Risk and Insurance, 73 (2006), 43-69.  doi: 10.1111/j.1539-6975.2006.00165.x.

[20]

E. R. Ulm, Analytic solution for return of premium and rollup guaranteed minimum death benefit options under some simple mortality laws, Astin Bulletin, 38 (2008), 543-563.  doi: 10.1017/S0515036100015282.

[21]

C. Yin, Y. Shen and Y. Wen, Exit problems for jump processes with applications to dividend problems, Journal of Computational and Applied Mathematics, 245 (2013), 30-52.  doi: 10.1016/j.cam.2012.12.004.

[22]

Z. Zhang and Y. Yong, Valuing guaranteed equity-linked contracts by Laguerre series expansion, Journal of Computational and Applied Mathematics, 357 (2019), 329-348.  doi: 10.1016/j.cam.2019.02.032.

[23]

Z. Zhang, Y. Yong and W. Yu, Valuing equity-linked death benefits in general exponential Lévy models, Journal of Computational and Applied Mathematics, 365 (2020), 112377, 18pp. doi: 10.1016/j.cam.2019.112377.

[24]

J. Zhou and L. Wu, Occupation times of refracted double exponential jump diffusion processes, Statistics and Probability Letters, 106 (2015), 218-227.  doi: 10.1016/j.spl.2015.07.023.

[25]

J. Zhou and L. Wu, The time of deducting fees for variable annuities under the state-dependent fee structure, Insurance: Mathematics and Economics, 61 (2015), 125-134.  doi: 10.1016/j.insmatheco.2014.12.008.

[26]

J. Zhou and L. Wu, Valuing equity-linked death benefits with a threshold expense strategy, Insurance: Mathematics and Economics, 62 (2015), 79-90.  doi: 10.1016/j.insmatheco.2015.03.002.

[27]

J. Zhou and L. Wu, The distribution of refracted Lévy processes with jumps having rational Laplace transforms, Journal of Applied Probability, 54 (2017), 1167-1192.  doi: 10.1017/jpr.2017.58.

Table 1.  Pricing up-and-in call options when $ K $ varies, $ B = 120 $ and $ L = 115 $.
$ \sigma = 0.1 $ $ \sigma = 0.2 $
$ \lambda $ $ K $ $ \mathcal{M}_{3} $ $ \mathcal{M}_{5} $ $ \mathcal{M}_{10} $ $ \mathcal{M}_{3} $ $ \mathcal{M}_{5} $ $ \mathcal{M}_{10} $
1 100 64.56585 64.55930 64.54628 68.75548 68.75253 68.74360
105 64.36554 64.35829 64.34406 68.60582 68.60241 68.59308
110 64.16854 64.16063 64.14525 68.45948 68.45562 68.44584
3 100 65.08980 65.08372 65.07131 69.01900 69.01623 69.00753
105 64.89483 64.88807 64.87452 68.87247 68.86926 68.86019
110 64.70320 64.69579 64.68116 68.72922 68.72559 68.71609
Table Options

Download as excel

$ \sigma = 0.1 $ $ \sigma = 0.2 $
$ \lambda $ $ K $ $ \mathcal{M}_{3} $ $ \mathcal{M}_{5} $ $ \mathcal{M}_{10} $ $ \mathcal{M}_{3} $ $ \mathcal{M}_{5} $ $ \mathcal{M}_{10} $
1 100 64.56585 64.55930 64.54628 68.75548 68.75253 68.74360
105 64.36554 64.35829 64.34406 68.60582 68.60241 68.59308
110 64.16854 64.16063 64.14525 68.45948 68.45562 68.44584
3 100 65.08980 65.08372 65.07131 69.01900 69.01623 69.00753
105 64.89483 64.88807 64.87452 68.87247 68.86926 68.86019
110 64.70320 64.69579 64.68116 68.72922 68.72559 68.71609
Table 2.  Pricing up-and-in call options when $ L $ varies, $ B = 120 $ and $ K = 100 $.
$ \sigma=0.1 $ $ \sigma=0.2 $
$ \lambda $ $ L $ $ \mathcal{M}_{3} $ $ \mathcal{M}_{5} $ $ \mathcal{M}_{10} $ $ \mathcal{M}_{3} $ $ \mathcal{M}_{5} $ $ \mathcal{M}_{10} $
1 105 64.58681 64.57969 64.56342 68.76237 68.75897 68.74997
125 64.53840 64.53193 64.52085 68.74569 68.74308 68.73608
145 64.48201 64.47513 64.46268 68.71998 68.71769 68.71446
3 105 65.10796 65.10130 65.08595 69.02559 69.02240 69.01370
125 65.06588 65.05994 65.04979 69.00962 69.00720 69.00034
145 65.01405 65.00780 64.99722 68.98490 68.98282 68.97977
Table Options

Download as excel

$ \sigma=0.1 $ $ \sigma=0.2 $
$ \lambda $ $ L $ $ \mathcal{M}_{3} $ $ \mathcal{M}_{5} $ $ \mathcal{M}_{10} $ $ \mathcal{M}_{3} $ $ \mathcal{M}_{5} $ $ \mathcal{M}_{10} $
1 105 64.58681 64.57969 64.56342 68.76237 68.75897 68.74997
125 64.53840 64.53193 64.52085 68.74569 68.74308 68.73608
145 64.48201 64.47513 64.46268 68.71998 68.71769 68.71446
3 105 65.10796 65.10130 65.08595 69.02559 69.02240 69.01370
125 65.06588 65.05994 65.04979 69.00962 69.00720 69.00034
145 65.01405 65.00780 64.99722 68.98490 68.98282 68.97977
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