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First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits
1. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
2. | School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China |
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.
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. |
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 |
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 |
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 |
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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