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doi: 10.3934/jimo.2022007
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Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model

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  July 2021 Revised  December 2021 Early access February 2022

Fund Project: The research of Zhimin Zhang was supported by the National Natural Science Foundation of China [grant numbers 11871121, 12171405], Natural Science Foundation Project of CQ CSTC [grant numbercstc2019jcyjmsxmX0004] and the Fundamental Research Funds for the Central Universities (project no.2020CDJSK02ZH03). The research of Wenguang Yu was supported by the Taishan Scholars Program of Shandong Province [project no. tsqn20161041]

In this paper, we investigate the valuation problem of equity-linked death benefits with a threshold expense structure. Specifically, a regime-switching Lévy process is used to describe the underlying asset price process, which is monitored periodically. The fees are assumed to be continuously deducted at some constant rate from the policyholder's account between the current and next monitoring times, if the account value is smaller than a pre-specified level at the current observation time point. Under the modified threshold expense structure, some explicit valuation expressions for life-contingent call options are derived by the Fourier cosine series expansion method. Numerical results demonstrate the accuracy and efficiency of our method.

Citation: Meiqiao Ai, Zhimin Zhang, Wenguang Yu. Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022007
References:
[1]

S. Asmussen, Applied Probability and Queues, $2^{nd}$ edition, Springer, New York, 2003.

[2]

T.-H. Bae and B.-W. Ko, On pricing equity-linked investment products with a threshold expense structure, The Korean Journal of Applied Statistics, 23 (2010), 621-633.  doi: 10.5351/KJAS.2010.23.4.621.

[3]

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

[4]

J. Buffington and R. Elliot, American options with regime switching models, J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.

[5]

Z. CuiJ. L. Kirkby and D. Nguyen, Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps, Insurance Math. Econom., 74 (2017), 46-62.  doi: 10.1016/j.insmatheco.2017.02.010.

[6]

Ł. Delong, Pricing and hedging of variable annuities with state-dependent fees, Insurance Math. Econom., 58 (2014), 24-33.  doi: 10.1016/j.insmatheco.2014.06.002.

[7]

F. Fang and C. W. Oosterlee, A novel pricing method for European options based on Fourier-cosine series expansions, SIAM J. Sci. Comput., 31 (2008/09), 826-848.  doi: 10.1137/080718061.

[8]

F. Fang and C. Oosterlee, Pricing early-exercise and discrete barrier options by fourier-cosine series expansions, Numerische Mathematik, 114 (2009), 27-62.  doi: 10.1007/s00211-009-0252-4.

[9]

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

[10]

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

[11]

H. U. GerberE. S. W. Shiu and H. Yang, Geometric stopping of a random walk and its applications to valuing equity-linked death benefits, Insurance Math. Econom., 64 (2015), 313-325.  doi: 10.1016/j.insmatheco.2015.06.006.

[12]

J. L. Kirkby and D. Nguyen, Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models, Ann. Finance, 16 (2020), 307-351.  doi: 10.1007/s10436-020-00366-0.

[13]

J. L. Kirkby and D. Nguyen, Equity-linked guaranteed minimum death benefits with dollar cost averaging, Insurance Math. Econom., 100 (2021), 408-428.  doi: 10.1016/j.insmatheco.2021.04.012.

[14]

B. Ko and T. Bae, Pricing guaranteed minimum death benefit contracts under the phase-type law of mortality, Lobachevskii J. Math., 36 (2015), 198-207.  doi: 10.1134/S1995080215020109.

[15]

A. W. Kolkiewicz and F. S. Lin, Pricing surrender risk in ratchet equity-index annuities under regime-switching Lévy processes, N. Am. Actuar. J., 21 (2017), 433-457.  doi: 10.1080/10920277.2017.1302804.

[16]

X. LiangC. C.-L. Tsai and Y. Lu, Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality, Insurance Math. Econom., 70 (2016), 150-161.  doi: 10.1016/j.insmatheco.2016.06.004.

[17]

X. S. Lin and K. P. Sendova, The compound Poisson risk model with multiple thresholds, Insurance Math. Econom., 42 (2008), 617-627.  doi: 10.1016/j.insmatheco.2007.06.008.

[18]

A. MacKayM. AugustyniakC. Bernard and M. Hardy, Risk management of policyholder behavior in equity-linked life insurance, Journal of Risk and Insurance, 84 (2017), 661-690.  doi: 10.1111/jori.12094.

[19]

A. C. Ng, The compound Poisson risk model with multiple thresholds, Insurance: Mathematics and Economics, 44 (2009), 315-324.

[20]

M. J. Ruijter and C. W. Oosterlee, Two-dimensional Fourier cosine series expansion method for pricing financial options, SIAM J. Sci. Comput., 34 (2012), B642–B671. doi: 10.1137/120862053.

[21]

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

[22]

G. TourN. ThakoorA. Q. M. Khaliq and D. Y. Tangman, COS method for option pricing under a regime-switching model with time-changed Lévy processes, Quant. Finance, 18 (2018), 673-692.  doi: 10.1080/14697688.2017.1412494.

[23]

N. Wan, Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion, Insurance Math. Econom., 40 (2007), 509-523.  doi: 10.1016/j.insmatheco.2006.08.002.

[24]

Y. Wang, Z. Zhang and W. Yu, Pricing equity-linked death benefits by complex Fourier series expansion in a regime-switching jump diffusion model, Appl. Math. Comput., 399 (2021), Paper No. 126031, 15 pp. doi: 10.1016/j.amc.2021.126031.

[25]

J. Xie and Z. Zhang, Finite-time dividend problems in a Lévy risk model under periodic observation, Appl. Math. Comput., 398 (2021), Paper No. 125981, 22 pp. doi: 10.1016/j.amc.2021.125981.

[26]

Z. Zhang, Approximating the density of the time to ruin via Fourier-cosine series expansion, Astin Bull., 47 (2017), 169-198.  doi: 10.1017/asb.2016.27.

[27]

Z. Zhang and E. C. K. Cheung, A note on a Lévy insurance risk model under periodic dividend decisions, J. Ind. Manag. Optim., 14 (2018), 35-63.  doi: 10.3934/jimo.2017036.

[28]

B. Zhang and C. W. Oosterlee, Efficient pricing of european-style Asian options under exponential Lévy processes based on Fourier cosine expansions, SIAM J. Financial Math., 4 (2013), 399-426.  doi: 10.1137/110853339.

[29]

Z. Zhang and Y. Yong, Valuing guaranteed equity-linked contracts by Laguerre series expansion, J. Comput. Appl. Math., 357 (2019), 329-348.  doi: 10.1016/j.cam.2019.02.032.

[30]

Z. Zhang, Y. Yong and W. Yu, Valuing equity-linked death benefits in general exponential Lévy models, J. Comput. Appl. Math., 365 (2020), 112377, 18 pp. doi: 10.1016/j.cam.2019.112377.

[31]

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

[32]

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

show all references

References:
[1]

S. Asmussen, Applied Probability and Queues, $2^{nd}$ edition, Springer, New York, 2003.

[2]

T.-H. Bae and B.-W. Ko, On pricing equity-linked investment products with a threshold expense structure, The Korean Journal of Applied Statistics, 23 (2010), 621-633.  doi: 10.5351/KJAS.2010.23.4.621.

[3]

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

[4]

J. Buffington and R. Elliot, American options with regime switching models, J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.

[5]

Z. CuiJ. L. Kirkby and D. Nguyen, Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps, Insurance Math. Econom., 74 (2017), 46-62.  doi: 10.1016/j.insmatheco.2017.02.010.

[6]

Ł. Delong, Pricing and hedging of variable annuities with state-dependent fees, Insurance Math. Econom., 58 (2014), 24-33.  doi: 10.1016/j.insmatheco.2014.06.002.

[7]

F. Fang and C. W. Oosterlee, A novel pricing method for European options based on Fourier-cosine series expansions, SIAM J. Sci. Comput., 31 (2008/09), 826-848.  doi: 10.1137/080718061.

[8]

F. Fang and C. Oosterlee, Pricing early-exercise and discrete barrier options by fourier-cosine series expansions, Numerische Mathematik, 114 (2009), 27-62.  doi: 10.1007/s00211-009-0252-4.

[9]

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

[10]

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

[11]

H. U. GerberE. S. W. Shiu and H. Yang, Geometric stopping of a random walk and its applications to valuing equity-linked death benefits, Insurance Math. Econom., 64 (2015), 313-325.  doi: 10.1016/j.insmatheco.2015.06.006.

[12]

J. L. Kirkby and D. Nguyen, Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models, Ann. Finance, 16 (2020), 307-351.  doi: 10.1007/s10436-020-00366-0.

[13]

J. L. Kirkby and D. Nguyen, Equity-linked guaranteed minimum death benefits with dollar cost averaging, Insurance Math. Econom., 100 (2021), 408-428.  doi: 10.1016/j.insmatheco.2021.04.012.

[14]

B. Ko and T. Bae, Pricing guaranteed minimum death benefit contracts under the phase-type law of mortality, Lobachevskii J. Math., 36 (2015), 198-207.  doi: 10.1134/S1995080215020109.

[15]

A. W. Kolkiewicz and F. S. Lin, Pricing surrender risk in ratchet equity-index annuities under regime-switching Lévy processes, N. Am. Actuar. J., 21 (2017), 433-457.  doi: 10.1080/10920277.2017.1302804.

[16]

X. LiangC. C.-L. Tsai and Y. Lu, Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality, Insurance Math. Econom., 70 (2016), 150-161.  doi: 10.1016/j.insmatheco.2016.06.004.

[17]

X. S. Lin and K. P. Sendova, The compound Poisson risk model with multiple thresholds, Insurance Math. Econom., 42 (2008), 617-627.  doi: 10.1016/j.insmatheco.2007.06.008.

[18]

A. MacKayM. AugustyniakC. Bernard and M. Hardy, Risk management of policyholder behavior in equity-linked life insurance, Journal of Risk and Insurance, 84 (2017), 661-690.  doi: 10.1111/jori.12094.

[19]

A. C. Ng, The compound Poisson risk model with multiple thresholds, Insurance: Mathematics and Economics, 44 (2009), 315-324.

[20]

M. J. Ruijter and C. W. Oosterlee, Two-dimensional Fourier cosine series expansion method for pricing financial options, SIAM J. Sci. Comput., 34 (2012), B642–B671. doi: 10.1137/120862053.

[21]

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

[22]

G. TourN. ThakoorA. Q. M. Khaliq and D. Y. Tangman, COS method for option pricing under a regime-switching model with time-changed Lévy processes, Quant. Finance, 18 (2018), 673-692.  doi: 10.1080/14697688.2017.1412494.

[23]

N. Wan, Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion, Insurance Math. Econom., 40 (2007), 509-523.  doi: 10.1016/j.insmatheco.2006.08.002.

[24]

Y. Wang, Z. Zhang and W. Yu, Pricing equity-linked death benefits by complex Fourier series expansion in a regime-switching jump diffusion model, Appl. Math. Comput., 399 (2021), Paper No. 126031, 15 pp. doi: 10.1016/j.amc.2021.126031.

[25]

J. Xie and Z. Zhang, Finite-time dividend problems in a Lévy risk model under periodic observation, Appl. Math. Comput., 398 (2021), Paper No. 125981, 22 pp. doi: 10.1016/j.amc.2021.125981.

[26]

Z. Zhang, Approximating the density of the time to ruin via Fourier-cosine series expansion, Astin Bull., 47 (2017), 169-198.  doi: 10.1017/asb.2016.27.

[27]

Z. Zhang and E. C. K. Cheung, A note on a Lévy insurance risk model under periodic dividend decisions, J. Ind. Manag. Optim., 14 (2018), 35-63.  doi: 10.3934/jimo.2017036.

[28]

B. Zhang and C. W. Oosterlee, Efficient pricing of european-style Asian options under exponential Lévy processes based on Fourier cosine expansions, SIAM J. Financial Math., 4 (2013), 399-426.  doi: 10.1137/110853339.

[29]

Z. Zhang and Y. Yong, Valuing guaranteed equity-linked contracts by Laguerre series expansion, J. Comput. Appl. Math., 357 (2019), 329-348.  doi: 10.1016/j.cam.2019.02.032.

[30]

Z. Zhang, Y. Yong and W. Yu, Valuing equity-linked death benefits in general exponential Lévy models, J. Comput. Appl. Math., 365 (2020), 112377, 18 pp. doi: 10.1016/j.cam.2019.112377.

[31]

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

[32]

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

Table 1.  The life-contingent call option with different observation interval $ \Delta $ and guaranteed level $ K $
$ \Delta $ $ L_1=L_2 $ $ K=95 $ $ K=110 $ $ K=125 $ $ K=140 $ $ K=155 $ $ K=170 $
1 $ 2^{9} $ 35.8286 31.2993 27.7919 25.0436 22.8343 21.0117
$ 2^{10} $ 35.8286 31.2993 27.7919 25.0436 22.8343 21.0117
0.1 $ 2^{11} $ 35.0826 30.4789 27.0734 24.4349 22.3055 20.5350
$ 2^{12} $ 35.0826 30.4789 27.0734 24.4349 22.3055 20.5350
0.01 $ 2^{12} $ 34.9980 30.3918 27.0016 24.3746 22.2527 20.4862
$ 2^{13} $ 34.9978 30.3915 27.0012 24.3743 22.2522 20.4859
C.O. 34.8256 30.3869 26.9982 24.3724 22.2516 20.4855
$ \Delta $ $ L_1=L_2 $ $ K=95 $ $ K=110 $ $ K=125 $ $ K=140 $ $ K=155 $ $ K=170 $
1 $ 2^{9} $ 35.8286 31.2993 27.7919 25.0436 22.8343 21.0117
$ 2^{10} $ 35.8286 31.2993 27.7919 25.0436 22.8343 21.0117
0.1 $ 2^{11} $ 35.0826 30.4789 27.0734 24.4349 22.3055 20.5350
$ 2^{12} $ 35.0826 30.4789 27.0734 24.4349 22.3055 20.5350
0.01 $ 2^{12} $ 34.9980 30.3918 27.0016 24.3746 22.2527 20.4862
$ 2^{13} $ 34.9978 30.3915 27.0012 24.3743 22.2522 20.4859
C.O. 34.8256 30.3869 26.9982 24.3724 22.2516 20.4855
Table 2.  Model parameters
$ \alpha(0)=1 $ BSM $ \sigma_1=0.1 $
Kou(DE) $ \sigma_1=0.1 $, $ \lambda_1=2 $, $ p_1=0.75 $, $ \eta_{11}=10 $, $ \eta_{12}=8 $
MJD $ \sigma_1=0.1 $, $ \lambda_1=1 $, $ \mu^J_{1}=-0.1 $, $ \sigma^J_{1}=0.05 $
MNJD $ \sigma_1=0.1 $, $ \lambda_1=2 $, $ p_1=0.6 $, $ \mu^J_{11}=-0.1 $, $ \mu^J_{12}=0.15 $, $ \sigma^J_{11}=0.1 $, $ \sigma^J_{12}=0.07 $
$ \alpha(0)=2 $ BSM $ \sigma_2=0.3 $
Kou(DE) $ \sigma_2=0.3 $, $ \lambda_2=1 $, $ p_2=0.25 $, $ \eta_{21}=8 $, $ \eta_{22}=7 $
MJD $ \sigma_2=0.3 $, $ \lambda_2=0.5 $, $ \mu^J_{2}=-0.12 $, $ \sigma^J_{2}=0.07 $
MNJD $ \sigma_2=0.3 $, $ \lambda_2=1 $, $ p_2=0.4 $, $ \mu^J_{21}=-0.12 $, $ \mu^J_{22}=0.12 $, $ \sigma^J_{21}=0.2 $, $ \sigma^J_{22}=0.09 $
$ \alpha(0)=1 $ BSM $ \sigma_1=0.1 $
Kou(DE) $ \sigma_1=0.1 $, $ \lambda_1=2 $, $ p_1=0.75 $, $ \eta_{11}=10 $, $ \eta_{12}=8 $
MJD $ \sigma_1=0.1 $, $ \lambda_1=1 $, $ \mu^J_{1}=-0.1 $, $ \sigma^J_{1}=0.05 $
MNJD $ \sigma_1=0.1 $, $ \lambda_1=2 $, $ p_1=0.6 $, $ \mu^J_{11}=-0.1 $, $ \mu^J_{12}=0.15 $, $ \sigma^J_{11}=0.1 $, $ \sigma^J_{12}=0.07 $
$ \alpha(0)=2 $ BSM $ \sigma_2=0.3 $
Kou(DE) $ \sigma_2=0.3 $, $ \lambda_2=1 $, $ p_2=0.25 $, $ \eta_{21}=8 $, $ \eta_{22}=7 $
MJD $ \sigma_2=0.3 $, $ \lambda_2=0.5 $, $ \mu^J_{2}=-0.12 $, $ \sigma^J_{2}=0.07 $
MNJD $ \sigma_2=0.3 $, $ \lambda_2=1 $, $ p_2=0.4 $, $ \mu^J_{21}=-0.12 $, $ \mu^J_{22}=0.12 $, $ \sigma^J_{21}=0.2 $, $ \sigma^J_{22}=0.09 $
Table 3.  COS method vs. MC for a call option with $ K = 100 $, $ B = 150 $ and $ \alpha(0) = 1 $
Model COS method MC method
$ n $ $ L_1=L_2=2^{10} $ $ L_1=L_2=2^{11} $ Mean $ 95\% $ C.I.
Value Time(sec) Value Time(sec) Value Time(sec) Lower Upper
30 52.6245 0.6301 52.6245 2.5990 52.8673 3.4275 52.5522 53.1824
BSM 60 62.2973 1.3866 62.2973 5.1029 62.7358 7.0263 62.0664 63.4052
30 60.9391 0.7907 60.9389 2.4777 60.8112 692.1551 60.1851 61.4373
Kou(DE) 60 68.6834 1.3303 68.6863 5.1685 68.4184 1457.9465 66.5529 70.2839
30 55.2039 0.6401 55.2039 2.6163 55.1863 1497.4727 54.8200 55.5526
MJD 60 64.3113 1.4141 64.3113 5.2043 64.313767 1955.5970 63.4624 65.1651
30 60.8012 0.7566 60.8012 2.6076 60.9323 1565.7978 60.3158 61.5489
MNJD 60 68.6062 1.4865 68.6068 5.2389 68.5280 3037.4160 66.6457 70.4103
Model COS method MC method
$ n $ $ L_1=L_2=2^{10} $ $ L_1=L_2=2^{11} $ Mean $ 95\% $ C.I.
Value Time(sec) Value Time(sec) Value Time(sec) Lower Upper
30 52.6245 0.6301 52.6245 2.5990 52.8673 3.4275 52.5522 53.1824
BSM 60 62.2973 1.3866 62.2973 5.1029 62.7358 7.0263 62.0664 63.4052
30 60.9391 0.7907 60.9389 2.4777 60.8112 692.1551 60.1851 61.4373
Kou(DE) 60 68.6834 1.3303 68.6863 5.1685 68.4184 1457.9465 66.5529 70.2839
30 55.2039 0.6401 55.2039 2.6163 55.1863 1497.4727 54.8200 55.5526
MJD 60 64.3113 1.4141 64.3113 5.2043 64.313767 1955.5970 63.4624 65.1651
30 60.8012 0.7566 60.8012 2.6076 60.9323 1565.7978 60.3158 61.5489
MNJD 60 68.6062 1.4865 68.6068 5.2389 68.5280 3037.4160 66.6457 70.4103
Table 4.  COS vs. MC for a call option with $ K = 100 $, $ B = 150 $ and $ \alpha(0) = 2 $
Model COS method MC method
$ n $ $ L_1=L_2=2^{10} $ $ L_1=L_2=2^{11} $ Mean $ 95\% $ C.I.
Value Time(sec) Value Time(sec) Value Time(sec) Lower Upper
30 55.9191 0.7598 55.9191 2.4770 55.6698 3.6186 55.3396 56.0000
BSM 60 65.4643 1.3866 65.4643 5.0556 65.0906 7.1603 64.3606 65.8206
30 63.0073 0.6424 63.0072 2.6167 62.5578 708.2445 61.8836 63.2319
Kou(DE) 60 70.6986 1.4507 70.6979 5.1633 70.5694 1440.9059 68.4339 72.7048
30 58.0972 0.7539 58.0972 2.6003 57.8536 934.2320 57.4455 58.2617
MJD 60 67.1014 1.2928 67.0997 5.2210 66.7409 1929.4853 65.7567 67.7250
30 62.8908 0.6188 62.8908 2.6428 62.7879 1562.6075 62.1114 63.4644
MNJD 60 70.6342 1.5680 70.6340 5.2169 70.1752 3154.4060 67.9776 72.3728
Model COS method MC method
$ n $ $ L_1=L_2=2^{10} $ $ L_1=L_2=2^{11} $ Mean $ 95\% $ C.I.
Value Time(sec) Value Time(sec) Value Time(sec) Lower Upper
30 55.9191 0.7598 55.9191 2.4770 55.6698 3.6186 55.3396 56.0000
BSM 60 65.4643 1.3866 65.4643 5.0556 65.0906 7.1603 64.3606 65.8206
30 63.0073 0.6424 63.0072 2.6167 62.5578 708.2445 61.8836 63.2319
Kou(DE) 60 70.6986 1.4507 70.6979 5.1633 70.5694 1440.9059 68.4339 72.7048
30 58.0972 0.7539 58.0972 2.6003 57.8536 934.2320 57.4455 58.2617
MJD 60 67.1014 1.2928 67.0997 5.2210 66.7409 1929.4853 65.7567 67.7250
30 62.8908 0.6188 62.8908 2.6428 62.7879 1562.6075 62.1114 63.4644
MNJD 60 70.6342 1.5680 70.6340 5.2169 70.1752 3154.4060 67.9776 72.3728
Table 5.  Life-contingent call option with different pre-specified level $ B $ and mortality models
BSM Kou(DE) MJD MNJD
$ \mathcal{M}_l $ $ B $ $ \alpha(0)=1 $ $ \alpha(0)=2 $ $ \alpha(0)=1 $ $ \alpha(0)=2 $ $ \alpha(0)=1 $ $ \alpha(0)=2 $ $ \alpha(0)=1 $ $ \alpha(0)=2 $
$ \mathcal{M}_1 $ 110 44.7548 47.6643 50.5271 52.6335 47.0423 49.4114 51.6029 53.5005
140 38.5904 42.3158 46.9095 49.2138 41.2790 44.5152 47.5186 49.8212
170 34.8883 38.3526 43.0069 45.2976 37.6470 40.7673 44.1626 46.4576
200 32.2941 35.4808 39.6674 41.8896 35.1186 38.0277 42.1979 44.4442
$ \mathcal{M}_2 $ 110 69.4464 71.5599 73.0547 74.7596 70.8398 72.5261 73.2677 74.7749
140 60.2182 63.4759 66.9787 69.0355 62.3668 65.1826 67.0595 69.0808
170 54.2824 57.3588 62.0305 64.0756 56.7456 59.5144 62.2488 64.3305
200 49.8613 52.7005 58.0956 60.0775 52.5483 55.1467 58.7869 60.8205
BSM Kou(DE) MJD MNJD
$ \mathcal{M}_l $ $ B $ $ \alpha(0)=1 $ $ \alpha(0)=2 $ $ \alpha(0)=1 $ $ \alpha(0)=2 $ $ \alpha(0)=1 $ $ \alpha(0)=2 $ $ \alpha(0)=1 $ $ \alpha(0)=2 $
$ \mathcal{M}_1 $ 110 44.7548 47.6643 50.5271 52.6335 47.0423 49.4114 51.6029 53.5005
140 38.5904 42.3158 46.9095 49.2138 41.2790 44.5152 47.5186 49.8212
170 34.8883 38.3526 43.0069 45.2976 37.6470 40.7673 44.1626 46.4576
200 32.2941 35.4808 39.6674 41.8896 35.1186 38.0277 42.1979 44.4442
$ \mathcal{M}_2 $ 110 69.4464 71.5599 73.0547 74.7596 70.8398 72.5261 73.2677 74.7749
140 60.2182 63.4759 66.9787 69.0355 62.3668 65.1826 67.0595 69.0808
170 54.2824 57.3588 62.0305 64.0756 56.7456 59.5144 62.2488 64.3305
200 49.8613 52.7005 58.0956 60.0775 52.5483 55.1467 58.7869 60.8205
Table 6.  The characteristic exponents $ \psi_j(\xi) $, and cumulants $ \gamma^j_k $
Model $\psi_j(\xi)$ Cumulants $\gamma^j_k$
BSM $\psi_j(\xi)=i\xi\mu_j-\frac{\sigma^2_j}{2}\xi^2$, $\mu_j=r-\frac{\sigma^2_j}{2}$, $\gamma^j_1=\mu_j$, $\gamma^j_2=\sigma^2_j$, $\gamma^j_4=0$
Kou(DE) $\psi_j(\xi)=i\xi\mu_j-\frac{\sigma^2_j}{2}\xi^2+\lambda_j \left(\frac{p_j\eta_{j1}}{\eta_{j1}-i\xi}+\frac{(1-p_j)\eta_{j2}}{\eta_{j2}+i\xi}-1\right)$
$\mu_j=r-\frac{\sigma^2_j}{2}-\lambda_j \left(\frac{p_j\eta_{j1}}{\eta_{j1}-1}+\frac{(1-p_j)\eta_{j2}}{\eta_{j2}+1}-1\right)$
$\gamma^j_1=\mu_j+\lambda_j\left(\frac{p_j}{\eta_{j1}}-\frac{(1-p_j)}{\eta_{j2}}\right)$
$\gamma^j_2=\sigma^2_j+2\lambda_j\left(\frac{p_j}{\eta_{j1}^2}+\frac{(1-p_j)}{\eta_{j2}^2}\right)$
$\gamma^j_4=24\lambda_j\left(\frac{p_j}{\eta_{j1}^4}+\frac{(1-p_j)}{\eta_{j2}^4}\right)$
MJD $\psi_j(\xi)=i\xi\mu_j-\frac{\sigma^2_j}{2}\xi^2+\lambda_j \left(\exp(i\xi\mu^J_j-\frac{{\sigma^J_j}^2}{2}\xi^2)-1\right)$
$\mu_j=r-\frac{\sigma^2_j}{2}-\lambda_j \left(\exp(\mu^J_j+\frac{{\sigma^J_j}^2}{2})-1\right)$
$\gamma^j_1=\mu_j+\lambda_j\mu^J_j$
$\gamma^j_2=\sigma^2_j+\lambda_j\left({\mu^J_j}^2+{\sigma^J_j}^2\right)$
$\gamma^j_4=\lambda_j\left({\mu^J_j}^4+6{\mu^J_j}^2{\sigma^J_j}^2+3{\sigma^J_j}^4\right)$
MNJD $\psi_j(\xi)=i\xi\mu_j-\frac{\sigma^2_j}{2}\xi^2+\lambda_j \left(p_j\exp(i\xi\mu^J_{j1}-\frac{{\sigma^J_{j1}}^2}{2}\xi^2)\right.$
$\left.+(1-p_j)\exp(i\xi\mu^J_{j2}-\frac{{\sigma^J_{j2}}^2}{2}\xi^2)-1\right)$
$\mu_j=r-\frac{\sigma^2_j}{2}-\lambda_j \left(p_j\exp(\mu^J_{j1}+\frac{{\sigma^J_{j1}}^2}{2})\right.$
$\left.+(1-p_j)\exp(\mu^J_{j2}+\frac{{\sigma^J_{j2}}^2}{2})-1\right)$
$\gamma^j_1=\mu_j+\lambda_j(p_j\mu^J_{j1}+(1-p_j)\mu^J_{j2})$
$\gamma^j_2=\sigma^2_j+\lambda_jp_j\left({\mu^J_{j1}}^2+{\sigma^J_{j1}}^2\right)$
$\qquad+\lambda_j(1-p_j)\left({\mu^J_{j2}}^2+{\sigma^J_{j2}}^2\right)$
$\gamma^j_4=\lambda_jp_j\left({\mu^J_{j1}}^4+6{\mu^J_j}^2{\sigma^J_j}^2+3{\sigma^J_j}^4\right)$
$+\lambda_j(1-p_j)\left({\mu^J_{j2}}^4+6{\mu^J_j}^2{\sigma^J_j}^2+3{\sigma^J_j}^4\right)$
Model $\psi_j(\xi)$ Cumulants $\gamma^j_k$
BSM $\psi_j(\xi)=i\xi\mu_j-\frac{\sigma^2_j}{2}\xi^2$, $\mu_j=r-\frac{\sigma^2_j}{2}$, $\gamma^j_1=\mu_j$, $\gamma^j_2=\sigma^2_j$, $\gamma^j_4=0$
Kou(DE) $\psi_j(\xi)=i\xi\mu_j-\frac{\sigma^2_j}{2}\xi^2+\lambda_j \left(\frac{p_j\eta_{j1}}{\eta_{j1}-i\xi}+\frac{(1-p_j)\eta_{j2}}{\eta_{j2}+i\xi}-1\right)$
$\mu_j=r-\frac{\sigma^2_j}{2}-\lambda_j \left(\frac{p_j\eta_{j1}}{\eta_{j1}-1}+\frac{(1-p_j)\eta_{j2}}{\eta_{j2}+1}-1\right)$
$\gamma^j_1=\mu_j+\lambda_j\left(\frac{p_j}{\eta_{j1}}-\frac{(1-p_j)}{\eta_{j2}}\right)$
$\gamma^j_2=\sigma^2_j+2\lambda_j\left(\frac{p_j}{\eta_{j1}^2}+\frac{(1-p_j)}{\eta_{j2}^2}\right)$
$\gamma^j_4=24\lambda_j\left(\frac{p_j}{\eta_{j1}^4}+\frac{(1-p_j)}{\eta_{j2}^4}\right)$
MJD $\psi_j(\xi)=i\xi\mu_j-\frac{\sigma^2_j}{2}\xi^2+\lambda_j \left(\exp(i\xi\mu^J_j-\frac{{\sigma^J_j}^2}{2}\xi^2)-1\right)$
$\mu_j=r-\frac{\sigma^2_j}{2}-\lambda_j \left(\exp(\mu^J_j+\frac{{\sigma^J_j}^2}{2})-1\right)$
$\gamma^j_1=\mu_j+\lambda_j\mu^J_j$
$\gamma^j_2=\sigma^2_j+\lambda_j\left({\mu^J_j}^2+{\sigma^J_j}^2\right)$
$\gamma^j_4=\lambda_j\left({\mu^J_j}^4+6{\mu^J_j}^2{\sigma^J_j}^2+3{\sigma^J_j}^4\right)$
MNJD $\psi_j(\xi)=i\xi\mu_j-\frac{\sigma^2_j}{2}\xi^2+\lambda_j \left(p_j\exp(i\xi\mu^J_{j1}-\frac{{\sigma^J_{j1}}^2}{2}\xi^2)\right.$
$\left.+(1-p_j)\exp(i\xi\mu^J_{j2}-\frac{{\sigma^J_{j2}}^2}{2}\xi^2)-1\right)$
$\mu_j=r-\frac{\sigma^2_j}{2}-\lambda_j \left(p_j\exp(\mu^J_{j1}+\frac{{\sigma^J_{j1}}^2}{2})\right.$
$\left.+(1-p_j)\exp(\mu^J_{j2}+\frac{{\sigma^J_{j2}}^2}{2})-1\right)$
$\gamma^j_1=\mu_j+\lambda_j(p_j\mu^J_{j1}+(1-p_j)\mu^J_{j2})$
$\gamma^j_2=\sigma^2_j+\lambda_jp_j\left({\mu^J_{j1}}^2+{\sigma^J_{j1}}^2\right)$
$\qquad+\lambda_j(1-p_j)\left({\mu^J_{j2}}^2+{\sigma^J_{j2}}^2\right)$
$\gamma^j_4=\lambda_jp_j\left({\mu^J_{j1}}^4+6{\mu^J_j}^2{\sigma^J_j}^2+3{\sigma^J_j}^4\right)$
$+\lambda_j(1-p_j)\left({\mu^J_{j2}}^4+6{\mu^J_j}^2{\sigma^J_j}^2+3{\sigma^J_j}^4\right)$
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