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November  2020, 16(6): 2603-2623. doi: 10.3934/jimo.2019072

Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level

Chao Xu 1, , Yinghui Dong 1,, , Zhaolu Tian 2,3, and  Guojing Wang 2,3,

1. 

Department of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China
2. 

College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China
3. 

Department of Mathematics and Center for Financial Engineering, Soochow University, Suzhou 215006, China

* Corresponding author: Yinghui Dong

Received  February 2018 Revised  March 2019 Published  November 2020 Early access  July 2019

Fund Project: The authors thank the anonymous referees for valuable comments to improve the earlier version of the paper. This work is supported by the NSF of Jiangsu Province (Grant No. BK20170064), the NNSF of China (Grant No. 11771320), QingLan Project, the scholarship of Jiangsu Overseas Visiting Scholar Program, Suzhou Key Laboratory for Big Data and Information Service (SZS201813) and the Graduate Innovation Program (Grant No. KYCX17-2059) of Jiangsu Province of China

In this paper, we investigate the valuation of dynamic fund protections under the assumption that the market value of the basic fund and the protection level follow regime-switching processes with jumps. The price of the dynamic fund protection (DFP) is associated with the Laplace transform of the first passage time. We derive the explicit formula for the Laplace transform of the DFP under the regime-switching, hyper-exponential jump-diffusion process. By using the Gaver-Stehfest algorithm, we present some numerical results for the price of the DFP.

Keywords: Dynamic fund protection, regime-switching, hyper-exponential jump-diffusion process, Laplace transform.
Mathematics Subject Classification: Primary: 91B25, 91G20; Secondary: 62P20.
Citation: Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072
References:
[1]

J. Buffington and R. J. Elliott, American options with regime switching, International Journal of Theoretical and Applied Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.  Google Scholar

[2]

C. C. Chang, Y. H. Lian and M. H. Tsay, Pricing dynamic guaranteed funds under a double exponential jump diffusion model, Academia Economic Papers, 40 (2012), 269-306.   Google Scholar

[3]

Y. H. Dong, Pricing dynamic guaranteed funds with stochastic barrier under Vasicek interest rate model, hinese Journal of Applied Probability and Statistics, 29 (2013), 237-245.   Google Scholar

[4]

Y. H. Dong, G. J. Wang and R. Wu, Pricing the zero-coupon bond and its fair premium under a structural credit risk model with jumps, Journal of Applied Probability, 48 (2011), 404-419.  doi: 10.1239/jap/1308662635.  Google Scholar

[5]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag, Berlin-Heidelberg-New York, 1994.  Google Scholar

[6]

R. J. Elliott, C. Leunglung and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432.  doi: 10.1007/s10436-005-0013-z.  Google Scholar

[7]

H. K. Fung and L. K. Li, Pricing discrete dynamic fund protections, North American Actuarial Journal, 7 (2003), 23-31.  doi: 10.1080/10920277.2003.10596115.  Google Scholar

[8]

H. U. Gerber and G. Pafumi, Pricing dynamic investment fund protection, North American Actuarial Journal, 4 (2000), 28-37.  doi: 10.1080/10920277.2000.10595894.  Google Scholar

[9]

H. U. Gerber and E. S. Shiu, From ruin theory to pricing reset guarantees and perpetual put options, Insurance: Mathematics and Economics, 24 (1999), 3-14.  doi: 10.1016/S0167-6687(98)00033-X.  Google Scholar

[10]

H. U. Gerber and E. S. Shiu, Pricing perpetual fund protection with withdrawal option, North American Actuarial Journal, 7 (2003), 60-77.  doi: 10.1080/10920277.2003.10596087.  Google Scholar

[11]

H. U. Gerber and E. S. W. Shiu, The time value of ruin in a Sparre Andersen model, North American Actuarial Journal, 9 (2005), 49-84.  doi: 10.1080/10920277.2005.10596197.  Google Scholar

[12]

X. Guo, Information and option pricings, Quantitative Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.  Google Scholar

[13]

J. D. Hamilton, Rational-expectations econometric analysis of changes in regime: An investigation of the terms tructure of interest rates, Journal of Economic Dynamics and Control, 12 (1998), 385-423.  doi: 10.1016/0165-1889(88)90047-4.  Google Scholar

[14]

M. R. Hardy, A regime-switching model of long-term stock returns, North American Actuarial Journal, 5 (2001), 41-53.  doi: 10.1080/10920277.2001.10595984.  Google Scholar

[15]

J. Imai and P. P. Boyle, Dynamic fund protection, North American Actuarial Journal, 5 (2001), 31-47.  doi: 10.1080/10920277.2001.10595996.  Google Scholar

[16]

Z. Jin, L. Y. Qian, W. Wang and R. M. Wang, Pricing dynamic fund protections with regime switching, Journal of Computational and Applied Mathematics, 297 (2016), 13-25.  doi: 10.1016/j.cam.2015.11.012.  Google Scholar

[17]

S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.   Google Scholar

[18]

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

[19]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model., Management Science, 50 (2004), 1178-1192.   Google Scholar

[20]

V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, Journal of Finance, 48 (1993), 1969-1984.  doi: 10.1111/j.1540-6261.1993.tb05137.x.  Google Scholar

[21]

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.  Google Scholar

[22]

H. Y. Wong and C. M. Chan, Lookback options and dynamic fund protection under multiscale stochastic volatility, Insurance: Mathematics and Economics, 40 (2007), 357-385.  doi: 10.1016/j.insmatheco.2006.05.006.  Google Scholar

[23]

H. Y. Wong and K. W. Lam, Valuation of discrete dynamic fund protection under Lévy processes, North American Actuarial Journal, 13 (2009), 202-216.  doi: 10.1080/10920277.2009.10597548.  Google Scholar

[24]

C. Xu and Y. H. Dong, Pricing dynamic fund protections under a stochastic boundary, Journal of Suzhou University of Science and Technology, 2 (2018), 21-25.   Google Scholar

show all references

References:
[1]

J. Buffington and R. J. Elliott, American options with regime switching, International Journal of Theoretical and Applied Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.  Google Scholar

[2]

C. C. Chang, Y. H. Lian and M. H. Tsay, Pricing dynamic guaranteed funds under a double exponential jump diffusion model, Academia Economic Papers, 40 (2012), 269-306.   Google Scholar

[3]

Y. H. Dong, Pricing dynamic guaranteed funds with stochastic barrier under Vasicek interest rate model, hinese Journal of Applied Probability and Statistics, 29 (2013), 237-245.   Google Scholar

[4]

Y. H. Dong, G. J. Wang and R. Wu, Pricing the zero-coupon bond and its fair premium under a structural credit risk model with jumps, Journal of Applied Probability, 48 (2011), 404-419.  doi: 10.1239/jap/1308662635.  Google Scholar

[5]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag, Berlin-Heidelberg-New York, 1994.  Google Scholar

[6]

R. J. Elliott, C. Leunglung and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432.  doi: 10.1007/s10436-005-0013-z.  Google Scholar

[7]

H. K. Fung and L. K. Li, Pricing discrete dynamic fund protections, North American Actuarial Journal, 7 (2003), 23-31.  doi: 10.1080/10920277.2003.10596115.  Google Scholar

[8]

H. U. Gerber and G. Pafumi, Pricing dynamic investment fund protection, North American Actuarial Journal, 4 (2000), 28-37.  doi: 10.1080/10920277.2000.10595894.  Google Scholar

[9]

H. U. Gerber and E. S. Shiu, From ruin theory to pricing reset guarantees and perpetual put options, Insurance: Mathematics and Economics, 24 (1999), 3-14.  doi: 10.1016/S0167-6687(98)00033-X.  Google Scholar

[10]

H. U. Gerber and E. S. Shiu, Pricing perpetual fund protection with withdrawal option, North American Actuarial Journal, 7 (2003), 60-77.  doi: 10.1080/10920277.2003.10596087.  Google Scholar

[11]

H. U. Gerber and E. S. W. Shiu, The time value of ruin in a Sparre Andersen model, North American Actuarial Journal, 9 (2005), 49-84.  doi: 10.1080/10920277.2005.10596197.  Google Scholar

[12]

X. Guo, Information and option pricings, Quantitative Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.  Google Scholar

[13]

J. D. Hamilton, Rational-expectations econometric analysis of changes in regime: An investigation of the terms tructure of interest rates, Journal of Economic Dynamics and Control, 12 (1998), 385-423.  doi: 10.1016/0165-1889(88)90047-4.  Google Scholar

[14]

M. R. Hardy, A regime-switching model of long-term stock returns, North American Actuarial Journal, 5 (2001), 41-53.  doi: 10.1080/10920277.2001.10595984.  Google Scholar

[15]

J. Imai and P. P. Boyle, Dynamic fund protection, North American Actuarial Journal, 5 (2001), 31-47.  doi: 10.1080/10920277.2001.10595996.  Google Scholar

[16]

Z. Jin, L. Y. Qian, W. Wang and R. M. Wang, Pricing dynamic fund protections with regime switching, Journal of Computational and Applied Mathematics, 297 (2016), 13-25.  doi: 10.1016/j.cam.2015.11.012.  Google Scholar

[17]

S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.   Google Scholar

[18]

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

[19]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model., Management Science, 50 (2004), 1178-1192.   Google Scholar

[20]

V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, Journal of Finance, 48 (1993), 1969-1984.  doi: 10.1111/j.1540-6261.1993.tb05137.x.  Google Scholar

[21]

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.  Google Scholar

[22]

H. Y. Wong and C. M. Chan, Lookback options and dynamic fund protection under multiscale stochastic volatility, Insurance: Mathematics and Economics, 40 (2007), 357-385.  doi: 10.1016/j.insmatheco.2006.05.006.  Google Scholar

[23]

H. Y. Wong and K. W. Lam, Valuation of discrete dynamic fund protection under Lévy processes, North American Actuarial Journal, 13 (2009), 202-216.  doi: 10.1080/10920277.2009.10597548.  Google Scholar

[24]

C. Xu and Y. H. Dong, Pricing dynamic fund protections under a stochastic boundary, Journal of Suzhou University of Science and Technology, 2 (2018), 21-25.   Google Scholar

Figure 1.  $ DFP_0 $ versus $ T $
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Figure 2.  $ DFP_0 $ versus $ F_0 $
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Figure 3.  $ DFP_0 $ versus $ K_0 $
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Figure 4.  $ DFP_0 $ versus $ a_{12} $
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Figure 5.  $ DFP_0 $ versus $ \lambda_{11} $
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Figure 6.  $ DFP_0 $ versus $ \sigma_1 $
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Figure 7.  For example, if $ k_{i1} = 1,i = 1,\cdots,m, $ $ k_{i2} = 1,i = 0,1,\cdots,m-1, k_{m2} = 2, $ then we have $ h(\tilde{\alpha}_{ij}-) = -\infty, h(\tilde{\alpha}_{ij}+) = +\infty. $ Therefore, there exists at least one root at each of the $ 2m $ intervals, $ (0,\tilde{\alpha}_{11}),(\tilde{\alpha}_{11},\tilde{\alpha}_{12}),(\tilde{\alpha}_{12},\tilde{\alpha}_{21}),\cdots,(\tilde{\alpha}_{m1},\tilde{\alpha}_{m2}) $ and there exists at least two roots at the interval $ (\tilde{\alpha}_{m2},+\infty). $
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