ZHANG Jiesong.A catastrophe shock model and the bond pricing[J].Journal of Shenzhen University Science and Engineering,2021,38(2):208-213.[doi:10.3724/SP.J.1249.2021.02208]





A catastrophe shock model and the bond pricing
淮北师范大学经济与管理学院, 安徽淮北 235000
ZHANG Jiesong
School of Economics and Management, Huaibei Normal University, Huaibei 235000, Anhui Province, P.R.China
probability theory catastrophe risk fractional compound Poisson process moment matching method generalized Pareto distribution insurance bond
In order to describe the memory effects of catastrophe events such as earthquake, typhoon and to transfer risk reasonably, we adopt a catastrophe impact model with power-law waiting time, so called the fractional compound Poisson process, to describe the underwriting risk of insurance companies and to study the pricing for catastrophe insurance-linked bonds. Firstly, we use the moment matching method to get the generalized Pareto-type approximation distribution for the cumulative losses. Then, the formula for bond price is derived under the CIR (Cox-Ingersoll-Ross) interest rate model. At last, numerical examples are used to verify the validity of the approximation for loss distribution. The results show that with the increase of memory parameter, the change trend of expected risk and bond price is opposite, and both of them may have a polymorphic trend of increasing, decreasing or both, which is closely related to the maturity level.


[1] 马超群, 马宗刚. 基于 Vasicek 和 CIR 模型的巨灾风险债券定价[J]. 系统工程, 2013, 31(9): 33-38.
MA Chaoqun, MA Zonggang. Pricing catastrophe risk bonds using the Vasicek and CIR models[J]. Systems Engineering, 2013, 31(9): 33-38.(in Chinese)
[2] 谢卓伦, 陈佳琰, 叶露. 中国大陆地区地震巨灾风险的分布拟合及债券定价[J]. 浙江理工大学学报, 2019, 42(1): 10-19.
XIE Zhuolun, CHEN Jiayan, YE Lu. Distribution fitting of earthquake catastrophe risk and bond pricing in mainland China[J]. Journal of Zhejiang Sci-Tech University, 2019, 42(1): 10-19.(in Chinese)
[3] 马宗刚, 马超群, 肖时松. 台风风暴潮债券定价——基于我国沿海1989-2015灾害数据[J]. 系统工程, 2017, 35(9): 18-26.
MA Zonggang, MA Chaoqun, XIAO Shisong. Pricing typhoon bonds based on storm surge disaster in coastal areas of China:based on the data from 1989 to 2015[J]. Systems Engineering, 2017, 35(9): 18-26.(in Chinese)
[4] GEMAN H, YOR M. Stochastic time changes in catastrophe option pricing[J]. Insurance: Mathematics and Economics, 1997, 21(3): 185-193.
[5] HEYDE C C, WANG D. Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims[J]. Advances in Applied Probability, 2009, 41(1): 206-224.
[6] 苏必豪, 李婧超. 经典风险模型中破产变量的联合分布[J].深圳大学学报理工版, 2019, 36(4): 419-423.
SU Bihao, LI Jingchao. The joint distribution of ruin related quantities in the classical risk model[J].Journal of Shenzhen University Science and Engineering, 2019, 36(4): 419-423.(in Chinese)
[7] BENSON D A, SCHUMER R, MEERSCHAERT M M. Recurrence of extreme events with power‐law interarrival times[J]. Geophysical Research Letters, 2007, 34(16): L16404.
[8] MUSSON R M W, TSAPANOS T, NAKAS C T. A power-law function for earthquake interarrival time and magnitude[J]. Bulletin of the Seismological Society of America, 2002, 92(5): 1783-1794.
[9] SALIM A, PAWITAN Y. Extensions of the Bartlett-Lewis model for rainfall processes[J]. Statistical Modelling, 2003, 3(2): 79-98.
[10] STOYNOV P, ZLATEVA P, VELEV D, et al. Modelling of major flood arrivals on Chinese rivers by switch-time processes[C]// The 3rd International Conference on Advances in Environment Research. Beijing: IOP Publishing, 2017: 012006.
[11] REPIN O N, SAICHEV A I. Fractional Poisson law[J]. Radiophysics and Quantum Electronics, 2000, 43(9): 738-741.
[12] BIARD R, SAUSSEREAU B. Fractional Poisson process: long-range dependence and applications in ruin theory[J]. Journal of Applied Probability, 2014, 51(3): 727-740.
[13] SCALAS E. A class of CTRWs: compound fractional Poisson processes[M]// KLAFTER J, LIM S C, METZLER R. Fractional dynamics. Hackensack, USA: World Scientific Publication, 2012: 353-374.
[14] MAHESHWARI A, VELLAISAMY P. On the long-range dependence of fractional Poisson and negative binomial processes[J]. Journal of Applied Probability, 2016, 53(4): 989-1000.
[15] WANG Ying, WANG Dehui, ZHU Fukang. Estimation of parameters in the fractional compound Poisson process[J]. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(10): 3425-3430.
[16] SCALAS E, VILES N. On the convergence of quadratic variation for compound fractional Poisson processes[J]. Fractional Calculus and Applied Analysis, 2012, 15(2): 314-331.
[17] ZHANG Jiesong. Optimal layer reinsurance for compound fractional Poisson model[J]. Discrete Dynamics in Nature and Society, 2019(16): 1-8.
[18] LASKIN N. Fractional Poisson process[J]. Communications in Nonlinear Science and Numerical Simulation, 2003, 8(3): 201-213.
[19] JARROW R A. The term structure of interest rates[J]. The Annual Review of Financial Economics, 2009, 1(1): 69-96.
[20] 谢赤, 吴雄伟. 基于 Vasicek 和 CIR 模型中的中国货币市场利率行为实证分析[J]. 中国管理科学, 2002, 10(3): 22-25.
XIE Chi, WU Xiongwei. An empirical analysis of the interest rate behavior in China’s monetary market using the Vasicek and CIR models[J]. Chinese Journal of Management Science, 2002, 10(3): 22-25.(in Chinese)
[21] COX S H, PEDERSEN H W. Catastrophe risk bonds[J]. North American Actuarial Journal, 2000, 4(4): 56-82.
[22] LEE J P, YU M T. Pricing default-risky CAT bonds with moral hazard and basis risk[J]. Journal of Risk and Insurance, 2002, 69(1): 25-44.
[23] BAI Lihua, CAI Jun, ZHOU Ming. Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting[J]. Insurance: Mathematics and Economics, 2013, 53(3): 664-670.
[24] JOHNSON M A, TAAFFE M R. Matching moments to phase distributions: mixtures of Erlang distributions of common order[J]. Stochastic Models, 1989, 5(4): 711-743.
[25] LINDSKOG F, MCNEIL A J. Common Poisson shock models: applications to insurance and credit risk modeling[J]. ASTIN Bulletin: The Journal of the IAA, 2003, 33(2): 209-238.
[26] BARGES M, COSSETTE H, LOISEL S, et al. On the moments of aggregate discounted claims with dependence introduced by a FGM copula[J]. ASTIN Bulletin: The Journal of the IAA, 2011, 41(1): 215-238.
[27] BEGHIN L, ORSINGHER E. Fractional Poisson processes and related planar random motions[J]. Electronic Journal of Probability, 2009, 14: 1790-1826.
[28] 王颖. 复合分数阶泊松过程的参数估计及应用[D]. 长春: 吉林大学, 2015.
WANG Ying. Parameter estimation and application of the fractional compound Poisson process[D]. Changchun: Jilin University, 2015.(in Chinese)


Foundation:Humanity and Social Science Youth Foundation of Ministry of Education (17YJC630212); Key Program of Natural Science Foundation of Anhui Higher Education Institution of China (KJ2019A0607)
Corresponding author:Associate professor ZHANG Jiesong.E-mail: j_s_zhang@126.com
Citation:ZHANG Jiesong. A catastrophe shock model and the bond pricing[J]. Journal of Shenzhen University Science and Engineering, 2021, 38(2): 208-213.(in Chinese)
基金项目:教育部人文社会科学研究青年基金资助项目(17YJC630 212); 安徽省高校自然科学研究重点项目资助项目(KJ2019A0607)
引文:张节松.一类巨灾冲击模型及其债券定价[J]. 深圳大学学报理工版,2021,38(2):208-213.
更新日期/Last Update: 2021-03-30