Modeling earthquake bond prices with correlated dual trigger indices and the approximate solution using the Monte Carlo algorithm

Riza Andrian Ibrahim; Sukono; Herlina Napitupulu; Rose Irnawaty Ibrahim; Riza Andrian Ibrahim; Sukono; Herlina Napitupulu; Rose Irnawaty Ibrahim

doi:10.3934/math.2025103

AIMS Mathematics

2025, Volume 10, Issue 2: 2223-2253. doi: 10.3934/math.2025103

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Research article

Modeling earthquake bond prices with correlated dual trigger indices and the approximate solution using the Monte Carlo algorithm

1.
Doctoral Program of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang 45363, Indonesia
2.
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang 45363, Indonesia
3.
Faculty of Science and Technology, Universiti Sains Islam Malaysia, Negeri Sembilan 71800, Malaysia

Received: 18 November 2024 Revised: 08 January 2025 Accepted: 13 January 2025 Published: 08 February 2025
MSC : 91B70, 91G30, 91G70, 91G80

Countries prone to earthquakes face increasing seismic activity, often resulting in losses that exceed national budgets. To mitigate these losses, earthquake bonds present a promising alternative funding source; however, pricing them is complex, requiring simultaneous accounting for financial and seismic risks. Therefore, this study aimed to model earthquake bond pricing. The model incorporates earthquake intensity to account for rising seismic activity. It also includes depth and maximum magnitude as correlated dual trigger indices, making the bonds more attractive to investors, as claims are generated if both events occur. These three factors were modeled together as a compound stochastic process. The bond price was then formulated using a risk-neutral pricing measure with a stochastic interest rate under the Cox-Ingersoll-Ross model. Since the model lacks a closed-form solution, we employed an algorithm based on the Monte Carlo method for estimation. Through this algorithm, we showed that bond prices for terms of one to six years follow a normal distribution. The use of stochastic interest rates becomes significant as the bond term increases. We also found that earthquake intensity and bond terms negatively correlate with bond prices, while annual coupons positively correlate. Additionally, including dual triggers lowers claim probability and increases the bond demand, but is compensated by higher prices. This study can assist issuers in pricing earthquake bonds based on earthquake severity-maximum magnitude, depth, and intensity-and aid geological institutions in estimating earthquake risk in observed areas.
- earthquake bond,
- price,
- intensity,
- depth,
- maximum magnitude,
- risk-neutral pricing measure,
- Monte Carlo method
Citation: Riza Andrian Ibrahim, Sukono, Herlina Napitupulu, Rose Irnawaty Ibrahim. Modeling earthquake bond prices with correlated dual trigger indices and the approximate solution using the Monte Carlo algorithm[J]. AIMS Mathematics, 2025, 10(2): 2223-2253. doi: 10.3934/math.2025103

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Abstract

Countries prone to earthquakes face increasing seismic activity, often resulting in losses that exceed national budgets. To mitigate these losses, earthquake bonds present a promising alternative funding source; however, pricing them is complex, requiring simultaneous accounting for financial and seismic risks. Therefore, this study aimed to model earthquake bond pricing. The model incorporates earthquake intensity to account for rising seismic activity. It also includes depth and maximum magnitude as correlated dual trigger indices, making the bonds more attractive to investors, as claims are generated if both events occur. These three factors were modeled together as a compound stochastic process. The bond price was then formulated using a risk-neutral pricing measure with a stochastic interest rate under the Cox-Ingersoll-Ross model. Since the model lacks a closed-form solution, we employed an algorithm based on the Monte Carlo method for estimation. Through this algorithm, we showed that bond prices for terms of one to six years follow a normal distribution. The use of stochastic interest rates becomes significant as the bond term increases. We also found that earthquake intensity and bond terms negatively correlate with bond prices, while annual coupons positively correlate. Additionally, including dual triggers lowers claim probability and increases the bond demand, but is compensated by higher prices. This study can assist issuers in pricing earthquake bonds based on earthquake severity-maximum magnitude, depth, and intensity-and aid geological institutions in estimating earthquake risk in observed areas.

References

[1]	S. Bhattacharya, J. K. Ray, S. Sinha, B. K. Sachdev, An scientific analysis on Japan's technology progress and success and how Covid-19 has affected the economic growth of the Third largest economy, GIIRJ, 9 (2021), 500–506.
[2]	A. Suwignyo, A tsunami-related life history of survivors in Banda Aceh, Indonesia and Sendai, Japan, JSP, 23 (2019), 120–134. https://doi.org/10.22146/jsp.49876 doi: 10.22146/jsp.49876
[3]	S. Widiyanto, D. Adi, R. V. Soans, Agent-based simulation disaster evacuation awareness on night situation in Aceh, J. Eng., 8 (2022), 36–43. https://doi.org/10.12962/j23378557.v8i1.a12799 doi: 10.12962/j23378557.v8i1.a12799
[4]	I. Towhata, G. Wang, Q. Xu, C. Massey, Coseismic landslides, Singapore: Springer, 2022. https://doi.org/10.1007/978-981-19-6597-5
[5]	J. Y. Shin, S. Chen, T. W. Kim, Application of Bayesian Markov Chain Monte Carlo method with mixed gumbel distribution to estimate extreme magnitude of tsunamigenic earthquake, KSCE J. Civ. Eng., 19 (2015), 366–375. https://doi.org/10.1007/s12205-015-0430-0 doi: 10.1007/s12205-015-0430-0
[6]	C. Davis, V. Keilis-Borok, V. Kossobokov, A. Soloviev, Advance prediction of the March 11, 2011 Great East Japan Earthquake: A missed opportunity for disaster preparedness, Int. J. Disast. Risk Re., 1 (2012), 17–32. https://doi.org/10.1016/j.ijdrr.2012.03.001 doi: 10.1016/j.ijdrr.2012.03.001
[7]	N. B. Parwanto, T. Oyama, Investigating the impact of the 2011 Great East Japan Earthquake and evaluating the restoration and reconstruction performance, J. Asian Public Polic., 8 (2015), 329–350. https://doi.org/10.1080/23307706.2015.1006764 doi: 10.1080/23307706.2015.1006764
[8]	A. K. C. Bhandari, O. Takahashi, Knowledge, attitude, practice and perceived barriers of natural disaster preparedness among Nepalese immigrants residing in Japan, BMC Public Health, 22 (2022), 492. https://doi.org/10.1186/s12889-022-12844-3 doi: 10.1186/s12889-022-12844-3
[9]	I. Demirdag, A.W. Nirwansyah, Unravelling the economic impacts, J. Reg. City Plan., 35 (2024), 21–43. https://doi.org/10.5614/jpwk.2024.35.1.2 doi: 10.5614/jpwk.2024.35.1.2
[10]	A. Laapo, D. N. Asih, D. Howara, H. Sultan, I. Abubakar, A. Wahid, et al., Estimated economic value of the lost coastal resources due to tsunami in September 2018 in Palu City, Indonesia, IOP Conf. Ser. Earth Environ. Sci., 575 (2020), 012054. https://doi.org/10.1088/1755-1315/575/1/012054 doi: 10.1088/1755-1315/575/1/012054
[11]	R. S. Purnamasari, A. Febriady, B. A. Wirapati, M. N. Farid, P. Milne, Y. Kawasoe, et al., Welfare tracking in the aftermath of crisis: The central Sulawesi disaster response, Jakarta: World Bank, 2021.
[12]	S. Shakya, S. Basnet, J. Paudel, Natural disasters and labor migration: Evidence from Nepal's earthquake, World Dev., 151 (2022), 105748. https://doi.org/10.1016/j.worlddev.2021.105748 doi: 10.1016/j.worlddev.2021.105748
[13]	J. Paudel, H. Ryu, Natural disasters and human capital: The case of Nepal's earthquake, World Dev., 111 (2018), 1–12. https://doi.org/10.1016/j.worlddev.2018.06.019 doi: 10.1016/j.worlddev.2018.06.019
[14]	R. B. Bista, Economics of community-based disaster management and household participation: Evidence of the Western Nepal, ARRUS J. Soc. Sci. Humanit., 2 (2022), 120–125. https://doi.org/10.35877/soshum722 doi: 10.35877/soshum722
[15]	S. P. Koenig, V. Rouzier, S. C. Vilbrun, W. Morose, S. E. Collins, P. Joseph, et al., Tuberculosis in the aftermath of the 2010 earthquake in Haiti, Bull. World Health Organ., 93 (2015), 498–502. https://doi.org/10.2471/BLT.14.145649 doi: 10.2471/BLT.14.145649
[16]	R. A. Ibrahim, Sukono, H. Napitupulu, R. I. Ibrahim, How to price catastrophe bonds for sustainable earthquake funding? A systematic review of the pricing framework, Sustainability, 15 (2023), 7705. https://doi.org/10.3390/su15097705 doi: 10.3390/su15097705
[17]	J. D. Cummins, O. Mahul, Catastrophe risk financing in developing countries, World Bank Publications, 2009.
[18]	W. K. Härdle, B. L. Cabrera, Calibrating CAT bonds for Mexican earthquakes, J. Risk Insur., 77 (2010), 625–650. https://www.jstor.org/stable/40783698
[19]	G. Deng, S. Liu, L. Li, C. Deng, Research on the pricing of global drought catastrophe bonds, Math. Probl. Eng., 2020 (2020), 3898191. https://doi.org/10.1155/2020/3898191 doi: 10.1155/2020/3898191
[20]	A. A. Zimbidis, N. E. Frangos, A. A. Pantelous, Modeling earthquake risk via extreme value theory and pricing the respective catastrophe bonds, ASTIN Bull., 37 (2007), 163–183. https://doi.org/10.2143/AST.37.1.2020804 doi: 10.2143/AST.37.1.2020804
[21]	J. Shao, A. Pantelous, A. D. Papaioannou, Catastrophe risk bonds with applications to earthquakes, Eur. Actuar. J., 5 (2015), 113–138. https://doi.org/10.1007/s13385-015-0104-9 doi: 10.1007/s13385-015-0104-9
[22]	Q. Tang, Z. Yuan, CAT bond pricing under a product probability measure with POT risk characterization, ASTIN Bull., 49 (2019), 457–490. https://doi.org/10.1017/asb.2019.11 doi: 10.1017/asb.2019.11
[23]	L. Hofer, M. A. Zanini, P. Gardoni, Risk-based catastrophe bond design for a spatially distributed portfolio, Struct Saf., 83 (2020), 101908. https://doi.org/10.1016/j.strusafe.2019.101908 doi: 10.1016/j.strusafe.2019.101908
[24]	H. K. Mistry, D. Lombardi, Pricing risk-based catastrophe bonds for earthquakes at an urban scale, Sci. Rep., 12 (2022), 9729. https://doi.org/10.1038/s41598-022-13588-1 doi: 10.1038/s41598-022-13588-1
[25]	W. Anggraeni, S. Supian, Sukono, N. A. Halim, Single earthquake bond pricing framework with double trigger parameters based on multi regional seismic information, Mathematics, 11 (2023), 689. https://doi.org/10.3390/math11030689 doi: 10.3390/math11030689
[26]	H. K. Mistry, D. Lombardi, A stochastic exposure model for seismic risk assessment and pricing of catastrophe bonds, Nat. Hazards, 117 (2023), 803–829. https://doi.org/10.1007/s11069-023-05884-4 doi: 10.1007/s11069-023-05884-4
[27]	L. Wei, L. Liu, J. Hou, Pricing hybrid-triggered catastrophe bonds based on copula-EVT model, Quant. Financ. Econ., 6 (2022), 223–243. https://doi.org/10.3934/QFE.2022010 doi: 10.3934/QFE.2022010
[28]	Y. E. Aghdam, A. Neisy, A. Adl, Simulating and pricing CAT bonds using the spectral method based on Chebyshev Basis, Comput Econ., 63 (2022), 423–435. https://doi.org/10.1007/s10614-022-10347-2 doi: 10.1007/s10614-022-10347-2
[29]	R. A. Ibrahim, Sukono, H. Napitupulu, R. I. Ibrahim, Earthquake bond pricing model involving the inconstant event intensity and maximum strength, Mathematics, 12 (2024), 786. https://doi.org/10.3390/math12060786 doi: 10.3390/math12060786
[30]	W. Anggraeni, S. Sudradjat, Sukono, N. A. Halim, Decomposition of disaster region using earthquake parameter and STDM distance: Catastrophe bond pricing single period, J. Adv. Res. Appl. Sci. Eng. Technol., 43 (2025), 171–200. https://doi.org/10.37934/araset.43.1.171200 doi: 10.37934/araset.43.1.171200
[31]	G. Reshetar, Pricing of multiple-event coupon paying CAT bond, Zurich: Swiss Banking Institute, 2008. https://doi.org/10.2139/ssrn.1059021
[32]	Sukono, R. A. Ibrahim, M. P. A. Saputra, Y. Hidayat, H. Juahir, I. G. Prihanto, et al, Modeling multiple-event catastrophe bond prices involving the trigger event correlation, interest, and inflation rates, Mathematics, 10 (2022), 4685. https://doi.org/10.3390/math10244685 doi: 10.3390/math10244685
[33]	Z. G. Ma, C. Q. Ma, Pricing catastrophe risk bonds: A mixed approximation method, Insur. Math. Econ., 52 (2013), 243–254. https://doi.org/10.1016/j.insmatheco.2012.12.007 doi: 10.1016/j.insmatheco.2012.12.007
[34]	Sukono, H. Napitupulu, Riaman, R. A. Ibrahim, M. D. Johansyah, R. A. Hidayana, A regional catastrophe bond pricing model and its application in Indonesia's provinces, Mathematics, 11 (2023), 3825. https://doi.org/10.3390/math11183825 doi: 10.3390/math11183825
[35]	H. Loubergé, E. Kellezi, M. Gilli, Using catastrophe-linked securities to diversify insurance risk: A financial analysis of Cat bonds, J. Insur. Issues., 22 (1999), 125–146. https://www.jstor.org/stable/41946177
[36]	R. A. Ibrahim, Sukono, Riaman, Estimation of the extreme distribution model of economic losses due to outbreaks using the POT method with Newton Raphson iteration, Int. J. Quant. Res. Model., 2 (2021), 37–45. https://doi.org/10.46336/ijqrm.v2i1.118 doi: 10.46336/ijqrm.v2i1.118
[37]	J. D. Cummins, M. A. Weiss, Convergence of insurance and financial markets: Hybrid and securitized risk-transfer solutions, J. Risk Insur., 76 (2009), 493–545. https://doi.org/10.1111/j.1539-6975.2009.01311.x doi: 10.1111/j.1539-6975.2009.01311.x
[38]	G. Woo, A catastrophe bond niche: multiple event risk, 2004.
[39]	S. Osaki, Poisson processes, In: Applied stochastic system modeling, Berlin: Springer, 1992. https://doi.org/10.1007/978-3-642-84681-6_3
[40]	J. C. Cox, J. E. Ingersoll, S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385–407. https://doi.org/10.2307/1911242 doi: 10.2307/1911242
[41]	P. Nowak, M. Romaniuk, Valuing catastrophe bonds involving correlation and CIR interest rate model, Comp. Appl. Math., 37 (2018), 365–394. https://doi.org/10.1007/s40314-016-0348-2 doi: 10.1007/s40314-016-0348-2
[42]	K. Kladivko, Maximum likelihood estimation of the Cox-Ingersoll-Ross process: The Matlab implementation.
[43]	G. Salvadori, C. D. Michele, N. T. Kottegoda, R. Rosso, Extreme in nature: An approach using copula, Dordrecht: Springer, 2007. https://doi.org/10.1007/1-4020-4415-1_1
[44]	S. H. Cox, H. W. Pedersen, Catastrophe risk bonds, North Am. Actuar. J., 4 (200), 56–82. https://doi.org/10.1080/10920277.2000.10595938 doi: 10.1080/10920277.2000.10595938

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