Modelo discreto aleatorio para calcular el índice de pérdidas desencadenante de los Cat Bonds
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dic 5, 2022
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Resumen
En este trabajo se propone un modelo discreto aleatorio que describe la evolución del índice de pérdidas desencadenante de los bonos sobre catástrofes. Para ello, se supone que la cuantía total de la catástrofe es la suma de la cuantía declarada de siniestros y de la cuantía de siniestros pendientes de declaración. Tras establecer una dinámica de la declaración de siniestros determinista, en la que la aleatoriedad del modelo reside únicamente en la ocurrencia o no de una catástrofe se introduce la aleatoriedad en el modelo substituyendo la tasa de nominal de declaración de siniestros por una variable aleatoria dicotómica que permite simular una declaración de siniestros lenta o rápida en cada periodo.
Keywords
cat bonds, incurred but not yet reported claims amount, nominal claims reporting rate, binomial modelbonos sobre catástrofes, cuantía de siniestros pendiente de declaración, tasa nominal de declaración de siniestros, modelo binomial
References
Aase, K. (1999). An Equilibrium Model of Catastrophe Insurance Futures and Spreads. Geneva Papers on Risk and Insurance Theory, 24(1), 69-96.
Aase, K. (2001). A Markov Model for the Pricing of Catastrophe Insurance Futures and Spreads. Journal of Risk and Insurance, 68(1), 25-50.
Alegre, A., Pérez-Fructuoso, M. J., y Devolder, P. (2003). Modèles discrets d’options sur risques catastrophiques. Belgian Actuarial Bulletin, 3(1), 28-32.
Baryshnikov, Y., Mayo, A. y Taylor, D.R. (2001). Pricing of Cat Bonds, Working Paper, Version October. Disponible en: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.202.9296&rep=rep1&type=pdf
Biagini, F., Bregman, Y. y Meyer-Brandis, T. (2008). Pricing of catastrophe insurance options written on a loss index with reestimation. Insurance: Mathematics and Economics, 43(2), 214-222.
Braun, A. (2011). Pricing catastrophe swaps: A contingent claims approach. Insurance: Mathematics and Economics, 49(3), 520-536.
Burnecki, K. (2005). Pricing catastrophe bonds in a compound non-homogeneous Poisson model with left truncated loss distributions. Presentation Wroclaw University of Technology. Disponible en: https://www.researchgate.net/publication/289106928_Pricing_of_Catastrophe_Bonds
Burnecki, K. y Kukla, G. (2003). Pricing of zero-coupon and coupon cat bonds. Applicationes Mathematicae, 30, 315- 324.
Cox, S. H. y Pedersen, H. (2000). Catastrophe Risk Bonds. North American Actuarial Journal, 4 (4), 56-82.
Cox, J., Ross, S. y Rubinstein, M. (1979). Option Pricing: a simplified approach. Journal of Financial Economics, 7, 229-263.
Cummins J. D. y Geman, H. (1995): Pricing Catastrophe Insurance Futures and Call Spreads: An Arbitrage Approach. Journal of Fixed Income, 4 (4), 46-57.
Chang, C.W., Chang, J.S.K. y Lu, W. (2010): Pricing catastrophe options with stochastic claim arrival intensity in claim time. Journal of Banking & Finance, 34 (1), 24-32.
Egami, M. y Young, V. (2008). Indifference prices of structured catastrophe (CAT) bonds. Insurance: Mathematics and Economics, 42(2), 771-778.
Embrechts, P. y Meister, S. (1997). Pricing insurance derivatives, the case of CAT futures. En Proceedings of the 1995 Bowles Symposium on Securitization of Insurance Risk, Georgia State University, Atlanta, Georgia. Society of Actuaries, Monograph M-FI97-1: 15-26.
Geman, H. y Yor, M. (1997). Stochastic time changes in catastrophe option pricing. Insurance: Mathematics and Economics, 21(3),185-193.
Jaimungal, S. y Wang, T. (2006). Catastrophe options with stochastic interest rates and compound Poisson losses. Insurance: Mathematics and Economics, 38(3), 469-483.
Jarrow, R.A. (2010). A simple robust model for Cat bond valuation. Finance Research Letters, 7, 72-79
Lai, V. S., Parcollet, M. y Lamond, B. F. (2014). The valuation of catastrophe bonds with exposure to currency exchange risk. International Review of Financial Analysis, 33(C), 243-252.
Lee, J. P. y Yu, M. T. (2002). Pricing default-risky Cat bonds with moral hazard and basis risk. Journal of Risk and Insurance, 69 (1), 25-44.
Lee, J. P. y Yu, M. T. (2007). Valuation of catastrophe reinsurance with catastrophe bonds, Insurance: Mathematics and Economics, 41 (2), 264-278.
Loubergé, H., Kellezi, E. y Gilli, M. (1999): Using Catastrophe-Linked Securities to Diversify Insurance Risk: A Financial Analysis of Cat Bonds. Journal of Insurance Issues, 22 (2), 125-146.
Muermann, A. (2003). Actuarially Consistent Valuation of Catastrophe Derivatives. Working Paper Series The Wharton Financial Institutions Center, 03-18, 2003. Disponible en: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.387.1798&rep=rep1&type=pdf
Nowak, P. y Romaniuk, M (2013). Princing and simulations of catastrophe Bonds. Insurance: Mathematics and Economics, 52(1), 18-28.
Pérez-Fructuoso, M. J. (2005). La titulizaciónn del riesgo catastrófico: descripción y análisis de los cat bonds (Bonos de Catástrofes). Revista Española de Seguros, 121, 75-92.
Pérez-Fructuoso, M. J. (2008). Modelling loss index triggers for CAT bonds: a continuous approach. Variance, 2(2), 253-265.
Pérez-Fructuoso, M. J. (2009). Elaborating a catastrophic loss index for insurance-linked securities (ILS): A continuous model. Asia-Pacific Journal of Risk and Insurance, 3(2), 1-13.
Pérez-Fructuoso, M. J. (2016a). Tarificación de derivados sobre catástrofes con desencadenantes de índices de pérdidas: modelo asintótico basado en un proceso de Wiener. Rect@, 17(1), 81-103.
Pérez-Fructuoso, M. J. (2016b). Modelo discreto cierto (y simple) para determinar una ratio de pérdidas por catástrofes subyacente de los derivados sobre seguros (insurance-linked derivatives, OLS). Revista Ibero-Latinoamericana de Seguros, 44, 185-205. http://dx.doi.org/10. 11144/Javeriana.ris44.mdcd
Pérez-Fructuoso, M. J. (2017). Tarificación de bonos sobre catástrofes (cat bonds) con desencadenantes de índices de pérdidas. Modelización mediante un proceso de Ornstein-Uhlenbeck. Revista de métodos Cuantitativos para la Economía y la Empresa, 24, 340-361.
Polacek, A. (2018). Catastrophe bonds: A primer and retrospective. [online] Chicago Fed Leeter, The Federal Reserve Bank of Chicago. Disponible en: https:// https://www.chicagofed.org/publications/chicago-fed-letter/2018/405
Shao, J., Pantelous, A., Papaioannou, A. D. (2015). Catastrophe risk bonds with applications to earthquakes. European Actuarial Journal, 5: 113-138
Unger, A.J.A. (2010). Pricing index-based catastrophe bonds: Part 1. Formulation and discretization issues using a numerical PDE approach. Computers & Geosciences, 36 (2), 139-149.
Zong-Gang, M. y Chao-Qun, M. (2013). Princing catastrophe risk Bonds: a mixed approximation method. Insurance: Mathematics and Economics, 52 (2), 243-254.
Wang, X. (2016). Catastrophe equity put option with target variance. Insurance: Mathematics and Economics, 71(C), 79-86.
Aase, K. (2001). A Markov Model for the Pricing of Catastrophe Insurance Futures and Spreads. Journal of Risk and Insurance, 68(1), 25-50.
Alegre, A., Pérez-Fructuoso, M. J., y Devolder, P. (2003). Modèles discrets d’options sur risques catastrophiques. Belgian Actuarial Bulletin, 3(1), 28-32.
Baryshnikov, Y., Mayo, A. y Taylor, D.R. (2001). Pricing of Cat Bonds, Working Paper, Version October. Disponible en: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.202.9296&rep=rep1&type=pdf
Biagini, F., Bregman, Y. y Meyer-Brandis, T. (2008). Pricing of catastrophe insurance options written on a loss index with reestimation. Insurance: Mathematics and Economics, 43(2), 214-222.
Braun, A. (2011). Pricing catastrophe swaps: A contingent claims approach. Insurance: Mathematics and Economics, 49(3), 520-536.
Burnecki, K. (2005). Pricing catastrophe bonds in a compound non-homogeneous Poisson model with left truncated loss distributions. Presentation Wroclaw University of Technology. Disponible en: https://www.researchgate.net/publication/289106928_Pricing_of_Catastrophe_Bonds
Burnecki, K. y Kukla, G. (2003). Pricing of zero-coupon and coupon cat bonds. Applicationes Mathematicae, 30, 315- 324.
Cox, S. H. y Pedersen, H. (2000). Catastrophe Risk Bonds. North American Actuarial Journal, 4 (4), 56-82.
Cox, J., Ross, S. y Rubinstein, M. (1979). Option Pricing: a simplified approach. Journal of Financial Economics, 7, 229-263.
Cummins J. D. y Geman, H. (1995): Pricing Catastrophe Insurance Futures and Call Spreads: An Arbitrage Approach. Journal of Fixed Income, 4 (4), 46-57.
Chang, C.W., Chang, J.S.K. y Lu, W. (2010): Pricing catastrophe options with stochastic claim arrival intensity in claim time. Journal of Banking & Finance, 34 (1), 24-32.
Egami, M. y Young, V. (2008). Indifference prices of structured catastrophe (CAT) bonds. Insurance: Mathematics and Economics, 42(2), 771-778.
Embrechts, P. y Meister, S. (1997). Pricing insurance derivatives, the case of CAT futures. En Proceedings of the 1995 Bowles Symposium on Securitization of Insurance Risk, Georgia State University, Atlanta, Georgia. Society of Actuaries, Monograph M-FI97-1: 15-26.
Geman, H. y Yor, M. (1997). Stochastic time changes in catastrophe option pricing. Insurance: Mathematics and Economics, 21(3),185-193.
Jaimungal, S. y Wang, T. (2006). Catastrophe options with stochastic interest rates and compound Poisson losses. Insurance: Mathematics and Economics, 38(3), 469-483.
Jarrow, R.A. (2010). A simple robust model for Cat bond valuation. Finance Research Letters, 7, 72-79
Lai, V. S., Parcollet, M. y Lamond, B. F. (2014). The valuation of catastrophe bonds with exposure to currency exchange risk. International Review of Financial Analysis, 33(C), 243-252.
Lee, J. P. y Yu, M. T. (2002). Pricing default-risky Cat bonds with moral hazard and basis risk. Journal of Risk and Insurance, 69 (1), 25-44.
Lee, J. P. y Yu, M. T. (2007). Valuation of catastrophe reinsurance with catastrophe bonds, Insurance: Mathematics and Economics, 41 (2), 264-278.
Loubergé, H., Kellezi, E. y Gilli, M. (1999): Using Catastrophe-Linked Securities to Diversify Insurance Risk: A Financial Analysis of Cat Bonds. Journal of Insurance Issues, 22 (2), 125-146.
Muermann, A. (2003). Actuarially Consistent Valuation of Catastrophe Derivatives. Working Paper Series The Wharton Financial Institutions Center, 03-18, 2003. Disponible en: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.387.1798&rep=rep1&type=pdf
Nowak, P. y Romaniuk, M (2013). Princing and simulations of catastrophe Bonds. Insurance: Mathematics and Economics, 52(1), 18-28.
Pérez-Fructuoso, M. J. (2005). La titulizaciónn del riesgo catastrófico: descripción y análisis de los cat bonds (Bonos de Catástrofes). Revista Española de Seguros, 121, 75-92.
Pérez-Fructuoso, M. J. (2008). Modelling loss index triggers for CAT bonds: a continuous approach. Variance, 2(2), 253-265.
Pérez-Fructuoso, M. J. (2009). Elaborating a catastrophic loss index for insurance-linked securities (ILS): A continuous model. Asia-Pacific Journal of Risk and Insurance, 3(2), 1-13.
Pérez-Fructuoso, M. J. (2016a). Tarificación de derivados sobre catástrofes con desencadenantes de índices de pérdidas: modelo asintótico basado en un proceso de Wiener. Rect@, 17(1), 81-103.
Pérez-Fructuoso, M. J. (2016b). Modelo discreto cierto (y simple) para determinar una ratio de pérdidas por catástrofes subyacente de los derivados sobre seguros (insurance-linked derivatives, OLS). Revista Ibero-Latinoamericana de Seguros, 44, 185-205. http://dx.doi.org/10. 11144/Javeriana.ris44.mdcd
Pérez-Fructuoso, M. J. (2017). Tarificación de bonos sobre catástrofes (cat bonds) con desencadenantes de índices de pérdidas. Modelización mediante un proceso de Ornstein-Uhlenbeck. Revista de métodos Cuantitativos para la Economía y la Empresa, 24, 340-361.
Polacek, A. (2018). Catastrophe bonds: A primer and retrospective. [online] Chicago Fed Leeter, The Federal Reserve Bank of Chicago. Disponible en: https:// https://www.chicagofed.org/publications/chicago-fed-letter/2018/405
Shao, J., Pantelous, A., Papaioannou, A. D. (2015). Catastrophe risk bonds with applications to earthquakes. European Actuarial Journal, 5: 113-138
Unger, A.J.A. (2010). Pricing index-based catastrophe bonds: Part 1. Formulation and discretization issues using a numerical PDE approach. Computers & Geosciences, 36 (2), 139-149.
Zong-Gang, M. y Chao-Qun, M. (2013). Princing catastrophe risk Bonds: a mixed approximation method. Insurance: Mathematics and Economics, 52 (2), 243-254.
Wang, X. (2016). Catastrophe equity put option with target variance. Insurance: Mathematics and Economics, 71(C), 79-86.
Cómo citar
Pérez Fructuoso, M. J. (2022). Modelo discreto aleatorio para calcular el índice de pérdidas desencadenante de los Cat Bonds. Revista Ibero-Latinoamericana De Seguros, 31(57). https://doi.org/10.11144/Javeriana.ris57.mdac
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Sección Técnico-económica
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