Document Type : Research Article

Authors

1 Allameh Tabatba&#039;i Univerisy

2 PHD in the Finance,Department of Finance and Banking, Faculty of Management &amp; Accounting, Allameh Tabataba&#039;i University, Tehran, Iran

3 Management and accounting, Allameh tabatabai&#039;e university

4 Finance and Banking, Faculty of Management &amp; Accounting, Allameh Tabataba&#039;i University, Tehran, Iran.

Abstract

Pricing catastrophe swap as an instrument for insurance companies risk management, has received trivial attention in the previous studies, but in most of them, damage severities caused by the disaster has been considered to be fixed. In this study, through considering jumps for modeling the occurrence of disasters as in Unger [32] and completing it through considering damages caused by natural disasters as stochastic, an integro-differential model was extracted to value catastrophe swap contracts. In determining the swap price changes, the Ito command was followed and to achieve the catastrophic swap model, the generalization of the Black and Scholes modeling method was used. [3]. With regard to the initial and boundary conditions, extracted model does not have an analytical solution; thus, its answer was approximated using the finite difference numerical method and the effect of considering the damage as stochastic on swap value was analyzed. In addition, the model and the extracted numerical solution were separately implemented on the data about the earthquake damage in the United States and Iran. The results showed that prices will experience a regular upward trend until damage growth, damage severities, and occurrence probability of a catastrophe are not so high that the buyer of the swap is forced to pay compensation to the swap’s seller. Of course, the prices will fall sharply as soon as they reach and cross the threshold.

Keywords

References
[1] Aase , K.K., A Markov model for the pricing of catastrophe insurance futures and spreads, Journal of Risk and Insurance ,68 (1), (2001), pp. 2549.
[2] Azizi, S. Neisy, A. , A New Approach in Geometric Brownian Motion Model. Fuzzy Information and Engineering and Decision, Advances in Intelligent Systems and Computing, vol 646. Springer.
[3] Black, F., Scholes, M., AThe pricing of options and corporate liabilities, Journal of Political Economy, 81 (3), (1973) ,pp. 637654.
[4] Biagini, F., Bregman, Y., & Meyer-Brandis, T., Pricing of catastrophe insurance options written on a loss index with re-estimation. Insurance, Mathematics and Economics, 43(2), (2008) ,pp. 214-222.
[5] Bjork, T. , Arbitrage theory in continuous time, Oxford University Press, Oxford, 3rd ed (2009).
[6] Borden, S., Sarkar, A., Securitizing property catastrophe risk, Current Issues in Economics and Finance, 2 (9), (1996) ,pp. 16.
[7] Braun, A., Pricing catastrophe swaps: A contingent claims approach. Insurance , Mathematics and Economics, 49(3), (2011) ,pp. 520-536.
[8] Canter, M.S., Cole, J.B., Sandor, R.L., Insurance derivatives: a new asset class for the capital markets and a new hedging tool for the insurance industry, Applied Corporate Finance, 10 (3), (1997) ,pp. 6981.
[9] Chang, C.W., Chang, J.S.K., Yu, M.-T., Pricing catastrophe insurance futures call spreads, Risk and Insurance, Vol. 63, No. 4, (1996) ,pp. 599-617
[10] Chang, C. W., Chang, J. S., & Lu, W., Pricing catastrophe options in discrete operational time, Insurance: Mathematics and Economics, 43(3), (2008) ,pp. 422-430.
[11] Chang, C. W., Chang, J. S., & Lu, W., Pricing catastrophe options with stochastic claim arrival intensity in claim time, Banking & Finance, 34(1), (2010) ,pp. 24-32.
[12] Cummins, J.D., CAT bonds and other risk-linked securities: state of the market and recent developments, Risk Management and Insurance Review ,11 (1), (2008) ,pp. 2347.
[13] Cummins, J.D., Geman, H., An Asian option approach to the valuation of insurance futures contracts, Review of Futures Markets, 13 (2), (1994) ,pp. 517557.
[14] Cummins, J.D., Geman, H., Pricing catastrophe insurance futures and call spreads: an arbitrage approach, Journal of Fixed Income, 4 (4), (1995) ,pp. 4657.
[15] Cummins, J. D., & Weiss, M. A., Convergence of insurance and  nancial markets: Hybrid and securitized risktransfer solutions, Risk and Insurance, 76(3), (2009) ,pp. 493-545.
[16] Deutsche Bank, 2006 , Deutsche Bank launches Event Loss Swaps to help clients hedge against disasters.,
[17] Egami, M., Young, V.R., Indifference prices of structured catastrophe (CAT) bonds, Insurance: Mathematics and Economics, 42 (2), (2008) ,pp. 771778.
[18] EM-DAT: The Emergency Events Database - Universite catholique de Louvain (UCL) - CRED - www.emdat.be, Brussels, Belgium., ,
[19] Geman, H., Yor, M., Stochastic time changes in catastrophe option pricing, Insurance: Mathematics and Economics, 21 (3), (1997) ,pp. 185193.
[20] Hardle, W. K., & Cabrera, B. L., Calibrating CAT bonds for Mexican earthquakes , Risk and Insurance, 77(3), (2010) ,pp. 625-650.
[21] Hirsa, Ali,, Computational Methods in Finance, Chapman and Hall/CRC, Financial Mathematics Series, Taylor & Francis (2012).
[22] Hull, J. C. , Options futures and other derivatives (10th Edition). , Pearson Education,(2018).
[23] Ionut florescu, Ruihua liu, Maria Cristina Mariana, Solutions to a partial integrodifferential parabolic system arising in the pricing of  nancial options in regime-switching jump diffusion models, Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 231, (2012) ,pp. 112.
[24] Lane, M.N., Pricing risk transfer transactions, Astin Bulletin 30 (2), (2000) ,pp. 259293.
[25] Lane, M., & Mahul, O., Catastrophe risk pricing: an empirical analysis, International Bank for Reconstruction and Development, The World Bank (2008)
[26] Lee, J. P., & Yu, M. T., Valuation of catastrophe reinsurance with catastrophe bonds, Insurance: Mathematics and Economics, 41(2), (2007) ,pp. 264-278.
[27] Macmillan Publishers Limited., Pool knowledge to stem losses from disasters, Nature, vol 522, 18 june(2015).
[28] Safaei, Maryam & Neisy, Abodolsadeh & Nematollahi, Nader, New Splitting Scheme for Pricing American Options Under the Heston Model, Computational Economics, Springer;Society for Computational Economics, vol. 52(2), (2018) ,pages 405-420, August.
[29] Muermann, A., Market price of insurance risk implied by catastrophe derivatives, North American Actuarial Journal, 12(3), (2008) ,pp. 221-227.
[30] Munich Re NatCatSERVICE, Natural catastrophes in the  rst half of 2018.,
[31] Neisy, A, Salmani, K., An inverse  nance problem for estimation of the volatility, Computational Mathematics and Mathematical Physics, Volume 53, Issue 1, (2013) ,pp 6377. Springer.
[32] Unger, A.J.A., Pricing index-based catastrophe bonds: Part 1: Formulation and discretization issues using a numerical PDE approach , Computers & Geosciences,36, 2, 2010, pp.139-149.
[33] Vickery, P., Skerlj, P., Lin, J., Twisdale, L., Jr, Young, M., & Lavelle, F., HAZUS-MH Hurricane model methodology. II: Damage and loss estimation, Natural Hazards Review,7, (2006) ,pp. 94103.
[34] Vranes, K., & Pielke Jr, R., Normalized earthquake damage and fatalities in the United States: 19002005, Natural Hazards Review, 10(3), (2009) ,pp. 84-101.
[35] Wu, Y. C., & Chung, S. L., Catastrophe risk management with counterparty risk using alternative instruments, Insurance: Mathematics and Economics, 47(2), (2010) ,pp. 234-245.
[36] Xu, Y., A Study of the Loss Distribution of Natural Disasters in Norway Comparing a Common Model with a Model Broken Down into Catastrophe Types, University of Oslo Library, (2016).
[37] Young, V.R., Pricing in an incomplete market with an affine term structure, Mathematical Finance, 14 (3), (2004) ,pp. 359381.
[38] Zolfaghari, M.R., and Campbell, K.W., A New Insurance Loss Model to Promote Catastrophe Insurance Market in India And Pakistan, The 14th World Conference on Earthquake Engineering October 12-17, Beijing, China (2008).