Catastrophe Risk Derivatives: A New Approach

ABSTRACT

The multiplication of disasters during the last two decades beside the urbanism expansion has made catastrophe claims grow dramatically. Against a priced reinsurance, catastrophe derivative products became ever more attractive to insurance companies. A robust pricing of these derivatives is based on an appropriate modeling of the loss index. The current study proposes a unique model that takes into account the statistical characteristics of the loss amount’s tails to assess its real distribution. Thus, unlike previous models, we elaborately do not make any assumption regarding the probability of jump sizes to facilitate the calculation of the option price but deduct it instead of using Extreme Value Theory. The core of our model is a jump process that allows later for loss amounts’ re-estimation. Using both the Esscher transform and the martingale approach, we present the price of a call option on the loss index in a closed form. Finally, to confirm the underpinning theory of the model, numerical examples are presented as well as an algorithm that can be used to derive the option prices in real time.

Cite this paper

M. Abdessalem and M. Ohnishi, "Catastrophe Risk Derivatives: A New Approach,"*Journal of Mathematical Finance*, Vol. 4 No. 1, 2014, pp. 21-34. doi: 10.4236/jmf.2014.41003.

M. Abdessalem and M. Ohnishi, "Catastrophe Risk Derivatives: A New Approach,"

References

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http://dx.doi.org/10.1016/j.insmatheco.2003.12.006

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[6] J. C. Duan and S. Giambastiani, “Faire Insurance Guaranty Premia in Presence of Risk Based Capital Regulations, Stochastic Interest Rate and Catastrophe Risk,” Journal of Banking and Finance, Vol. 28, No. 10, 2005, pp. 2435-2454.

http://dx.doi.org/10.1016/j.jbankfin.2004.08.012

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[13] F. Esscher, “On the Probability Function in the Collective Theory of Risk,” Skandinavisk Aktuarietidskrift, Vol. 15, No. 3, 1932, pp. 175-195.

http://dx.doi.org/10.1080/03461238.1932.10405883

[14] U. H. Gerber and E. S. W. Shiu, “Martingale Approach to Pricing Perpetual American Options,” ASTIN Bulletin, Vol. 24, No 2, 1994, pp. 1995-220. http://dx.doi.org/10.2143/AST.24.2.2005065

[15] H. R. Schradin, “PCS Catastrophe Insurance Options a New Instrument for Managing Catastrophe Risk,” Zeitschrift für die gesamte Versicherungswissenschaft, Vol. 83, 1994, pp. 633-682.

[16] F. Biagini, Y. Bergman and T. Meyer-Brandis, “Pricing of Catastrophe Insurance Options under Immediate Loss Reestimation,” Journal of Applied Probability, Vol. 45, No. 3, 2008, pp. 831-845. http://dx.doi.org/10.1239/jap/1222441832

[17] R. C. Merton, “Option Prices When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 125-144. http://dx.doi.org/10.1016/0304-405X(76)90022-2

[1] H. Cox, J. Fairchild and H. Pederson, “Valuation of Structured Risk Management Products,” Insurance: Mathematics and Economics, Vol. 34, No. 2, 2004, pp. 259-272.

http://dx.doi.org/10.1016/j.insmatheco.2003.12.006

[2] H. Geman and M. Yor, “Stochastic Time Change in CAT Option Pricing,” Insurance: Mathematics and Economics, Vol. 21, No. 2, 1997, pp. 185-193. http://dx.doi.org/10.1016/S0167-6687(97)00017-6

[3] A. Muerman, “Actuarially Consistent Valuation of CAT Derivatives,” Working Paper of the Wharton Financial Institution Center, 2003, 3-18.

[4] S. Jaimungal and T. Wang, “Catastrophe Options with Stochastic Interest Rates and Compound Poisson Losses,” Insurance: Mathematics and Economics, Vol. 38, No. 3, 2006, pp. 469-483. http://dx.doi.org/10.1016/j.insmatheco.2005.11.008

[5] A. Dasios and J.-W. Jang, “Pricing of Catastrophe Reinsurance and Derivatives Using the Cox Process with Shot Noise Intensity,” Finance and Stochastics, Vol. 7, No. 1, 2003, pp. 73-95. http://dx.doi.org/10.1007/s007800200079

[6] J. C. Duan and S. Giambastiani, “Faire Insurance Guaranty Premia in Presence of Risk Based Capital Regulations, Stochastic Interest Rate and Catastrophe Risk,” Journal of Banking and Finance, Vol. 28, No. 10, 2005, pp. 2435-2454.

http://dx.doi.org/10.1016/j.jbankfin.2004.08.012

[7] T. Fujita, N. Ishimura and D. Tanaka, “An Arbitrage Approach to the Pricing of Catastrophe Options Involving the Cox Process,” Hitotsubashi Journal of Economics, Vol. 49, No. 2, 2008, pp. 67-74.

[8] H. Kanamori, “Earthquake Prediction: An Overview,” IASPEI Handbook of Earthquake and Engineering Seismology, 2000.

[9] S. Coles, “An Introduction to Statistical Modelling of Extreme Values,” Springer Series in Statistics, 2001.

http://dx.doi.org/10.1007/978-1-4471-3675-0

[10] I. A. Fraga and C. Neves, “Extreme Value Distributions,” International Encyclopedia of Statistical Science, 2011.

http://dx.doi.org/10.1007/978-3-642-04898-2_246

[11] M. Evans, N. Hastings and J. Peacock, “Statistical Distributions,” 3rd Edition, Wiley, New York, 2000.

[12] N. L. Johnson, S. Kotz and A. W. Kemp, “Univariate Discrete Distributions”, 2nd Edition, John Wiley and Sons, New York, 1992.

[13] F. Esscher, “On the Probability Function in the Collective Theory of Risk,” Skandinavisk Aktuarietidskrift, Vol. 15, No. 3, 1932, pp. 175-195.

http://dx.doi.org/10.1080/03461238.1932.10405883

[14] U. H. Gerber and E. S. W. Shiu, “Martingale Approach to Pricing Perpetual American Options,” ASTIN Bulletin, Vol. 24, No 2, 1994, pp. 1995-220. http://dx.doi.org/10.2143/AST.24.2.2005065

[15] H. R. Schradin, “PCS Catastrophe Insurance Options a New Instrument for Managing Catastrophe Risk,” Zeitschrift für die gesamte Versicherungswissenschaft, Vol. 83, 1994, pp. 633-682.

[16] F. Biagini, Y. Bergman and T. Meyer-Brandis, “Pricing of Catastrophe Insurance Options under Immediate Loss Reestimation,” Journal of Applied Probability, Vol. 45, No. 3, 2008, pp. 831-845. http://dx.doi.org/10.1239/jap/1222441832

[17] R. C. Merton, “Option Prices When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 125-144. http://dx.doi.org/10.1016/0304-405X(76)90022-2