×

Catastrophe insurance derivatives pricing using a Cox process with jump diffusion CIR intensity. (English) Zbl 1417.91273

Summary: We propose an analytical pricing method for stop-loss reinsurance contracts and catastrophe insurance derivatives using a Cox process with jump diffusion Cox-Ingersoll-Ross (CIR) intensity. The expected payoff of these contracts is expressed by the Laplace transform of the integration of the jump diffusion CIR process and the first moment of the aggregate loss. To confirm that the proposed analytical formula provides stable and accurate insurance derivative prices, we simulate them using a full Monte Carlo method compared to those obtained from its theoretical expectation. It shows that it is much faster way to obtain them than the full Monte Carlo method. We also conduct sensitivity analysis by changing the relevant parameters in the loss intensity providing their figures.

MSC:

91B30 Risk theory, insurance (MSC2010)
91G20 Derivative securities (option pricing, hedging, etc.)
60J75 Jump processes (MSC2010)
91G60 Numerical methods (including Monte Carlo methods)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albrecher, H.; Asmussen, S., Ruin probabilities and aggregate claims distributions for shot noise Cox processes, Scandinavian Actuarial Journal, 2, 2, 86-110, (2006) · Zbl 1129.91022
[2] Asmussen, S., Ruin Probabilities, (2000), Singapore: World Scientific, Singapore
[3] Bakshi, G.; Madan, D., Spanning and derivative-security valuation, Journal of Financial Economics, 55, 205-238, (2000)
[4] Bartlett, M. S., The spectral analysis of point processes, Journal of the Royal Statistical Society, 25, 264-296, (1963) · Zbl 0124.08504
[5] Basu, S.; Dassios, A., A Cox process with log-normal intensity, Insurance: Mathematics and Economics, 31, 2, 297-302, (2002) · Zbl 1055.91038
[6] Bremaud, P., Point Processes and Queues: Martingale Dynamics, (1981), Berlin, Heidelberg, New York: Springer, Berlin, Heidelberg, New York · Zbl 0478.60004
[7] Cai, N.; Kou, S. G., Option pricing under a mixed-exponential jump diffusion model, Management Science, 57, 11, 2067-2081, (2011)
[8] Cai, N.; Kou, S., Pricing Asian options under a hyper-exponential jump diffusion model, Operations Research, 60, 1, 64-77, (2012) · Zbl 1241.91111
[9] Carr, P.; Madan, D., Option valuation using the fast Fourier transform, Journal of Computational Finance, 2, 4, 61-73, (1999)
[10] S. P. D’Arcy & R. W. Gorvett (1999) Pricing catastrophe risk: Could CAT futures have coped with Andrew? Casualty Actuarial Society, https://www.casact.org/pubs/dpp/dpp99/99dpp59.pdf.
[11] Chung, S. F.; Wong, H. Y., Analytical pricing of discrete arithmetic Asian options with mean reversion and jumps, Journal of Banking and Finance, 44, 7, 130-140, (2014)
[12] Cox, D. R., Some statistical methods connected with series of events, Journal of the Royal Statistical Society B, 17, 129-164, (1955) · Zbl 0067.37403
[13] Cox, J.; Ingersoll, J.; Ross, S., A theory of the term structure of interest rates, Econometrica, 53, 2, 385-407, (1985) · Zbl 1274.91447
[14] Dassios, A.; Jang, J., Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity, Finance and Stochastics, 7, 1, 73-95, (2003) · Zbl 1039.91038
[15] Duffie, D.; Filipovic, D.; Schachermayer, W., Affine processes and applications in finance, The Annals of Applied Probability, 13, 3, 984-1053, (2003) · Zbl 1048.60059
[16] Grandell, J., Doubly Stochastic Poisson Processes, (1976), Berlin, Heidelberg, New York: Springer, Berlin, Heidelberg, New York · Zbl 0339.60053
[17] Grandell, J., Aspects of Risk Theory, (1991), Berlin, Heidelberg, New York: Springer, Berlin, Heidelberg, New York · Zbl 0717.62100
[18] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6, 327-343, (1993) · Zbl 1384.35131
[19] Jang, J., Jump diffusion processes and their applications in insurance and finance, Insurance: Mathematics and Economics, 41, 1, 62-70, (2007) · Zbl 1119.91054
[20] Jang, J.; Fu, G., Transform approach for operational risk management: VaR and TCE, Journal of Operational Risk, 3, 2, 45-61, (2008)
[21] Jang, J.; Fu, G., Measuring tail dependence for aggregate collateral losses using bivariate compound shot-noise Cox/Poisson process, Applied Mathematics, 3, 12, 2191-2204, (2012)
[22] Kim, B.; Wee, I. S., Pricing of geometric Asian options under Heston’s stochastic volatility model, Quantitative Finance, 14, 10, 1795-1809, (2014) · Zbl 1402.91792
[23] Macci, C.; Torrisi, G. L., Risk processes with shot noise Cox claim number process and reserve dependent premium rate, Insurance: Mathematics and Economics, 48, 1, 134-145, (2011) · Zbl 1233.91152
[24] Report of 2009 Victorian Bushfires Royal Commission (2010), Final Report Summary, Parliament of Victoria, Australia.
[25] Serfozo, R. F., Conditional Poisson processes, Journal of Applied Probability, 9, 288-302, (1972) · Zbl 0237.60026
[26] Zheng, W.; Yuen, C.; Kwok, Y., Recursive algorithms for pricing discrete variance options and volatility swaps under time-changed Levy processes, International Journal of Theoretical and Applied Finance, 19, 2, 1650011, (2016) · Zbl 1337.91132
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.