Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T04:43:50.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Pricing of Catastrophe Insurance Options Under Immediate Loss Reestimation

Part of: Stochastic processes Miscellaneous applications of functional analysis Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Francesca Biagini ,
Yuliya Bregman  and
Thilo Meyer-Brandis
Francesca Biagini*
Affiliation:
Ludwig-Maximilians Universität München
Yuliya Bregman*
Affiliation:
Ludwig-Maximilians Universität München
Thilo Meyer-Brandis*
Affiliation:
University of Oslo
*
∗Postal address: Department of Mathematics, Ludwig-Maximilians Universität München, Theresienstrasse 39, D-80333 Munich, Germany.
∗Postal address: Department of Mathematics, Ludwig-Maximilians Universität München, Theresienstrasse 39, D-80333 Munich, Germany.
∗∗∗∗Postal address: CMA, University of Oslo, Postbox 1035, Blindern, Norway. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We specify a model for a catastrophe loss index, where the initial estimate of each catastrophe loss is reestimated immediately by a positive martingale starting from the random time of loss occurrence. We consider the pricing of catastrophe insurance options written on the loss index and obtain option pricing formulae by applying Fourier transform techniques. An important advantage is that our methodology works for loss distributions with heavy tails, which is the appropriate tail behavior for catastrophe modeling. We also discuss the case when the reestimation factors are given by positive affine martingales and provide a characterization of positive affine local martingales.

Keywords

Catastrophe insurance optionloss indexFourier transformoption pricing formulaheavy tailpositive affine martingale

MSC classification

Secondary: 46N30: Applications in probability theory and statistics 60G07: General theory of processes 91B30: Risk theory, insurance
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 3 , September 2008 , pp. 831 - 845
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Biagini, F., Bregman, Y. and Meyer-Brandis, T. (2008). Pricing of catastrophe insurance options written on a loss index. To appear in Insurance Math. Econom. CrossRefGoogle Scholar
[2] Christensen, C. V. (1999). A new model for pricing catastrophe insurance derivatives. CAF Working Paper Series No. 28.Google Scholar
[3] Çinlar, E., Jacod, J., Protter, P. and Sharpe, M. J. (1980). Semimartingales and Markov processes. Prob. Theory Relat. Fields 54, 161–219.Google Scholar
[4] Corless, R. M. et al. (1996). On the Lambert W function. Adv. Comput. Math. 5, 329–359.CrossRefGoogle Scholar
[5] Duffie, D., Pan, J. and Singleton, K. (2000). Transform analysis and asset pricing for affine Jump diffusions. Econometrica 68, 1343–1376.CrossRefGoogle Scholar
[6] Duffie, D., Filipovic, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.CrossRefGoogle Scholar
[7] Filipović, D. (2001). A general characterization of one factor affine term structure models. Finance Stoch. 5, 389–412.CrossRefGoogle Scholar
[8] Kavazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 36–54.Google Scholar
[9] Königsberger, K. (1993). Analysis 2. Springer, Berlin.Google Scholar
[10] Leipnik, R. B. (1991). On lognormal random variables. I. The characteristic function. J. Austral. Math. Soc. Ser. B 32, 327–347.CrossRefGoogle Scholar
[11] Möller, M. (1996). Pricing PCS-options with the use of Esscher-transforms. AFIR Colloquium Nürnberg, October 1–3 1996. Available at http://www.actuaries.org/AFIR/colloquia/Nuernberg/papers.cfm.Google Scholar
[12] Mürmann, A. (2001). Pricing catastrophe insurance derivatives. Financial Markets Group Discussion Paper 400. Available at http://ssrn.com/abstract=31052.Google Scholar
[13] Mürmann, A. (2003). Actuarially consistent valuation of catastrophe derivatives. Working paper. Available at http://knowledge.wharton.upenn.edu/muermann/.Google Scholar
[14] Mürmann, A. (2006). Market price of insurance risk implied by catastrophe derivatives. Working paper. Available at http://irm.wharton.upenn.edu/muermann/.Google Scholar
[15] Rolski, T., Schmidli, H., Schmidt, V. and Teugles, J. L. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
[16] Schmidli, H. (2001). Modeling PCS options via individual indices. Working paper 187, Laboratory of Actuarial Mathematics, University of Copenhagen.Google Scholar
[17] Schradin, H. R. (1996). PCS Catastrophe insurance options—a new instrument for managing catastrophe risk. Contribution to the 6th AFIR International Colloquium Nürnberg, October 1–3 1996. Available at http://www.actuaries.org/AFIR/colloquia/Nuernberg/papers.cfm.Google Scholar
[18]Sigma Nr. 3 (2001). Capital Market Innovation in the Insurance Industry. Swiss Re, Zürich.Google Scholar
[19]Sigma Nr. 7 (2006). Securitization—New Opportunities for Insurers and Investors. Swiss Re, Zürich.Google Scholar
[20]Sigma Nr. 2 (2007). Natural Catastrophes and Man-Made Disasters in 2006. Swiss Re, Zürich.Google Scholar