基于Esscher变换的巨灾指数期权定价与数值模拟

中国管理科学 ›› 2014, Vol. 22 ›› Issue (1): 20-28.

基于Esscher变换的巨灾指数期权定价与数值模拟

程铖¹, 石晓军², 张顺明²

1. 北京航空航天大学经济管理学院, 北京 100191;
2. 中国人民大学财政金融学院, 北京 100872

收稿日期:2011-12-29 修回日期:2012-10-10 出版日期:2014-01-20 发布日期:2014-01-20
基金资助:
国家自然基金资助项目（71172014）；国家杰出青年科学基金项目（70825003）

Catastrophe Index Options Pricing Using Esscher-Transformation and Numerical Simulation

CHENG Cheng¹, SHI Xiao-jun², ZHANG Shun-ming²

1. School of Economics and Management, Beijing University of Aeronautics and Astronautics, Beijing 100191, China;
2. School of Finance, Renmin University of China, Beijing 100872, China

Received:2011-12-29 Revised:2012-10-10 Online:2014-01-20 Published:2014-01-20

摘要/Abstract

摘要： 巨灾指数期权是最重要的巨灾衍生工具之一，在我国有很好的发展前景。但巨灾指数期权在我国推广的一个主要技术障碍是，在信息较少的情况下，如何对巨灾指数期权进行快速的定价。本文提出了一种基于Esscher变换的巨灾指数期权定价的解析表达公式，区别于以往文献采用亚式期权或随机时间变化的方法。这个方法的优势在于能够反映巨灾指数的跳跃性、两部性（损失期和延展期）、上界性特点。同时，Esscher 变换的无套利等价性也赋予该方法坚实的理论基础，有较好的延展性，可以使用多种分布过程。首先，具体给出漂移泊松、漂移伽马和维纳过程条件下的巨灾指数期权定价公式。通过数值模拟分析结果与Black-Scholes公式结果及巨灾指数历史数据的对比，认为基于漂移伽马过程的定价结果能更好地反映巨灾指数的特点。最终，指出了巨灾指数的开发和本文提出的方法在中国具有很好的应用前景。

关键词: 巨灾指数期权, Esscher变换, 漂移伽马过程

Abstract: Catastrophe index options are kinds of the most important catastrophe derivatives at present and hence have large potential in China. However, a major obstacle in the usage of this instrument in China is how to price it under given limited market information as there is not yet an actively traded market for it in China. A closed-form formula of the catastrophe index options pricing based on Esscher transformation is proposed in this paper which has a sound theoretical foundation. Three major features of the catastrophe index: jumping, two periods and two caps can be captured by the proposed method. Moreover, the flexibility of Esscher transformation allaws the method to apply to various distributions. Thus formulas of catastrophe index options pricing are obtained under shifted Poisson, shifted Gamma and Wiener processes respectively in this paper. Simulation part compares the different formulas with the standard Black-Scholes theorem as well as historical PCS catastrophe loss indices, which indicates that the shifted Gamma process is a good candidate for the implied stochastic process of the catastrophe index options pricing. The exploration of catastrophe index option and application of our method is of practical relevance in China.

Key words: catastrophe options, Esscher transformation, Shifted Gamma process

中图分类号:

程铖, 石晓军, 张顺明. 基于Esscher变换的巨灾指数期权定价与数值模拟[J]. 中国管理科学, 2014, 22(1): 20-28.

CHENG Cheng, SHI Xiao-jun, ZHANG Shun-ming. Catastrophe Index Options Pricing Using Esscher-Transformation and Numerical Simulation[J]. Chinese Journal of Management Science, 2014, 22(1): 20-28.

参考文献

[1] 中国保监监督管理委员会."5.12"汶川特大地震保险理赔工作基本完成[EB/OL] .(2009-05-11). http://www.circ.gov.on/tabid/106/InfoID/100457/frtid/3871/Default.aspx.
[2] Cummins J D, German H. An Asian option approach to the valuation of insurance futures contracts[R]. Working Papers, Wharton Financial Institutions, 1993.
[3] Cummins J D, German H. Pricing catastrophe insurance futures and call spreads: An arbitrage approach[J]. Journal of Fixed Income, 1995, 4(March):46-57.
[4] Chang W C, Chang S K J, Lu W. Pricing and hedging catastrophe-linked risk in discrete time[J]. Insurance: Mathematics and Economics, 2008, 43: 422-430.
[5] German H, Yor M. Stochastic time changes in catastrophe option pricing[J]. Insurance: Mathematics and Economics, 1997, 21: 185-193.
[6] Chang W C, Chang S K J, Yu M. Pricing catastrophe insurance futures call spreads: A randomized operational time approach[J]. Journal of Risk and Insurance, 1996, 63(4):599-617.
[7] Biagini F, Bregman Y, Meyer-Brandis T. Pricing of catastrophe insurance options written on a loss index with reestimation[J]. Insurance: Mathematics and Economics, 2008, 43(2): 214-222.
[8] Wu Yangche, Chung S L. Catastrophe risk management with counterparty risk using alternative instruments[J]. Insurance: Mathematics and Economics. 2010, 47(2):234-245.
[9] Aase K K. An equilibrium model of catastrophe insurance futures and spreads[J]. The Geneva Papers on Risk and Insurance Theory, 1999, 24(1), 69-96.
[10] Aase K K. A Markov model for the pricing of catastrophe insurance futures and spreads[J]. Journal of Risk and Insurance, 2001, 68(1):25-50.
[11] Young V R. Pricing in an incomplete market with an affine term structure[J]. Mathematical Finance, 2004, 14(3): 359-381.
[12] Young V R, Zariphopoulou T. Pricing dynamic insurance risks using the principle of equivalent utility[J]. Scandinavian Actuarial Journal, 2002, 4:246-279.
[13] Hobson D, Henderson V. Utility indifference pricing-an overview[M]//Carmona R. Indifference pricing: Theory and applications Princeton: Princeton University Press, 2009.
[14] Lim T, Quenez M C. Portfolio optimization in a defaults model under full/partial information[J]. Quantitative Finance, 2010.
[15] Elliott R J, Siu T K. A stochastic differential game for optimal investment of an insurer with regime switching[J]. Quantitative Finance, 2011, 11(3): 365-380.
[16] Chen An, Pelsser A, Vellekoop M. Modeling non-monotone risk aversion using SAHARA utility functions[J]. Journal of Economic Theory, 2011, 146(5):2075-2092.
[17] Ikefuji M, Laeven R J A, Magnus J R, et al. Expected utility and catastrophic risk in a stochastic economy-climate model[R]. CentER Discussion Paper Series, 2010.
[18] 尚勤, 秦学志, 周颖颖, 巨灾死亡率债券定价模型研究[J].系统工程学报, 2010, 25(2):203-208.
[19] 丁波, 巴曙松.中国地震巨灾期权定价机制研究[J].中国管理科学, 2010, 18(15): 34-39.
[20] Buhlmann H. An economic premium principle[J]. ASTIN Bulletin, 1980, 11(1): 52-60.
[21] Mürmann A. Pricing catastrophe insurance derivatives[R]. Discussion Paper, Financial Markets Group, 2001.
[22] Mürmann A. Actuarially consistent valuation of catastrophe derivatives[J]. Working paper, 2003.
[23] Mürmann A. Market price of insurance risk implied by catastrophe derivatives[J]. North American Actuarial Journal, 2006, 12(3), 221-227.
[24] 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[J]. Journal of Applied Corporate Finance, 1997, 10(3), 69-83.
[25] Davis M H A. Option pricing in incomplete markets[M]//Dempster M A H, Pliska S R. Mathematics of derivatives securities. Cambridge: Cambridge University Press, 1997.
[26] Froot K A. The market for catastrophe risk: A clinical examination[J]. Journal of Finance Economics, 2001, 60, 529-571.
[27] Hoyt R E, McCullough K A. Catastrophe insurance options: Are they zero-beta assets?[J]. Journal of Insurance Issues, 1999, 22(2), 147-163.
[28] Beard R E, Pentikinen T, Pesonen E. Risk theory: The stochastic basis of insurance[M]. London: Chapman and Hall, 1984.
[29] Beekman J A. Two stochastic processes[J]. Stockholm: Almqvist & Wiksell, 1974.
[30] Breiman L. Probability[M]. Menlo Park, Calif.: Addison-Wesley, 1968.
[31] Feller W. An introduction to probability theory and its applications[M]. New York: Wiley, 1971.
[32] Huebner S S. Foundation monograph series[M]. Homewood, Ill.: Irwin, 1979.
[33] Jensen J L. Saddle point approximations to the distribution of the total claim amount in some recent risk models[J]. Scandinavian Actuarial Journal, 1991, (2):154-68.
[34] Seal H L. The stochastic theory of a risk business[J]. New York: Wileym, 1969.