基于Esscher变换的巨灾指数期权定价与数值模拟

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。

Catastrophe Index Options Pricing Using Esscher-Transformation and Numerical Simulation

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.