馬可夫跳躍擴散模型下巨災保險商品之定價

Pricing Catastrophe Insurance Products in Markov Jump Diffusion Models

林士貴(Shih-Kuei Lin) ； 徐守德(David Shyu) ； 張嘉倩(Chia-Chien Cahng)

《財務金融學刊》 16卷2期 (2008/06) Pp. 1-33

https://doi.org/10.6545/JFS.2008.16(2).1

摘要

過去文獻假設巨災事件到達率服從普瓦松過程，然而此過程固定頻率之假設並不適用於實際巨災資料，因此本文提出馬可夫調整並瓦松過程以捕捉實際巨災事件到達率過程。在此過程下，狀態服從齊一的馬可夫鏈，此過程可退化為Cummins and Genman（1993，1995）、Chang，Chang and Yu（1996）、Geman and Yor（1997）以及Vaugirard（2003a，2003b）之模型。本文應用馬可夫跳躍擴散模型推導出巨災期貨買權、巨災PCS價差買權與巨災債券之封閉解。在實分析方面，我們利用1950年至2004年PCS指數與颶風事件發生次數之資料，檢定馬可夫調和普瓦松過程與普瓦松過程之評價巨災商品配適能力。在數值分析方面，顯示巨災保險商品與巨災事件跳躍次數、跳躍規模的標準差、跳躍規模的平均數之間的關係。

關鍵字

馬可夫調整普瓦松過程 ； 馬可夫跳躍擴散模型 ； 巨災期貨買權 ； 巨災PCS價差買權 ； 巨災債券

並列摘要

For catastrophic events, the assumption that catastrophe claims occur in terms of the Poisson process is inadequate as it has constant intensity. This article proposes Markov Modulated Poisson process to model the arrival process for catastrophic events. Under this process, the underlying state is governed by a homogenous Markov chain, and it is the generalization of Cummins and Geman (1993. 1995). Chang, Chang, and Yu (1996). Geman and Yor (1997) and Vaugirard (2003a, 2003b). We apply Markov jump diffusion model to derive pricing formulas for catastrophe insurance products, included catastrophe futures call option, catastrophe PCS call spread and catastrophe bond. We use the data of PCS index and the annual number of hurricane events during 1950 to 2004 to test the quality of the filling under the MMPP and the PP. The numerical analysis shows how the catastrophe insurance products prices are related to jump rate of catastrophe events. standard deviation of jump size, and mean of jump size.

並列關鍵字

Markov modulated Poisson process ； Markov jump diffusion model ； Futures eall option ； Catastrophe PCS call spread ； Catastrophe bond

參考文獻

Abate, J.,W. Whitt(1992).Numerical Inversion of Probability Generating Functions.Operations Research Letters.12,245-251.

Canter, M.,J. B. Cole,R. L. Sandor(1997).Insurance Derivatives: A New Asset Class for the Capital Markets and A New Fledging Tool for the Insurance Industry.Journal of Applied Corporate Finance.10(3),69-83.

Chang, C.,J. Chang,M.-T, Yu(1996).Pricing Catastrophe Insurance Futures Call Spreads: A Randomized Operational Time Approach.Journal of Risk and Insurance.63,599-617.

Cox, S. H.,R. G. Schwebach(1992).Insurance Futures and fledging Insurance Price Risk.Journal of Risk and Insurance.59,628-644.

Wharton School Center for Financial Institutions, Working Paper