壽命連結商品是保險公司用於規避人們壽命延長（長壽風險）或偶然巨災事件發生（死亡率風險）的工具，這些商品利用證券化的手法將可能的損失移轉給投資人。瑞士再保險公司於2003年底推出的Swiss Re mortality bond 就是個相當成功的例子，也是本文用於訂價之商品。文章中引入了均衡定價模型，假設死亡率指數服從Transformed Gamma分配，且假設投資人的期末財富、邊際效用及標的資產的期末價值以特定的形式表達，經由這三層假設得到風險中立評價關係式。因此，以無風險利率折現的期望報酬便可視為壽命連結商品的價格，進一步也可推導出類似Black-Scholes 公式的封閉解，建構於Transformed Gamma分配下的債券參考價格便可得知。

Due to the frequent catastrophes all over the world in recent years, it causes the extreme mortality rate and gives the insurance companies and reinsurance companies a pound. In order to transfer the mortality systematic risk to capital markets, the mortality-linked securities were issued. The issuance of Swiss Re mortality bond in the end of 2003 year is an example. In this paper, we assume the mortality rate has a transformed gamma distribution and the security is priced by an equilibrium method in the discrete time economy. Furthermore, the risk neutral valuation relationship (RNVR) is obtained. The price of any life-related security is the sum of expected payoff discounted by risk-free rate. Finally, under the restricted conditions of the investor’s preference, the distribution of the mortality and wealth, we can obtain the closed-form solution of the mortality-linked securities and we take the Swiss Re mortality bond as numerical example.

摘要 ii
Abstract iii
Table of Contents iv
List of Figures v
List of Tables vi
1 Introduction 1
2 Literature Review 3
2.1 The Mortality Bond 3
2.2 The Mortality Process 5
2.3 The Transformed Gamma Distribution 6
3 Pricing Model 9
3.1 Equilibrium Pricing Model 9
3.2 Security Price 11
3.3 The Closed-form Solutions for Option Price 14
4 Parameter Estimation and Valuation 16
4.1 The Simulation of Mortality Index 16
4.2 Parameter Estimation by Transformed Gamma Distribution 17
4.3 Mortality Bond Valuation 22
5 Conclusion 25
Appendix 26
Appendix A: Prove the pricing kernel and asset-specific pricing kernel 26
Appendix B. Prove Corollary 1 28
Appendix C. Derive log gamma option pricing formula 28
Appendix D. Derive Weibull option pricing formula 29
Reference 31

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