本論文涵蓋三個研究主題。我們首先簡介選擇權評價的理論演進與方法，接著引入後續研究主題的相關文獻。第一個主題是推導在GARCH-Lévy過程下的歐式選擇權評價，並做台股指數選擇權的實證分析，在與其它普遍使用的模型做比較，結果顯示GARCH- Lévy過程在樣本內有好預測，但是以三的損失函數來度量樣本外績效，則無一模型在所有價內外水準皆有最好表現，價平時以VG模型較好，價內外則是NGARCH模型。

第二個主題是在多元normal inverse Gaussian模型下去評價兩個組合式選擇權，我們推得幾何組合式選擇權的解析評價公式，對此公式偏微分得到避險參數。同理，也得到算術平均組合式選擇權的漸近評價公式，和它的避險參數。此類選擇權可應用來評價連結多元標的保險契約價值。蒙地卡羅數值模擬結果確定本公式的準確度。

第三個主題是研究巨災保險商品的價值。我們檢視巨災的資料發現雖然它無法被預測，但是卻顯現出某些規則，例如，地震的能量釋放會遞延下次的發生，週期2至7年的聖嬰現象，對全球海洋、氣候及熱浪有著循環與季節性的影響。我們採用機制轉換複合卜松過程來描述在不同自然狀態下的巨災損失與發生過程。 大部份的巨災保險契約都有理賠條件或債務豁免條款。因此，我們推導障礙選擇權的評價公式，並且以此為基礎去評價巨災保險商品。巨災資料的實證分析確認在模描述巨災過程時，機制轉換複合卜松過程優於單一卜松過程。敏感度分析顯示不同的參數對價格的影響。

This dissertation includes three topics. We first introduce the statistical properties of option pricing and literature related to the following topics. The first topic focuses on GARCH processes with Lévy innovation and their empirical analysis on TAIEX index options. Comparing to other popular option pricing models, the results show that the GARCH-Lévy processes fit well in-sample data. However, according to out-of- sample performances from three loss functions, there is no best model across all moneynesses. Although the VG option pricing model performs well at the money, the NGARCH option pricing model for in-the-money or out-of-the-money contracts is better than others.

The second topic studies two baskets options under a multivariate normal inverse Gaussian model. The value of a geometric basket option can be expressed as an analytic-form formula and then its hedge ratios can be obtained from the partial derivatives of its pricing formula. Similarly, the value of an arithmetic basket option can be expressed as an analytic-form formula and then its hedge ratios can be obtained from the partial derivatives of its pricing formula. These options can be further applied to price related products, for example, multifund unit-linked insurance contracts. The numerical result supports the internal consistency of our closed-form analytical expressions for two basket options on two assets.

The third topic studies the valuation of catastrophe insurance products. We survey the data about catastrophe events. Catastrophe occurrences can be forecasted, yet appear to have some rules, for example, the energy released by an earthquake can delay the next occurrence. Moreover, the 2-7 year cycle or pattern referred to as ENSO is a frequent natural reminder about the complex influences of the global ocean, atmosphere, and continental heat budget cycling and seasonality. The regime-switching compound Poisson process can be adopted to describe the jump-diffusion process under different states and thus be incorporated into the catastrophe loss or claim dynamics under different natural environments. Most catastrophe insurance contracts have provisions on some triggers to make loss claims or debt-forgiveness. Thus, we derive the pricing formulas of trigger options and then pricing catastrophe insurance products using an option-based method. The empirical evidences show that the regime-switching Poisson process in the RJSD model fits better than the pure Poisson process in a jump diffusion model in describing the arrival rates of great natural disasters over 1950-2006. We can further extend it to enough states to fit CAT arrival better, and then price other catastrophe insurance products more exactly.

TABLE OF CONTENTS
List of Tables
List of Figures
Abstract
Chapter 1: Introduction………………………………………….…….…..….…...1
1.1 Background……………………………………….….….……………1
1.2 Objectives……………………………………….…………………....6
1.3 Framework…………………………………….…………….………..7
Chapter 2: Literature review…………………………………………….…………8
2.1 Literature related to GARCH-Lévy processes…………..………..9
2.2 Literature related to basket options………………………...……...9
2.3 Literature related to catastrophe products……………………..…11
Chapter 3: GARCH-Lévy processes and an empirical analysis on TAIEX
Index options………………………………………………………….16
3.1 The time-changed Lévy process……………...…………………..16
3.2 The GARCH

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