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研究生:余欣庭
研究生(外文):HSIN-TING YU
論文名稱:馬可夫轉換之跳躍擴散模型應用於選擇權訂價與投資組合保險
論文名稱(外文):Markov-Switching Jump-Diffusion Models Applied to Option Pricing and Portfolio Insurance
指導教授:繆維中繆維中引用關係
指導教授(外文):Daniel Wei-Chung Miao
口試委員:繆維中
口試日期:2012-07-14
學位類別:博士
校院名稱:國立臺灣科技大學
系所名稱:財務金融研究所
學門:商業及管理學門
學類:財務金融學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
語文別:英文
論文頁數:102
中文關鍵詞:厚尾狀態轉換跳躍擴散模型間斷的卜松過程轉換的卜松過程選擇權訂價投資組合保險
外文關鍵詞:heavy tailregime-switchingjump-diffusion modelinterrupted Poisson processswitched Poisson processoption pricingportfolio insurance
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本文目標係在呈現馬可夫轉換之跳躍擴散模型為選擇權訂價與投資組合保險之必要工具,其能對權益指數波動與市場風險提供適切描述,同時捕捉資產報酬的兩項顯著特性:厚尾與波動率叢聚。在實證分析支持權益指數跳躍行為往往較常發生於風險時期,而較少發生在正常時期的基礎上,本文假設驅動跳躍的卜松分配係由兩狀態的馬可夫鏈所支配。此假設包含了對莫頓的跳躍擴散模型之兩個延伸建議,據以反應市場狀態變遷下的非時間同質性跳躍,讓跳躍可間斷地發生或隨著循環結構而改變,進而產生在這些兩狀態下的不同資產報酬動態。其中之一是間斷的卜松過程,解析在間斷的跳躍擴散模型下之選擇權訂價與模型本身的時間非同質特性;另外一項是以轉換的卜松過程作為本文的一般化議題。
本文從計算矩陣指數函數為基礎對建議的模型作了完整的分析,並特別就根本的開與關之馬可夫鏈推導了條件狀態機率之遞迴公式解,這些解析結果足以作為在正常與風險狀態下選擇權訂價的快速演算法。數值釋例也說明了時間非同質性是如何影響報酬分配、選擇權價格與波動性微笑,特別是加上了歐米茄測度與投資組合保險的應用。這些在不同狀態所看到的明顯不同態樣,指出了時間同質模型的不足,也證明本文模型的用途。因此,本文藉由提供選擇權訂價與投資組合保險一個切實與解析的應用方法,對隨機模型學術作出貢獻。
The goal of this thesis is to show that the Markov-switching jump-diffusion models are an essential tool for option pricing and portfolio insurance, and that they provide an adequate description of index price fluctuations and market risks. The benefit is that the Markov-switching jump-di usion models simultaneously capture two salient features in asset returns: heavy tailness and volatility clustering. On the basis of an empirical analysis where jumps are found to happen much more frequently in risky periods than in normal periods, we assume that the Poisson process for driving jumps are governed by a two-state Markov chain. This assumption consists of two proposed extensions of Mertons jump-diffusion model to reflect the time inhomogeneity caused by changes of market states. This makes jumps happen interruptedly or change with the cyclical structure, and helps to generate different dynamics under these two states. One of them focus on interrupted Poisson process; option pricing in interrupted jump-diffusion model and its model feature of time inhomogeneity. The other one take switched Poisson process as our generalized theme.
We provide a full analysis for the proposed models based on computing matrix exponential and particularly derive the recursive formulas for the conditional state probabilities of the underlying on-off Markov chain. These analytical results lead to a fast algorithm that can be implemented to determine the prices of European options under normal and risky states. Numerical examples are given to demonstrate how time inhomogeneity influences return distributions, option prices, and volatility smiles. In particular, numerical interpretations of the omega measure and portfolio insurance are further provided. The contrasting patterns seen in different states indicate the insufficiency of using time-homogeneous models and justify the use of the proposed models. Therefore, this work contributes to the stochastic modeling literature by providing a realistic and analytical approach for the option pricing and application problems.
Abstract i
Acknowledgements iii
List of Figures vii
List of Tables ix
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Review of Option Pricing and Portfolio Insurance 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Empirical Properties of Asset Returns . . . . . . . . . . . . . . . . . . . . . 9
2.3 Modeling Beyond the GBM . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 A brief review of jump-diffusion model . . . . . . . . . . . . . . . . . 14
2.3.2 Another potentials of compound Poisson process aside from the lognormal
assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Portfolio Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Option-based portfolio insurance (OBPI) strategy . . . . . . . . . . . 19
2.4.2 Constant proportion portfolio insurance (CPPI) strategy . . . . . . . 20
2.4.3 Measuring portfolio insurance performance . . . . . . . . . . . . . . . 21
3 Interrupted Jump-Diffusion Model Applied to Option Pricing 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Empirical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 Conditional return distributions under different market states . . . . 24
3.2.2 Testing jumps in different states . . . . . . . . . . . . . . . . . . . . . 30
3.3 Interrupted Jump Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.1 Modeling interrupted jumps with a two-state Markov chain . . . . . . 35
3.3.2 Model analysis and pricing algorithm . . . . . . . . . . . . . . . . . . 36
3.4 Applications to Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Effects on the distribution of R(t) . . . . . . . . . . . . . . . . . . . . 48
3.4.2 Effects on option prices . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.3 Effects on volatility smile . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.4 A more comprehensive numerical study . . . . . . . . . . . . . . . . . 54
3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Switched Jump-Diffusion Model Applied to Portfolio Insurance 58
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 A More Comprehensive Study to Discuss Jumps in Both States . . . . . . . 59
4.2.1 Conditional return distributions under jump periods . . . . . . . . . . 60
4.2.2 Heaviness under jump periods . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Switched Jump-Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.1 Model analysis and pricing algorithm . . . . . . . . . . . . . . . . . . 67
4.3.2 IJD model as a special case of SJD model . . . . . . . . . . . . . . . 70
4.3.3 IJD model as a boundary case of SJD model . . . . . . . . . . . . . . 72
4.4 Applications to Portfolio Insurance . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.1 The preliminary set up . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.2 Numerical analysis of OBPI as a generalized CPPI . . . . . . . . . . 77
4.4.3 Numerical comparisons on the omega measure of S(t) . . . . . . . . . 80
4.4.4 Numerical comparisons on the omega measure of OBPI . . . . . . . . 83
4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Conclusions 88
5.1 Summaries of the proposed Models . . . . . . . . . . . . . . . . . . . . . . . 88
5.2 Potential Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A Technical Details 91
A.1 Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.2 The Counts and Period Length in Both States . . . . . . . . . . . . . . . . . 96
Bibliography 97
Copyright Agreement 102
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