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Author: 簡憶茹
Yi-Ju Chien
Thesis Title: 寇氏跳躍-擴散模型的不同變化延伸之研究
A Study on Different Extended Variations on Kou’s Jump-Diffusion Model
Advisor: 繆維中
Wei-Chung Miao
林昌碩
Xenos, Chang-Shuo Lin
Committee: 韓傳祥
Chuan-Hsiang Han
陳俊男
Chun-Nan Chen
Degree: 碩士
Master
Department: 管理學院 - 財務金融研究所
Graduate Institute of Finance
Thesis Publication Year: 2019
Graduation Academic Year: 108
Language: 英文
Pages: 66
Keywords (in Chinese): 跳躍-擴散模型1F1 函數快速傅立葉 轉換法(FFT)常態-伽瑪混合模型常態-伽瑪函數
Keywords (in other languages): jump-diffusion model,, 1F1 function, Mixture Exponential as Gamma, Fast Fourier Transform(FFT), Normal-Gamma function
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  • 布雷克-休斯選擇權定價公式在過去的幾十年當中被廣泛的應用; 然而,這模型卻無
    法完美地捕捉到真實的金融市場跳躍的狀況。因而莫頓提出了另一種模型來證明選擇
    權和衍生品市場中確實存在“波動度微笑”。為了捕捉跳躍的真實情況,本論文研究了
    延伸寇氏的跳躍擴散模型,並對其進行了三種延伸,包括:(1)常態-伽瑪混合分佈描
    述跳躍事件,1F1 函數推導到概率密度函數;(2)混合伽瑪分佈可以描繪在一個短暫時
    間裡發生的跳躍事件:(3)快速傅立葉變換提供我們高效率又快速的定價工具, 對於機
    率密度函數較為複雜的模型可以快速的得到累積密度函數。在研究這些延伸模型時,
    本文將討論這些模型有效率的特點及其限制。


    Black-Scholes option pricing formula has been widely used in the past decades;
    however, this model cannot perfectly capture the real situation of financial market jump events. Therefore, Merton proposed another model to prove that there is indeed a "volatility smile" in the options and derivatives markets. In order to capture the real situation of jumping, this thesis studies the extension of Kou’s jump-diffusion model and makes three extensions to it, including: (1) Shift Normal-Gamma jump-diffusion model depicts the jump events, and 1F1 function is introduced here to function as probability density function; (2) Mixture Exponential as Gamma distribution can capture one jump event in brief time. The double Normal-Gamma jump-diffusion model uses mixture Exponential limited to shape parameter < “1” ; (3) Fast Fourier Transformation (FFT) provides us with a high efficient and effective tools for the option pricing. In the study of these extended models, this thesis will discuss the characteristics of these extended models and their limitations.

    Recommendation Letter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Approval Letter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Abstract in Chinese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Model 1 : SNGJD - Shift Normal-Gamma Jump-Diffusion Model . . . . . . 6 2.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Conclusion Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Model 2 : 1-DNGJD - Double Normal-Gamma Jump-Diffusion Model by Using Mixtures Exponential as Gamma Model . . . . . . . . . . . . . . . . 22 3.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Conclusion Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Model 3 : DGJD - Double Normal-Gamma Jump-Diffusion Model by Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.3 Conclusion Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5 Conclusions and Future work . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Appendix A: Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Appendix B: Matlab Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Letter of Authority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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