寇氏跳躍-擴散模型的不同變化延伸之研究｜國立臺灣科技大學博碩士論文系統

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研究生：	簡憶茹 Yi-Ju Chien
論文名稱：	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
指導教授：	繆維中 Wei-Chung Miao 林昌碩 Xenos, Chang-Shuo Lin
口試委員:	韓傳祥 Chuan-Hsiang Han 陳俊男 Chun-Nan Chen
學位類別：	碩士 Master
系所名稱：	管理學院 - 財務金融研究所 Graduate Institute of Finance
論文出版年：	2019
畢業學年度：	108
語文別：	英文
論文頁數：	66
中文關鍵詞：	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
外文關鍵詞：	jump-diffusion model,, 1F1 function, Mixture Exponential as Gamma, Fast Fourier Transform(FFT), Normal-Gamma function
相關次數：	點閱：237 下載：1
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查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

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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