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研究生： 翁新傑
Hsin-Chieh Wong
論文名稱： 離散監測跳躍擴散模型之跨界問題與財務應用
Boundary Crossing Problem under Discrete Monitored Jump-Diffusion Models with Finance Applications
指導教授： 傅承德
Cheng-Der Fuh
口試委員:
學位類別： 博士
Doctor
系所名稱： 理學院 - 統計研究所
Graduate Institute of Statistics
論文出版年： 2017
畢業學年度： 105
語文別： 英文
論文頁數： 99
中文關鍵詞： 跨界 、過衝 、連續校正 、拉普拉斯轉換 、離散選擇權 、跳躍擴散模型 、市場化
外文關鍵詞： boundary crossing, overshoot, continuity correction, Laplace transform, discrete options, jump-diffusion models, marketability
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    • 在本論文中，我們研究了混合指數跳躍擴散模型下跨越平坦邊界之首次通過時間。我們建立一個對於平坦的通過時間及其停止時間的過程之共同分配的連續校正，以便在跨越時間離散監視交叉事件。跨界問題的連續性校正有廣泛的應用，包括離散屏障選項，離散回溯選項，以及一些關於財務分析的離散監控。將Keener 的方法從偏微分方程延拓到偏積分微分方程來完成這項工作。與現有文獻不同，新提出的校正量與擴散部分和跳躍部分皆有相關。數值結果表明，我們的結果確實提高了近似性能，特別是當監測頻率低，邊界在起始點附近時。當將這種新方法應用於離散障礙選擇權和回溯選擇權的定價時，將獲得類似的結論。

    In this dissertation, we study first passage times of crossing a flat boundary under mixed exponential jump diffusion models. We establish a continuity correction for the joint distribution of a flat passage time and its stopped underlying process when the event of crossing is monitored only discretely. Continuity correction for boundary crossing problem is motived by a wide class of applications, including discrete barrier options, discrete lookback options, and some on discrete monitoring of financial analysis. The work is done by extending the Keener’s approach from partial differential equations to partial integro-differential equations. Unlike the exiting literature, the newly proposed correction amounts (regarding the boundary level) are different across different scenarios and related to both the diffusion and jumps parts. Numerical results indicate that such our results do improve the approximation performance especially when the monitoring frequency is low and the boundary is nearby. Similar conclusions are obtained when applying this new method to the pricing of a discrete barrier option and lookback option.

    中文摘要.................................................i Abstract...............................................ii 誌謝..................................................iii 1 Introduction........................................1 2 Problem Formulation and Methodology...................4 2.1 The model...........................................4 2.2 First passage times.................................6 2.3 PIDE approach and principle of smooth fit...........7 3 Main Results.........................................10 3.1 Barrier option pricing.............................11 3.2 Special case: Double-exponential jump-diffusion model (DEM) of Kou (2002)....................................12 4 Proof of Theorem 1...................................17 4.1 Error estimation...................................17 4.2 An approximation for Ef^1..........................21 4.3 Combining Ef^0 and Ef^1............................26 5 Numerical Analyses...................................28 5.1 MGF................................................28 5.2 Joint probability..................................37 5.3 Barrier option pricing.............................44 6 The Finance Application: Marketability 56 6.1 The upper bound under the MEM......................59 6.2 Longstaff’s model vs. our result...................67 6.3 A Comparison.......................................68 6.4 Concluding Remarks.................................69 7 Conclusion...........................................71 Appendix...............................................73 A Overshoot problem....................................73 B Two estimates of supplementary.......................79 C Proofs of Corollary 1 to 3...........................81 Bibliography...........................................85

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