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| 研究生： |
方麒豪 Fang, Chi-Hao |
|---|---|
| 論文名稱： |
選擇權偏微分方程之數值分析: 有限差分法及類神經網路法之應用 Numerical Analysis of Option Partial Differential Equations: Applications of Finite Difference and Neural Networks Methods |
| 指導教授： |
許順吉
Sheu, Shuenn-Jyi 林士貴 Lin, Shih-Kuei |
| 口試委員: |
姜祖恕
Chiang, Tzuu-Shuh 江彌修 Chiang, Mi-Hsiu 陳亭甫 Chen, Ting-Fu 許順吉 Sheu, Shuenn-Jyi 林士貴 Lin, Shih-Kuei |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2022 |
| 畢業學年度： | 110 |
| 語文別： | 中文 |
| 論文頁數： | 39 |
| 中文關鍵詞： | 類神經網路 、有限差分法 、Merton 偏積分微分方程 、Black- Scholes 偏微分方程 、歐式選擇權價格 |
| 外文關鍵詞： | Neural Networks, Finite Difference, Merton PIDE, Black-Scholes PDE, European Call Option Price |
| DOI URL： | http://doi.org/10.6814/NCCU202201023 |
| 相關次數： | 點閱：145 下載：38 |
| 分享至: |
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M. Raissi et al.(2019) 首先提出使用監督式學習方法用於求解偏微分方程。他們著重於有封閉解的偏微分方程並且使用封閉解與預測值的差距作為類神經網路的損失函數於訓練中。Lu et al.(2019) 提出更有效率的演算法用於求解多種類型的偏微分方程，包含正演問題以及反演問題。本文將著重於觀察歐式選擇權的類神經網路預測值行為與封閉解的差距並且跟有限差分法進行比較。
M. Raissi et al.(2019) first proposed a supervised learning method for solving partial differential equations. They focused on partial differential equations that have closed form solutions and used the difference between closed form solutions and neural network outputs as loss function for training. Lu et al.(2019) presented an efficient algorithm for solving several types of partial differential equations, including forward problem and inverse problems. This dissertation aims at observing the behaviour of European call option prices predicted by neural networks and comparing it with closed form price.
中文摘要 i
Abstract ii
Contents iii
List of Tables v
List of Figures
1 Introduction 1
2 Literature Review 3
2.1 Finite Difference 3
2.2 Neural Network 3
3 Methodology 5
3.1 Black-Scholes Partial Differential Equation 5
3.2 Finite Difference for Black-Scholes PDE 7
3.3 Merton Partial Integro-Differential Equation 8
3.4 Finite Difference for Merton PIDE 12
3.5 Neural Network for Black-Scholes PDE 15
4 Numerical Results 20
4.1 Finite Difference 20
4.1.1 BS PDE 20
4.1.2 Merton PIDE 21
4.2 Neural Networks 27
4.2.1 BS PDE 27
5 Conclusions 29
References 30
A Derivation of Black-Scholes Option Price Formula 31
B Derivation of Merton Jump Diffusion Model Option Price Formula 33
C More Figures and Tables 39
Cont, R., & Voltchkova, E. (2006). A finite difference scheme for option pricing in jump diffusion and exponential lévy models. SIAM Journal on Numerical Analysis, 43(4), 1596–1626. https://doi.org/10.1137/S0036142903436186
Cybenko, G. (1989). Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems, 2, 303–314. https://doi.org/10.1007/BF02551274
Leshno, M., Lin, V. Y., Pinkus, A., & Schocken, S. (1993). Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6(6), 861–867. https://doi.org/10.1016/S0893-6080(05)80131-5
Lu, L., Meng, X., Mao, Z., & Karniadakis, G. E. (2021). Deepxde: A deep learning library for solving differential equations. SIAM Review, 63(1), 208–228. https://doi.org/10.1137/19M1274067
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378(1), 686–707. https://doi.org/10.1016/j.jcp.2018.10.045
Schwartz, E. (1977). The valuation of warrants: Implementing a new approach. Journal of Financial Economics, 4, 79–93. https://doi.org/10.1016/0304-405X(77)90037-X
