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題名 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
作者 方麒豪
Fang, Chi-Hao
貢獻者 許順吉<br>林士貴
Sheu, Shuenn-Jyi<br>Lin, Shih-Kuei
方麒豪
Fang, Chi-Hao
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
Neural Networks
Finite Difference
Merton PIDE
Black-Scholes PDE
European Call Option Price
日期 2022
上傳時間 1-Aug-2022 18:13:19 (UTC+8)
摘要 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.
參考文獻 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
30
描述 碩士
國立政治大學
應用數學系
109751007
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0109751007
資料類型 thesis
dc.contributor.advisor 許順吉<br>林士貴zh_TW
dc.contributor.advisor Sheu, Shuenn-Jyi<br>Lin, Shih-Kueien_US
dc.contributor.author (Authors) 方麒豪zh_TW
dc.contributor.author (Authors) Fang, Chi-Haoen_US
dc.creator (作者) 方麒豪zh_TW
dc.creator (作者) Fang, Chi-Haoen_US
dc.date (日期) 2022en_US
dc.date.accessioned 1-Aug-2022 18:13:19 (UTC+8)-
dc.date.available 1-Aug-2022 18:13:19 (UTC+8)-
dc.date.issued (上傳時間) 1-Aug-2022 18:13:19 (UTC+8)-
dc.identifier (Other Identifiers) G0109751007en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/141183-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 109751007zh_TW
dc.description.abstract (摘要) 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。zh_TW
dc.description.abstract (摘要) 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.en_US
dc.description.tableofcontents 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		zh_TW
dc.format.extent 654331 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0109751007en_US
dc.subject (關鍵詞) 類神經網路zh_TW
dc.subject (關鍵詞) 有限差分法zh_TW
dc.subject (關鍵詞) Merton 偏積分微分方程zh_TW
dc.subject (關鍵詞) Black- Scholes 偏微分方程zh_TW
dc.subject (關鍵詞) 歐式選擇權價格zh_TW
dc.subject (關鍵詞) Neural Networksen_US
dc.subject (關鍵詞) Finite Differenceen_US
dc.subject (關鍵詞) Merton PIDEen_US
dc.subject (關鍵詞) Black-Scholes PDEen_US
dc.subject (關鍵詞) European Call Option Priceen_US
dc.title (題名) 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zh_TW
dc.title (題名) Numerical Analysis of Option Partial Differential Equations: Applications of Finite Difference and Neural Networks Methodsen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) 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
30		zh_TW
dc.identifier.doi (DOI) 10.6814/NCCU202201023en_US