Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning

Many problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDEs) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, thi...

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Main Authors: Xiangdong Liu, Yu Gu
Format: Article
Language:English
Published: MDPI AG 2023-06-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/11/12/2658
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author Xiangdong Liu
Yu Gu
author_facet Xiangdong Liu
Yu Gu
author_sort Xiangdong Liu
collection DOAJ
description Many problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDEs) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, this paper applies the deep learning algorithm to solve a class of high-dimensional nonlinear partial differential equations with jump terms and their corresponding backward stochastic differential equations (BSDEs) with jump terms. Using the nonlinear Feynman-Kac formula, the problem of solving this kind of PDE is transformed into the problem of solving the corresponding backward stochastic differential equations with jump terms, and the numerical solution problem is turned into a stochastic control problem. At the same time, the gradient and jump process of the unknown solution are separately regarded as the strategy function, and they are approximated, respectively, by using two multilayer neural networks as function approximators. Thus, the deep learning-based method is used to overcome the “curse of dimensionality” caused by high-dimensional PDE with jump, and the numerical solution is obtained. In addition, this paper proposes a new optimization algorithm based on the existing neural network random optimization algorithm, and compares the results with the traditional optimization algorithm, and achieves good results. Finally, the proposed method is applied to three practical high-dimensional problems: Hamilton-Jacobi-Bellman equation, bond pricing under the jump Vasicek model and option pricing under the jump diffusion model. The proposed numerical method has obtained satisfactory accuracy and efficiency. The method has important application value and practical significance in investment decision-making, option pricing, insurance and other fields.
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spelling doaj.art-dfc390aac4d5493a8af67d63cdcd8fdc2023-11-18T11:27:59ZengMDPI AGMathematics2227-73902023-06-011112265810.3390/math11122658Study of Pricing of High-Dimensional Financial Derivatives Based on Deep LearningXiangdong Liu0Yu Gu1Department of Statistics and Data Science, Jinan University, Guangzhou 510632, ChinaDepartment of Statistics and Data Science, Jinan University, Guangzhou 510632, ChinaMany problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDEs) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, this paper applies the deep learning algorithm to solve a class of high-dimensional nonlinear partial differential equations with jump terms and their corresponding backward stochastic differential equations (BSDEs) with jump terms. Using the nonlinear Feynman-Kac formula, the problem of solving this kind of PDE is transformed into the problem of solving the corresponding backward stochastic differential equations with jump terms, and the numerical solution problem is turned into a stochastic control problem. At the same time, the gradient and jump process of the unknown solution are separately regarded as the strategy function, and they are approximated, respectively, by using two multilayer neural networks as function approximators. Thus, the deep learning-based method is used to overcome the “curse of dimensionality” caused by high-dimensional PDE with jump, and the numerical solution is obtained. In addition, this paper proposes a new optimization algorithm based on the existing neural network random optimization algorithm, and compares the results with the traditional optimization algorithm, and achieves good results. Finally, the proposed method is applied to three practical high-dimensional problems: Hamilton-Jacobi-Bellman equation, bond pricing under the jump Vasicek model and option pricing under the jump diffusion model. The proposed numerical method has obtained satisfactory accuracy and efficiency. The method has important application value and practical significance in investment decision-making, option pricing, insurance and other fields.https://www.mdpi.com/2227-7390/11/12/2658deep learningbackward stochastic differential equationnonlinear Feynman-Kac formulahigh dimensional PDEderivatives pricingneural network
spellingShingle Xiangdong Liu
Yu Gu
Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning
Mathematics
deep learning
backward stochastic differential equation
nonlinear Feynman-Kac formula
high dimensional PDE
derivatives pricing
neural network
title Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning
title_full Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning
title_fullStr Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning
title_full_unstemmed Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning
title_short Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning
title_sort study of pricing of high dimensional financial derivatives based on deep learning
topic deep learning
backward stochastic differential equation
nonlinear Feynman-Kac formula
high dimensional PDE
derivatives pricing
neural network
url https://www.mdpi.com/2227-7390/11/12/2658
work_keys_str_mv AT xiangdongliu studyofpricingofhighdimensionalfinancialderivativesbasedondeeplearning
AT yugu studyofpricingofhighdimensionalfinancialderivativesbasedondeeplearning