A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model

Huang, Jian; Cen, Zhongdi; Le, Anbo

doi:10.1155/2013/651573

Cited by 5 publications

(3 citation statements)

References 37 publications

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“…In finance, nonlocal PDEs are used, e.g., in jump-diffusion models for the pricing of derivatives where the dynamics of stock prices are described by stochastic processes experiencing large jumps [3,[12][13][14][15][16][17][18]. Penalty methods for pricing American put options such as in Kou's jump-diffusion model [19,20], considering large investors where the agent policy affects the assets prices [15,21], or considering default risks [22,23] can further introduce nonlinear terms in nonlocal PDEs. In economics, non-local nonlinear PDEs appear, e.g., in evolutionary game theory with the so-called replicator-mutator equation capturing continuous strategy spaces [24][25][26][27][28] or in growth models where consumption is non-local [29].…”

Section: Introductionmentioning

confidence: 99%

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Boussange,

Becker,

Jentzen

et al. 2023

Partial Differ. Equ. Appl.

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Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. Non-local nonlinear PDE models can accurately capture these phenomena, but traditional numerical approximation methods are infeasible when the considered non-local PDE is high-dimensional. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman–Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

show abstract

Section: Introductionmentioning

confidence: 99%

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Boussange,

Becker,

Jentzen

et al. 2023

Partial Differ. Equ. Appl.

View full text Add to dashboard Cite

show abstract

“…In finance, non-local PDEs are used, e.g., in jump-diffusion models for the pricing of derivatives where the dynamics of stock prices are described by stochastic processes experiencing large jumps [74,22,65,1,15,90,28,26]. Penalty methods for pricing American put options such as in Kou's jump-diffusion model [58,42], considering large investors where the agent policy affects the assets prices [5,1], or considering default risks [83,55] can further introduce nonlinear terms in non-local PDEs. In economics, non-local nonlinear PDEs appear, e.g., in evolutionary game theory with the so-called replicator-mutator equation capturing continuous strategy spaces [79,62,50,3,4] or in growth models where consumption is nonlocal [6].…”

Section: Introductionmentioning

confidence: 99%

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Boussange¹,

Becker²,

Jentzen³

et al. 2022

Preprint

View full text Add to dashboard Cite

Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. In order to properly capture these phenomena non-local nonlinear PDE models are frequently employed in the literature. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions.In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman-Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the nonlocal term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

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“…Therefore we need to use a numerical approximation. To obtain an approximation of the option value, one can compute a solution of the BS equations (1) and (2) using a finite difference method (FDM) [2][3][4][5][6][7][8], finite element method [9][10][11], finite volume method [12][13][14], a fast Fourier transform [15][16][17], and also their optimal BC [18].…”

Section: Introductionmentioning

confidence: 99%

Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

Jeong

Seo²,

Hwang

et al. 2015

Discrete Dynamics in Nature and Society

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We briefly review and investigate the performance of various boundary conditions such as Dirichlet, Neumann, linear, and partial differential equation boundary conditions for the numerical solutions of the Black-Scholes partial differential equation. We use a finite difference method to numerically solve the equation. To show the efficiency of the given boundary condition, several numerical examples are presented. In numerical test, we investigate the effect of the domain sizes and compare the effect of various boundary conditions with pointwise error and root mean square error. Numerical results show that linear boundary condition is accurate and efficient among the other boundary conditions.

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A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model

Cited by 5 publications

References 37 publications

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

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