A highly parallel Black–Scholes solver based on adaptive sparse grids

Heinecke, Alexander; Schraufstetter, Stefanie; Bungartz, Hans‐Joachim

doi:10.1080/00207160.2012.690865

Cited by 16 publications

(23 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3. Afterwards we repeat our numerical experiments shown in [7] in order to demonstrate the correctness of our performance improvements in Sect. 4 and finally we conclude our work in Sect.…”

Section: Introduction and Related Workmentioning

confidence: 92%

“…Since we violate this smoothness assumption at the payoff-kink, we "resolve" this issue by using local refinements to full-grid-like substructures. For details, we refer to [3,7].…”

Section: Discretization With Sparse Gridsmentioning

confidence: 99%

“…Last but not least we employ a spatially adaptive discretization in the sub-domain of the payoff's kink. All of these different transformations can be briefly summarized (please refer to [7] for details):…”

Section: Applied Transformations To the Black-scholes Equationmentioning

confidence: 99%

“…As they are a measurement for the second partial derivatives of the interpolated function [1], we use them for our spatial adaptivity strategies [3,7].…”

Section: Discretization With Sparse Gridsmentioning

confidence: 99%

“…In this work we extend our research presented in [2,3,7]. Here, the Black-Scholes PDE is discretized with finite elements on spatially adaptive sparse grids in order to optimally adopt to a given option that should be priced.…”

Section: Introduction and Related Workmentioning

confidence: 98%

See 4 more Smart Citations

Many-core architectures boost the pricing of basket options on adaptive sparse grids

Heinecke

Jepsen

Bungartz

2013

Proceedings of the 6th Workshop on High Performance Computational Finance

Self Cite

View full text Add to dashboard Cite

In this work, we present a highly scalable approach for numerically solving the Black-Scholes PDE in order to price basket options. Our method is based on a spatially adaptive sparse-grid discretization with finite elements. Since we cannot unleash the compute capabilities of modern many-core chips such as GPUs using the complexity-optimal Up-Down method, we implemented an embarrassingly parallel direct method. This operator is paired with a distributed memory parallelization using MPI and we achieved very good scalability results compared to the standard Up-Down approach. Since we exploit all levels of the operator's parallelism, we are able to achieve nearly perfect strong scaling for the Black-Scholes solver. Our results show that typical problem sizes (5 dimensional basket options), require at least 4 NVIDIA K20X Kepler GPUs (inside a Cray XK7) in order to be faster than the Up-Down scheme running on 16 Intel Sandy Bridge cores (one box). On a Cray XK7 machine we outperform our highly parallel Up-Down implementation by 55X with respect to time to solution. Both results emphasize the competitiveness of our proposed operator.

show abstract

“…3. Afterwards we repeat our numerical experiments shown in [7] in order to demonstrate the correctness of our performance improvements in Sect. 4 and finally we conclude our work in Sect.…”

Section: Introduction and Related Workmentioning

confidence: 92%

“…Since we violate this smoothness assumption at the payoff-kink, we "resolve" this issue by using local refinements to full-grid-like substructures. For details, we refer to [3,7].…”

Section: Discretization With Sparse Gridsmentioning

confidence: 99%

Section: Applied Transformations To the Black-scholes Equationmentioning

confidence: 99%

“…As they are a measurement for the second partial derivatives of the interpolated function [1], we use them for our spatial adaptivity strategies [3,7].…”

Section: Discretization With Sparse Gridsmentioning

confidence: 99%

Section: Introduction and Related Workmentioning

confidence: 98%

See 3 more Smart Citations

Many-core architectures boost the pricing of basket options on adaptive sparse grids

Heinecke

Jepsen

Bungartz

2013

Proceedings of the 6th Workshop on High Performance Computational Finance

Self Cite

View full text Add to dashboard Cite

show abstract

Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation

2015

Self Cite

View full text Add to dashboard Cite

This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid discretizations of the PDEs, which allow us to cope with the curse of dimensionality to some extent. We then derive reduced models with proper orthogonal decomposition. Our numerical results with the Black-Scholes model show that sufficiently accurate results are achieved while gaining speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options. For the Heston model, results are presented for a single-asset option that leads to a two-dimensional pricing problem, where we achieve significant speedups with our model reduction approach based on high-fidelity sparse grid models.

show abstract

DGM: A deep learning algorithm for solving partial differential equations

Sirignano

Spiliopoulos

2018

Journal of Computational Physics

1,748

1,261

View full text Add to dashboard Cite

High-dimensional PDEs have been a longstanding computational challenge. We propose to solve highdimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points. The algorithm is tested on a class of high-dimensional free boundary PDEs, which we are able to accurately solve in up to 200 dimensions. The algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE and Burgers' equation. The deep learning algorithm approximates the general solution to the Burgers' equation for a continuum of different boundary conditions and physical conditions (which can be viewed as a high-dimensional space). We call the algorithm a "Deep Galerkin Method (DGM)" since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions. In addition, we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic PDEs. Deep learning and high-dimensional PDEsHigh-dimensional partial differential equations (PDEs) are used in physics, engineering, and finance. Their numerical solution has been a longstanding challenge. Finite difference methods become infeasible in higher dimensions due to the explosion in the number of grid points and the demand for reduced time step size. If there are d space dimensions and 1 time dimension, the mesh is of size O d+1 . This quickly becomes computationally intractable when the dimension d becomes even moderately large. We propose to solve high-dimensional PDEs using a meshfree deep learning algorithm. The method is similar in spirit to the Galerkin method, but with several key changes using ideas from machine learning. The Galerkin method is a widely-used computational method which seeks a reduced-form solution to a PDE as a linear combination of basis functions. The deep learning algorithm, or "Deep Galerkin Method" (DGM), uses a deep neural network instead of a linear combination of basis functions. The deep neural network is trained to satisfy the differential operator, initial condition, and boundary conditions using stochastic gradient descent at randomly sampled spatial points. By randomly sampling spatial points, we avoid the need to form a mesh (which is infeasible in higher dimensions) and instead convert the PDE problem into a machine learning problem.DGM is a natural merger of Galerkin methods and machine learning. The algorithm in principle is straightforward; see Section 2. Promising numerical results are presented later in Section 4 for a class *

show abstract

A highly parallel Black–Scholes solver based on adaptive sparse grids

Cited by 16 publications

References 32 publications

Many-core architectures boost the pricing of basket options on adaptive sparse grids

Many-core architectures boost the pricing of basket options on adaptive sparse grids

Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation

DGM: A deep learning algorithm for solving partial differential equations

Contact Info

Product

Resources

About