• Corpus ID: 228373783

The Deep Parametric PDE Method: Application to Option Pricing

  title={The Deep Parametric PDE Method: Application to Option Pricing},
  author={Kathrin Glau and Linus Wunderlich},
  journal={arXiv: Computational Finance},
We propose the deep parametric PDE method to solve high-dimensional parametric partial differential equations. A single neural network approximates the solution of a whole family of PDEs after being trained without the need of sample solutions. As a practical application, we compute option prices in the multivariate Black-Scholes model. After a single training phase, the prices for different time, state and model parameters are available in milliseconds. We evaluate the accuracy in the price… 
Neural networks-based algorithms for stochastic control and PDEs in finance
This paper presents machine learning techniques and deep reinforcement learningbased algorithms for the efficient resolution of nonlinear partial differential equations and dynamic optimization problems arising in investment decisions and derivative pricing in financial engineering, and compares the different schemes illustrated by numerical tests on various financial applications.
Option Pricing Techniques
It is found that on nearly all the supervised learning problems, the generalised highway architecture outperforms its counterparts in terms of MSE relative to computation time and on the semi­supervised learning problem the performance of the DGM network remained consistent when increasing the dimensionality of the problem.
Studying first passage problems using neural networks: A case study in the slit-well microfluidic device
The results illustrate that the neural network method is synergistic with the time-integrated Smoluchowski model: together, these are used to construct continuous mappings from key physical inputs to key output metrics (mean mean passage time and effective mobility).
Differential learning methods for solving fully nonlinear PDEs
Compared to existing methods, the addition of a differential loss function associated to the gradient, and augmented training sets with Malliavin derivatives of the forward process, yields a better estimation of the PDE’s solution derivatives, in particular of the second derivative, which is usually dfficult to approximate.
Self-adaptive loss balanced Physics-informed neural networks


Unbiased deep solvers for parametric PDEs
Several deep learning algorithms for approximating families of parametric PDE solutions are developed that are robust with respect to quality of the neural network approximation and consequently can be used as a black-box in case only limited a priori information about the underlying problem is available.
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations
This work proposes a new method for solving high-dimensional fully nonlinear second-order PDEs and shows the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 200-dimensional G-Brownian motion.
Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning
It is shown that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region and that the proposed method does not suffer from the curse of dimensionality, distinguishing it from almost all standard numerical methods for P DEs.
Pricing options and computing implied volatilities using neural networks
A data-driven approach, by means of an Artificial Neural Network, to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods.
Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning
The main goals of this paper are to elucidate the features, capabilities and limitations of DGM by analyzing aspects of its implementation for a number of different PDEs and PDE systems.
Solving parametric PDE problems with artificial neural networks
This work proposes using neural network to parameterise the physical quantity of interest as a function of input coefficients and demonstrates the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.
DGM: A deep learning algorithm for solving partial differential equations
Deep backward schemes for high-dimensional nonlinear PDEs
The proposed new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs) rely on the classical backward stochastic differential equation (BSDE) representation of PDEs and provide error estimates in terms of the universal approximation of neural networks.
Low-Rank Tensor Approximation for Chebyshev Interpolation in Parametric Option Pricing
The core of the method is to express the tensorized interpolation in tensor train (TT) format and to develop an efficient way, based on tensor completion, to approximate the interpolation coefficients.
Model Reduction and Neural Networks for Parametric PDEs
A neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation is developed.