Related papers: The Deep Parametric PDE Method: Application to Opt…
In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution…
Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man made complex systems. In particular, parabolic PDEs are a fundamental tool to determine fair prices of financial…
Pricing financial derivatives, in particular European-style options at different time-maturities and strikes, means a relevant problem in finance. The dynamics describing the price of vanilla options when constant volatilities and interest…
We study an algorithm which has been proposed by Chinesta et al. to solve high-dimensional partial differential equations. The idea is to represent the solution as a sum of tensor products and to compute iteratively the terms of this sum.…
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs),…
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
In this article, we study the rate of convergence of prices when a model is approximated by some simplified model. We also provide a method how explicit error formula for more general options can be obtained if such formula is available for…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
We present the Deep Picard Iteration (DPI) method, a new deep learning approach for solving high-dimensional partial differential equations (PDEs). The core innovation of DPI lies in its use of Picard iteration to reformulate the typically…
We consider a Black-Scholes type equation arising on a pricing model for a multi-asset option with general transaction costs. The pioneering work of Leland is thus extended in two different ways: on the one hand, the problem is…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…
We propose the Compound BSDE method, a fully forward, deep-learning-based approach for solving a broad class of problems in financial mathematics, including optimal stopping. The method is based on a reformulation of option pricing problems…
We develop an unsupervised deep learning method to solve the barrier options under the Bergomi model. The neural networks serve as the approximate option surfaces and are trained to satisfy the PDE as well as the boundary conditions. Two…
The main aim of this study is to introduce a 2-layered Artificial Neural Network (ANN) for solving the Black-Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed…
In this paper, we present a randomized extension of the deep splitting algorithm introduced in [Beck, Becker, Cheridito, Jentzen, and Neufeld (2021)] using random neural networks suitable to approximately solve both high-dimensional…
This paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma process. The continuous time Variance Gamma process is approximated by a discrete time Markov chain with the same…
The Black-Scholes option pricing model remains a cornerstone in financial mathematics, yet its application is often challenged by the need for accurate hedging strategies, especially in dynamic market environments. This paper presents a…
The approach that allows find European option price on the assumption of hedging at discrete times is proposed. The routine allows find the option price not for lognormal distribution functions of underlying asset only but for wide enough…