Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black-Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.

Solving stochastic differential equations and Kolmogorov equations by means of deep learning

Numerical pricing of options using high-order compact finite difference schemes

Solving stochastic differential equations and Kolmogorov equations by means of deep learning.

A fast high-order finite difference algorithm for pricing American options

In this paper we present an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing American and European option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. The analysis is performed within the framework of the vertical method of lines, where the spatial discretization is formulated as a Petrov–Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems. We establish the stability and an error bound for the solutions of the fully discretized system. Numerical results are presented to validate the theoretical results.

Convergence of a fitted finite volume method for the penalized Black–Scholes equation governing European and American Option pricing

This paper introduces a fast numerical method for computing American option pricing problems governed by the Black--Scholes equation. The treatment of the free boundary is based on some properties of the solution of the Black--Scholes equation. An artificial boundary condition is also used at the other end of the domain. The finite difference method is used to solve the resulting problem. Computational results are given for some American call option problems. The results show that the new treatment is very efficient and gives better accuracy than the normal finite difference method.

https://www.jstor.org/stable/pdfplus/4101191.pdf

Xiaonan Wu

Papers

A Fast Numerical Method for the Black--Scholes Equation of American Options