Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks

Beatriz Salvador,Cornelis W Oosterlee,Remco Van Der Meer

doi:10.3390/math9010046

Abstract

Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black–Scholes equation, whereas for the American option we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements). In the short note, previously published, a brief introduction to this work was given, where some ideas to price vanilla options by ANNs were presented, and only European options were addressed. In the current work, the methodology is introduced in much more detail.

Highlights

The interest in machine learning techniques, due to the remarkable successes in different application areas, is growing exponentially
Classical problems in financial option pricing have been addressed with artificial neural networks
A new unsupervised learning methodology is introduced to solve the option value problems based on the partial differential equations (PDEs) formulation

Summary

Introduction

The interest in machine learning techniques, due to the remarkable successes in different application areas, is growing exponentially. ANNs are learning systems based on a collection of artificial neurons that constitute a connected network [1]. Such systems “learn” to perform tasks, generally without being programmed with task-specific rules. The neurons are organized in multiple layers; The input layer receives external data, the output layer produces the final result. Many different financial problems have been addressed with machine learning, like stock price prediction, where ANNs are trained to detect patterns in historical data sets to predict future trends [3,4], or bond rating predictions, see [5,6,7]

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Methods

Conclusion