  Artificial neural networks (ANNs) have recently also been applied to solve
partial differential equations (PDEs). In this work, the classical problem of
pricing European and American financial options, based on the corresponding PDE
formulations, is studied. 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).
