Validation of Financial Options Models Using Neural Networks with Invariance to Fourier Transform

Abstract

It is known that numerical solution of the Black–Scholes PDE enables to compute with precision the values of financial options, within a finite time horizon. It is also known that solutions to the option pricing problem can be obtained in closed form using Fourier methods, such as the Fast Fourier Transform, the expansion in Fourier-cosine series or the expansion in Fourier–Hermite series. In this chapter, modeling of financial options’ dynamics is performed, using a neural network with 2D Gauss–Hermite basis functions that remain invariant to Fourier transform. Knowing that the Gauss–Hermite basis functions satisfy the orthogonality property and remain unchanged under the Fourier transform, subjected only to a change of scale, one has that the considered neural network provides the spectral analysis of the options’ dynamics model. Actually, the squares of the weights of the output layer of the neural network denote the spectral components for the monitored options’ dynamics. By observing changes in the amplitude of the aforementioned spectral components one can have also an indication about deviations from nominal values, for parameters that affect the options’ dynamics, such as interest rate, dividend payment and volatility. Moreover, since specific parametric changes are associated with amplitude changes of specific spectral components of the options’ model, isolation of the distorted parameters can be also performed.