FAST CALCULATION OF INTEGRAL CONVOLUTION TYPE OPERATORS IN OPTION ESTIMATION PROBLEMS IN LEVY MODELS

An approximate algorithm for calculating integral operators of the convolution type that arise when evaluating barrier options in Levy models by the Wiener–Hopf method is constructed. Additionally, the question of the possibility of applying machine learning methods (artificial neural networks) to the approximation of a special type of integrals, which are a key element in the construction of approximate formulas for the Wiener–Hopf integral operators under consideration, is investigated. The main idea is to decompose the price function into a Fourier series and transform the integration contour for each term of the Fourier series. As a result, we obtain a set of typical integrals that depend on Wiener–Hopf factors, but do not depend on the price function, while the most computationally expensive part of the numerical method is reduced to calculating these integrals. Since they need to be calculated only once, and not at each iteration, as was the case in standard implementations of the Wiener–Hopf method, this will significantly speed up calculations. Moreover, a neural network can be trained to calculate typical integrals. The proposed approach is especially effective for spectrally one-sided Levy processes, for which explicit Wiener–Hopf factorization formulas are known. In this case, we obtain computationally convenient formulas by integrating along the section. The main advantage of including neural networks in a computational scheme is the ability to perform calculations on an uneven grid. Such a hybrid numerical method will be able to successfully compete with classical methods of computing convolutions in similar tasks using the fastFourier transform. Computational experiments show that neural networks with one hidden layer of 20 neurons are able to effectively cope with the tasks of approximating the auxiliary integrals under consideration.

Wiener–Hopf factorization, Levy processes, integral convolution operators, integral transformations, numerical methods, machine learning, computational financial mathematics