Fast Calculation of Integral Convolution Operators in Problems of Evaluating Options in Lévy’s Models

An approximate algorithm for calculating integral convolution operators that arise when estimating barrier options in Lévy models using the Wiener–Hopf method is constructed. Additionally, the possibility of applying machine learning methods (artificial neural networks) to approximating a special type of integrals, which are a key element in the construction of approximate formulas for the considered Wiener–Hopf integral operators, is studied. The main idea is to expand the price function in the 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 the Wiener–Hopf factors but are independent of the price function; then, the most computationally expensive part of the numerical method is reduced to calculating these integrals. Since they only need to be calculated once, rather than at each iteration as was the case in standard implementations of the Wiener–Hopf method, this will significantly speed up the calculations. Moreover, a neural network can be trained to calculate typical integrals. The proposed approach is especially efficient for spectrally one-sided Lévy processes for which explicit Wiener–Hopf factorization formulas are known. In this case, we obtain formulas convenient for calculations by integrating along the cut. The main advantage of including neural networks in the computational scheme is the ability to perform calculations on a nonuniform grid. Such a hybrid numerical method can successfully compete with classical methods for calculating convolutions in similar problems using the fast Fourier transform. Computational experiments show that neural networks with one hidden layer of 20 neurons are able to efficiently cope with the problems of approximating the auxiliary integrals under consideration.