Fourier pricing of two-asset options: a comparison of methods

Fourier methods form an integral part in the universe of option pricing due to their speed, accuracy and diversity of use. Two types of methods that are extensively used are fast Fourier transform (FFT) methods and the Fourier-cosine series expansion (COS) method. Since its introduction the COS method has been seen to be more efficient in terms of rate of convergence than its FFT counterparts when pricing vanilla options; however limited comparison has been performed for more exotic options and under varying model assumptions. This paper will expand on this research by considering the efficiency of the two methods when applied to spread and worst-of rainbow options under two different models - namely the Black-Scholes model and the Variance Gamma model. In order to conduct this comparison, this paper considers each option under each model and determines the number of terms until the price estimate converges to a certain level of accuracy. Furthermore, it tests the robustness of the pricing methodologies to changes in certain discretionary parameters. It is found that although under the Black-Scholes model the COS method converges in fewer terms than the FFT method for both spread options (32 versus 128 terms) and the rainbow options (64 versus 512 terms), this is not the case under the more complex Variance Gamma model where the terms to convergence of both methods are similar. Both the methodologies are generally robust against changes in the discretionary variables; however, a notable issue appears under the implementation of the FFT methodology to worst-of rainbow options where the choice of the truncated integration region becomes highly influential on the ability of the method to price accurately. In sum, this paper finds that the improved speed of the COS method against the FFT method diminishes with a more complex model - although the extent of this can only be determined by testing for increasingly complex characteristic functions. Overall the COS method can be seen to be preferable from a practical point of view due to its higher level of robustness

Pricing options and computing implied volatilities using neural networks

This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods. With ANNs being universal function approximators, this method trains an optimized ANN on a data set generated by a sophisticated financial model, and runs the trained ANN as an agent of the original solver in a fast and efficient way. We test this approach on three different types of solvers, including the analytic solution for the Black-Scholes equation, the COS method for the Heston stochastic volatility model and Brent's iterative root-finding method for the calculation of implied volatilities. The numerical results show that the ANN solver can reduce the computing time significantly

Pricing american-style options under the Heston model using the Cos approximation

Tese de mestrado em Matemática Financeira, Universidade de Lisboa, Faculdade de Ciências, 2020Nesta tese, apresenta-se o método COS: um método numérico para avaliação de opções, baseado na expansão em série de cossenos de Fourier. Foi proposto por Fang e Oosterlee, e pode ser eficazmente utilizado para avaliar opções de estilo Bermudiano subjacentes a processos de Lévy ou ao modelo de volatilidade estocástica de Heston. O ingrediente principal do método é a relação próxima entre a função característica e os coeficientes da expansão em série de cossenos de Fourier da função densidade. Ao considerar o modelo de Heston, o problema de avaliação bidimensional é abordado combinando a ideia anterior com regras de quadratura de ordem superior no domínio da log-variância.In this thesis, we present the COS method: an option pricing numerical method based on Fourier cosine series expansions. It was proposed by Fang and Oosterlee, and it can be used to effectively price Bermudan options under Lévy processes or the Heston stochastic volatility model. The method’s key ingredient is the close relationship between the characteristic function and the series coefficients of the Fourier cosine expansion of the density function. Under the Heston model, the two-dimensional pricing problem is dealt by combining the prior Fourier cosine series insight with high-order quadrature rules in the log-variance dimension