In this presentation we will explain how we can solve linear, semi-linear as well as nonlinear partial differential equations by the concept of backward stochastic differential equations and Fourier cosine expansions. We will discuss the highly efficient pricing of financial options in the Fourier context by means of the COS method. Particularly, we also present a new jump-diffusion process, the Heston-Queue-Hawkes (HQH) model, combining the well-known Heston model and the recently introduced Queue-Hawkes (Q-Hawkes) jump process. Like the Hawkes process, the HQH model can capture the effects of self-excitation and contagion of stock prices. However, since the characteristic function of the HQH process is known in closed-form, Fourier-based fast pricing algorithms can be fully exploited with this model. In the final part of the presentation, we will give a brief overview of our experiences with artificial neural networks (ANNs) in finance. We will outline the use of neural networks for the calibration of a financial asset price model in the context of financial option pricing. To provide an efficient calibration framework, a data-driven approach is proposed to learn the solutions of financial models and to reduce the corresponding computation time significantly. Specifically, fitting model parameters is formulated as training hidden neurons within a machine-learning framework.
Kees Oosterlee has been working on computational problems in financial mathematics since 2000. After his PhD, in 1993, he worked at the GMD (now Fraunhofer SCAI) in Sankt Augustin, Germany, until 2002, and at TU Delft. Since 2007 he has been a part-time professor in Delft and also a scientist at the Centrum Wisknude & Informatica (CWI) in Amsterdam, where he was also a member of the management team. Oosterlee is co-author of two books ("Multigrid" 2001, and "Mathematical Modeling and Computation in Finance", 2019), and many scientific publications. Methods he co-developed include: the COS method, SWIFT (Shannon Wavelet Inverse Fourier Transform method) , SGBM (Stochastic Grid Bundling Method), SCMC (Stochastic Collocation Monte Carlo Method), and the Seven-League scheme (7L). Machine learning in Finance is a recent research interest.