Deep Learning Reduced Basis Method for High-Dimensional Parametric Partial Differential Equations in Finance

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The financial sector is challenged by the digital revolution, utmost competitiveness, and serious risks along with advancing regulatory and accounting requirements, which leads to highly complex computational problems. A prominent example is the computation of the capital reserve where counterparty credit risks have to be incorporated as a direct regulatory response to the causes of the last financial crisis. This is of fundamental importance for the financial system. To develop adequate computational methods for option pricing and risk management we develop a new approach combining deep learning with well-understood numerical techniques, which have already proven highly trustworthy for complex real-world problems in engineering, where risk control is taken very seriously. To be more precise, we develop an offline-online method for parameter-dependent partial differential equations. Offline, the complex problem is processed with the help of machine learning to prepare an efficient solver. Online, this solver can be called in real-time for all parameters. This way, we will bring together the efficiency of deep learning with the reliability and economic interpretability of numerical algorithms. Thus we will contribute to the applicability of machine learning in the financial sector, where an intuitive understanding of the results is as crucial as their liability.High-dimensionality is intrinsic in finance and resulting computational problems are traditionally solved by Monte Carlo simulations. For urgent tasks such as real-time risk monitoring, uncertainty quantification and credit value adjustments, this approach is computationally too expensive. Lacking appropriate computational tools, ad hoc simplifications are the current market standard, yielding uncontrollable operational risk. The new combination of model order reduction for parametric (nonlinear) partial differential equations with deep learning allows us to break the curse of dimensionality. While classical discretisations are infeasible, deep neural networks allow for efficient approximations of high-dimensional functions. While with machine learning, results are efficiently evaluated, but difficult to interpret, we gain economic interpretability. The performance of the novel techniques will be evaluated and compared and their capabilities will be shown on realistic examples.