Supervised deep learning in computational finance

Abstract

Mathematical modeling and numerical methods play a key role in the field of quantitative finance, for example, for financial derivative pricing and for risk management purposes. Asset models of increasing complexity, like stochastic volatility models (local stochastic volatility, rough volatility based on fractional Brownian motion) require advanced, efficient numerical techniques to bring them successfully into practice. When computations take too long, an involved asset model is not a feasible option as practical considerations demand a balance between the model’s accuracy and the time it takes to compute prices and risk management measures. In the big data era, typical basic computational tasks in the financial industry are often involved and computationally intensive due to the large volumes of financial data that are generated nowadays. Besides the traditional numerical methods in financial derivatives pricing in quantitative finance (like partial differential equation (PDE) discretization and solution methods, Fourier methods, Monte Carlo simulation), recently deep machine learning techniques have emerged as powerful numerical approximation techniques within scientific computing. Following the so-called Universal Approximation Theory, we will employ deep neural networks for financial computations, either to speed up the solution processes or to solve highly complicated, highdimensional, problems in finance. Particularly, we will employ supervised machine learning techniques, based on intensive learning of so called labeled information (input-output relations, where sets of parameters form the input to a neural network, and the output to be learned is a solution to a financial problem).