Robust financial calibration: a Bayesian approach for neural stochastic differential equations

Christa Cuchiero, Eva Flonner and Kevin Kurt

Abstract

The paper presents a Bayesian framework for the calibration of financial models using neural stochastic differential equations (neural SDEs), for which we also formulate a global universal approximation theorem based on Barron-type estimates. The method is based on the specification of a prior distribution on the neural network weights and a well-chosen likelihood function. The resulting posterior distribution can be seen as a mixture of different classical neural SDE models yielding robust bounds on the implied volatility surface. A methodology for learning the change of measure between the risk-neutral and the historical measure is necessary to take into consideration both historical financial time series data and option price data. The key ingredient for the robust numerical optimization of our neural networks is a Langevin-type algorithm, commonly used in the Bayesian approaches to draw posterior samples.