Calibration of local-stochastic volatility models with neural networks

During the last twenty years several models have been proposed to improve the classic Black-Scholes framework for equity derivatives pricing. Recently a new model has been proposed: Local-Stochastic Volatility Model (LSV). This model considers volatility as the product between a deterministic and a stochastic term. So far, the model choice was not only driven by the capacity of capturing empirically observed market features well, but also by the computational tractability of the calibration process. This is now undergoing a big change since machine learning technologies offer new perspectives on model calibration. In this thesis we consider the calibration problem to be the search for a model which generates given market prices and where additionally technology from generative adversarial networks can be used. This means parametrizing the model pool in a way which is accessible for machine learning techniques and interpreting the inverse problems a training task of a generative network, whose quality is assessed by an adversary. The calibration algorithm proposed for LSV models use as generative models so-called neural stochastic differential equations (SDE), which just means to parameterize the drift and volatility of an Ito-SDE by neural networks.