Robust option pricing with Neural SDEs

点击翻译按钮获取中文摘要

1. Introduction: Mathematical modelling is ubiquitous in the financial industry and drives key decision processes. Every model provides only a crude approximation to reality and the risk of using an inadequate model is usually hard to detect and quantify. Model uncertainty is, hence, an essential part of mathematical modelling and is particularly important in mathematical finance and economics, where one cannot base models on well-established physical laws. Nevertheless, standard modelling practices in finance rarely address this model uncertainty. One such paradigm, where the model risk is central to its philosophy, is the robust finance approach. Currently, machine learning techniques are opening doors to different ways of robust and data-driven model selection mechanisms. However, most machine learning models are still considered to be so-called "black-boxes" as individual parameters do not have meaningful interpretation. We therefore plan to focus on combining classical modelling with deep learning techniques in the context of option pricing. Until recently, model complexity was undesirable, amongst other reasons, for increasing the computational effort required to perform, in particular calibration, but also pricing and risk calculations. With greater uptake of machine learning methods and greater computational power more complex models can now be used. In our approach, we let the data dictate the model, while still keeping a strong prior on the model form. This is achieved by using stochastic differential equations (SDEs) for the model dynamics, but instead of choosing a fixed parametrization for the model SDEs we allow the drift and diffusion to be given by over-parametrised neural networks. We refer to these as Neural SDEs. These are shown to not only provide a systematic framework for model selection, but also, quite remarkably, to produce robust estimates on the derivative prices. Here, the calibration and model selection are done simultaneously. Since the neural SDE model is overparametrised, there is a large pool of possible models and the training algorithm selects a model. 2. Alignment to EPSRC research areas: This project falls within the EPSRC 'Statistics and applied probability' research area. Presented methodology combines classical probabilistic techniques from stochastic and probabilistic modelling and novel machine learning approaches from data-science, more specifically -- deep neural networks. We kept our focus on applied probability in combination with robust statistics and artificial intelligence, which is in line with the proposed research area. We emphasise this approach has applications well beyond just option pricing and, more broadly, finance. It covers any scenario including modelling processes with inherit randomness and known values or measurements (or functions thereof) at different points in time. Such applications include problems in data analytics, healthcare modelling and medical statistics, artificial intelligence and uncertainty quantification etc.3. Collaboration: The project was carried out jointly with the Alan Turing Institute (ATI) and University of Edinburgh, more particularly with David Siska and the Programme Director for Finance and Economics at ATI, Lukasz Szpruch and members of his research team Marc Sabate Vidales and Patryk Gierjatowicz.