Robust option pricing with Neural SDEs

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.

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs