Calibrating financial market models via optimal transport

Any financial market model, before it can used in practice, needs to calibrated so that it replicates and accounts for the structure of observed market data. The classical Black-Scholes model assumes that a stock evolves with a constant volatility function, however by inverting the analytic formula for a call option and using observed market prices, one can compute the "implied volatility" surface. This surface typically displays skews and smiles, so making a constant volatility assumption a clearly unrealistic one. Dupire introduced a formula for so-called "local volatility" where the volatility function depends on time and the stock price as well. The formula links the price of a European call option to the volatility function. However, it requires access to market prices at a continuum of strikes and maturities. Further, in a setting of stochastic interest rates, it also requires the whole term structure of the interest rates. Since, in reality, only finitely many data points are ever available, some form of interpolation is required. Many techniques, parametric or not, have been used in the past to help with this crucial interpolation task. In this project, we propose to employ optimal transport techniques. We use the dynamical formulation of optimal transport, that is where our model probability measures are constrained to be semimartingale measures, and we aim to minimise a given convex cost function that will penalise deviations from the structure of whichever model we choose to use. In addition, we enforce matching conditions in the market data and use known analytical formulae to change the model parameters to match the market data. This naturally leads to a PDE formulation of a minimisation problem, so a duality argument is carried out, and the dual problem is attained, with some adaptation to the duality argument in [1] required. The resulting problem will require the numerical solution of an HJB equation to compute the optimal parameters via a policy iteration method.

We aim to apply the techniques used in [1] and [2] on Local-Stochastic Volatility calibration and the joint calibration of SPX/VIX where the interest rate was assumed to be zero, and extend them to the setting of stochastic interest rates. Later on, a joint calibration with the (LIBOR/EURIBOR)-market model in a multi-curve setting could be the objective. This could be done sequentially, that is we calibrate a market model first, then using that calibrate the underlying separately. Or we could consider jointly calibrating the market model and the underlying - which will either require making the market model depend on the underlying itself or by modifying the cost function to have a penalty that forces the market model to jointly calibrate. The extra layer of complexity will present numerical challenges as adding more state variables, inherited from interest rates market model, will add dimensions to the resulting HJB equation and therefore render the PDE numerically unsolvable through methods generally used in two dimensions. Thus, another objective is to figure out if there is some structure that can be used in the market model to solve such a PDE or what techniques can be applied to numerically solve the problem there - avenues of investigation currently include neural network approximations.

This project is interdisciplinary in that it brings together techniques from optimal transport, financial model calibration, numerics for non-linear PDEs and potentially machine learning. It builds on previous works but extends them into novel directions.

This project falls into the EPSRC "Operational Research" research area and has involvement from BNP Paribas.
References:
[1] Ivan Guo, Gregoire Loeper, & Shiyi Wang. "Calibration of local-stochastic volatility models by optimal transport". arXiv preprint arXiv:1906.06478 (2019).
[2] Ivan Guo et al. "Joint modelling & calibration of SPX and VIX by optimal transport". Available at SSRN 356899 (2020)

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
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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
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