Calibrating financial market models via optimal transport

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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)