We discuss the problem of calibrating a math finance model to market data. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton–Jacobi–Bellman equations arising from the dual formulation. Our algorithm draws parallels with the one devised by Avellaneda et al. (1997) in the context of entropy minimisation.

The method is tested on both simulated data and market data. We first address the joint calibration problem of SPX options and VIX options or futures, a problem known to be difficult. Numerical examples show that our approach can handle the data well and produces a LV model which is accurately calibrated to SPX options, VIX options and VIX futures simultaneously. We then consider joint calibration to interest rates products and SPX options. Finally, we also comment on analytically tractable calibrations giving raise to the Arithmetic and Geometric Benamou-Brenier problems, and discuss the prospect of extending the methodology to cover American options.

Based on joint works with Ivan Guo, Benjamin Joseph, Gregoire Loeper and Shiyi Wang.

Bio: Jan Obloj is a Professor of Mathematics at the University of Oxford's Mathematical Institute, an Official Fellow of St John's College Oxford and a member of the Oxford-Man Institute of Quantitative Finance. Before coming to Oxford, he was a Marie Curie Post-Doctoral Fellow at Imperial College London and he holds PhD from University Paris VI and Warsaw University. He is the current President of the Bachelier Finance Society and is Fellow of the Institute of Mathematical Statistics. He has a general interest in mathematics of randomness. Most of his research sits at the crossroads of various fields, including: probability theory, statistics, mathematical finance, operations research, optimal transportation and data science. His main focus is on robustness of the modelling pathways from input out outputs, ways to understand and quantify it and his research spans the spectrum from theoretical foundations of robust pricing and hedging paradigm in mathematical finance, to practical questions of building fast generic ways to approximate adversarial robustness of deep neural networks.