abstract: 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. Time permitting, we will also 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.