Martingale Benamou-Brenier: arthimetic and geometric Bass martingales

Optimal transport (OT) proves to be a powerful tool for non-parametric calibration: it allows us to take a favourite (non-calibrated) model and project it onto the space of all calibrated (martingale) models. The dual side of the problem leads to an HJB equation and a numerical algorithm to solve the projection. However, in general, this process is costly and leads to spiky vol surfaces. We are interested in special cases where the projection can be obtained semi-analytically. This leads us to the martingale equivalent of the seminal fluid-dynamics interpretation of the optimal transport (OT) problem developed by Benamou and Brenier. Specifically, given marginals, we look for the martingale which is the closest to a given archetypical model. If our archetype is the arithmetic Brownian motion, this gives the stretched Brownian motion (or the Bass martingale), studied previously by Backhoff-Veraguas, Beiglbock, Huesmann and Kallblad (and many others). Here we consider the financially more pertinent case of Black-Scholes (geometric BM) reference and show it can also be solved explicitly. In both cases, fast numerical algorithms are available.

Based on joint works with Julio Backhoff, Benjamin Joseph and Gregoire Leoper.  

This talk reports a work in progress. It will be done on a board.