Jan Obloj - Optimal Transport and Robustness in Math Finance and beyond
For first year CDT in Mathematics of Random Systems students
Robust approach to mathematical finance focuses on the classical questions – pricing, hedging, risk management, optimal investment – but aims to understand and quantify how the decisions are affected by model uncertainty. I will introduce some such problems and present various methods used to provide answers. In particular, I will link pricing-hedging problems to (martingale) optimal transport problems. I will then focus on the latter and discuss various application of optimal transport tools in mathematical finance, optimization and statistics. In the second part of the talk, I will consider sensitivity of a generic stochastic optimization problem to model uncertainty. I will capture model uncertainty using Wasserstein balls around the postulated model and derive explicit formulae for the first order correction to both the value function and the optimizer. I will showcase many applications of such results and present two follow-up projects: one in devising a variant of stochastic gradient descent algorithm for deep NN training which is ensures robustness to adversarial data examples, and another in deriving robust (non-parametric) version of the classical vega and other sensitivities (in option pricing and hedging).
