# Martingale Optimal Transport

Mathematically a contingent claim is a deterministic function of a future random price of the price of a traded stock. This function determines the random future pay-off of the claim and if traded liquidly, finance theory dictates that its price is given as its expected value with respect to an unknown risk-neutral measure. This probability measure is the risk-adjusted distribution of the stock price process. In practice, one determines this measure by making model assumptions on the stochastic dynamics of the stock price process. The famous Black & Scholes formula is derived after assuming that the stock process is a geometric Brownian motion. An alternate approach is to estimate the risk-neutral measure by using the option prices. This is an inverse problem and if a model is prescribed then it reduces to a calibration problem. Martingale Optimal Transport is, on the other hand, a model-free approach. It uses the option data to estimate the risk-adjusted future distributions of the stock price at some future values. The volatility option, VIX, traded at the Chicago Board of Exchange is an example of this approach. In this context, one may see the time evolution of the stock distribution as a transport and optimality searches for the worst-cases. Additionally, as the stock is liquidly traded over time, its discounted price process must be a martingale. Hence, one needs to restrict the transport plans to have this property.

In this talk, I will introduce this problem and prove several related convex duality results.

*Prof. Soner works on stochastic optimal control and decisions under uncertainty with applications in mathematical finance. He rejoined Princeton in 2019 from ETH Zurich where he was the Chair of the Department Mathematics. He holds a Ph.D. in Applied Mathematics from Brown University.*