# martingale-transport

# Optimal martingale transport

## Description:

Optimal transport theory has important applications in finance, more specifically in option pricing theory. Financial derivatives may depend on several underlying assets; this is the case of spread options, for instance, or of basket options. The standard Black-Scholes-Merton theory of option pricing says that if there is a liquid market of vanilla options on a single underlying, then the risk-neutral distribution of the underlying can be recovered from the option prices; and we can therefore obtain a unique price associated with any more complicated single-underlying option. However, in the case of an option on two underlying, the market prices on the single-name options do not imply the joint distribution of two such assets, and one can then define no-arbitrage bounds, which corresponds to the cheapest and most expensive prices of the option that is consistent with the market. These bounds formulate as a Monge-Kantorovich problem, and the dual problem ensures that they correspond to the most expensive sub-replicating (lower bound) and the cheapest super-replicating portfolio (upper bound).

In a number of cases, the two underlying quantities are not the value of two assets at the same date in time, but the price of the same asset at two different dates in the future. There is then an important further restriction on the joint distribution of these assets: they should be the margins of a martingale. Computing the bounds of the option prices leads then to a variant of Monge-Kantorovich theory, where one looks the optimal coupling that is a martingale. This further constraint yields a supplementary term in the dual formulation, which has an interesting interpretation in terms of sub/super-replicating portfolio: the portfolio is not only made of calls and puts at the two maturities (static hedging), but also allows for rebalancing at the earlier maturity, allowing for dynamic hedging.

Moving beyond the static problem, there are interesting dynamic formulation of the problem. In particular, one may consider among the set of semi-martingales that start at a given distribution and end up at a given distribution, those who minimize a the time integral of the expectation of a Lagrangian that depends on the drift and diffusions parameters. This nicely extends the Benamou-Brenier dynamic formulation of optimal transport, and can provide interesting insights on particular solutions to the Skorohod embedding problem.

## My co-authors:

Guillaume Carlier, Pierre Henry-Labordere, Nizar Touzi.

## Presentation slides:

Presentation slides can be found here.

## References:

Guillaume Carlier, and Alfred Galichon (2012). Exponential convergence for a convexifying equation (2012). * Control, Optimization and Calculus of Variations *18(3), pp. 611–620. Available here.

Pierre Henry-Labordere, Alfred Galichon, and Nizar Touzi (2014). A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options.

*24 (1), pp. 312-336. Available here.*

**Annals of Applied Probability**