Luca Nenna (Université Paris-Saclay) : Risk management via optimal transport.

Abstract : In a variety of problems in operations research, a variable of interest $b=b(x_1,x_2,\dots,x_d)$ depends on several underlying random variables, whose individual distributions are known but whose joint distribution is not. A natural example arises in finance, when one considers the payout of a derivative depending on several underlying assets. An estimate of the distribution of the asset values themselves can often be inferred from the prices of vanilla call and put options for a wide range of strike prices ; since these options are widely traded, their prices are readily available. However, estimating the price of the joint distribution would require prices of a wide range of derivatives with payouts depending on all the variables, which are typically much scarcer. In political science, $b$ might represent the outcome of an election and the $x_i$ vote shares in different regions.
Metrics used in risk management depend on the distribution of the output variable $b$, and therefore, in turn, on the joint distribution of the $x_i$. A natural problem is therefore to determine bounds on these metrics ; that is, to maximize the given metric over all possible joint distributions of the $x_i$ with known marginal distributions. In this talk, we show that for a large class of metrics, the maximization can in fact be formulated as a traditional multi-marginal optimal transport problem with $d+1$ marginals : the given marginals distributions of the $x_i$ as well as another distribution arising from the particular form of the risk metric. In particular, in the special but important case of conditional value at risk, the problem further reduces to a multi-marginal partial transport problem on the $d$ original distributions.
When the underlying variables $x_i$ are all one dimensional, this allows us to derive a nearly explicit characterization of solutions in a substantial class of relevant problems, and facilitates the use of a very broad range of computational methods for optimal transport problems for others.