The Theory of Optimal Transport and Quantitative Finance in Simple Terms

The theory of optimal transport focuses on a problem initially formulated in 1781 by the French mathematician Gaspard Monge (portrait below). In his work titled "Théorie des déblais et des remblais", Monge laid the foundations for this theory by seeking to determine the most efficient way to transport mass from one point to another while minimizing the associated costs.

The optimal transport problem can be described as follows: given two probability distributions, a source μ(describing the initial distribution of mass) and a destination ν (describing the desired final distribution), how can the mass be transported from the source to the destination in a way that minimizes the total cost?

More formally, let's imagine that we wish to move a certain amount of mass from one set to another. The cost of transporting mass from a point x in the source to a point y in the destination is given by a cost function c(x, y). The goal would then be to find a "transport plan" that describes the amount of mass transported between each pair of points (x, y) and that minimizes the total cost.

A key tool for solving this problem is the notion of a transport measure or "transport plan," denoted as π. This measure describes the proportion of mass transported between each point x of the source distribution μ and each point  y of the target distribution ν. The problem then becomes minimizing the integral of the cost function weighted by this transport plan:

C(μ, ν) = inf ( ∫ c(x, y) dπ(x, y) )where the infimum (*) is taken over all transport plans π that respect the marginal (**) distributions μ and ν (i.e., the total quantities of transported mass must match the initial and final distributions).

Calibrating financial models involves adjusting a model to align with observed data. The Wasserstein distances (***) could be used as an error measure to compare the theoretical price distribution of a financial model with observed market prices. By minimizing this distance, one could adjust the model parameters to best fit the real data.

The theory of optimal transport could also be applied to the problem of portfolio hedging under uncertainty. When there are uncertainties regarding future price scenarios, minimizing the Wasserstein distance between the predicted return distribution of a hedged portfolio and the actual market distribution could help determine optimal hedging strategies.

(*) Infimum: The greatest lower bound of a set.

(*) Marginal: The marginal distribution (or marginal) refers to the distribution of a single variable within a set of multiple variables.

(***)The Wasserstein distance, is also known as the "Earth Mover's Distance (EMD)". It measures the "work" required to transform one pile of earth into another, taking into account the weight of the earth moved and the distance traveled.