# Martingale Optimal Transport in the Discrete Case Via Simple Linear Programming Techniques

## TL;DR

This paper introduces a linear programming approach to solve martingale optimal transport problems with discrete marginals, providing simple proofs and algorithms for constructing optimal transport plans in model-independent finance.

## Contribution

It demonstrates that discrete marginals allow reduction to linear programs and proves the optimality of left-monotone transport plans under the Spence Mirrlees condition.

## Key findings

- Linear programs efficiently solve the transport problem with discrete marginals.
- Left-monotone transport plans are proven optimal under the Spence Mirrlees condition.
- An illustrative example demonstrates the practical application of the approach.

## Abstract

We consider the problem of finding consistent upper price bounds and super replication strategies for exotic options, given the observation of call prices in the market. This field of research is called model-independent finance and has been introduced by Hobson 1998. Here we use the link to mass transport problems. In contrast to existing literature we assume that the marginal distributions at the two time points we consider are discrete probability distributions. This has the advantage that the optimization problems reduce to linear programs and can be solved rather easily when assuming a general martingale Spence Mirrlees condition. We will prove the optimality of left-monotone transport plans under this assumption and provide an algorithm for its construction. Our proofs are simple and do not require much knowledge of probability theory. At the end we present an example to illustrate our approach.