Arbitrage and Geometry

TL;DR

This paper explores the concept of arbitrage through geometric and algebraic perspectives, linking it to Farkas' Lemma and analyzing the probability of arbitrage opportunities in random payoff matrices.

Contribution

It introduces a geometric interpretation of arbitrage, establishes its equivalence to Farkas' Lemma, and investigates the probability of arbitrage in random scenarios.

Findings

Geometric interpretation of arbitrage opportunities.

Equivalence between Arbitrage Theorem and Farkas' Lemma.

Analysis of arbitrage probability in random payoff matrices.

Abstract

This article introduces the notion of arbitrage for a situation involving a collection of investments and a payoff matrix describing the return to an investor of each investment under each of a set of possible scenarios. We explain the Arbitrage Theorem, discuss its geometric meaning, and show its equivalence to Farkas' Lemma. We then ask a seemingly innocent question: given a random payoff matrix, what is the probability of an arbitrage opportunity? This question leads to some interesting geometry involving hyperplane arrangements and related topics.