Asymptotic Exponential Arbitrage and Utility-based Asymptotic Arbitrage in Markovian Models of Financial Markets

TL;DR

This paper investigates conditions under which investors can achieve exponentially growing profits with high probability in Markovian financial models, using ergodic theory and large deviations, and explores the connection to utility-based arbitrage.

Contribution

It establishes new conditions for exponential arbitrage in Markovian models and links asymptotic arbitrage to utility maximization, extending previous results with novel probabilistic techniques.

Findings

Existence of exponentially growing profit opportunities with probability tending to 1.

Use of ergodic theory and large deviations to prove arbitrage results.

Discussion of utility-based asymptotic arbitrage and its relation to exponential arbitrage.

Abstract

Consider a discrete-time infinite horizon financial market model in which the logarithm of the stock price is a time discretization of a stochastic differential equation. Under conditions different from those given in a previous paper of ours, we prove the existence of investment opportunities producing an exponentially growing profit with probability tending to $1$ geometrically fast. This is achieved using ergodic results on Markov chains and tools of large deviations theory. Furthermore, we discuss asymptotic arbitrage in the expected utility sense and its relationship to the first part of the paper.