Statistical Arbitrage in the Black-Scholes Framework

TL;DR

This paper demonstrates the existence of statistical arbitrage opportunities within the Black-Scholes model by analyzing specific trading strategies and deriving conditions based on the Sharpe ratio, supported by simulations.

Contribution

It provides analytical formulas for trading profit metrics and establishes no-arbitrage conditions related to the stock's Sharpe ratio in the Black-Scholes framework.

Findings

Existence of statistical arbitrage in Black-Scholes model

Derived formulas for expected profit, variance, and loss probability

No-arbitrage condition linked to the Sharpe ratio

Abstract

In this study we prove the existence of statistical arbitrage opportunities in the Black-Scholes framework by considering trading strategies that consists of borrowing from the risk free rate and taking a long position in the stock until it hits a deterministic barrier level. We derive analytical formulas for the expected value, variance, and probability of loss for the discounted cumulative trading profits. No-statistical arbitrage condition is derived for the Black-Scholes framework, which imposes a constraint on the Sharpe ratio of the stock. Furthermore, we verify our theoretical results via extensive Monte Carlo simulations.