  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.

Mathematical Finance;Pricing of Securities