Barrier-Based Statistical Arbitrage in the Black–Scholes Model

Summary

The study analyzes a strategy in the Black–Scholes framework that borrows at the risk-free rate and holds a long stock position until the price reaches a deterministic barrier. It derives analytical expressions for the expected value, variance, and probability of loss of discounted cumulative trading profits. These quantities describe both the strategy’s average outcome and aspects of its risk.

The authors also derive a condition for the absence of statistical arbitrage, expressed as a constraint on the stock’s Sharpe ratio. They report checking the theoretical results with Monte Carlo simulations. The document presents a mathematical result within the Black–Scholes assumptions; it does not establish that the strategy remains profitable under real market frictions, changing parameters, or other market models. The excerpt supplies no numerical simulation findings or implementation details, so it supports understanding the framework rather than assessing real-world returns.

Key ideas

• The strategy borrows at the risk-free rate and holds a long stock position until a deterministic price barrier is reached.
• The analysis derives the discounted profits’ expectation, variance, and probability of loss.
• A no-statistical-arbitrage condition constrains the stock’s Sharpe ratio.
• Monte Carlo simulations are used to check the theoretical results.
• The findings are framed within the Black–Scholes model and do not establish real-market profitability.

Tags

# Statistical Arbitrage in the Black-Scholes Framework


# Statistical Arbitrage in the Black-Scholes Framework









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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