Whether Zero-Cost Arbitrage Implies Negative-Cost Arbitrage

Summary

The document asks whether a zero-cost arbitrage opportunity necessarily implies that a negative-cost arbitrage also exists. It explores the question in a simple two-asset binomial market containing a riskless bond and a risky stock, assuming traders may take fractional long or short positions. Two informal constructions are proposed: shorting the stock and investing the proceeds when the bond is cheaper, or borrowing slightly more than needed and using most of the funds to buy the stock in the opposite price ordering.

These examples motivate the conjecture that negative initial outlay with a positive later payoff can be engineered, but the author does not establish that result. The document provides no formal proof, complete case analysis, or discussion of the conditions under which the proposed trades remain arbitrage after financing, constraints, or payoff requirements are considered. Its contribution is a question and a preliminary line of reasoning, not a demonstrated general theorem.

Key ideas

• The document asks whether a zero-cost arbitrage implies an arbitrage with negative initial cost.
• It tests the idea informally in a binomial market with a bond and a risky stock.
• The proposed constructions rely on fractional positions and, in one case, short selling.
• The examples do not constitute a proof or establish the claim for general markets.

Tags

# Does zero cost arbitrage imply the existence of negative cost arbitrage?


# Does zero cost arbitrage imply the existence of negative cost arbitrage?












I've been wondering, if there exists a zero-cost arbitrage trading strategy in some market, does that also mean that there also has to be a negative-cost arbitrage trading strategy in the same market?

I did a small experiment with a simple market of two assets, the riskless one and a risky one, in a binomial tree setting. It seems that, if you are allowed to buy/short fractional amounts of either asset it is always possible to engineer a strategy in which you have negative initial outlay and positive payoff later (even if it is a worse strategy than the zero-cost strategy).

Scenario 1: Bond is cheaper than the Stock

- Short the stock, Invest proceeds (negative cost)

Scenario 2: Stock is cheaper than the Bond

- Borrow slightly more money than you need, buy the stock with almost all of the money.(negative cost)

However, I'm not really sure this small example covers all scenarios. If my observation is true, how should I approach verifying it in a more rigorous manner?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.