Arbitrage Conditions and Why a Short Position Alone Is Not Risk-Free Profit

Summary

The document presents a two-date definition of arbitrage: a portfolio must have nonpositive initial value, nonnegative value at the later date in every outcome, and a strictly positive chance of a gain. It questions why both value conditions are needed, using a single asset priced at 3 initially and either 1 or 2 later. A short position has negative value at the later date under the stated portfolio-value convention, so it does not meet the definition as written.

The example highlights the distinction between a portfolio’s marked value and the complete cash flows from opening and closing a short. A sound arbitrage analysis must account for short-sale proceeds and the cash account or other financing arrangements; the document does not specify these. Its example therefore raises a useful modeling question but does not establish that the short is an arbitrage under the stated definition. The answer would depend on how the position and its associated cash flows are represented.

Key ideas

• An arbitrage requires a nonpositive initial value and a nonnegative terminal value across outcomes, with a possible strict gain.
• A short position in the asset has negative terminal marked value in the example.
• Short-sale proceeds and financing cash flows must be included when evaluating the trade’s total payoff.
• The example’s portfolio representation is insufficient to conclude that the short is an arbitrage.

Tags

# Definition of Arbitrage


# Definition of Arbitrage












Definition. An arbitrage is a portfolio $H$ ∈ $R^n$ such that

• $H · P_0 ≤ 0 ≤ H · P_1$ almost surely, and

• $P(H · P_0 = 0 = H · P_1) < 1$.

where $P_0$ and $P_1$ $\in R^n$ represent the prices at time $t=0,1$ respectively.

Now, my question is why do we need the first condition. Suppose there is only one asset A which at time $t=0$ costs $3$. Then, at $t=1$ we have $P(A=2)=\frac{1}{2}$ and $P(A=1)=\frac{1}{2}$. The portfolio $H=-1$ should be an arbitrage because it yields certain profit with no risk attached but it isn't because $H \cdot P_1<0$.

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