Two Formulations of Arbitrage in a One-Step Binomial Model

Summary

The document compares two definitions of arbitrage in a one-step binomial market containing a risk-free asset and a risky asset. One definition requires a zero-cost portfolio whose payoff is never negative and is positive with nonzero probability. The other allows a portfolio with negative initial cost so long as its next-step payoff is never negative.

The responses distinguish the cases: the zero-cost definition excludes riskless, cost-free opportunities with a possible gain, while the negative-cost formulation also excludes receiving money upfront while facing no chance of a loss. A second response questions whether the definitions are equivalent, noting that one permits a gain only with some positive probability whereas the other requires a nonnegative payoff with certainty after receiving the initial payment. The exchange raises a definitional distinction but does not provide a formal proof or settle all assumptions needed to compare the formulations.

Key ideas

• One arbitrage definition starts with zero cost and requires a nonnegative payoff with a possible gain.
• Another permits a negative initial cost while requiring a nonnegative next-step payoff with certainty.
• The second formulation includes cases where an investor receives money upfront without risking a loss.
• The responses differ on equivalence, and the document gives no formal proof resolving the issue.

Tags

# Equivalent formulations of the No-Arbitrage Principle


# Equivalent formulations of the No-Arbitrage Principle












Let us consider two definitions of the arbitrage opportunity, for simplicity focusing on the one-step binomial model with one risk-free and one risky asset. We define an arbitrage opportunity to be either a portfolio $V_t$ such that

- $V_0 = 0$, $V_1 \geq 0$ and $V_1 > 0$ with non-zero probability

or a portfolio such that

- $V_0 < 0$ and $V_1 \geq 0$ with probability one.

Then we can formulate the No-Arbitrage Principle using (1) or (2). Which one is stronger? Any help would be appreciated.

## Answer by ir7 (score 4)

https://quant.stackexchange.com/a/83592

Definition 1 excludes portfolios where one invests nothing, has no risk of loss, but could profit.

Definition 2 excludes portfolios where one can get paid and is essentially guaranteed not to lose any money at the next step.

Basically, Definition 2 covers extreme arbitrages too, not just the common sense ones.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/83730

Hmm I don’t see how these are equivalent. A portfolio satisfying (2) makes money with probability one, whereas a portfolio satisfying (1) makes money with a probability p which only has to satisfy 0<p<=1.

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