How Arbitrage and Dominance Relate in Option Markets
Summary
The exchange explores whether the absence of arbitrage implies the absence of a dominant trading strategy. Its main lesson is that the relationship depends on precise definitions and market constraints. An answer uses put-call parity to illustrate how restricting short sales can affect the comparison between option and underlying positions, and argues that absence of dominance is tied to equilibrium assumptions that are not identical to the no-arbitrage condition.
The responses also distinguish a dominant hedge or long position from an arbitrage opportunity, noting that pricing relationships can reveal arbitrage even when no single position dominates alternatives. However, the discussion offers no fully worked numerical example, and its claims are presented informally. The put-call parity illustration is limited and depends on assumptions such as the treatment of interest rates and short selling. Readers should use it as a conceptual prompt to define dominance, arbitrage, and market constraints carefully, rather than as a complete proof of their relationship.
Key ideas
- No-arbitrage and absence of dominant strategies are distinct conditions whose relationship depends on definitions and assumptions.
- Short-sale constraints can change how option pricing inequalities relate to arbitrage and dominance.
- Put-call parity provides a way to discuss pricing inconsistencies between options and the underlying asset.
- The exchange distinguishes a dominant position from an arbitrage opportunity but does not give a formal proof.
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Full text
# Arbitrage and dominant strategies # Arbitrage and dominant strategies If there is no arbitrage there is no dominant trading strategy, but there may be arbitrage opportunities even if there are no dominant trading strategies. Could you explain this statement and bring an example? ## Answer by Drew (score 1) https://quant.stackexchange.com/a/15869 Please clarify rigorously what you mean by each term. It is not true that no dominance is a consequence of no arbitrage. Think of the put-call parity: $C-P=S-K$, assuming $r=0$ since it's inconsequential. If there is no short selling then we can have: $C-P \geq S-K$ without arbitrage but No Dominance would not hold. If you think very deeply about this, and I am assuming conventional meanings since no definition is given, then no arbitrage can be the consequence of a single trader taking huge positions and removing arb opportunities from the market. But ND is something that will occur only if the asset prices correspond to an equilibrium process in some economy. Conventionally, $ND \implies NA$, but don't confuse this for risk-neutral valuation. For that to be the true value of an asset, we need $ND$ in some economy. ## Answer by JTHouseCat (score -1) https://quant.stackexchange.com/a/15888 If there are no arbitrage opportunities there is no dominant hedge or long position. Why would there be an arbitrage opportunity if everything was priced correctly? There may be arbitrage opportunities even if there are no dominant hedges or long positions. Put-call parity shows arbitrage opportunities of badly priced options regardless of long position mispricing. Using put-call parity, only, to try and explain this is a bit limiting, and I would like to warn you about people who offer answers only relating to put-call parity.
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