Equivalent Martingale Measures and the Meaning of Fair Price

Summary

The document raises a foundational question from mathematical finance: why does the existence of an equivalent martingale measure for discounted stock prices rule out arbitrage, and what does a fair price mean? It frames the issue in a finite-outcome setting, where an equivalent probability measure changes probabilities without changing which outcomes are possible, and asks how the martingale condition connects to fair valuation.

The document provides no answer, proof, example, or trading method. It is useful as a prompt for studying the fundamental theorem of asset pricing, but readers will need another source to learn the precise assumptions and argument. In particular, the claim is stated without discussing market completeness, admissible trading strategies, or technical conditions on prices and portfolios, all of which matter when interpreting the no-arbitrage equivalence.

Key ideas

• An equivalent martingale measure is presented as the condition associated with an arbitrage-free market.
• The question asks how a martingale valuation measure gives meaning to a fair price.
• The document poses the issue but does not supply a derivation or resolve it.
• The stated equivalence needs model and trading assumptions to be interpreted precisely.

Tags

# Fair price and no arbitrage


# Fair price and no arbitrage












The market is arbitrage-free iff there exists an equivalent martingale measure for the discounted price process of the stock.

So in a world with a finite amount of possible outcomes $\Omega$ that follow the probability distribution $P$, where we can change $P$ by an equivalent probability measure that is a martingale, there are no arbitrage opportunities.

Now this martingale measure is supposed to model a 'fair game/price'. Could someone please shed some light on why a 'fair price' implies there is no arbitrage? What is a 'fair price'?

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