From No Free Lunch with Vanishing Risk to Martingale Measures

Summary

The exchange asks what conditions ensure that an equivalent pricing measure makes discounted asset prices true martingales, rather than only local martingales. Its answer distinguishes the setting in the fundamental theorem of asset pricing: under the stated locally bounded semimartingale framework, the no-free-lunch-with-vanishing-risk condition yields an equivalent local martingale measure. The response says boundedness of the asset process allows the stronger conclusion of an equivalent martingale measure.

The discussion points to Delbaen and Schachermayer’s work as the relevant theoretical reference, while another answer challenges whether the formal definition of arbitrage matches how traders use the term. This is a short exchange, not a full statement or proof of the theorem. The boundedness claim is stated briefly and should be read in the context of the precise assumptions in the cited result; the excerpt does not define all market conditions or clarify distinctions among possible boundedness notions.

Key ideas

• No-free-lunch-with-vanishing-risk is associated with an equivalent local martingale measure under locally bounded semimartingale assumptions.
• A stronger equivalent martingale measure conclusion is linked in the answer to bounded asset processes.
• True martingales and local martingales are distinct conditions in mathematical asset pricing.
• The exchange provides a reference and a brief claim, but omits the theorem’s full assumptions and proof.

Tags

# Equivalent (true) Martingale Measures and no-arbitrage conditions


# Equivalent (true) Martingale Measures and no-arbitrage conditions












I hope this is the correct site for this question, as it is rather theoretical...

In their famous paper, Delbaen and Schachermayer proved that the No Free Lunch with Vanishing Risk condition is equivalent to the existence of the Equivalent Local Martingale Measure. Are there any stronger no-arbitrage type conditions that guarantee that this measure is a true martingale measure (i.e. that all discounted asset prices are true martingales as opposed to merely local ones)?

I would be grateful for (academic) references.

## Answer by lehalle (score 2, accepted)

https://quant.stackexchange.com/a/3963

(If I remember well,) the local nature of the equivalent measure in the NFLVR theory comes from the fact that the market $S$ is a locally bounded semi-martingale. If it is bounded, you obtain an equivalent martingale measure.

Should be in A general version of the fundamental theorem of asset pricing, by Freddy Delbaen and Walter Schachermayer (thanks to Richard's remark, your answer seem to be theorem 1.1 of the paper).

## Answer by Keith A. Lewis (score 1)

https://quant.stackexchange.com/a/15692

Their definition of arbitrage is not what a trader would call arbitrage. See http://kalx.net/ftapd.pdf for the details.

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