Equivalent Martingale Measures and Strictly Local Martingales
Summary
The document asks how equivalent martingale measures (EMMs) differ from equivalent local martingale measures (ELMMs) in Brownian financial-market models, and whether existence of an ELMM guarantees an EMM. It highlights that an EMM is also a local martingale measure, while the reverse implication is not automatic: a local martingale can be strict rather than a true martingale.
The response points to stochastic-volatility examples in which natural candidate measures are only strictly local martingale measures, while actual martingale measures can also exist. It also references general integral tests for determining when EMMs or ELMMs exist. These examples and tests show why the distinction matters when connecting measure changes to no-arbitrage claims.
The document does not derive the conditions or state a general theorem resolving every model. Its value is to flag that the terminology carries mathematical substance and direct readers toward technical sources for the precise existence criteria.
Key ideas
- Every equivalent martingale measure is a local martingale measure, but the converse requires care.
- A local martingale may be strict and fail to be a true martingale.
- Stochastic-volatility models can have natural candidate measures that are only strictly local martingale measures.
- General integral tests can help assess the existence of EMMs and ELMMs.
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Full text
# Equivalent local martingale measure vs. equvalent martingale measure in a Brownian setup # Equivalent local martingale measure vs. equvalent martingale measure in a Brownian setup Assume you have the standard financial market built up of a Brownian motion. I have seen some books say that an equivalent local martingale measure imples no arbitrage, and some say that an equivalent martingale measure implies no arbitrage. These statements are not contradictory one is just stricter than the other. But I am wondering, why do some state it with the "local" and some not? Is there some difference here when you include "local"? Is there some meaning behind it, or do they just simplify when they exclude it? My last qustion is: An equivalent martingale measure is ofcourse also a local one, but is the existence of a equivalent local martingale measure equivalent to an existence of a martingale measure? That is, does the existence of a local martingale measure imply the existence of an equivalent martingale measure? ## Answer by ir7 (score 1) https://quant.stackexchange.com/a/66048 The examples provided by Sin in their article Complications with Stochastic Volatility Models might help to answer your questions. I'm transcribing the abstract below: > We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what it is usually assumed in the finance literature. We also show the existence of martingale measures, however, and give explicit examples. And this technical article (No arbitrage in continuous financial markets, by Criens) covers general integral tests for the existence and non-existence of EMM and ELMM (e.g., Theorem 3.1).
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