Why Risk-Neutral Measures Must Be Equivalent to Real-World Measures
Summary
The document explains why an equivalent martingale measure (EMM) is used in pricing theory. Under the stated one-period setup, a pricing measure must be a probability measure and must make discounted portfolio values satisfy the martingale condition. Equivalence adds the requirement that the pricing and real-world measures assign zero probability to the same events. This preserves which outcomes are possible when changing measures for valuation.
The answers connect this condition to arbitrage: whether an opportunity exists should not depend on which probability measure is used to describe outcomes. In a finite discrete model with positive probability assigned to every state, equivalence is natural when state prices are also positive. The explanation is conceptual rather than a proof of the fundamental theorem of asset pricing, and it does not discuss technical conditions for continuous-time markets or cases with states assigned zero probability. Its focus is the role of shared null events, not the derivation of a particular pricing measure.
Key ideas
- An equivalent martingale measure makes discounted portfolio values martingales while remaining a probability measure.
- Equivalence means the real-world and pricing measures share the same probability-zero events.
- Sharing null events preserves which outcomes are considered possible when changing measures.
- Positive state prices in a finite model support the use of an equivalent measure.
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# Equivalent Martingale (/Risk Neutral) Measure Conditions
# Equivalent Martingale (/Risk Neutral) Measure Conditions
I am trying to understand EMM's and wanted to understand why we use EMM's over just martingale measures. The way we define EMM's is (for a simple one period model):
Given a probability measure $\mathbb{P},$
- $\mathbb{Q}$ must be a probability measure,
- $\mathbb{E}^{\mathbb{Q}}[\bar{V_1}]=\bar{V_0}$ where $V_i$ is the portfolio value at times $0 \text{ and } 1$,
- $\mathbb{Q}\sim \mathbb{P}$ in the sense that they both have the same null sets.
I have seen the usefulness of assumption two and the first assumption seems like a core requirement to the framework, but I would like to know more about why the third assumption is useful. I have been told that it is useful when transfering statements about being $\mathbb{P}$ almost surely and $\mathbb{Q}$ almost surely.
For reference, we define a martingale measure as just the first two conditions.
## Answer by user34971 (score 4, accepted)
https://quant.stackexchange.com/a/49486
The concept of arbitrage must be independent of the probability measure. This means that only measures equivalent to the real-world measure, where you can actually measure arbitrage, are permissible.
## Answer by Kevin (score 2)
https://quant.stackexchange.com/a/49485
Two measures are said to equivalent if they have the same null sets. For finance, this means that both measures agree on which events can happen and which events can't. So, all this definitiion really imposes is that the real world measure and the risk-neutral measure agree on what events can take place and which ones can't.
In a simple discrete space setting, you typically assume $\mathbb{P}$ to be a strictly positive measure (well, unless $\emptyset$ obviously). i.e. $\mathbb{P}[\{\omega\}]>0$ for all $\omega\in\Omega$. So basically, you don't include any impossible states of world. This also means that statements hold for all states and not just $\mathbb{P}$-as. As you know, you can define the risk-neutral measure $\mathbb{Q}$ via Arrow-Debreu prices - and they should obviously positive as well. So, $\mathbb{Q}$ is a strictly posiitve measure as well and thus $\mathbb{P}\sim\mathbb{Q}$ is kind of natural.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.