Completeness, Replicable Claims, and Uniqueness of Pricing Measures
Summary
This exchange gives intuition for the link between market completeness and a unique equivalent martingale measure. In a complete market, contingent claims can be replicated by trading available assets. No-arbitrage then fixes the price of each claim at the cost of its replicating strategy. If two equivalent pricing measures assigned different expected values to a replicable claim, they would imply different prices for the same payoff, contradicting that unique replication price.
The response contrasts this with an incomplete market, where some claims cannot be replicated and no-arbitrage may permit a range of prices. Multiple equivalent pricing measures can then be consistent with the traded asset prices. The explanation is conceptual rather than a proof in a specified model, and its compact phrasing does not develop technical conditions such as the market’s time structure or admissible strategies. It addresses why completeness implies uniqueness through pricing, rather than detailing a finite-state example or the converse direction of the theorem.
Key ideas
- In a complete market, every contingent claim can be replicated using traded assets.
- No-arbitrage ties a replicable claim’s price to the cost of its replicating strategy.
- Two pricing measures that give different values to a replicable claim would imply inconsistent prices.
- Incomplete markets may allow multiple no-arbitrage prices and equivalent martingale measures.
- The response offers intuition but does not provide a formal proof or model-specific assumptions.
Tags
Full text
# A financial market is complete if and only iff there exists a unique equivalent martingale measure
# A financial market is complete if and only iff there exists a unique equivalent martingale measure
Do you have any intuition behind the following theorem :
> A financial market is complete if and only iff there exists a unique equivalent martingale measure.
I understand the easier version of the theorem. Now I m trying to understand the dynamics of this theorem.
In particular, I can't understand why there must be only one unique. Can someone explain it to me ? I don't mind you take the simplest example, I just want to understand how the proof works in easy example.
The explaination that I have for now is that
$X_1$ is replicable, iff it has a unique AFP.
This is because here we can reproduce the pay off of $X_1$ by a combination of other strategies. Then, because the price of other instruments is fixed, the AFP must be unique.
Now having a unique AFP makes $ E^\mathbb Q [ X_1 ]$ constant accross all equivalent martingale measures. I don't understand this part . Is it simply because changing the measure doesn't change the outcome of a constant ?
What do you think of that?
## Answer by siou0107 (score 1, accepted)
https://quant.stackexchange.com/a/50069
Basically, if a contingent claim is replicable its value today is the value of the replicating strategy.
If you suppose that the market is complete and that there are two equivalent pricing measures $\mathbb{Q}^1$ and $\mathbb{Q}^2$, the price of a claim $A$ is given either by $\mathbb{E}^{\mathbb{Q}^1}\left( A \right)$ or by $\mathbb{E}^{\mathbb{Q}^2}\left( A \right)$. But because $A$ is replicable, by NA there can only be a unique price, that of the replicating strategy. For that, you must have $\mathbb{Q}^1=\mathbb{Q}^2$.
If the market was not complete, there would be a range of NA prices, and thus several (even an infinity) of equivalent pricing measures.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.