# 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?

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 $$\mathbb{Q}^1=\mathbb{Q}^2$$.
• The general definition of an equivalent martingale measure is: a measure that is equivalent to the physical probability measure $\mathbb{P}$ and under which asset prices relative to a certain numéraire (which can be any strictly positive asset price: the cash account, the ZC bond, the annuity, a stock price...) are martingales. The primary example is the so-called risk-neutral measure: the one under which asset prices expressed in terms of the cash account are martingales. You use such probability measures to price instruments, hence the term "pricing measure". – siou0107 Dec 3 '19 at 8:57
• The equivalence is necessary for NA to hold: if a claim paying off in a $\mathbb{P}$-impossible event has a positive price (i.e., it is not $\mathbb{Q}$-impossible, with $\mathbb{Q}$ your pricing measure), just sell it and wait. Conversely, if a claim has a null price (i.e. it is $\mathbb{Q}$-impossible) but pays off in an event that has a positive real probability (under the physical measure $\mathbb{P}$), just buy it and wait (classical definition of an arbitrage). – siou0107 Dec 3 '19 at 16:56