# 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?

## 1 Answer

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.

• Can you please detail what an equivalent pricing measure is Commented Dec 3, 2019 at 8:00
• 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". Commented Dec 3, 2019 at 8:57
• Two prob measures are equivalent if they assign zero probability to the same set of events, they may assign different non-zero probabilities to other events, but they agree what events have zero prob. Commented Dec 3, 2019 at 14:22
• 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). Commented Dec 3, 2019 at 16:56