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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?

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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.

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  • $\begingroup$ Can you please detail what an equivalent pricing measure is $\endgroup$
    – Marine Galantin
    Commented Dec 3, 2019 at 8:00
  • $\begingroup$ 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". $\endgroup$
    – siou0107
    Commented Dec 3, 2019 at 8:57
  • $\begingroup$ 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. $\endgroup$
    – Alex C
    Commented Dec 3, 2019 at 14:22
  • $\begingroup$ 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). $\endgroup$
    – siou0107
    Commented Dec 3, 2019 at 16:56

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