Prove that a market is arbitrage free

Question

The question is based on a one period model. Let a market be arbitrage free, and then let a security $X$ be added to it. Denote $P(X)$ as the price of this security at $t=0$. The security has the property that:

$$A=\text{inf}\{E(\phi X)|E(\phi X)<\infty,\phi \in \Phi\}$$ $$B=\text{sup}\{E(\phi X)|E(\phi X)<\infty,\phi \in \Phi\}$$

$A<B$, $P(X)\in (A,B)$. Where $\Phi$ is the set of state space deflators. I want to show that the market is arbitrage free.

Any tips on how to prove this result? And what is $E(X)$ even supposed to mean here? I suppose it's the expectation of the price on period 1?

Attempt ($Q$ denotes quantity of a specific security. $S(t)$ the price at $t$):

Let value of a portfolio: $\sum^n_{i=1}Q_i S_i(0)=P(X)=E(\phi X)$

Therefore there exists a deflators $\phi_1,\phi_2$:

$$E(X\phi_1)\leq\sum^n_{i=1}Q_iE(\phi_1 S_i(0))=E(\phi X)=\sum^n_{i=1}Q_iE(\phi_2 S_i(0))\leq E(X\phi_2)$$

Now assume that in every case:

$$\sum^n_{i=1}Q_i S_i(1)\geq X$$

And there is a possibility: $$\sum^n_{i=1}Q_i S_i(1)> X$$

This is impossible by the above, but how better show it?

Answer 1

It would involve comparison of the payoff and price of security X to the prices and payoffs of all the securities that are in the market now. Ignoring the difference between strong and weak arbitrage for ease of exposition: The price of this security X should be higher than the prices of all the securities whose payoffs are lower than the payoff of this security and should be lower than the prices of all securities whose payoffs are higher. You can set up two linear program problems to establish the arbitrage free price range: lower and upper bounds (A and B in your case).

The above problem is a variation of the above but is expressed in terms the state price deflators/density. Essentially if the market is not complete, then there would be many state price deflators (same thing as the price bounds in the first para). If one knows all these deflators, then one can find the one that gives the min price and the one that gives the max price. The corresponding prices are the lower and upper bounds. If the price of the security lies in the range, then there is no arbitrage.

Why is that so? It is because $ A<B $ implies that security X is not redundant (otherwise A and B should be equal). We know the prices and payoffs of the existing securities (when you say market is arbitrage free) but we know only the payoff of X. So what price should we assign to X. From the known market prices we can only infer a range of prices for this security as we have multiple deflators and we don’t know which one will the market use to price this asset (if market were complete then there would be no ambiguity). We cannot therefore construct an arbitrage portfolio if the price of X is within these bounds. If we construct the arbitrage portfolio using the deflator associated with A, then it would involve buying at A and selling at P (buy cheap sell expensive), but if the true deflator turned out to be B, then the arbitrage portfolio will make a loss instead of a profit. You might ask why we just try to market the security to get a price quote? Valid question but then this problem would not exist as we would then have a unique deflator (hopefully!) and A=P=B.

Re your other question: $E[ \phi X]$ means expected value of the deflated payoff under the probability associated with each of the deflator, same thing as price.

Hope this helps.