The Paradox of Arbitrage

The no arbitrage principle makes sense but when the time dimension is considered on how two identical assets trading at different venues can reach no arbitrage pricing there seam to be a paradox given that there is no party at either venue who can fix price fixing, i.e., the arbitrage argument for two assets some times seams paradoxical when there are two trading venues and trading is competitive (no body can set the price) and the time to converge is considered.

Lets consider two trading venues with supplies of identical products. If there is a miss pricing, arbitragers would buy form one venue cheaper and trader on the other. The supplier with the higher price would see less demand and would lower his price decreasing until the prices converge. But still the supplier can maintain his price or have a different price at this trading venue depending on whether he want to fix the price or his output. Any new information on the asset would create fluctuation in demand and the output would fluctuate if price is kept constant. Alternatively the price can be varied to meet a target demand but it is practically not possible.

The equation would be:

dX = [α (X – Y) + μ] dt

dY = - [β (X – Y) + μ] dt

Now let’s assume that there are many market plays now hold stocks of the assets. If there is counterparty, a trade would happen in the market. On new information on the asset, the supply and demand forces would vary randomly.

The equations

dX = [α (X – Y) + μ] dt + σXdW₁

dY = - [β (X – Y) + μ] dt + σYdW₂

If information dissipation and analysis is imperfect correl(dW₁, dW₂) < 1

In this case at least intuitively, it seam very difficult to imagine that there be no arbitrage opportunities. If the distance widens the are pulled back, but the random movements would make them drift in different directions. In a time interval they will not be the same. In the case that there is transport cost, with a certain probability there would be a arbitrage opportunity.

I am trying to work it out or trying to figure out if I overlooked something.

Best regards, Suminda Sirinath Salpitikorala Dharmasena