Do NGDP futures markets already exist?

Unfortunately forex markets don't know they're NGDP markets.

Scott Sumner's big idea is that we should have a high-volume, liquid NGDP futures (prediction) market and use that to determine monetary policy. He frequently suggests that the existence of such a market would help understand macroeconomic conditions (for example, here with Australia).
Now I am not as convinced about the utility of prediction markets [1], but I had the random thought the other day while I was on a flight to New Mexico: Scott's high-volume, liquid futures market exists and is called the forex market.
The exchange rate between two countries is effectively (proportional to) a ratio of their aggregate demands [4]. The ratio of the price of a dollar to the price of a Euro depends not just on the supply, but the demand for both currencies -- the exchange rate is a general equilibrium solution, not partial equilibrium [2]. That is to say, you can look at the exchange rates between each pair of currencies and get a measure the relative NGDP of the two countries. Since exchange rate data is typically available immediately, that gives a forecast of the NGDP number that comes out usually a month (first estimate) to three months (third estimate) after the quarter it represents. You could measure any given currency against a basket of currencies to approximate an absolute measure.
Is there are problem with this?

Yes. Yes, there is.

Forex markets are notoriously volatile (see e.g. the Dornbusch overshooting model). And one of the reasons, at least from the information equilibrium perspective [3], is that it seems traders might have a sign error in their mental model leading to corrections that go the wrong way at first and only gradually return to a normal level (see the picture at the top of this post which is from [5]). For example, in a supply and demand diagram an expansion of the monetary base leads to a price drop for a currency. However, in general equilibrium an expansion of the monetary base is (usually) accompanied by a (relative) expansion in the economy. That is to say additional money isn't printed unless there is demand for it. And demand may rise a little bit, just the right amount or too much -- leading to excess inflation, trend inflation or below trend inflation/deflation.
It is possible to fix this by convincing forex markets that they are NGDP futures markets. But that is dependent on the "wrong model" theory [3, 5] accounting for most of the volatility.

Update + 10 min

I should add that this is relevant. Potentially exchange rates are measuring long run NGDP ratios in the same way P/E ratios measure long run returns which are volatile.
References (from this blog)

[1] Is the market intelligent?
[2] What do exchange rates measure?
[3] Is market monetarism wrong because the market is wrong?
[4] Exchange rates and monetary policy
[5] Exchange rates and irrational markets