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I read an interview today with Stephane Coquillaud.

He talked about this idea of formulating a data set of the G5 currencies as a pentahedron. The obvious benefit is the fact that there is more information contained in the data and you'll no longer be testing on marginal price movements of single pairs but rather on the entire major currency spectrum.

He mentioned the other benefit of complexifying the data set from a simple 2 dimensional time series to a 5 dimensional cascade of multiple currency pairs is that the complexity itself reduces the tendency of over fitting parameters in back-tests.

Has anyone here attempted anything like this? I'm having a hard time wrapping my head around the data structure needed for this pentahedron "message" and how to calculate it from a collection of 2 dimensional currency pair time series. In the past, I've built a relative currency strength indicator that took incremental market returns from currency pairs and adjusted the relative strength of each individual currency component of the pair. For example if EUR/USD goes up 2%, then I add 2% to EUR and subtract 2% from USD and factor in the movement from over 24 pairs. However, I don't think this is quite the same thing. Suggestions and insight are wholly welcome!

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Since pentahedrons are 3d shapes, but there is no reason to think currencies live in a 3d world, you can just treat the 'pentahedron' as a weighted node graph of the 5 currencies. A graph edge from one currency to another represents an exchange of those two currencies. So in the same fashion as usual vectors, I can go from currency A to C via B by executing AC,CB.

If we also know some cross-currency prices for AC, we can construct a triangle ABC where we know the price of each edge. In geometry, we get the hypotenuse from the sides via Pythagoras, but the price of AC = AB/BC (or multiplied, depending on the quoting). So you need to do some translation to get from cross currency pricing to the geometry/topology of the graph, but the basic idea is there; if you update the price for a particular trade, that alters the graph in ways that imply changes to other prices.

I think the nub of the problem, though, is to know how 'stiff' the edges are. For example, suppose EUR/DKK is liquid, EUR/TRY less so, and DKK/TRY much less so; so seeing a change in EUR/TRY probably means that TRY is moving, not EUR, so you expect DKK/TRY to change rather than EUR/DKK. Do you pin USD and EUR in space, and let the rest of the geometry move around them?

You can probably perform some kind of dimensional reduction like PCA on the 5d structure to see the main axes of variance, but since none of the currencies is entirely dependent on the others, there will always be extra factors that prevent you eliminating any dimensions entirely; every government and domestic market is a source of information, and thus an unpredictable input into the currency's martingale.

Ultimately, it's about what you want to measure. If you have a reason to pick out a specific weighted basket of currencies, then you can just price that basket to see movements. In finance there are no absolutes, only relative values. So you have to have a reason to look at a particular basket; there is no independent view of the market, and neither do you need one.

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I know of computational geometry used in finance mostly for optimization problems but never anything of the sort like this.. –  pyCthon Oct 15 '12 at 16:03
Very interesting...How would you maintain equilateral triangles to adjust the change in price(assuming you made right triangles to utilize Pythagoras, I could be way off base) - How do you keep normalizing the edges as the currencies move? From your explanation I'm visualizing a structure that could theoretically begin as a pentahedron but begins to collapse as soon as prices begin to move. –  jeff m Oct 15 '12 at 16:47
The point of the first paragraph was that I don't think it's an actual pentahedron in 3d space, because that assumes there are only 3 degrees of freedom (dimensions) in 5 currencies. In truth there are at least 5. So think more about graph theory and less about geometry; arbitrage forms alternate paths on the graph. –  Phil H Oct 16 '12 at 12:21
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