US Canadian Dollar Exchange Rate Prediction

January 13th, 2012 and December 19th, 2011 are the nearest inflection dates of significant importance. Gann date cycles indicate that December 19th is next expected cycle low. Approaching Dec 19th the USD/CAD is expected to be equal to or less than 1.0137. The price level of 1.0137 is the theoretical "Brick Wall" for our outlook. If the brick wall does not break, we expect a continuation of the existing Gann Price Trends as the USD/CAD rallies up to either Mid 1.05's or 1.0811 by Jan 13th 2012. If the bricks begin to crumble, we expect a continuation of the existing Gann Time Trends as the USD/CAD slides down to .9769 by Jan 13th, 2011.

 A reader lacking the trader mentality will think I have determined price will either go up or down by Jan 13th. Potential skepticism is caused by the inability to determine how to profit from the price outlook. Secondly,  skepticism is a result of the lack of understanding as to how your clients can hedge risk based on this outlook. Some of my successful spec trades involve trading in the direction of the "windfall gain" while acknowledging the limitations of that strategy. The earlier you can acknowledge when the potential limitation occurs, the quicker you can realize profit. The "windfall Gain" in this scenario occurs if prices break below 1.00 (or 1.0137). With a break below 1.00 an acceleration of the existing trend is expected to occur and we can price in a downward "Market Shock" scenario as the USD/CAD falls to .9769. If a test of 1.0137 reveals strength then I can cover USD/CAD shorts or  Sell EUR/USD. With certainty we can determine that when the USD/CAD is below 1.00 the primary functions remain linear. Above 1.00, the Gann prices form an exponential relationship.

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Some functions, when differentiated, give a result which can be written in terms of the original function. Perhaps the simplest example is the exponential function, f(x) = e^x. If we differentiate this function we get e^x again, that is

Another example of a function like this is the reciprocal function, g(x) = 1/x. If we differentiate this function we will see that

Other functions may not have the above property, but their derivative may be written in terms of functions like those above. For example if we take the function h(x) = e^xlog(x) then we see

Functions like these form the links in a so-called Pfaffian chain. Such a chain is a sequence of functions, say f₁, f₂, f₃, etc., with the property that if we differentiate any of the functions in this chain then the result can be written in terms of the function itself and all the functions preceding it in the chain (specifically as a polynomial in those functions and the variables involved). So with the functions above we have that f, g, h is a Pfaffian chain.
A Pfaffian function is then just a polynomial in the functions appearing in a Pfaffian chain and the function argument. So with the Pfaffian chain just mentioned, functions such as F(x) = x³f(x)² − 2g(x)h(x) are Pfaffian.

References

• Khovanskii, A. G. (1991), Fewnomials, Princeton, NJ: Princeton University Press, ISBN 0821845470 .