Risk Latte - Parabolas in Financial Markets

Parabolas in Financial Markets

Team latte
May 22, 2008

Commenting on the relentless rise of the crude oil prices, Richard E. Cripps, chief market strategist for Stifel Nicolaus said yesterday: "There's almost a parabolic rise going on."*

These days, like those days in the late nineties, one hears the term "parabolic rise" quite often in the financial press, especially with regard to the commodity prices. The blogsphere and financial press is replete with analyst quotes that talk about parabolic rise of the commodities market, crude oil prices, gold prices and the like.

A year or so back, analysts talked a lot about the "parabolic rise" of the Chinese stock market.** If one looks into Technical Analysis methodologies in some detail, he will find a term called "Parabolic SAR" technique to look at stock prices.

Just as in physics and engineering (as well as nature around us) parabolas and parabolic shapes are very common in financial markets (stocks, bonds, commodities, etc.). These shapes manifest themselves prominently when the markets are in steep rise or steep fall. It is then that analysts start talking about "parabolic rise" or "parabolic fall".

Mathematically, a parabola is conic section, a curve which follows a certain generalized equation. It is actually a set of all points in the  xy  plane equidistant from a given line  L and a given point  F (focus) not on the line. A very common form of parabola, which is found in many applications in quantitative finance, is given by the equation in  xy  plane as:

y = ax2 + bx + c

In the above,  a, b and  c  are arbitrary constants. Let's look at two examples from the stock/commodity markets. If you look at the weekly closing prices of NASDAQ stock market between Jan 1997 and March 2000 you'll see a parabolic fit to the prices

From the above graph we see that the price history fits a parabolic shape. The polynomial equation that fits the data is given by a parabola:

y = 0.032 x2 - 228.39 x + 4000000

Here, the constants are:  a = 0.0032,  b = - 2228.39  and  c = 4000000. The goodness of fit given by R2 statistic is very high suggesting that the parabola fits the price data very well. In the equation,  R2 = 0.919.

Another example would be the PHLX Gold & Silver sector Index. The PHLX Gold/Silver SectorSM (XAUSM) is a capitalization-weighted index composed of 16 companies involved in the gold and silver mining industry. XAU was set to an initial value of 100 in January 1979. If we take the weekly closing prices of this index between January 1995 and May 2008 we will see a parabolic shape fitting the price history.

Here again we see an extremely good fit of a parabolic shape to the observed index values. The polynomial equal is a generalized parabola given by:

y = 0.00002 x2 - 1.2772 x + 23612

The coefficients (constants) are  a = 0.00002,  b = -1.2772  and  c = 23612. The goodness of fit is:  R2 = 0.86

If you take any asset market during times of boom (great optimism or rapid rise of price due to any reason) or crisis (crash) you would be able to find parabolic shapes in the price history. An interesting point is that since parabolas are conic sections they have turning points (points of inflexion) which can signal a change in direction. Many a times a parabolic rise in price of an asset is marked by a steep fall (though this may be intuitively obvious, tracing the curve may be an interesting observation).


* http://biz.yahoo.com/ap/080521/wall_street.html
** http://www.marketoracle.co.uk/Article838.html
*** http://mathworld.wolfram.com/Parabola.html

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