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I am analyzing a financial instrument that has seven prices. The seven prices each have an expiration of 1 month, 2 month, 3 month etc. I would like to measure the curvature of the curve. I know how to easily calculate the slope of the curve, but how do I calculate the curvature? Here is an example of one series:

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Thank you,

Todd
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  • Try to fit a polynomial curve to the data, then try to compute its derivative, that will essentially give you a 'slope function' –  Feb 02 '16 at 14:53
  • I will get the coefficients from the polynomial curve. Then use those coefficients to calculate the slope? – Todd Feb 02 '16 at 15:04
  • If you use a second-degree polynomial, $y = ax^2 + bx + c$, the coefficient of the $x^2$ term is proportional to the rate of change of the slope. That's one way to measure curvature; there are others, but it's not clear why any other way would apply here. What do you think "curvature" could tell you about the prices of your instrument? – David K Feb 02 '16 at 15:18
  • Thanks David. Well the short end of the curve can have a negative slope sometimes while the long end of the curve can have a positive slope sometimes. Calculating the slope of the line does not make sense. I figured getting an idea of the curvature would be better. The line can be straight sometimes and sometimes look very odd. When it looks odd it provides information about the stock market. That is what I am trying to get after. How to measure it when one end is sloping down and the other is straight or up. – Todd Feb 02 '16 at 17:08

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For plane curve $y=f(x)$, $$\kappa = \frac{y''}{(1+y'^{2})^{3/2}} $$ For small $h$, $$y'=f'(x) \approx \frac{f(x+h)-f(x-h)}{2h} $$ $$y''=f''(x) \approx \frac{f(x+h)-2f(x)+f(x-h)}{h^{2}} $$

Ng Chung Tak
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