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Newton Raphson Iteration Procedure





The Newton-Raphson iteration procedure is designed to solve an equation of the form f(x) = 0 by extrapolating the local derivative to find the next estimate of the root.


It starts with a guess of the solution: x = x0. It then produces successively better estimates of the solution: x = x1, x = x2, x = x3,.... by using the following formula:



Equation Template


xi = current known x-value
xi+1 = next x-value used as a 'candidate' solution
f(xi) = g(xi) - TV
g(xi) = value of calculated function at xi
TV = target value of g(xi)
f'(xi) = first derivative (slope) of the function with respect to xi


Convergence Criteria

Usually x2 or x3 is close to the true solution. However in order to get the desired accuracy level the procedure may iterate up to 40 times.

The desired accuracy level varies depending on the scale of x0. For option valuations, base line accuracy is set at ten decimal places for x0 = 1, and decreases as the order of magnitude of x0 increases. For example, if x0 = 10 then the accuracy is set to nine decimal places; if x0 = 100 then the iteration procedure will converge to eight decimal places; and so on. For bond valuations, the desired accuracy level is nine decimal places regardless if the PPH is greater or less than $100.

If after 40 iterations the value is not within the desired accuracy level, then the entire process will start again with another estimate of x0. This will occur up to five times, and if the procedure still fails to converge then an error message will be displayed.

In most cases the procedure will converge within 3 and 8 iterations.



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