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The premium on a put option may be estimated very easily using the following equation that I have discovered, which is fairly accurate compared to BS when strike price is on the x-axis and premium is on the y-axis: $2\sigma e^{-\frac{\left|k-S\right|}{4\sigma}}\ +\left|\frac{k-S}{2}\right|+\frac{k-S}{2}$ For a call option, reflect the equation in the y axis ...

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I built something like you attempted here to estimate yield curves for various sectors using Bbg bond quotes years ago in VBA. For the calibration of the parameters I used Nelder Mead. It doesn't need the first and second derivative or estimators of it such as Newton-Ralphson and such.

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When we worked with that model several years go, we used Differential Evolution and it worked very well. See Calibrating the Nelson-Siegel-Svensson Model. At least in the standard version, a best-of-many gradient searches (with random initial values) also worked well. See A Note on 'Good Starting Values' in Numerical Optimisation. If you were willing to use ...

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