Looking at a technical paper for converting European Normal Vol back into LogNormal Vol but missing some logic, was hoping you can help.
We start with this 1.2(a):
Run into some issues when F->K and use for those cases use 1.4(a,b):
Now we want to Invert the above so that if given a Normal Vol we may recover Black(Lognormal) Vol using 1.5-1.7:
Here is the question: in (1.7) We have the circled term; this is what we are solving for so if we knew it we would not need use at as our Guess in NR algorithm. I ran the model for the cases where |F-K|/K >= .005 and it works fine; but am not sure how to implement the |F-K|/K < .001 Case. I suppose I can simply put a fixed value in the guess of .001 for Black Vol but this does not seem correct as we are trying to find a good guess for it by using; it would be a guess of a guess and arbitrary and may not guarantee convergence. Are we supposed to run the Newton Scheme twice, once to find out initial guess in (1.7) and again to solve (1.5) what is the correct way to use this?
Using the derivative approximation for Abs((f-K)/K) < .0001 causes the the newtwon scheme to fail to converge; As an example for F =3679.75 ; K=3680; Abs((f-K)/K) = 6.79E-5 < .001. The two derivative estimations are wildy different HPrime1 = 3679.874998 where as other derivative estimate HPrime2 =0.000271748. Is there a typo, what is wrong with the first derivative estimates that cause the netwon scheme to not converge and how can I alter this?
Please let me know what I am missing. Thanks.
Ref: http://janroman.dhis.org/finance/Norm%2 ... ormvol.pdf