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You could probably do it with Matlab or Mathematica, but I do not know how to use them. Thanks ahead of time.Where:
Standard Normal Distribution Function =
Really? Just to check: you know all the parameters in BS formula except [$]\sigma[$] and you want to find the latter?I agree, but no one seems to be able to answer the question. It's not even in the CFA review manual (levels 1-3). I'm trying to create a shortcut for my VIX model. There are easy ways to compute it without solving for it but if you solve for it, it simplifies the program dramatically.
OK:Yes correct.
Mathematically, there are various types of functions that have been given names: "elementary" functions like [$]\sin x[$] and "special" functions like the cumulative normal function in the Black-Scholes formula. What you are trying to do is called "root finding": finding a solution [$]x = x_0[$] that satisfies an equation of the general form: [$]f(x)=0[$]. The problem is that, even when [$]f(x)[$] can be expressed entirely in terms of "named" functions (elementary or special), the root [$]x_0[$] cannot be. It requires a numerical approach, such as Newton's method, as has been explained to you. This is so common a problem that root-finding numerical methods are widely available in existing software. But if you have to "roll your own", then you need to open up an introductory calculus text and learn Newton's method.Mathematically, what is the issue when you try to solve for it?.
We discussed this in the iv thread AFAIR (maybe the post was deleted?There is also a rational approximation with absolute error 1e-3 .. 1e-4
https://www.researchgate.net/profile/Mi ... 000000.pdf
). Rational approximation is better than polynomial approx and it will probably not be robust enough. But their ideas is very good.. bracket the solution (rough) and the a few NR iterations. You gotta knew the interval!
) expansions. In general, most 'tactical' methods/tricks will break, as can be seen from the papers. I vaguely remember that someone mentioned it was an ill-posed problem. So, then we need to consider some regularisation process.
I will add a p.s. that, while Newton's method or other root-finding numerical methods are not too complicated, writing a truly robust implied volatility routine is actually a fair amount of work. You will discover this after you initially get some numerics working and then later find that your routine fails because of some new data that you did not anticipate handling. Good luck!
