Is there a good closed-form approximation for Black-Scholes implied volatility?

Question

While the solution for IV can certainly be reached using numerical search methods, I wonder if a high precision closed-form approximation exists.

For example, there is a very robust (precise within 10^-12) approximation for Bachelier IV (paper / SSRN), but is there anything similar for Black'76 and/or Black-Scholes?

Answer 1

The method described in Hallerbach (2004) always worked well for me.

We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [JBaF, 1996] estimator, has substantially higher approximation accuracy and extends over a wider region of moneyness.

Answer 2

Let's Be Rational uses exactly two iterations to give full machine accuracy for all inputs. It can be viewed as a three-stage analytical formula if you like.

The code is free to download at www.jaeckel.org.

Rgds, Peter

Answer 3

There are some other references:

• Li and Lee (2009) [download] An adaptive successive over-relaxation method for computing the Black–Scholes implied volatility

• Stefanica and Radoicic (2017) An Explicit Implied Volatility Formula

Related discussions on the implied volatility inversion:

• How can the implied volatility be calculated?

• What is an efficient method to find implied volatility?

For the normal (or Bachelier) implied volatility, there's an improvement to Choi et al (2009) [paper / SSRN] mentioned in the question:

• Fabien Le Floc'h (2016)

Answer 4

Jaeckel has a paper "Let's be rational" in which he "show how Black’s volatility can be implied from option prices with as little as two iterations to maximum attainable precision on standard (64 bit floating point) hardware for all possible inputs.".

I guess it doesn't qualify as closed-form for you, though one might argue that having to apply a deterministic algorithm twice to get accurate answer with machine precision, sort of is.

FWIW, I've not tried to implement what Jaeckal did in "Let's be rational" yet, but I have implemented his previous paper "By implication", which always worked well for me (but relies on a root-search without any guarantees on how quickly it converges).

Answer 5

Peter Jaeckel methods from the papers mentioned are the industry standard used by most practitioners to get IV.

In addition in practice the article you mention is probably of very little use because the analytic approximation you refer to in the SSRN paper needs both call and put price to extract the implied vol however usually only one of the 2 instruments is liquid when you are not close to ATM.