Approaches to Black–Scholes Implied Volatility Inversion
Summary
The discussion surveys ways to recover implied volatility from Black–Scholes option prices when a closed-form inverse is unavailable or an approximation is desired. It cites Hallerbach’s estimator as more accurate and applicable across a wider moneyness range than the Corrado–Miller estimator. It also describes Jaeckel’s “Let’s Be Rational” as a highly accurate method using two iterations, and lists other approaches, including an adaptive numerical method and an explicit formula. Root-search methods are mentioned as another practical route.
The thread provides references and practitioner claims rather than a common benchmark or independent numerical comparison. It distinguishes an analytical approximation from a short deterministic iterative procedure, which may not meet a strict definition of closed form. A respondent also cautions that an approximation requiring both call and put prices may be less useful away from at-the-money strikes, where one side can be illiquid. Method choice therefore depends on accuracy, inputs, and market conditions.
Key ideas
- The discussion compares approximations and numerical procedures for Black–Scholes implied volatility inversion.
- Hallerbach’s estimator is reported to improve accuracy and moneyness coverage over Corrado–Miller.
- Jaeckel’s method is described as reaching machine precision with two iterations.
- An approximation that needs both call and put prices may be constrained when one side is illiquid.
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Full text
# Is there a good closed-form approximation for Black-Scholes implied volatility? # Is there a good closed-form approximation for Black-Scholes implied volatility? 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 by Jakøb H. (score 8, accepted) https://quant.stackexchange.com/a/14707 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 by Peter Jaeckel (score 8) https://quant.stackexchange.com/a/31371 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 by jaehyukchoi49 (score 5) https://quant.stackexchange.com/a/32594 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 by Bram (score 3) https://quant.stackexchange.com/a/14710 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 by Ezy (score 2) https://quant.stackexchange.com/a/42195 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.
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