We all know if you back out of the Black Scholes option pricing model you can derive what the option is "implying" about the underlyings future expected volatility.

Is there a simple, closed form, formula deriving Implied Volatility (IV)? If so can you could you direct me to the equation?

Or is IV only numerically solved?

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    $\begingroup$ I found this one via Google: Implied Volatility Formula $\endgroup$ – chrisaycock Apr 17 '13 at 2:28
  • $\begingroup$ yea, saw that one too. Newton method was used here. am I right? But how is IV calculated? Anyone here use a standard procedure? $\endgroup$ – jessica Apr 17 '13 at 2:30
  • $\begingroup$ Jaeckel has a paper for a more efficient method of backing out the implied vol here - it includes a link to the source code. $\endgroup$ – will Jul 22 '16 at 9:08
  • $\begingroup$ Please refer to this 2016-17 article by Jaeckel : jaeckel.000webhostapp.com/ImpliedNormalVolatility.pdf It has been mentioned above in a comment, but that link is broken $\endgroup$ – Nishant Jul 11 '19 at 7:44

The Black-Scholes option pricing model provides a closed-form pricing formula $BS(\sigma)$ for a European-exercise option with price $P$. There is no closed-form inverse for it, but because it has a closed-form vega (volatility derivative) $\nu(\sigma)$, and the derivative is nonnegative, we can use the Newton-Raphson formula with confidence.

Essentially, we choose a starting value $\sigma_0$ say from yoonkwon's post. Then, we iterate

$$ \sigma_{n+1} = \sigma_n - \frac{BS(\sigma_n)-P}{\nu(\sigma_n)} $$

until we have reached a solution of sufficient accuracy.

This only works for options where the Black-Scholes model has a closed-form solution and a nice vega. When it does not, as for exotic payoffs, American-exercise options and so on, we need a more stable technique that does not depend on vega.

In these harder cases, it is typical to apply a secant method with bisective bounds checking. A favored algorithm is Brent's method since it is commonly available and quite fast.


Brenner and Subrahmanyam (1988) provided a closed form estimate of IV, you can use it as the initial estimate:

$$ \sigma \approx \sqrt{\cfrac{2\pi}{T}} . \cfrac{C}{S} $$

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    $\begingroup$ If you could embed the link to the article in your answer, it would be great. $\endgroup$ – SRKX Apr 17 '13 at 9:24
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    $\begingroup$ What are the definitions of T,C and S ? I'm guessing T is the Duration of the option-contract, C is the theoretical Call-value and S is the Strike-price, correct ? $\endgroup$ – Nick Oct 9 '13 at 12:49
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    $\begingroup$ No, S is the current price of the underlying. However the approximation by Brenner and Subrahmanyam works best for at the money options, hence the difference should be small in that case. $\endgroup$ – jcfrei May 9 '14 at 14:29
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    $\begingroup$ @Dominique (S = Spot price of the underlying, a.k.a. current price) $\endgroup$ – Franck Dernoncourt Jul 27 '17 at 18:43
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    $\begingroup$ The formula is based on the ATM price under normal model approximation. See quant.stackexchange.com/a/1154/26559 for further detail. $\endgroup$ – Jaehyuk Choi Aug 4 '18 at 15:19

It is a very simple procedure and yes, Newton-Raphson is used because it converges sufficiently quickly:

  • You need to obviously supply an option pricing model such as BS.
  • Plug in an initial guess for implied volatility -> calculate the the option price as a function of your initial iVol guess -> apply NR -> minimize the error term until it is sufficiently small to your liking.
  • the following contains a very simple example of how you derive the implied vol from an option price: http://risklearn.com/estimating-implied-volatility-with-the-newton-raphson-method/

  • You can also derive implied volatility through a "rational approximation" approach (closed form approach -> faster), which can be used exclusively if you are fine with the approximation error or as a hybrid in combination with a few iterations of NR (better initial guess -> less iterations). Here a reference: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=952727


There are some references on this topic. You may find them helpful.

Peter Jaeckel has articles named "By Implication (2006)" and "Let's be rational (2013)"

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

  • $\begingroup$ Do you know if Li & Lee (2009) provide their code somewhere? $\endgroup$ – Guilherme Salomé Aug 15 '17 at 1:32
  • $\begingroup$ Probably not... $\endgroup$ – Jaehyuk Choi Aug 18 '17 at 6:15
  • $\begingroup$ This is the best answer since jaeckel method is the industry standard implementation for european IV calculation $\endgroup$ – Ezy Dec 28 '18 at 9:00

To get IV I do the following: 1) change sig many times and calculate C in BS formula every time. That can be done with OIC calculator All other parameters are kept constant in BS call price calculations. The sig that corresponds to C value closest to the call market value is probably right. 2) without OIC calculator for every chosen sig I am using old approach: calculate d1, d2, Nd1, Nd2 and BS option value. Again calculated BS value closest to the market value probably correspond to correct IV.


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