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"An Improved Estimator For Black-Scholes-Merton Implied Volatility" by Hallerbach (2004) (link to article) provides an equation (Eq. 24, Page 13, and below) for the implied volatility of a call option. What would be the equation for a put option? Thank you!

$$\sigma\sqrt{T} = \frac{\sqrt{2\pi}}{2(S+X)}\Biggl[2C+X-S+\sqrt{(2C+X-S)^2-1.85\frac{(S+X)(X-S)^2}{\pi\sqrt{XS}}}\Biggl]$$

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    $\begingroup$ If you are considering European options (as it seems to be the case in the paper) then it should be the same (for the same strike level) by absence of arbitrage, see call-put parity. $\endgroup$ – Quantuple Aug 4 '17 at 16:50
  • $\begingroup$ @Quantuple I agree that the implied volatility should be the same (at the same $S$, $X$ and $T$) but I think a different equation is required for a put option. Here, $C$ is the value of a call option which, for the same strike level, would be different than the value of a put option $P$. So while keeping $S$, $X$ and $T$ constant, simply replacing $C$ with $P$ will yield a different implied volatility. $\endgroup$ – R.G. Aug 8 '17 at 16:12
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    $\begingroup$ if you know the forward price and the discount factor, use call-put parity (i.e. $C-P=DF(F-K)$) to replace $C$ by $P+DF(F-K)$. It is not clear from your question what your inputs are unfortunately. $\endgroup$ – Quantuple Aug 8 '17 at 16:55
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    $\begingroup$ In this equation, $X = Ke^{-rt}$. I worked through this and found that the equation for a put option is $$\sigma\sqrt{T} = \frac{\sqrt{2\pi}}{2(S+X)}\Biggl[2P+S-X+\sqrt{(2P+S-X)^2-1.85\frac{(S+X)(S-X)^2}{\pi\sqrt{XS}}}\Biggl]$$ $\endgroup$ – R.G. Aug 8 '17 at 17:15
  • $\begingroup$ Glad you found your answer (although I do not see the influence of dividends which should appear through the forward but again maybe you consider a non dividend paying equity underlying) $\endgroup$ – Quantuple Aug 8 '17 at 17:44
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I would look to these papers below by Dan Stefanica et al. Very easy to code and yields better results.

An Explicit Implied Volatility Formula

Tighter Bounds for Implied Volatility

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    $\begingroup$ Peter jaeckel has an open source black implied vol function that I find to be very good, you can find it by googling "let's be rational". It's a cpp package that compiles to a dll and exports the implied vol function. (i know many will assume this a trivial problem, but the solution in the mentioned link is almost certainly more efficient than whatever solution you have used if it's based on brent/bisection). $\endgroup$ – will Aug 9 '17 at 22:30

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