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I'm trying to calculate the implied volatility for out-of-the-money options, and to a lesser extent, in-the-money options. Most of the literature estimations I could find for implied volatility were for at-the-money options.

In other words, given $C(s,t)$, $S$, and $Ke^{-r(T-t)}$, related by:

$$C(s,t) = SN(d_1) - N(d_2)Ke^{-r(T-t)}$$ $$d_1 = \frac{1}{\sigma\sqrt{T-t}}\left(\log(S/K)+\left(r+\frac{\sigma^2}{2}\right)\left(T-t\right)\right)$$ $$d_2 = d_1 - \sigma\sqrt{T-t}$$

I'm trying to calculate $\sigma$. My preliminary investigations have revealed no closed-form solution, so I've resolved to a numerical approximation instead, but I haven't found any literature results on this approximation.

I would be glad if anyone could refer me to any useful approximations or other results. Other comments on this are also welcome as it's a tricky topic.

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    $\begingroup$ Newton's method, or the more elementary bi-section method, which you can find in any math finance book. $\endgroup$
    – Gordon
    Commented Apr 13, 2016 at 20:42
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    $\begingroup$ Strictly speaking the CDF $N(d_{1,2})$ is already not in closed form. $\endgroup$
    – Olaf
    Commented Apr 14, 2016 at 8:16
  • $\begingroup$ Are you looking specifically at OTM options? There is a general question on IV computations here. $\endgroup$
    – SRKX
    Commented Apr 15, 2016 at 3:20
  • $\begingroup$ If you use R then this link could be useful. This is an article that contains a code to calculate IV using bi-section method r-bloggers.com/the-only-thing-smiling-today-is-volatility $\endgroup$
    – tosik
    Commented Apr 15, 2016 at 6:09

3 Answers 3

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Peter Jaeckel has written various papers on this. "by implication" and "Let's be rational" are the most recent ones. He also provides code on his website www.jaeckel.org.

(Note: the question asked for literature.)

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Look on Google for Asymptotic behavior of Implied Volatility Near Infinity

you will find results like :

$$I(K) \stackrel{K\to\infty}{=} \sqrt{\frac{2}{T}}\left(\sqrt{\ln \frac{K}{C(K)}}-\sqrt{\ln\frac{1}{C(K)}}\right) +\text{O}_{K\to \infty}\left(\frac{\ln\ln\frac{1}{C(K)}}{\sqrt{\ln\frac{1}{C(K)}}}\right)$$

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  • $\begingroup$ What is exactly going to infinity? $\endgroup$
    – SmallChess
    Commented Apr 14, 2016 at 15:10
  • $\begingroup$ strike goes to $\infty$ $\endgroup$
    – M. Jeunesse
    Commented Apr 14, 2016 at 15:31
  • $\begingroup$ this is an approximation for a deep out-of-the money call option $\endgroup$
    – M. Jeunesse
    Commented Apr 14, 2016 at 15:40
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One more reference that I know is

Li and Lee (2009) [download]

An adaptive successive over-relaxation method for computing the Black–Scholes implied volatility

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