Numerical Methods and Asymptotics for Implied Volatility
Summary
The document concerns recovering implied volatility from option prices when options are out of the money, where a direct closed-form inversion of the Black–Scholes price is unavailable. It points readers toward numerical methods and published work, including an adaptive successive over-relaxation approach and methods designed to compute Black–Scholes implied volatility robustly.
It also mentions asymptotic results for implied volatility at extreme strikes, illustrating how volatility can be approximated from option prices in the far tail. These references indicate that the appropriate technique depends on the region of the volatility surface and numerical stability needs. The note is a collection of pointers rather than a derivation or comparison: it does not lay out algorithms, implementation details, performance tests, or guidance on selecting a method for a particular market or option dataset.
Key ideas
- Implied volatility from an option price generally requires numerical inversion of the Black–Scholes formula.
- Published methods include adaptive successive over-relaxation and specialized robust inversion techniques.
- Asymptotic formulas can describe implied volatility behavior at very large strikes.
- The document lists references but does not compare their accuracy, speed, or implementation requirements.
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Full text
# How to calculate Implied Volatility for out-of-the-money options?
# How to calculate Implied Volatility for out-of-the-money options?
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.
## Answer by Mark Joshi (score 6)
https://quant.stackexchange.com/a/25432
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.)
## Answer by M. Jeunesse (score 3)
https://quant.stackexchange.com/a/25419
Look on Google for Asymptotic behavior of Implied Volatility Near Infinity
you will find results like :
$$I(K) \stackrel{K\to\infty}{=} \sqrt{\frac{2}{T}}\left(\sqrt{\ln \frac{K}{C(K)}}-\sqrt{\ln\frac{1}{C(K)}}\right) +\text{O}_{K\to \infty}\left(\frac{\ln\ln\frac{1}{C(K)}}{\sqrt{\ln\frac{1}{C(K)}}}\right)$$
## Answer by jaehyukchoi49 (score 2)
https://quant.stackexchange.com/a/32531
One more reference that I know is
Li and Lee (2009) [download]
An adaptive successive over-relaxation method for computing the Black–Scholes implied volatilityShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.