Skip to content
All library documents

Why Deep In-the-Money Options Can Have Unstable Implied Volatility

Article Quant Q&A · Author: UmaN
Read the original

Summary

This discussion examines why numerical implied volatility routines can behave poorly for deep in-the-money options under Black–Scholes. The question compares bisection and Brent root-finding and describes a short-dated call whose calculated premium changes so little with volatility that many volatility inputs appear to produce essentially the same value. In such a case, recovering a unique implied volatility is numerically ill-conditioned.

The response directs readers to work on efficient implied-volatility calculation that explains the difficulty with in-the-money options. It suggests solving from out-of-the-money options as a practical approach for the questioner’s purposes. The exchange does not derive analytical bounds or give a diagnostic algorithm, and the suggestion is not presented as a universal rule. It offers a reference and a workaround, rather than a full comparison of solvers or evidence across option prices and market conditions.

Key ideas

  • Deep in-the-money option prices may be relatively insensitive to volatility in some cases.
  • Weak price sensitivity can make implied volatility difficult to identify numerically.
  • Bisection and Brent methods cannot resolve a unique value when the pricing relationship is poorly conditioned.
  • The answer recommends using out-of-the-money options as a practical alternative.
  • The discussion does not provide analytical bounds or a complete diagnostic method.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Implied Volatility Calculation for Deep In The Money Calls, Numerical Issues


# Implied Volatility Calculation for Deep In The Money Calls, Numerical Issues












I have two implementations for finding the implied volatility under Black-Scholes formula. One is bisection and the other is brent's method. (I know Newton-Raphson is popular due to speed and will support this as well later...).

My question seems to be related to this question: Lower bound of ITM Calls when computing Implied Volatility

Basically, I notice that the implied volatility calculation breaks down for deep in the money call options and probably deep in the money put options (same for deep out of the money options of both types?).

Example arguments:

```
double r = 0.05;
double T = 0.00274;    
double S = 50.0;    
double K = 1.0;
```

For these arguments, it seems any volatility value will do as the option premium from BS is always ~49.00015... and so a unique implied volatility cannot be found?

I would like to provide some method that can provide a fast check on the input that would ideally check analytical bounds for when bisection/brent will give back sound answers and when the input arguments belong to a "degenerate" case.

Are such bounds well-known? Any link to an article available would be appreciated.

## Answer by UmaN (score 3, accepted)

https://quant.stackexchange.com/a/16302

As pointed out in the comment, the answer here: What is an efficient method to find implied volatility? provides:

Link to http://www.jaeckel.org/ and in particular the "By Implication" paper: http://www.pjaeckel.webspace.virginmedia.com/ByImplication.pdf .

An explanation of the problem with in-the-money options and implied volatility is provided in that paper.

Working with out-of-the-money options seems to be the easiest, most reasonable approach and will work for my purposes.

Of course if anyone has additional insights, please share them.

Shown 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.