Interpreting ATM Limits in Local-to-Implied Volatility Formulas

Summary

The document asks why formulas connecting local volatility to implied volatility appear to give zero volatility when the underlying price equals the strike. That apparent result would make the equations seem unusable for at-the-money options, and values near that point seem similarly implausible to the questioner.

The reply explains that the equations relate a non-observable local volatility function to implied volatility inferred from call and put prices. It also states that the formulas are well defined as the underlying price approaches the strike, so direct substitution at equality should not be taken as evidence that the relationship fails. The discussion gives no derivation of the limiting value or worked numerical example, and points readers to a separate answer for further detail. Its main lesson is to evaluate the limit of a formula at ATM rather than infer its behavior from an expression that may be indeterminate there.

Key ideas

• The equations connect local volatility, which is not directly observed, with implied volatility inferred from option prices.
• The at-the-money case should be assessed through the limit as the underlying price approaches the strike.
• A zero-looking expression at equality does not by itself establish that the formula is unusable.
• The brief reply does not show the limiting derivation or a numerical example.

Tags

# Of what use is this implied volatility formula?


# Of what use is this implied volatility formula?












From a paper I am reading, it is written

These equations do not make any sense. If $s = k$, i.e. if we are pricing ATM options, then this volatility is identically zero, hence useless.

How am I to make sense of these formulas? Even if $s \approx k$, we get a value close to zero, and hence again nonsensical implied volatilities.

## Answer by Ezy (score 1)

https://quant.stackexchange.com/a/42189

these 2 equations (one for the normal model and the other for the lognormal model) link the non-observable, local volatility diffusion functional (sigma) to the implied volatility of the observable call/put prices.

As indicated in the comments, those equations are well defined in the limit s->k

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.