ATM Call Delta as Volatility Approaches Zero

Summary

The discussion addresses why an at-the-money call can appear to have a delta near one when volatility is close to zero. One reply offers an intuition based on positive carry: with a risk-free rate above the dividend yield, a spot-at-the-money option can be in the money relative to the forward when uncertainty vanishes. This depends on the model’s carry assumptions, so it is not a universal explanation for every definition of at-the-money.

Another reply focuses on how delta is measured at the strike boundary. A small upward price bump can produce a one-sided delta of one as volatility tends to zero, while a downward bump gives zero; a centered delta is one half. This distinction helps explain why plotted values may differ even for the same option. The document gives conceptual explanations rather than derivations or numerical tests, and its observations depend on the bump convention and option assumptions.

Key ideas

• At zero volatility, positive carry can make a spot-at-the-money call in the money relative to the forward.
• A one-sided upward price bump can give an ATM call delta of one as volatility tends to zero.
• A downward bump at the strike boundary can instead give a delta of zero.
• A centered delta at the boundary is one half under the convention discussed.
• Delta readings near zero volatility depend on bump convention and model assumptions.

Tags

# ATM call option delta with low volatility


# ATM call option delta with low volatility












I put together some charts to understand how option greeks change. Can someone please explain why the delta of an ATM call option is 1 when vol is close to zero?

I get that an increase in vol for OTM or ITM options will increase the chance that they expire ITM or OTM. But what is the explanation for how the delta of an ATM option changes with volatility?

## Answer by Lliane (score 2, accepted)

https://quant.stackexchange.com/a/42855

I don't know how complex is your model, but the intuitive answer is that if you have 0 vol and a positive drift (interest rate > dividend rate) your option is actually "in the money forward."

Interestingly an ATM option with extremely high volatility will also have a delta close to 1. That's a classic interview question btw

## Answer by Sebapi (score 1)

https://quant.stackexchange.com/a/42872

We need to distinguish between centered delta and non-centered values like $\delta^+$ obtained using a small positive bump in price.

When $\sigma \rightarrow 0 $, $\delta^+ = 1$, $\delta^- = 0$.

So your graph looks consistent with $\delta^+$, the delta for a small positive perturbation.

The centered delta for an ATM call is $\delta=1/2$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.