What is the delta of an at-the-money European call option with respect to volatility?

Question

Question: What is the delta of an at-the-money European call option with respect to volatility?

Note that $$\frac{\partial\Delta}{\partial\sigma} = N'(d_1) \frac{\partial d_1}{\partial\sigma} = N'(d_1) \frac{- d_2}{\sigma} = \frac{-N'(d_1)d_2}{\sigma}$$ where $N(\cdot)$ is the CDF of the standard normal distribution. I am not able to deduce anything from this equation.

This QFSE post states that higher volatility for in-the-money option will have lower delta whereas higher volatility for out-of-the-money options will have higher delta.

Based on this website, it seems that higher volatility will lead to $\Delta = 0.5.$ But I am not able to show this.

Answer 1

Based on your computation, you can observe that the $N’$ term is always positive, between 0 and 0.4. As $\sigma$ is always positive, you can focus on the $-d_2$ term. When $d_2 > 0$, i.e. call is ITM, delta has a negative sensitivity to volatility ; conversely for OTM call. That is in line with your remark.

Answer 2

In the following, I am assuming the BS73 model and I assume that "ATM" means

$$ S = Xe^{-r\tau} $$ The pricing formula for a European call then becomes $$ \tag{1} O\propto N\left(+\frac{1}{2}\sigma\sqrt{\tau}\right)-N\left(-\frac{1}{2}\sigma\sqrt{\tau}\right) $$ times some scaling factor which is irrelevant for our purpose. Clearly, $$ Vega\equiv\frac{\partial O}{\partial \sigma}=\frac{1}{2}\sqrt{\tau} \cdot{} n\left(\frac{1}{2}\sigma\sqrt{\tau}\right)+\frac{1}{2}\sqrt{\tau} \cdot{} n\left(-\frac{1}{2}\sigma\sqrt{\tau}\right) $$ Leading us to $$ \tag{2} \frac{\partial O}{\partial \sigma}=\sqrt{\tau}\frac{e^{-\frac{1}{2}\left(0.5\sigma\sqrt{\tau}\right)^2}}{\sqrt{2\pi}} $$ Thus:

• For longer maturities, the Vega is larger than for smaller maturities
• For all practical purposes (i.e. $IV<75\%$, $\tau<1yr$, you can approximate the ATM Vega to $$ \tag{3*} Vega \approx \sqrt{\frac{\tau}{2\pi}} $$