# Why is Vega meaningful only for options which have single-signed gammas

I have been reading Wilmott Frequently Asked Question book and this was mentioned that Vega is not useful when measuring risk for options that have gammas changing signs such as Digital option or Barrier option. In particular, even though Vega is 0 when spot is around strike level, it is where the option is most sensitive to volatility. Unfortunately, it was not explained in details. Can anyone please elaborate on this ?

In the Black-Scholes model the price of a binary option is

$$B = e^{-r(T-t)}N(d_2)$$

with

$$d_2 = \frac{\log(\frac{S}{K})-\frac12 \sigma^2 (T-t)}{\sigma\sqrt{T-t}}$$

Differentiation with respect to $\sigma$ gives our our volatility risk, or vega

$$\frac{\partial B}{\partial\sigma} = e^{-r(T-t)} N^\prime(d_2)\frac{d_2+\sigma\sqrt{T-t}}{\sigma}$$

Therefore, if we happen to have

$$d_2 = -\sigma\sqrt{T-t}$$

or equivalently

$$S = Ke^{-\frac12 \sigma^2(T-t)}$$

Then the apparent risk is zero. Of course the instant any of these parameters, especially $\sigma$ or $S$, changes you will find yourself with considerable volatility risk.