# 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.

• Could you share a link to the paper? – Daneel Olivaw Oct 14 '18 at 23:29
• – Doe Oct 14 '18 at 23:33
• As $s\rightarrow k$ both the numerator and the denominator go to zero. L'hospital rule is probably needed here... – noob2 Oct 15 '18 at 1:00
• @noob2 is spot on. By l'Hospital rule, for the first definition: $$\lim_{s \to k} \nu = \lim_{s \to k} \frac{s-k}{\int_k^s \sigma(u)^{-1} du} = \lim_{s \to k} \frac{1}{\sigma(s)^{-1}} = \lim_{s \to k} \sigma(s) = \sigma(k)$$ – Quantuple Oct 15 '18 at 7:31