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From a paper I am reading, it is written enter image description here

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.

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  • $\begingroup$ Could you share a link to the paper? $\endgroup$ – Daneel Olivaw Oct 14 '18 at 23:29
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    $\begingroup$ See: janroman.dhis.org/finance/SABR/ZABR%20Andreasen.pdf $\endgroup$ – Doe Oct 14 '18 at 23:33
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    $\begingroup$ As $s\rightarrow k$ both the numerator and the denominator go to zero. L'hospital rule is probably needed here... $\endgroup$ – noob2 Oct 15 '18 at 1:00
  • $\begingroup$ @noob2 is spot on. By l'Hospital rule, for the first definition: $$ \lim_{s \to k} \nu = \lim_{s \to k} \frac{s-k}{\int_k^s \sigma(u)^{-1} du} = \lim_{s \to k} \frac{1}{\sigma(s)^{-1}} = \lim_{s \to k} \sigma(s) = \sigma(k)$$ $\endgroup$ – Quantuple Oct 15 '18 at 7:31
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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

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