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I recently found this article where towards the end the author describes a method to rebase cap volatilities.

Their method works like this: for a fixed strike assume that you are given the implied forward cap volatility for 1 year against 3M Libor (denoted with $\sigma_{3M}(0,1)$) and you want to find the implied forward cap volatility for 1 year against 6M Libor (denoted with $\sigma_{6M}(0,1)$). The author suggests to set $$ \sigma_{6M}(0,1) = \sigma_{3M}(0,1) \cdot \frac{\text{SwapRate}_{3M}(0,1)}{\text{SwapRate}_{6M}(0,1)}, $$ where $\text{SwapRate}_{3M}(0,1)$ is the swap rate for a one year swap with quaterly payments and $\text{SwapRate}_{6M}(0,1)$ is the swap rate for a one year swap with semi-annual payments.

My question is: what is the idea behind this method? I assume that the equality holds under certain assumptions, but I couldn't figure out which.
Or does anyone know other rebasing methods and/or can provide literature?

Thank you.

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This says that Gaussian volatility $\approx$ Log Normal volatility $\times$ ATM strike is constant across tenors, which would essentially hold if you assume that the basis between tenors is deterministic, or at least much less volatile then rates themselves.

Note that since rates became negative log normal models for caps/floors/swaptions are not much used anymore and have been replaced by displaced log normal model, so that particular rebasing method would now look like $$ \sigma_{6M} = \sigma_{3M} \frac{\text{SwapRate}_{3M} + \text{displacement}}{\text{SwapRate}_{6M} + \text{displacement}} $$

Another simple rebasing method along the same lines is to assume that ATM volatility and skew/smile parameters (e.g. SABR parameters) are constant across tenors. These are all "cooking recipes" with at best empirical justification, but necessary as cap/floors liquid quotations are usually found only for one specific tenor per currency/expiry(e.g. 3M for USD, 3M for EUR under 2Y, 6M for EUR above 2Y, ...)

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  • $\begingroup$ Thank you for the answer. I only work with Gaussian volatilities. Could the same procedure be justified also if $\sigma_{.M}$ denotes implied Gaussian volatility? $\endgroup$ – Cettt Feb 28 '18 at 8:19
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    $\begingroup$ If the sigmas are Gaussian volatilities already then the same rebasing approach would simply say $\sigma_{6M} = \sigma_{3M}$. Just make sure, in the spirit of the method, that you are working relative to ATM for each tenor, i.e. $\sigma^{ATM}_{6M} = \sigma^{\text{ATM}}_{3M}$, $\sigma^{\text{ATM} \pm x \text{ bps}}_{6M} = \sigma^{\text{ATM} \pm x \text{ bps}}_{3M}$. Hope this helps. $\endgroup$ – Antoine Conze Feb 28 '18 at 8:27

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