Rebasing Cap Volatility Across Tenors with Swap Rates
Summary
The document describes a practical way to translate an at-the-money cap volatility quoted for one reference tenor into an estimate for another. The proposed scaling keeps the product of Gaussian volatility and the relevant swap rate approximately constant, so the new volatility is adjusted by the ratio of the two swap rates. This relies on the assumption that the basis between the tenors is deterministic or has much less volatility than interest rates themselves.
For markets where rates may be negative, the discussion suggests using displaced swap rates in the scaling relationship. It also mentions alternatives, such as assuming at-the-money volatility or smile parameters remain constant across tenors. These methods are presented as empirical rules of thumb, not exact identities or arbitrage-free transformations. Their usefulness is partly practical because cap and floor volatility quotes may be available for only one tenor at a given expiry and currency; the document does not provide validation data or quantify the errors these approximations can produce.
Key ideas
- Scale cap volatilities between tenors using the ratio of their swap rates under a Gaussian-volatility approximation.
- The scaling assumes tenor basis risk is negligible or relatively stable.
- A displacement can adapt the relationship to markets with negative rates.
- Assuming volatility or smile parameters are unchanged across tenors offers other approximate methods.
- These rebasing rules are empirical approximations rather than exact pricing results.
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Full text
# Rebasing of Cap Volatilities
# Rebasing of Cap Volatilities
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.
## Answer by Antoine Conze (score 2, accepted)
https://quant.stackexchange.com/a/38528
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, ...)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.