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Estimating 6M EUR Volatility from 3M Cap and Floor Volatility

Article Quant Q&A · Author: Adam
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Summary

The discussion addresses whether 3M flat cap and floor volatilities can be converted into 6M volatilities for EUR using forecast and discount curves. It explains that the conversion is not uniquely determined by the 3M curve: the basis between 3M and 6M rates and its correlation with the 3M curve are important inputs, so a curve-only transformation cannot guarantee an arbitrage-free result.

Without those correlations, the problem becomes empirical and requires assumptions. A multi-factor interest-rate model fitted to both 3M and 6M curves offers a theoretically structured approach, but it still rests on modeling assumptions rather than removing uncertainty. The discussion provides guidance rather than a specific calibration procedure or data-backed estimate, and does not supply the missing correlation information needed to determine a unique conversion.

Key ideas

  • A 3M volatility curve alone does not determine an arbitrage-free 6M volatility curve.
  • The 3M–6M basis and its correlation with the 3M curve are central to the conversion.
  • When correlation data are unavailable, the conversion is an empirical problem that requires assumptions.
  • A multi-factor interest-rate model can fit both curves, but its results still depend on model assumptions.

Tags

Fixed incomeVolatilityDerivatives pricing
Full text
# Transforming 3M volatilities into 6M volatilities in EUR forecast curves


# Transforming 3M volatilities into 6M volatilities in EUR forecast curves












I have implemented a stripping algorithm to extract forward volatilities from cap/floor flat volatilities for different currencies. I am however struggling a bit when implementing a method to convert the first 2 year 3M flat vols into 6M flat vols for the EUR currency.

The information I have available is the forecast curves (1M, 3M, 6M) and discount curve.

I have found some papers detailing a solution that involves the calculation of the correlation coefficient between different 3M forward rates, but that is unfortunately information that is not available to me. I have also found a paper from Bloomberg ("Bloomberg Volatility Cube" by Zhang and Wu) with some standard formulas to do the conversion, but they do not take into account the basis spread, which to me it seems key in doing the conversion.

Any help or guidance from professional practitioners would be very helpful.

Thanks in advance!

## Answer by ExIR (score 1)

https://quant.stackexchange.com/a/45650

You won't get a arbitrage-free estimate of 6M curve volatility based on 3m curve. That point is actually a relief -- meaning that if you make reasonable assumptions, it should be fine. The basis is all that matters here along with the correlation between the basis and the 3m curve. It becomes an empirical problem. Having said those, if you have to have something that's theoretically sound, you may come up with multi-factor interest rate model that fit into both 3m and 6m curves. Many shops think this is more appealing (good sell to risk managers), but in reality it just turns one assumption into another assumption.

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.