Matching Swaption Volatility Surfaces to the Underlying Swap Tenor

Summary

The document discusses which instruments should be used to build a swaption volatility surface when calibrating to a particular underlying swap. The answer recommends matching the swaption underlying to the swap being calibrated, so that the forward rate used in the pricing relationship corresponds to the target instrument. Using swaptions based on a different floating-rate tenor can capture the dynamics of that other tenor instead.

The response allows for converting between tenors if the analytics support it, but notes that such conversions can require a convexity adjustment based on covariance between the tenor spreads. This makes market liquidity alone an insufficient basis for selecting calibration instruments. The discussion is conceptual and does not give a conversion formula, data requirements, or details of a specific pricing framework; the appropriate construction depends on the instruments and adjustments supported by the model.

Key ideas

• Swaption calibration instruments should generally have underlyings that match the swap being calibrated.
• The forward rate in the pricing relationship must correspond to the target underlying.
• Using a different floating tenor can introduce dynamics from that tenor instead.
• Tenor conversion may require a convexity adjustment tied to spread covariance.

Tags

# Volatility surface for Swaptions


# Volatility surface for Swaptions












I understand the volatility surface for swaption is built using implied vols of ATM swaptions. I had a question on the instruments that are used. Should the instruments used change depending on the terms of the underlying swap? For example, if the underlying is a 6M LIBOR floating swap, then should we use instruments referencing 6M LIBOR to construct the vol surface? My understanding was that we will use the most liquid instruments which is swaps referencing 3M LIBOR.

## Answer by Kiann (score 3)

https://quant.stackexchange.com/a/44777

Yes, in general, you should match the swaptions such that the underlying in the swaptions match what you are trying to calibrate. Consider, that swaption(lognormal vols) = Annuity * function(F, K, vol, time). The F, is the forward and it has to match the underlying instrument.

If you do not, you are capturing the dynamics of a different instruments. Unless, your analytics has the conversion mathematics to shift from one tenor (such as 3m-float) to another tenor (such as your 6mth-float). Even then, these kind of conversions usually require a convexity adjustment that takes in the covariance of the 3m-6mth tenor spreads.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.