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Arbitrage Checks for ATM Swaption Volatility Surfaces

Article Quant Q&A · Author: toto
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Summary

The discussion asks whether ATM swaption volatilities indexed by expiry and tenor must satisfy no-arbitrage conditions. It explains that there are no direct no-arbitrage restrictions across such quotes because each swaption is written on a different underlying forward swap rate. Consequently, a collection of ATM volatilities cannot be assessed like options on one common underlying by comparing their values alone.

The answer distinguishes market arbitrage from compatibility with a chosen interest-rate model. For example, a Hull–White calibration to coterminal swaptions may fail if fitting a quote would require negative instantaneous short-rate variance over a time interval. That failure indicates a limitation of the model’s ability to represent the observed configuration; it does not establish an arbitrage. The discussion points to research on arbitrage-free swaption-cube construction, but gives no construction method or empirical demonstration, so further reading is needed to implement a surface test.

Key ideas

  • ATM swaption quotes across different expiries and tenors refer to different underlying swaps, so they have no direct cross-quote no-arbitrage condition.
  • A model’s inability to fit a set of quotes does not by itself imply that the quotes permit arbitrage.
  • Hull–White calibration can fail when fitting would require negative instantaneous variance in a time bucket.
  • Arbitrage-free swaption-cube construction is identified as a topic for further study.

Tags

Fixed incomeOptionsArbitrageVolatility
Full text
# Free Arbitrage conditions in ATM swaption surfaces


# Free Arbitrage conditions in ATM swaption surfaces












I'm wondering how can we check free arbitrage conditions in ATM swaptions surfaces since we only have access to Expiry, Tenor and volatility? Can someone help me please, i didn't find any article about it. N.B:I'm working on python

## Answer by Antoine Conze (score 6)

https://quant.stackexchange.com/a/64115

There are no no-arbitrage conditions on ATM vols of swaptions with different expiries/tenors, because the underlying swaps forward rates are different instruments. There are conditions however for these vols to be compatible with specific IR models. For instance when calibrating a Hull & White model on a set of coterminal swaptions, it sometimes happens that the models fails to fit a specific coterminal, because that would require a negative short rate instantaneous variance on the corresponding time bucket. It does not imply that the coterminal vols are not arbitrage free, but rather that the Hull & White model can only fit a limited set of market configurations.

## Answer by Marco B (score 1)

https://quant.stackexchange.com/a/77574

See Johnson, Simon and Nonas, Bereshad, Arbitrage-Free Construction of the Swaption Cube (January 5, 2009). Available at SSRN: https://ssrn.com/abstract=1330869 or http://dx.doi.org/10.2139/ssrn.1330869

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.