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Adapting SVI Moneyness for Negative Swap Rates

Article Quant Q&A · Author: math
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Summary

The document considers fitting the SVI volatility model to swaption implied volatilities when forward swap rates can be negative. Standard SVI formulations use log-moneyness, based on the ratio of strike to forward, which is undefined when those rates are negative. The response suggests matching the moneyness variable to the volatility convention: use the strike-forward difference for normal volatility dynamics.

Two alternatives are also described: shift both strike and forward by a chosen floor before taking their log ratio, or add an exponent parameter to obtain a shifted SABR-style specification. The answer recommends starting with the additive difference for the stated normal-volatility use case. It does not present empirical fitting results, formal theoretical constraints, or a detailed SVI calibration procedure; the suggestions are starting points for experimentation, and the shifted approaches may require transforming volatility conventions.

Key ideas

  • Log-moneyness based on strike divided by forward is undefined for negative rates.
  • For normal swaption volatility, strike minus forward is a compatible moneyness measure to test.
  • A shifted log-moneyness can be formed by adding a floor to both strike and forward.
  • An exponent parameter offers a shifted SABR-like alternative.
  • The response gives modeling suggestions rather than empirical validation or a proof of constraints.

Tags

Fixed incomeOptionsVolatilityDerivatives pricing
Full text
# SVI negative rates


# SVI negative rates












I've used the SVI model in the past for equity option which worekd quite well. I came across a post on Wilmott where someone said hes using SVI for swaption as well. I would like to test the model and fit it to swaption implied volatitilities (normal). However, there are markets where the forward swap rate goes negative and in the original paper Gatheral and Jacquier they use the moneyness $\log(\frac{K}{F_0})$, where $F_0$ is the forward swap rate. This is not defined for negative rates. Are you aware of any study for the SVI for fixed income? How else can we deal with negative rates in this case?

Since the model is not really based on any assumption on the underlying like in a SABR model we could change this. However, I'm not sure if there is any theoretical constraints on doing this.

## Answer by Kiwiakos (score 7, accepted)

https://quant.stackexchange.com/a/26238

I would say that $\log K/F$ points towards a log-normal type model. If I were you I would experiment with the moneyness defined as $K-F$ instead. This would make it consistent with normal dynamics.

An alternative would be to define an 'interest rate floor', say $L=-200bp$ and take relative changes relative to that rather than zero, ie define moneyness as $\log (K-L)/(F-L)$. This would correspond to shifted log-normal.

Or put an exponent $\beta$ as well and end up with a shifted Sabr.

But for what you describe I'd start with the first suggestion since you work with normal vols in the first place. The other two would require vol transformations to make it work as far as I can see.

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