SVI negative rates

Question

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 1

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.