A Quantitative Perspective

In this video blog, Dr. Alexandre Antonov, Senior Vice President of Quantitative Research at Numerix, discusses how the recent development of the Free Boundary SABR model for option pricing is a natural and efficient extension of the classical SABR model.

He explores how the Free Boundary SABR model is especially effective in the low and negative interest rate environments recently seen in Europe and Japan.  Alexandre also highlights how the Free Boundary SABR Model is able to overcome some of the limitations presented by the Shifted SABR Model.

Weigh in and continue the conversation on Twitter @nxanalytics, LinkedIn, or in the comments sections.

Video transcript:

Jim Jockle (Host): Hi welcome to Numerix Video Blog, your expert source for derivatives trends and topics. I’m your host Jim Jockle.

In the current interest rate environment, especially in Europe and Japan, we are seeing deposit rates below zero, and market rates in negative territory. As a result, financial institutions are finding themselves challenged on many fronts, in particular in terms of their option pricing models.

As this is becoming critically important, joining me today to discuss this is Alexandre Antonov, Senior Vice President of Quantitative Research at Numerix. Welcome Alexandre.

Alexandre Antonov (Guest): Thank you, Jim. Thank you for inviting me.

Jockle: Thank you. So question, when dealing with things like – negative strikes and forwards for short maturity caps and swaptions, how does this negative rate issue impact the volatility surfaces?

Antonov: Actually, the difference is twofold. The first thing is that we should change the quotation of the swaptions. The second thing, we should change the interpolation. So, the first issue is technically not that complicated, instead of making the quotation in terms of lognormal volatility, which we all get used to when the rates are positive. We should change option quotations for zero and negative strikes because they do not work for the lognormal (Black-Scholes) volatility.

There are three ways to do that, first is to quote option prices themselves, in dollar value, the second one is to pick up the normal volatility, which is always valid even for negative strikes, the third thing is to use the shifted, lognormal, volatility. Sometimes people do the shifted lognormal volatility with a fixed shift. For example, the current shift for Swiss Franc is 2%. So we expect the rates to be up to -2%. So, as I said it is not a very complicated thing, it is just a variation around the Black Scholes model, but technically it’s quite easy.

The issue with the swaption volatility interpolation is more complicated. The thing is our usual interpolation tool, the classical SABR model, doesn’t interpolate the strikes to the negative values.

Moreover, its behavior is quite bad for small strikes: if your strike is small, you can get non-arbitrage-free solutions. For example, the probability density function can be is negative, which is impossible. To address these complications, people start using the shifted SABR model for the interpolation.

Jockle: So, in terms of a solution, there are many qualities people look for in a forward rate models to fit to the market quotes being an important, but it’s not exclusive. So, is the market starting to see the Shifted SABR approach as being best practice?

Antonov: The market does it, because it does not have another choice. Actually, the shifted SABR model is not that good because everything is based on the manual selection of this shift. So it is an extra, a non-calibrated parameter in the volatility interpolation, and it’s not really clear how they chose it. We could of course pick up the shift from the volatility surface quotes, but it’s not the best solution. Actually, there are a lot of drawbacks for the shifted SABR model.

The first is the danger of non-smooth behavior of the parameters when the shift changes. Indeed, imagine tomorrow the rates are even more negative, or, for example, they become positive. This will force us to recalibrate the model to change the shift which stayed the same for several months. As a consequence, it will make a jump into the model prices, into the Greeks, and into the risk. As far as the swaption volatility is a very low basic stone for the pricing, any instrument will depend on that. So the whole book will jump, so it’s not really well appreciated on the market.

A less important drawback is that the swaption prices are limited from above due to the limitation of this strike from below. So it is a danger that that shifted SABR cannot maintain market prices near the shift position.

Jockle: So, how does this compare to the Free Boundary SABR approach that you and your colleagues have proposed?

Antonov: Our free SABR model is much more natural generalization of the classical SABR model. The logic is following. Imagine we have a classical SABR model. When a scenario (a Monte Carlo path) touches zero, we absorb it and set it to zero; that is why we call it “absorbing boundary SABR”. In our free boundary SABR model, we let the scenario (the path) crossing the zero boundary, without any absorption. Now it is a more natural implementation in the sense that we don’t have another parameter, as a shift, which we don’t calibrate.

Moreover, the free SABR has the same behavior as the classical absorbing SABR model for a small times around at-the-money strikes. Thus, it fits quite well the traders intuition: the parameters of the free boundary SABR will not deviate that far from what they get used to having with the classic SABR. But the free SABR will go to negative values without any problems and it has a quite smooth dependence.

The Free Boundary SABR has a nice analytical solution which is described in details in our paper. Obviously, it is quite important for calibration. We have derived an exact solution for zero correlation and very accurate approximation for non-zero correlation. This gives us a fast calibration, with the same number of parameters with respect to the initial SABR, so everything is just very good! 

Jockle: Well Alexandre, thank you so much for sharing that. And for those interested, Alexandre’s paper is available at numerix.com. You can go there and get that via download. So Alexandre, we’re going to be closely monitoring the negative rates issue, and we hope you’ll join us again on the video blog as this issue continues to evolve. 

Antonov:  Thank you, my pleasure.

Jockle: And for those who want to stay in touch with us, please feel free to follow us on LinkedIn or on Twitter @nxanalytics. And check back regularly for news, blogs, videos, research as well as technical papers. Thank you so much for joining.