Calibration and pricing using the free SABR model
Features of the free SABR model
The SABR model has become the dominant tool for smile-interpolations in the interest-rate world
The SABR model has become the dominant tool for smile-interpolations in the interest-rate world owing to two distinct features: Firstly, the fact it is a stochastic volatility model and can therefore fit the volatility smile, and, secondly, the fact that it allows for an approximate closed-form formula that expresses the implied volatility (Black or Bachelier) in terms of the model’s parameters.
Under the negative-rate environment the SABR model as well as the traditional Black model cannot work. The reason is that by-design the models expect forwards and strikes to be strictly positive. An exception error appears whenever this is the case.
In order to circumvent this exception problem the market has introduced a “shift” in both the forward rate and the strike. The shift is such that, when added to the strike and the forward, the relevant mathematical quantities remain well-defined.
An alternative approach to handle pricing of interest-rate derivatives in the negative-rate environment is the introduction of new models that can by-design handle negative rates. One such approach is the free SABR model by Antonov et al.1.
In this article we examine some of the features of this model and investigate its similarities to the traditional SABR model.
1 A Antonov, M Konikov and M Spector, “The Free Boundary SABR: Natural Extension to Negative Rates”, available at ssrn.com