Mixing SABR Models for Negative Rates

In the current low-interest-rate environment, extending option models to negative rates has become an important issue. In our previous paper, we introduced the Free SABR model, which is a natural and an attractive extension to the classical SABR model. In spite of its advantages over the Shifted SABR, the Free SABR option pricing formula is based on an approximation. Although this approximation is very good, it cannot guarantee the absence of arbitrage.In this article, we build on an exact option pricing formula for the normal SABR with a free boundary and an arbitrary correlation. First, we derive this formula in terms of a 1D integral, which is suitable for fast calibration. Next, we apply the formula as a control variate to the Free SABR to improve the accuracy of its approximation, especially for high correlations. Finally, we come up with a Mixture SABR model, which is a weighted sum of the normal and free zero-correlation models. This model is guaranteed to be arbitrage free and has a closed-form solution for option prices. Added degrees of freedom also allow the Mixture SABR model to be calibrated to a broader set of trades, in particular, to a joint set of swaptions and CMS payment. We demonstrate this capability with a wide set of numerical examples.