Stability of the SABR model

SABR as dominant market model

Since its inception the SABR model has become the dominant market model for interest-rate derivatives. It owes its popularity to two main factors: Firstly, it models both the underlying forward rate and its volatility. This is an essential element in order for any model to reproduce the volatility smile. The second element is the derivation by Hagan et al of an approximate closed-form formula for the implied volatility in terms of the four SABR parameters. This is a key ingredient for quick calibration of the model to the market.

Weaknesses of the SABR model

The closed form SABR formulas¹, however, come with a number of disadvantages. A well-known weakness is the fact that the probability density function of the forward rate becomes negative for very low strikes. This is an artefact of the various asymptotic expansions that lead to the closed-form formula. This weakness becomes particularly important in the current negative-rate environment where derivatives are traded at negative or low strikes. The literature in the area of “fixing” the SABR model is quite rich, ranging from “quick and dirty”-type solutions, where a reasonably-behaving tail is attached to the body of the SABR PDF, to more elaborate variations of the SABR model.

The stability of the SABR parameters

For risk-management purposes a common question concerning the SABR model is about the stability of its parameters: An undesirable feature would be to have jumps in the SABR parameters across expiries or across valuation dates which would trigger other risk-management actions.

In this document, we present some visualizations concerning the evolution of the parameters. We also compare two versions of the SABR model encountered in practice, one that fixes the parameter beta to a positive value (typically to beta=1/2) and a second one that fixes beta to the value of zero.

¹ See, for example, Hagan et al “Managing Smile Risk” Wilmott Magazine (7/2002), or, Berestycki et al. “Computing the implied volatility in stochastic volatility models” Comm. Pure Appl. Math., 57 1352, 2004