On the Estimation of the SABR Model’s Beta Parameter: The Role of Hedging in Determining the Beta Parameter

Abstract

The stochastic alpha-beta-rho (SABR) model has become one of the most popular models for real-world interest rate behavior, but not all of its four parameters are equally easy to estimate. The elasticity of the variance beta parameter determines the shape of the returns density, with a beta of 0 implying normal shocks; 1 implying lognormal shocks; and a value between 0 and 1 implying a constant elasticity of variance process. But fitting the model to market returns is not easy, because the data do not discriminate well among different beta values even though the differences are important for hedge design. This article proposes optimizing on a different metric for goodness of fit: hedge performance. Minimizing hedging error as a function of beta in addition to minimizing the error in calibrating to the market volatility smile substantially improves the ability to pin down the beta value. The method can be tuned to reflect the user’s desired trade-off between hedging versus pricing accuracy.