The SABR Model: Modelling the Volatility Smile

Annika Abildgaard Lund

Studenteropgave: Kandidatafhandlinger

Abstract

This thesis conducts an empirical analysis of the ability of the shifted SABR model, the normal SABR model, and free-boundary SABR model to manage risks in interest rate markets across different market conditions. We place a heavy focus on model calibration, parameter stability, and risk sensitivities to ensure the models are practically viable, and generally take a practitioner approach to the analysis. We calibrate each model using both Antonov’s exact solution and one of Hagans approximation formulas. We do this across a wide range of model parameterisations to determine the optimal fit to our data. The Antonov normal SABR was quickly discarded due to calibration issues, while the free-boundary SABR was discarded due to a stickiness at zero making it unable to accurately model negative rates. In general, calibrating the model to prices is found to be impractical due to a very long calibration time relative to the fast and relatively accurate Hagan approximation. The normal SABR model calibrated in a normal calibration space using the Hagan approximation generally gives the most stable parameters. Still, the overarching conclusion is that a shifted SABR with s = 5% and β = 1 calibrated in a shifted Black calibration space using the Hagan approximation has the best performance both in-sample and out-of-sample, while risk measures also conveniently can be calculated. The main limitation of this model is a relatively higher pricing error for deep OTM and short-term options. In addition, we observe a slight decrease in performance for the model when rates become negative, but overall, the we conclude that it is the best suited model for empirically modelling the volatility smile across different interest rate environments.

UddannelserCand.merc.fin Finance and Investments, (Kandidatuddannelse) Afsluttende afhandling
SprogEngelsk
Udgivelsesdato2023
Antal sider82
VejledereAnders Bjerre Trolle