How to understand the SABR model parameters
SABR model - Stochastic Alpha, Beta, Rho
The SABR model is a stochastic volatility model which, unlike Black-Scholes, models the implied volatility smile. It is the standard model in the financial industry and heavily used in the interest rate derivatives market.
This article describes how to actually understand each SABR model parameter.
Beta
The Beta parameter (elasticity parameter) determines the stochastic model used to model the underlying process.
β = 0: models the underlying as a Normal distribution
β = 1: models the undelying as a Lognormal distribution
0 < β < 1: models the underlying as in between Normal and Lognormal
Typically 0 < β < 1 is the market standard for modelling interest rate derivatives.
Rho
The rho parameter represents the correlation between the underlying and its volatility (e.g. correlation between interest rates and volatility).
A larger rho means that rate are more correlated with volatilty, i.e, volatilty increases as rates increase.
You can see this effect in the figure below. For larger values of rho, the implied volatilty surface is more skewed (it is more upward sloping).
Alpha (also known as the vol of vol)
The vol of vol represents the volatility of the implied volatility across different strikes. A larger vol of vol means that implied volatilties are more volatile across different strikes.
The vol of vol parameter controls the curvature of the implied volatility smile.
You can see in the figure below that a larger vol of vol parameter produces a volatily smile which is more curved.
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