What is the SABR Model? Managing the Volatility Surface

The SABR model is the industry standard for interest rate derivatives. Learn how the Beta parameter influences the shape of the volatility smile.

I analyzed the shift in the EURIBOR cap surface using the SABR Model to understand why my hedging ratios were failing during a parallel yield curve shift. SABR (Stochastic Alpha, Beta, Rho) is a stochastic volatility model primarily used by practitioners in fixed income to capture the Volatility Smile.

The SABR model assumes that the forward rate $F_t$ follows a stochastic process with volatility driven by another stochastic process: $dF_t = \sigma_t F_t^{\beta} dW_t$ $d\sigma_t = \nu \sigma_t dZ_t$ where $W_t$ and $Z_t$ are correlated Brownian motions with correlation coefficient $\rho$. The parameter $\beta$ controls the elasticity of volatility with respect to the underlying price, allowing the model to fit different shapes of the Volatility Smile.

The power of SABR lies in its ability to separate the 'at-the-money' volatility level from the 'skew' (governed by $ ho$) and the 'smile curvature' (governed by $ u$). This is vital for interest rate swaptions, where the relationship between the forward rate and its volatility is non-linear.

The primary risk with SABR is the 'model break' at very low or negative interest rates, where the power-law assumption for $eta$ creates numerical instabilities.

Calibrate the SABR parameters to a current swaption grid and check if the $ ho$ value correctly reflects the current market sentiment toward rate cuts.