Recalibrating SABR Parameters for Swaption Volatility

Summary

The document addresses which SABR parameters may need recalibration when at-the-money swaption volatility changes. It describes the model through a forward price process and a stochastic volatility process, with three parameters highlighted: alpha, beta, and rho. Alpha controls volatility of volatility, beta governs the forward-price dependence and is associated with skew, and rho represents the correlation between forward-price and volatility shocks.

The response connects these parameters to the shape of the implied volatility surface, including smile richness away from at-the-money strikes and the direction of skew. It recommends recalibrating all three parameters, but gives no calibration procedure, market data example, or rules for deciding whether to hold any parameter fixed. It notes that closed-form solutions are available only for certain beta choices, specifically zero or one, which constrains model implementation. The discussion is therefore a concise parameter overview rather than a complete swaption calibration guide.

Key ideas

• SABR represents forward price and volatility as stochastic processes linked by correlated shocks.
• Alpha is the volatility of volatility parameter and is constrained to be nonnegative.
• Beta controls the forward-price dependence and is associated with skew.
• Rho captures the correlation between forward-price and volatility movements.
• The response advises recalibrating alpha, beta, and rho, while offering no detailed calibration routine.

Tags

# Recalibrating SABR parameters for Swaption ATM volatility


# Recalibrating SABR parameters for Swaption ATM volatility












I understand that the ATM volatility of Swaption moves quite frequently and the SABR will need to be recalibrated. Which parameter should I recalibrate?

Is there any financial meanings why we only recalibrate on certain parameters?

Thanks.

## Answer by rrg (score 2)

https://quant.stackexchange.com/a/31576

You will need to recal alpha beta and rho:

\begin{align*} dF_{t}&=\sigma _{t}F_{t}^{\beta }\,dW_{t}\\ d\sigma _{t}&=\alpha \sigma _{t}^{{}}\,dZ_{t}\\ \end{align*} Where $$dW_{t}dZ_{t}=\rho dt$$

- alpha, volvol, lognormal vol of vol param sigma, alpha >= 0.

- beta, skew, closed form soln only if in set {0,1}

- rho, correlation coefficient between two stochastic state variables forward price F and volatility of fwd price, sigma.

Parameters describe smile (richness of out of the money options) and the skew (whether implied vol is upward or downward sloping as a function of strike).

Take a look at Matlab's implementation, which discusses two methods based on closed form, https://www.mathworks.com/help/fininst/calibrating-the-sabr-model.html?s_tid=gn_loc_drop

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.