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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.

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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

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