What are the SABR model parameters and how are they interpreted?

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AcadiFi · Accepted Answer

SABR (Stochastic Alpha Beta Rho) by Hagan et al. models the forward F and its volatility alpha with two correlated Brownian motions: dF = alpha * F^beta * dW1, d(alpha) = nu * alpha * dW2, corr(dW1, dW2) = rho. The four parameters have clean interpretations: - alpha (level): instantaneous vol of the forward at t=0. Controls ATM vol. - beta (elasticity): 0 for normal (Bachelier-like), 1 for lognormal, 0.5 for CIR-like. Fixes the backbone of how ATM vol moves with the forward. - rho (correlation): negative for equity (skew down-side), positive for some FX pairs. Controls skew. - nu (vol of vol): controls curvature / smile. The famous Hagan approximation gives closed-form implied vol sigma_B(K, F) as a function of these four parameters, making calibration fast. Typically traders fix beta based on market convention (0.5 for swaptions, 1 for equity FX) and solve for alpha, rho, nu to match three points on the smile. ```mermaid flowchart LR A[SABR] --> B[alpha
ATM Vol] A --> C[beta
Backbone] A --> D[rho
Skew] A --> E[nu
Smile] ``` A calibration example: Juniper Rates Desk calibrates 3M-into-10Y swaptions daily. beta = 0.5 is fixed, and (alpha, rho, nu) fit ATM-25-delta-75-delta in closed form. On a given day they might get alpha = 0.008, rho = -0.28, nu = 0.42. When swap rates rally, rho-nu product dictates how the ATM vol moves. The SABR arbitrage issue at low strikes (Hagan formula goes negative in the density) is a known FRM exam point — fixed via shifted SABR or arbitrage-free SABR extensions (Antonov).

What are the SABR model parameters and how are they interpreted?

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