Implied and Local Volatility Dynamics in the SABR Model

The stochastic α β ρ (SABR) model [1] provides a parsimonious parametrization of the implied volatility surface, based on a singular perturbation series of a particularly simple stochastic volatility model. The popularity of the SABR model is mainly due to its overwhelming empirical success, but also because it is extraordinary easy to calibrate. In this Demonstration, the asymptotic refinements of [3] are used. The local volatility surface is calculated exactly along the famous Dupire equation [2] in terms of implied volatility.

THINGS TO TRY

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DETAILS

The SABR model [1] is based on the simplest possible stochastic volatility model for forward prices

d(Subscript[F, i])=(Subscript[σ, t])(Subscript[F, t]^β)d(Subscript[W, t]),

d(Subscript[σ, t])=ν (Subscript[σ, t])d(Subscript[Z, t]),

under the forward measure, where (Subscript[W, t]) and (Subscript[Z, t]) are Brownian motions with d(Subscript[W, t])d(Subscript[Z, t])=ρ dt and (Subscript[σ, 0])=α. In detail, α≥0 is the initial volatility, 0≤β≤1 is the CEV-exponent (CEV stands for constant elasticity of variance), ν≥0 is the volatility of volatility (volvol), and -1≤ρ≤1 is the correlation between the two sources of random fluctuations.

Because the SABR model is the result of a perturbation series, there is a term of order O((ε^2)), which may or may not be included. The corrections due to this term are usually about 1%.

References

[1] P. S. Hagan, D. Kumar, A. S. Lesniewski, and D. E. Woodward, "Managing Smile Risk," Wilmott Magazine, September 2002 pp. 84–108.

[2] B. Dupire, "Pricing with a Smile," Risk, 7, 1994 pp. 18–20.

[3] J. Oblój, "Fine-Tune Your Smile: Correction to Hagan et al.," Wilmott Magazine, May 2008.

Thomas Mazzoni

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