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Quantitative Finance · Begrippenlijst

Wat is Backbone?

Definition 11.5 Derivatives and Volatility · Hoofdstuk 11 — SABR and Smile Dynamics

The backbone of a smile model is the curve traced by the at-the-money implied volatility as the forward moves, all other state variables fixed. In SABR it is σATM(F)≈αFβ−1\sigma_{\mathrm{ATM}}(F)\approx\alpha F^{\beta-1}: flat for β=1\beta=1, falling like 1/F1/F for β=0\beta=0.

SABR dynamics. Left: the at-the-money volatility as the forward moves, for three values of with set to give 20% at a 3% forward. Right: the smile at three forwards (=0.5, =-0.6, =0.5): it moves with the forward, its minimum travelling to the right as the forward rises. Data: the chapter’s code.
Figure 11.2. SABR dynamics. Left: the at-the-money volatility as the forward moves, for three values of β\beta with α\alpha set to give 20% at a 3% forward. Right: the smile at three forwards (β=0.5\beta=0.5, ρ=−0.6\rho=-0.6, ν=0.5\nu=0.5): it moves with the forward, its minimum travelling to the right as the forward rises. Data: the chapter’s code.

Voorbeelden

Example 11.4 (SABR on an equity smile)

With β=1\beta=1 (a lognormal backbone), SABR fitted to the one-year smile of chapter 9’s surface returns α=0.194\alpha=0.194, ρ=−0.68\rho=-0.68, ν=0.72\nu=0.72, with a root-mean-square error of 0.07 volatility point and at most 0.15 point, at the 70 strike (Figure 11.1). One expiry, three free parameters, a good fit: SABR is a smile interpolator as much as a model.

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