SABR with Circangle Upgrades

SABR

SABR (Hagan and colleagues, 2002) is the market standard for interest-rate and commodity smiles. The forward and its volatility move together, with correlation ρ, and Hagan’s formula gives the Black volatility at each strike. The circangle reading is that SABR already has two radii, hidden in ρ, and that its key term is an arc on the circangle at k = 1.

Option

Forward price

Strike

Time to expiry (years)

Interest rate (%)

Volatility level α

Exponent β

Correlation ρ

Vol of vol ν

Circangle upgrades

Circangle geometryShows ρ as the drivers’ ellipse, with k² = 2|ρ|/(1 + |ρ|), the joint-move chance from its area, and Hagan’s x(z) computed as an sc⁻¹ arc at k = 1. The price is unchanged. Two radii ExperimentalPrices each strike with Black–Scholes–Robilliard in place of Black, splitting SABR’s volatility into a downside scale b and an upside scale a, with (a + b)/2 equal to SABR’s volatility. Suggested in the book but not worked through there.

Downside / upside ratio b/a

At 1 the price is SABR’s exactly.

Brent example: illustrative inputs, not live market data. Book example: the figures in Circangle: π is Obsolete, Part 4 (σ_B = 20.11% and 8.0095 at the money).

SABR with Black

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SABR with two radii

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Volatility smile

Black implied volatility at each strike.

The circangle inside ρ

One-standard-deviation region of the two Brownian drivers: the forward across, its volatility up.

The formulas

With ℓ = ln(F/X) and , Hagan’s volatility is

with , and the price is Black’s formula at that volatility.

The two drivers’ correlation matrix has eigenvalues 1 + |ρ| and 1 − |ρ|, so their one-standard-deviation region is a circangle with semi-axes √(1 ± |ρ|) and . The square root in x(z) is length measured with that matrix, and x(z) is its integral, an arc at the k = 1 end of the circangle dictionary:

SABR’s volatility dynamics are Brownian motion on the hyperbolic plane, and z/x(z) is chord over arc in that geometry. The share of the ellipse in each quadrant is the chance of each joint move, for the forward falling as its volatility rises when ρ is negative, against ¼ with no correlation.

The two-radius upgrade is experimental: it puts the circangle’s two scales into the terminal distribution on top of SABR’s own skew, so a ratio above 1 adds downside weight beyond what ρ gives. It has not been tested against market quotes. Derivations are in Circangle: π is Obsolete, Part 4.

Use of the models is acceptance that they are for educational purposes only. See the disclaimer.