Valuation models

SABR smile

Alpha, rho and nu fitted to a volatility smile with beta fixed, or given; Hagan's implied vol at the strike, Black-76 price, sticky-SABR delta, vega, rho and nu risk, shifted SABR for rates.

Inputs

Results

Explain this result

Get a plain-English reading of the numbers above and what drives them.

Preview: rule-based reading. AI-assisted explanations run on the analytics core.

Methodology

Conventions

  • The SABR smile at the option's expiry gives the vol for its strike; Black-76 on the forward prices it.
  • Alpha, rho and nu not given are fitted to the surface at seven strikes around the forward; beta is fixed.
  • A shift is added to forward and strike for low or negative rates. Delta holds alpha fixed (sticky SABR).

References

  • Hagan, Kumar, Lesniewski and Woodward (2002), Managing Smile Risk

Formulas

Dynamics
dF = σ F^β dW₁, dσ = ν σ dW₂, dW₁ dW₂ = ρ dt
Vol
σ(K) ≈ α / (FK)^((1−β)/2) · z / x(z) · [1 + (…) T]