SABR Stochastic Volatility Model & Smile Lab

Simulate Patrick Hagan et al.'s (2002) SABR model. Calibrate CEV elasticity exponent (β), correlation (ρ), and volatility-of-volatility (ν) to reconstruct the complete Black-76 implied volatility smile, backbone dynamics, and option pricing surface.

Market Calibration Presets:

Model Parameters

Pts / %
Forward price or par swap rate under the forward measure.
Pts / %
Option strike for single-contract valuation and Greek extraction.
Years
%

SABR Diffusion Parameters

0.200
Controls the baseline level of volatility.
1.00
CEV backbone: 0.0 = Normal (Bachelier), 0.5 = CIR, 1.0 = Lognormal (Black).
-0.30
Skew slope: negative for equity/rates crash risk, positive for commodity supply shock.
0.40
Smile curvature / tail fatness: higher values create deeper smile wings.
SABR Implied Volatility σᵇ(K)
20.45%
ATM Vol: 20.00%
Black-76 Call Option Price
7.72
Δ: 0.539 • ν: 38.45
Black-76 Put Option Price
7.72
Δ: -0.422 • Γ: 0.019
Backbone Level & Skew Slope
-0.060% / pt
ATM Backbone: 20.00%

Quantitative Visualizer

SABR Smile σᵇ(K) ATM Benchmark Selected Strike (K)

Hagan et al. (2002) Asymptotic Black Implied Volatility Formula

σᵇ(F₀, K) ≈ [ α / (F₀K)^((1-β)/2) ] · [ 1 + ((1-β)²/24)ln²(F₀/K) + ((1-β)⁴/1920)ln⁴(F₀/K) ]⁻¹ · (z / χ(z)) · [ 1 + ( ((1-β)²/24)α²/(F₀K)^(1-β) + (1/4)ρβνα/(F₀K)^((1-β)/2) + ((2-3ρ²)/24)ν² ) T ]

Where z = (ν/α) (F₀K)^((1-β)/2) ln(F₀/K) and χ(z) = ln[ ( √(1 - 2ρz + z²) + z - ρ ) / (1 - ρ) ]. At the money (F₀ = K), the term z / χ(z) collapses to 1.

Sensitivity Grid 1: Correlation (ρ) vs. Vol-of-Vol (ν)

Evaluates 90% OTM Put Implied Volatility σᵇ(0.90 F₀), illustrating downside tail risk and crash protection pricing.

Sensitivity Grid 2: Elasticity (β) vs. Forward Shift (ΔF)

Evaluates ATM Implied Volatility across market regime moves, quantifying the backbone effect.

Institutional Mechanics of the SABR Model

1. The Two-Factor Stochastic Dynamics

The standard Black-Scholes and Black-76 models assume geometric Brownian motion with constant volatility. In reality, market quotes for interest rate swaptions, caps/floors, and equity options exhibit distinct smiles and skews. The SABR model addresses this by making volatility stochastic under the forward risk-neutral measure:

  • Forward Diffusion: dF_t = α_t F_t^β dW_t^F (stochastic forward asset or swap rate).
  • Volatility Diffusion: dα_t = ν α_t dW_t^α (lognormal stochastic volatility driver).
  • Innovation Correlation: d⟨W^F, W^α⟩_t = ρ dt (leverage effect and skew control).

2. Decoupling the Backbone (β) from the Smile (ρ, ν)

A critical advantage of SABR over local volatility models is its ability to decouple how volatility shifts when the market moves (the backbone) from the smile's shape at any single moment:

  • Backbone (β): When rates rise, local volatility models predict the smile shifts in a fixed deterministic direction, often causing severe hedging miscalculations. SABR's parameter β isolates this baseline relationship.
  • Vol-of-Vol (ν): Governs smile curvature. In low-interest rate environments, ν often rises as interest rate uncertainty expands.
  • Correlation (ρ): Generates the skew slope, essential for pricing out-of-the-money options accurately.

SABR Model Mastery Quiz

Frequently Asked Questions

The SABR (Stochastic Alpha, Beta, Rho) model, introduced by Patrick Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward in 2002, is a continuous-time two-factor model that captures the market volatility smile and skew in interest rate derivatives (swaptions, caps/floors), FX, and commodities. Unlike the constant volatility Black-76 model, SABR models the forward asset price F_t and its volatility alpha_t as correlated diffusions, providing an analytical asymptotic formula for Black implied volatilities across all strikes.

Using singular perturbation theory and heat kernel expansion methods on Riemannian manifolds, Hagan et al. derived a closed-form formula approximating the Black implied volatility sigma_B(F0, K) as a function of forward price F0, strike K, expiry T, and SABR parameters (alpha, beta, rho, nu). The formula decomposes the volatility into an ATM term, a logarithmic strike moneyness term modulated by z/chi(z), and higher-order time corrections O(T).

The elasticity parameter beta (between 0 and 1) governs the relationship between the forward rate and its volatility level, creating the 'volatility backbone'. When beta = 1 (lognormal), the ATM volatility remains invariant to forward rate shifts. When beta = 0 (normal), ATM volatility moves inversely with forward rates (sigma_ATM proportional to 1/F). Setting beta = 0.5 yields CIR-like square root dynamics. In fixed income markets, beta is frequently fixed based on historical backbone behavior (or regulatory standards) while alpha, rho, and nu are calibrated to market smiles.

The correlation parameter rho controls the slope or skew of the volatility curve: negative rho creates a downward-sloping skew where out-of-the-money puts trade at higher implied volatilities than calls (typical in equities and rates), while positive rho creates an upward skew. The vol-of-vol parameter nu governs the convexity or curvature of the smile: higher nu deepens the U-shaped wings, pricing higher probabilities into extreme market tails.

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