SABR Model

The instantaneous volatility level at time zero. Alpha sets the overall height of the volatility smile.

• Acts as a scaling factor for the entire at-the-money volatility term structure
• Higher alpha values shift the entire smile upward across all strikes
• Directly observable from at-the-money option prices
• In calibration, alpha is typically the first parameter fitted to match ATM implied volatility

Example: In a low-rate FX environment, alpha might be 0.08 (8% volatility), while in a stressed credit market it could exceed 0.50.

The constant elasticity of variance exponent that determines the distribution of the forward rate. Beta controls the backbone dynamics — how ATM volatility moves with the underlying.

• β = 0: Normal model (absolute diffusion, rates can go negative)
• β = 0.5: Square-root model (CIR-type dynamics)
• β = 1: Log-normal model (proportional diffusion, rates stay positive)
• Values between 0 and 1 interpolate between normal and log-normal behavior

Market convention: FX markets typically use β = 1, while interest rate markets often calibrate β or fix it at 0.5 for CMS pricing.

The correlation coefficient between the forward rate process and its stochastic volatility process. Rho controls the steepness and direction of the volatility skew.

• ρ < 0: Negative correlation produces a downward-sloping skew (higher IV for low strikes) — typical in equity and FX markets where falling prices increase uncertainty
• ρ > 0: Positive correlation produces an upward-sloping skew — observed in commodity markets during supply shocks
• ρ = 0: Symmetric smile with no directional skew
• The magnitude of rho determines how pronounced the skew becomes as strikes move away from ATM

Calibration note: Rho and nu are often jointly calibrated as they both influence the wings of the smile.

The volatility of the stochastic volatility process itself. Nu controls the convexity of the smile — how much the wings curve upward.

• Higher nu values produce a more pronounced smile curvature at extreme strikes
• Nu governs the kurtosis of the implied distribution: larger values create fatter tails
• In the limit ν → 0, the model collapses to a standard CEV process with no smile
• This parameter is crucial for pricing out-of-the-money options where smile convexity dominates

Practical insight: Nu is often the most difficult parameter to calibrate stably because it requires liquid quotes across a wide range of strikes to pin down the curvature.

Fitting the SABR model to market data follows a systematic process to ensure a smooth, arbitrage-free volatility surface.

Step-by-step approach:

• Step 1: Fix beta based on prior beliefs about the forward rate distribution or calibrate it historically
• Step 2: For each expiry, calibrate alpha, rho, and nu to minimize the squared error between model and market implied volatilities
• Step 3: Interpolate parameters across expiries to build a full term structure
• Step 4: Validate the surface for static arbitrage (butterfly and calendar spread conditions)

Common practice: Many desks fix β = 1 for FX options and β = 0.5 for interest rate swaptions, then calibrate the remaining three parameters per tenor.

The foundational backbone of the SABR model. The beta (β) parameter controls the distribution of the forward rate.

• β = 0: Assumes a normal (Gaussian) distribution, suitable for interest rates that can go negative.
• β = 0.5: Assumes a square-root diffusion, approximating a non-central chi-squared distribution (CIR model).
• β = 1: Assumes a lognormal distribution, the classic Black-Scholes assumption.

This parameter directly determines the backbone of the volatility smile and how ATM volatility moves with the underlying.

Unlike local volatility models where variance is deterministic, SABR treats volatility itself as a random diffusion governed by the alpha (α) and vol-of-vol (ν) parameters.

• Alpha (α): The initial volatility level. It sets the overall height of the smile.
• Vol-of-Vol (ν): Controls the curvature of the smile. A higher ν increases the implied volatility for far out-of-the-money options, capturing the fat tails observed in market data.

This stochastic nature allows SABR to fit the volatility smile dynamically without needing a full recalibration of the local volatility surface.

The rho (ρ) parameter captures the correlation between the forward rate and its instantaneous volatility. This is the primary driver of the volatility skew.

• ρ < 0: Negative correlation. As the forward rate drops, volatility spikes. This generates the classic downward-sloping skew seen in equity and FX markets, where downside puts carry a premium.
• ρ = 0: Zero correlation. The smile is symmetric, with no directional bias.
• ρ > 0: Positive correlation. As the forward rate rises, volatility increases. This is typical in commodity markets where supply shocks drive prices and uncertainty higher together.

The interaction of ρ with ν determines the overall shape of the smile.

The closed-form approximation that made SABR practical for trading floors. Patrick Hagan derived an analytical formula for implied volatility as a function of strike.

Key components of the expansion:

• Leading Order: Captures the CEV backbone and the basic skew.
• Correction Terms: Account for the vol-of-vol (ν) and correlation (ρ) effects to second-order accuracy.

This formula allows for instantaneous pricing of vanilla options without Monte Carlo simulation, enabling real-time calibration to market quotes. The approximation is most accurate near-the-money and for short-to-medium maturities.

The original Hagan expansion suffers from a well-known flaw: it can generate negative probability densities for very low strikes and long maturities, violating no-arbitrage conditions.

Modern fixes include:

• Normal SABR (β=0): Shifts to a normal backbone to handle negative rates, common in post-2008 interest rate markets.
• Mixture SABR: Combines multiple SABR parameter sets to fit the entire surface without arbitrage.
• Free-boundary SABR: Modifies the process to absorb at zero, preventing negative forward rates.

These extensions ensure the model remains theoretically sound for pricing exotic derivatives that depend on the entire risk-neutral density.

Fitting SABR to observable market quotes involves minimizing the error between model and market implied volatilities across strikes for a single maturity.

Typical calibration workflow:

• Step 1: Fix β to a pre-determined value based on the asset class (e.g., β=0.5 for interest rates).
• Step 2: Optimize α, ν, and ρ simultaneously to minimize the root-mean-square error (RMSE) of implied vols.
• Step 3: Repeat for each maturity slice independently.

This piecewise calibration creates a term structure of SABR parameters, which can then be interpolated to price exotic options with path-dependent features.