Local vs. Stochastic Volatility Models

Choosing between local and stochastic volatility models is like selecting between GPS navigation and weather forecasting—local vol tells you exactly where you are on the surface, while stochastic vol predicts how that surface might move. Each approach has distinct calibration requirements, hedge behaviors, and use cases for exotic pricing.

Model Definitions and Dynamics

Dupire Local Volatility

Dynamics: dS = μS dt + σ(S,t) S dW

Volatility σ(S,t) is a deterministic function of spot and time, extracted from the implied volatility surface.

Calibration: Fit exactly to all vanilla option prices. The local vol surface is derived from the market surface using Dupire's formula.

Key property: Matches all vanilla prices by construction. No calibration error for Europeans.

Heston Stochastic Volatility

Dynamics: dS = μS dt + √v S dW₁ dv = κ(θ - v) dt + σᵥ√v dW₂ Corr(dW₁, dW₂) = ρ

Volatility v follows its own mean-reverting process.

Parameters:

• κ (kappa) ≈ 1.0-3.0: Speed of mean reversion
• θ (theta) ≈ 0.04-0.10: Long-run variance (vol² of 20-30%)
• σᵥ (vol of vol) ≈ 0.3-0.8: Volatility of variance
• ρ (rho) ≈ -0.7 to -0.3: Spot-vol correlation (typically negative for equities)
• v₀: Initial variance

Calibration: Fit to ATM term structure and skew. Typically leaves some calibration error at wings.

SABR Model

Dynamics: dF = σF^β dW₁ dσ = ασ dW₂ Corr(dW₁, dW₂) = ρ

Used primarily for interest rate and FX options.

Parameters:

• α (alpha) ≈ 0.1-0.5: Initial volatility level
• β (beta) ≈ 0.5-1.0: CEV exponent (controls backbone)
• ρ (rho) ≈ -0.5 to 0.5: Forward-vol correlation
• ν (nu) ≈ 0.3-0.8: Vol of vol

Model Comparison

Calibration Inputs and Tolerance

Heston calibration workflow:

1. Collect market data: ATM vols for 1M, 3M, 6M, 1Y; 25Δ put/call skew
2. Set initial parameters: κ=2.0, θ=0.04, σᵥ=0.4, ρ=-0.6, v₀=0.04
3. Optimize: Minimize sum of squared errors between model and market vols
4. Tolerance: Accept if RMSE < 0.5 vols across calibration set

Calibration error example:

RMSE = 0.54 vols. Within acceptable tolerance for most applications.

Hedge Path Implications

Local vol hedge behavior: When spot moves, the model follows the pre-specified σ(S,t) surface. Delta hedging produces:

• Deterministic P/L from gamma
• No vega from realized vol changes
• May under-hedge if realized surface differs from calibrated

Stochastic vol hedge behavior: When spot moves, volatility can move independently. This creates:

• Vega P/L from vol moves
• Correlation effects (ρ < 0 means vol rises when spot falls)
• More realistic hedge performance for vol-sensitive products

Practical impact: For a barrier option near the barrier, local vol may predict different delta than Heston. The "correct" model depends on how the market actually behaves—if vol typically jumps when spot approaches barriers, stochastic vol better captures this.

Runtime and Infrastructure Cost

For real-time trading, SABR's speed is advantageous. For overnight batch pricing of exotics, Heston's accuracy justifies higher runtime.

Engine Selection Triggers

Use this checklist to select the appropriate model:

• Use Local Vol when: Pricing barriers, digitals, or products sensitive to spot path at specific levels; require exact calibration to vanilla market; infrastructure supports PDE solvers
• Use Heston when: Pricing variance-dependent products (variance swaps, vol swaps); need forward-starting option pricing; want vol correlation effects in hedges
• Use SABR when: Pricing rate or FX options; need fast calibration and repricing; smile dynamics matter more than smile fitting
• Avoid all models when: Product has complex path-dependence requiring full Monte Carlo with jumps or regime switches

Next Steps

For building and interpreting the surface these models calibrate to, see Implied Volatility Surface Basics.

To validate model calibration and performance, review Model Calibration and Validation.