Stochastic Local Volatility (SLV) Model

Local Vol

Combines Dupire local vol with Heston stochastic vol

Combines local volatility with stochastic variance for the best of both worlds: exact fit to the implied volatility surface (via local vol from SABR) and realistic volatility dynamics (via Heston-style stochastic vol). The effective volatility is the product of the local volatility surface and the square root of the stochastic variance factor.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Local volatility surface (from SABR)
Stochastic variance factor
Variance mean reversion speed
Long-run variance level
Vol-of-vol
Correlation

Key Assumptions

  • •Exact fit to implied vol surface via local vol component
  • •Realistic forward smile dynamics via stochastic vol
  • •Effective volatility = local vol × √(stochastic variance)
  • •Combines Dupire local vol with Heston stochastic vol
  • •Industry standard for exotic option pricing

Reference

Lipton, A. (2002). "The Vol Smile Problem." Risk Magazine. See also: Dupire (1994), Heston (1993).

Model Hierarchy

Built on simpler models
SimplerStandard

Dupire

Local Vol
SimplerStandard

Heston

Stochastic Vol
SimplerBaseline

GBM

Diffusion

Complexity increases from left to right. More complex models capture additional market phenomena but require more parameters and may be harder to estimate.

Empirical Context (SPY Returns)

How SLV relates to observed return properties

SPY options require both exact calibration (from local vol) and realistic dynamics (from stochastic vol). SLV combines both: the local vol surface from SABR ensures perfect IV fit, while the Heston-style stochastic factor provides realistic smile evolution. This is the industry standard for exotic pricing.

What SLV captures

  • Exact IV surface fit
  • Realistic forward smile dynamics
  • Combined local + stochastic vol

What it cannot capture

  • Requires careful calibration
  • Additional complexity vs pure SV
View full empirical analysis of SPY returns

Estimation Data

Select ticker for P-measure estimation

SPY

Related Models

Dupire

Local Vol

A deterministic volatility model where the instantaneous volatility depends on both spot price and time: σ_loc(S, t). Unlike stochastic volatility mod...

Compare with Dupire →

Heston

Stochastic Vol

Models volatility as a mean-reverting stochastic process correlated with returns. Captures volatility clustering (high vol follows high vol), leverage...

Compare with Heston →
View all stochastic models

Run Simulation

Watch the Stochastic Local Volatility (SLV) Model in action. Adjust parameters and observe how the price path evolves in real-time.

Charts:

Price Path Simulation

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Log Returns

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Model

Parameters

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Simulation

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Disclaimer: These simulations are for educational purposes only. They demonstrate the behavior of mathematical models and should not be used for trading decisions. Real market dynamics are significantly more complex than any single stochastic model can capture.