Markovian Projection (Gyöngy Lemma)

Local Vol

Projects any stochastic volatility model to local volatility

Projects any stochastic volatility model to an equivalent local volatility surface using Gyöngy's lemma. The projected local volatility is the conditional expectation of instantaneous variance given the spot price: σ²_loc(S, t) = E[σ²(t) | S_t = S]. The resulting local vol model preserves the marginal distributions of the original model at each time.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Projected local volatility
Reference model variance process
Reference model price process
Projected local vol price
Conditional expectation given spot

Key Assumptions

  • •Preserves marginal distributions at each time t
  • •Converts any SV model to local vol equivalent
  • •Uses kernel regression for conditional expectations
  • •Path-dependent properties differ from reference model
  • •Useful for comparing models and calibration

Reference

Gyöngy, I. (1986). "Mimicking the One-Dimensional Marginal Distributions of Processes Having an Itô Differential." Probability Theory and Related Fields.

Model Hierarchy

Built on simpler models
SimplerStandard

Dupire

Local Vol
SimplerBaseline

GBM

Diffusion
Extended by more complex models
More ComplexComplex

SLV

Local Vol

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 Gyöngy relates to observed return properties

Gyöngy projection maps stochastic volatility models to local volatility equivalents that match marginal distributions. For SPY, this allows comparing SV model implications to pure local vol. The projected model preserves vanilla option prices but not path-dependent payoffs.

What Gyöngy captures

  • Marginal distributions from any reference model
  • Local vol equivalent of SV models

What it cannot capture

  • Path-dependent properties of reference model
  • True stochastic vol dynamics
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 Markovian Projection (Gyöngy Lemma) in action. Adjust parameters and observe how the price path evolves in real-time.

Charts:

Price Path Simulation

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Log Returns

Log returns will appear here

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.