How does the Dupire local volatility model work?

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AcadiFi · Accepted Answer

The Dupire model assumes volatility is a deterministic function of spot and time: sigma(S, t). It's the unique diffusion consistent with today's vanilla smile surface...

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sigma_local^2(K, T) = (dC/dT + rK * dC/dK) / (0.5 * K^2 * d2C/dK^2)

Sonya Brynjolfsdottir at Reykjavik Derivatives Research calibrates this surface daily on 240 vanilla quotes across strikes and maturities. Once calibrated, she prices exotics by simulating the SDE dS = rS dt + sigma(S,t) S dW.

Advantages:
- Perfectly reprices all vanillas by construction
- Single-factor model, fast Monte Carlo
- Deterministic vol — no additional random source

Weaknesses:
- Forward smile is flat: the model predicts tomorrow's smile is flatter than today's, contradicting reality
- Barrier options and cliquets are mispriced because they depend on forward smile dynamics
- Calibration is unstable in wings where there are few vanilla quotes — noise blows up the second derivative

Local vol is appropriate for European-style exotics sensitive to the marginal distribution at each maturity (like digital options, power options). For path-dependent exotics, stochastic volatility or mixed local-stochastic models are preferred. Sonya's desk uses LSV — a hybrid where local vol ensures smile calibration while a stochastic factor governs smile dynamics. It's the industry standard for serious exotic books.

How does the Dupire local volatility model work?

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