Local Volatility (Dupire)
Also known as: Deterministic Volatility Function, DVF
Dupire's local volatility model (1994) is a deterministic framework that extracts a term and strike-dependent volatility function from market option prices. Unlike constant volatility, local volatility perfectly fits the observed implied volatility smile and is implemented via finite difference methods for European and American option pricing.
Key highlights
- Perfect fit to market prices: by construction, matches all observed European option prices
- No smile extrapolation needed: the implied smile is naturally embedded in the local volatility function
- Computationally efficient: PDE-based pricing (Crank-Nicolson) is fast for European and American options
- Intuitive: the volatility surface is directly observable from market quotes
Intuition
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How it works
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When to use it
Use local volatility when you need to price exotic or American options consistent with a smile. It is essential in single-asset equity and FX markets. However, local volatility overshoots volatility of volatility; it is less suitable for products where path-dependent volatility clustering matters, and it poorly predicts forward skew evolution.
Strengths & limitations
- Perfect fit to market prices: by construction, matches all observed European option prices
- No smile extrapolation needed: the implied smile is naturally embedded in the local volatility function
- Computationally efficient: PDE-based pricing (Crank-Nicolson) is fast for European and American options
- Intuitive: the volatility surface is directly observable from market quotes
- Sticky strike assumption: local volatility does not match market reality when the spot moves significantly
- Forward skew prediction: the model typically under-estimates volatility of volatility and underprices long-dated volatility derivatives
- American option sensitivity: highly sensitive to numerical discretization and boundary conditions
- Path dependency: the model assumes volatility is a function of spot level only, ignoring volatility clustering and mean reversion
Common pitfalls
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Applications
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Frequently asked
How does local volatility relate to implied volatility?
Implied volatility is the constant number that, when plugged into Black-Scholes, recovers the market price of a single option. Local volatility is a function sigma_loc(S, t) such that the SDE dS = r S dt + sigma_loc(S, t) S dW prices all options consistently. In general, the smile of local volatility at a fixed maturity is flatter than the implied smile because local volatility averages over future paths.
Why does local volatility give wrong prices for long-dated options?
Local volatility assumes volatility depends only on spot level, not on history. In reality, volatility exhibits clustering and mean reversion. When the spot moves, local volatility changes abruptly; but in the real world, volatility drifts more slowly. Over long horizons, this causes local volatility to significantly underestimate volatility-of-volatility, leading to mispricing of variance and volatility derivatives.
How do I compute local volatility from market prices numerically?
Interpolate the option price surface smoothly (e.g., cubic splines in log-strike and log-time), then compute derivatives numerically (finite differences or automatic differentiation). Apply Dupire's formula to extract sigma_loc. Regularization or smoothing of the surface is essential to avoid spurious oscillations.
Can local volatility handle negative spot prices?
No, local volatility is a single-factor spot model designed for positive asset prices (equities, FX). For rates that can go negative, use shift-adjusted models or stochastic volatility models like SABR.
Sources
- 1.Dupire, B. (1994). Pricing with a smile. Risk Magazine, 7(1), 18-20.
- 2.Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. John Wiley & Sons.
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Cite this page
ScholarGate. (2026, June 3). Local Volatility (Dupire). ScholarGate. https://scholargate.app/quantitative-finance/local-volatility