Local Volatility Model: From Option Prices to State-Dependent Volatility

Learn how the Dupire local volatility model turns an arbitrage-consistent European option surface into a state-dependent diffusion, and why exact calibration does not ensure realistic smile dynamics.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

A local volatility model sets instantaneous volatility as a deterministic function of the current underlying price and time. In a simple risk-neutral equity model, dS_t=(r-q)S_tdt+σ_loc(S_t,t)S_tdW_t. Unlike Black-Scholes with one constant volatility, a properly calibrated local volatility diffusion can reproduce the current prices of European options across strikes and maturities.

That calibration property is about today’s risk-neutral marginal distributions, not a forecast of realized volatility or tomorrow’s implied-volatility surface. The model supplies one internally consistent path process for pricing and hedging, but its deterministic state dependence can generate future smile movements unlike those observed in the market.

Dupire calibration mechanics

Let C(K,T) denote the time-zero price of a European call with strike K and maturity T. With constant continuously compounded rate r and continuous dividend yield q, Dupire’s forward equation implies, wherever the denominator is positive:

σ_loc²(K,T)=[∂_T C+(r-q)K∂_K C+qC]/[0.5K²∂²_KK C].

The curvature ∂²_KK C is proportional to the discounted risk-neutral terminal density. Call prices must therefore be decreasing and convex in strike, while maturity slices must satisfy the appropriate carry-adjusted calendar constraints. Sparse or inconsistent quotes can make the inferred variance unstable or negative.

Production calibration normally starts with synchronized option quotes, discount factors, forwards, dividends, and contract conventions. A smooth surface is fitted under static-arbitrage controls before derivatives are taken. The resulting local-volatility grid is then interpolated, bounded, and checked by repricing the calibration options.

Local volatility is not the quoted implied volatility at the same strike and maturity. Implied volatility is the constant Black-Scholes input that matches one option price; local volatility is an instantaneous diffusion coefficient whose evolution generates the full set of calibrated marginal distributions.

Illustrative Dupire calculation

At K=$100 and T=1 year, suppose a smoothed call surface gives C=$8, ∂_T C=10 dollars per year, ∂_K C=-0.45, and ∂²_KK C=0.018 per dollar. Let r=5% and q=2%. The numerator is:

10+(0.05-0.02)×100×(-0.45)+0.02×8=8.81.

The denominator is 0.5×100²×0.018=90. Local variance is therefore 8.81/90=0.0979, so local volatility is √0.0979=31.3%.

This is a numerical illustration, not a market estimate. If a noisier fit changed the curvature from 0.018 to 0.012 while every other input stayed fixed, the inferred volatility would rise to √(8.81/60)=38.3%. The sensitivity shows why cleaning, no-arbitrage smoothing, stable differentiation, and out-of-sample checks matter.

Calibration and model-risk checklist

Use synchronized executable quotes, forwards, discount factors, dividends, contract terms, and settlement conventions.
Remove crossed, stale, zero-bid, and economically inconsistent observations under documented rules.
Enforce strike monotonicity, strike convexity, and appropriate carry-adjusted calendar constraints before differentiating.
Test sensitivity to interpolation, smoothing penalties, quote weights, grid spacing, boundary conditions, and extrapolation.
Reprice every calibration vanilla and report errors in price units as well as implied-volatility points.
Compare hedge behavior with Black-Scholes and stochastic-volatility alternatives under spot and smile shocks.
Stress jumps and gaps because a continuous diffusion cannot hedge an instantaneous discontinuity.
Treat future surface dynamics as a model assumption, not a consequence guaranteed by today’s fit.
Use extra caution for barriers and other path-dependent claims whose values depend on the modeled path law.
Recalibrate when the surface changes and separate market P&L, carry, hedge error, and recalibration effects.
Limit extrapolation in sparse wings and long maturities because numerical derivatives amplify small input errors.
Account separately for liquidity, transaction costs, discrete hedging, early exercise, and counterparty or settlement terms.

Common misconceptions

“Local volatility equals implied volatility at each point.” They are different objects with different definitions.
“Exact calibration proves the model is true.” Many models fit today’s vanillas while implying different paths and hedges.
“The Dupire formula can be applied directly to raw quotes.” It requires smooth derivatives and arbitrage-consistent inputs.
“A complete diffusion removes all risk.” Jumps, discrete hedging, liquidity, and model error remain.
“More surface detail always improves accuracy.” Differentiation can amplify noisy and illiquid quotes.
“The model forecasts volatility.” Calibration extracts a risk-neutral pricing function, not a real-world forecast.
“Models matching all vanillas must price every exotic equally.” Their path laws and future smile dynamics can differ.
“Negative local variance is a valid market signal.” It usually identifies inconsistent data, fitting, derivatives, or assumptions.