Dupire Local Volatility: From an Option Surface to a Diffusion

For educational purposes only; not investment advice.

Direct answer

Dupire local volatility is a deterministic function of asset level and time inferred from an arbitrage-free surface of European option prices. It asks which instantaneous diffusion rate σ_loc(S,t) would make one continuous, risk-neutral price process reproduce today’s call prices across strikes and maturities.

It is a calibration model, not a forecast of realized volatility or stock direction. Matching today’s vanilla surface can be exact in theory while tomorrow’s smile dynamics, exotic-option prices, and hedge results remain wrong.

Model and formula

Under continuous risk-free rate r and dividend yield q, a local-volatility process can be written

dS_t = (r−q)S_t dt + σ_loc(S_t,t)S_t dW_t.

Let C(K,T) be a sufficiently smooth European call-price surface. One common Dupire form is

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

The denominator contains the call’s second strike derivative, which is linked to the discounted risk-neutral terminal density. Call prices must therefore be decreasing and convex in strike, and the surface must obey appropriate calendar constraints. Formula variants depend on discounting, forwards, dividends, and coordinates; inputs and units must match the chosen derivation.

Local volatility is not implied volatility. Implied volatility is the single Black-Scholes input that reproduces one option price. Local volatility is the instantaneous diffusion coefficient used when a simulated path reaches one state (S,t). Under a pure local-volatility model, that state has one prescribed volatility; a stochastic-volatility model can have several variance states at the same spot and time.

Illustrative derivative calculation

At K=$100 and T=0.5 years, suppose a smooth, arbitrage-free call surface gives:

C=7.00, ∂_T C=8.50 per year.
∂_K C=−0.45, ∂²_K C=0.018 per dollar squared.
r=4% and q=1%, continuously compounded.

The numerator is 8.50 + (0.04−0.01)×100×(−0.45) + 0.01×7.00 = 7.22. The denominator is 0.5×100²×0.018 = 90. Thus σ_loc²≈7.22/90≈0.0802, and σ_loc≈28.3%.

These derivatives are illustrative fitted-surface values, not differences to take directly from adjacent option-chain quotes. Market quotes are discrete, asynchronous, and noisy. A second strike derivative amplifies small errors; a negative or near-zero denominator can generate negative, infinite, or unstable local variance.

A defensible workflow cleans synchronized quotes, estimates discount factors and forwards, fits a smooth surface without butterfly or calendar arbitrage, differentiates that surface, applies boundary controls, and reprices the input options by PDE or simulation. Repriced values should return near executable market ranges, not merely match selected midpoints.

Calibration checklist

Use European options or a justified Europeanization procedure; American equity options contain early-exercise value.
Infer forwards consistently from rates, dividends, and put-call parity before defining moneyness.
Fit prices or total variance across multiple strikes and maturities with explicit no-static-arbitrage constraints.
Check strike monotonicity and convexity, calendar ordering, positive density, and positive local variance.
Compare fits to bid and ask, remove stale or crossed quotes, and document extrapolation in sparse wings.
Perturb one quote, forward, dividend, knot, interpolation rule, and grid spacing to measure instability.
Reprice all calibration instruments and hold out some strikes or maturities for validation.
Compare PDE and Monte Carlo outputs, especially near short maturities and boundaries.
Treat barrier and other path-dependent prices as model-dependent even when vanilla repricing is exact.
Evaluate hedge behavior under alternative smile dynamics, stochastic volatility, and jumps.

Common misconceptions

“Local volatility is the IV at that strike.” The two quantities have different definitions and need not match.
“Exact vanilla calibration proves the model is correct.” It establishes static consistency, not dynamic truth.
“Raw option-chain values can be inserted into the formula.” Smoothing and arbitrage controls are essential before differentiation.
“Negative local variance describes the market.” It usually signals bad inputs, arbitrage, interpolation, or numerical instability.
“The recovered surface predicts future realized volatility.” It is risk-neutral and calibrated to current option prices.
“One calibrated model gives the correct barrier price.” Path behavior and smile dynamics create material model risk.