The Dupire
Local Volatility Model.

A diffusion that fits the whole smile.

Black-Scholes assumes a single constant volatility, which cannot reproduce the smiles and skews observed on real option markets. Stochastic volatility models such as Heston introduce an extra source of randomness to capture these effects, at the cost of an incomplete market and a harder calibration problem.

Dupire (1994) took a different route: keep the underlying as a one-factor diffusion, as in Black-Scholes, but let the volatility depend deterministically on the current spot level and on time. This single generalization is enough, in principle, to reproduce every European option price quoted on the market, for every strike and maturity, exactly.

The local volatility diffusion.

Under the risk-neutral measure \(\mathbb{Q}\), the underlying price \(S_t\) is assumed to follow:

Unlike Black-Scholes, the diffusion coefficient \(\sigma_{\mathrm{loc}}(S,t)\) is a deterministic function of the spot level and time, rather than a constant. Unlike Heston, there is still only one source of randomness driving the process: the model remains a complete market, with a unique replicating hedge, while gaining enough flexibility to match an entire volatility surface.

The forward Dupire equation.

Instead of the backward Black-Scholes PDE in \((t,S)\), Dupire works with the price of a European call \(C(K,T)\) as a function of strike \(K\) and maturity \(T\), for a fixed valuation date and spot. Applying the Fokker-Planck (forward Kolmogorov) equation to the transition density of \(S_T\) yields the Dupire forward PDE:

This equation propagates option prices forward in maturity for every strike simultaneously, in contrast with the backward equation, which propagates a single option's price backward in time. This is what makes the forward equation the natural tool for extracting local volatility from an entire quoted surface in one pass.

Dupire's local volatility formula.

Rearranging the forward PDE gives a closed-form expression for the local variance in terms of market-observable call prices:

The numerator is the calendar-spread sensitivity of the option price, corrected for carry, and the denominator is proportional to the butterfly sensitivity, i.e. the risk-neutral density \(\partial^2 C/\partial K^2\) implied by the market. Local volatility is therefore fully determined, at every point of the surface, by the local slope in maturity and the local convexity in strike of the quoted call prices.

Expressing local volatility from implied volatility.

Market quotes are given in Black-Scholes implied volatility \(\sigma_{\mathrm{BS}}(K,T)\) rather than in raw call prices, which is both easier to interpolate and numerically better behaved. Using log-moneyness \(k=\ln(K/F(T))\), the local variance can be written directly in terms of the implied volatility surface:

This form, due to Gatheral and others, is the one typically implemented in practice: it converts a smoothly interpolated implied volatility surface directly into a local volatility surface, without ever differentiating the noisier raw price data.

Local volatility as a conditional expectation.

A key result, also due to Dupire, is that local volatility is the risk-neutral conditional expectation of the instantaneous variance, given that the underlying is at level \(K\) at time \(T\):

This holds for any underlying diffusion or stochastic volatility process consistent with the market smile, not only for the local volatility model itself. It is what justifies describing local volatility as the market-implied average of instantaneous variance paths, rather than as a model of the true dynamics of volatility.

Numerical pitfalls and model limitations.

From the quoted smile to a diffusion surface.