Dupire Equation

Unlike the backward Black-Scholes PDE, the Dupire equation evolves forward from the present (t=0) to expiration (t=T). This forward nature allows calibration directly from the current option price surface without solving an inverse problem. The state variable is the option price as a function of strike (K) and maturity (T), with the underlying spot price fixed at initiation.

The equation provides an explicit formula for local volatility σ(K, T) in terms of observable market quantities:

σ²(K, T) = (∂C/∂T + (r - q)K ∂C/∂K + qC) / (½ K² ∂²C/∂K²)

This requires computing the first and second derivatives of the call price surface with respect to strike, and the first derivative with respect to maturity. The denominator involves the risk-neutral density via the Breeden-Litzenberger formula.

The Dupire equation guarantees that the resulting local volatility surface is arbitrage-free by construction, provided the input option prices satisfy basic no-arbitrage conditions:

• Butterfly spread positivity: ∂²C/∂K² ≥ 0 ensures non-negative risk-neutral density
• Calendar spread monotonicity: ∂C/∂T ≥ 0 prevents negative forward variance
• Strike monotonicity: ∂C/∂K bounded between -e⁻qT and 0 Violations in market data must be repaired before calibration.

Local volatility σ(S, t) is a deterministic function of the underlying price and time, not a separate stochastic process. This distinguishes it from stochastic volatility models like Heston. The function is calibrated to exactly reproduce the market prices of all vanilla options. However, the resulting surface dynamics (how the smile moves with spot) are fully determined and may not match empirical sticky-strike or sticky-delta behavior.

Practical implementation requires careful handling of numerical differentiation. The second derivative ∂²C/∂K² amplifies noise in sparse or bid-ask-bounced option data. Common stabilization techniques include:

• Spline interpolation with smoothing penalties
• Regularization via Tikhonov methods
• Implied volatility space transformation before differentiation
• Arbitrage-free interpolation using monotone convex splines Poor numerical treatment leads to unstable or negative local variance estimates.

The Dupire equation is the adjoint of the Black-Scholes backward PDE. Mathematically, it is equivalent to the Fokker-Planck (forward Kolmogorov) equation for the transition probability density of the underlying asset. This duality means that calibrating local volatility is equivalent to finding the diffusion coefficient that makes the model's marginal distributions match those implied by market option prices at each maturity.

The direct output of the Dupire Equation. It is a deterministic function σ(S, t) of the underlying asset price and time, uniquely calibrated to exactly reproduce the market prices of all vanilla options. Unlike stochastic volatility models, local volatility assumes a single, instantaneous volatility for every possible future state, creating a complete and arbitrage-free model for pricing exotic derivatives.

The three-dimensional input required to solve the Dupire Equation. It represents implied volatility as a function of strike price and time to expiration. The Dupire formula uses a continuum of these prices to derive the unique local volatility surface. Key features include:

• Volatility Smile: Higher IV for OTM options
• Volatility Skew: Asymmetry in IV across strikes
• Term Structure: IV variation across expirations

Mathematical constraints that must be satisfied for the Dupire Equation to produce a valid, positive local volatility. These prevent static arbitrage in the input surface:

• Butterfly arbitrage: Ensures risk-neutral density is non-negative by requiring call prices to be convex in strike
• Calendar arbitrage: Ensures call prices are monotonically increasing in time to expiration A surface violating these conditions will yield imaginary local volatility values.

A fundamental distinction in quantitative finance. The Black-Scholes PDE is backward in time—you know the payoff at expiration and solve for the price today. The Dupire Equation is forward in time—you know today's option prices across all strikes and solve for the future evolution of local volatility. This forward nature makes it ideal for calibrating models to the entire market smile simultaneously.

An alternative to the local volatility framework. Models like Heston and SABR treat volatility as a separate random process with its own source of risk. While local volatility is a perfect fit to the static surface, stochastic models capture the dynamic evolution of the smile better. In practice, LVS (Local-Stochastic Volatility) hybrids combine both to achieve a perfect static fit and realistic forward dynamics.