What causes the volatility smile in equity options?

The smile reflects four overlapping market features: (1) fat-tailed returns, actual returns exhibit kurtosis above what the normal distribution implies, so OTM strikes carry tail-risk premium; (2) leverage effect, equity volatility rises when prices fall, producing left-skewed implied distributions; (3) supply-demand imbalance, institutional hedgers buy OTM puts for portfolio insurance, bidding up downside premium; (4) jump risk around scheduled events. Models like SABR, Heston, Jump Diffusion, and Variance Gamma each capture different combinations of these features, which is why their smile shapes differ from each other.

How does model calibration to the listed surface work?

Calibration finds the model parameters that minimize the squared error between model-computed prices and listed market prices across a chosen strike-tenor grid. For Heston, the five parameters (κ, θ, ξ, ρ, v₀) are jointly fit; for SABR, the three parameters (α, ρ, ν, with β fixed) are fit per-expiration. The optimization is typically Levenberg-Marquardt on the log-IV space (more stable than fitting prices directly). The platform calibrates daily after market close and exposes the resulting parameter sets, the residual error per option, and the implied IV surface from the calibrated model; this lets users see where each model is fitting and where it's straining.

Which pricing model should I use for which situation?

Black-Scholes for fast pre-trade pricing, IV computation, and as the reference for divergence checks. Heston for surface-aware pricing where the smile and term structure matter jointly. SABR for fitting per-expiration smiles cleanly. Local Volatility when you need calibration-perfect repricing of the listed surface. Jump-diffusion (Merton, Kou, Bates) for options around earnings or known event jumps. Variance Gamma for fat-tailed Lévy returns. Binomial trees or PDE for American-exercise pricing. Monte Carlo for path-dependent exotics and risk-neutral probability distributions. The platform exposes all of these so you can compare.

Why do different options pricing models give different prices for the same option?

Each model makes different assumptions about how the underlying moves. Black-Scholes assumes constant volatility and continuous paths, producing a single fair value per option. Heston adds stochastic volatility, capturing the smile that Black-Scholes can't reproduce. Jump-diffusion models (Merton, Kou, Bates) add discontinuous price jumps for event risk. Variance Gamma uses pure-jump Lévy dynamics with explicit fat tails. Local Volatility fits the listed surface exactly by construction. The 'correct' price depends on which assumptions best match the underlying's actual dynamics, and the gap between two models is itself priced information about which assumptions the market is paying for.

Local Volatility (Dupire) - Implied Vol Surface

Last reviewed: May 17, 2026 by Options Analysis Suite Research.

What Is Local Volatility?

Local volatility (LV) is a model where the instantaneous volatility is a deterministic function of spot and time: σ_local = σ(S, t). Introduced by Bruno Dupire in 1994, the local-volatility model is the unique extension of Black-Scholes that, by construction, fits the entire observed volatility surface exactly, given a continuous surface of European call prices, Dupire's formula recovers the local volatility function that produces those prices.

This makes LV the canonical "calibration-perfect" reference: by construction, every listed European option is repriced at its market quote. The cost is that LV is a deterministic function, with no stochastic component to the volatility itself. As a result, LV produces prices that are correct for vanilla expiry payoffs but distort the path-dependent dynamics that exotic options care about.

Dupire's Formula

σ²_local(K, T) = (∂C/∂T + (r − q)·K·∂C/∂K + q·C) / (½·K²·∂²C/∂K²). Each derivative is taken on the surface of European call prices C as a function of strike K and expiration T. In practice the derivatives are estimated from a smoothed implied-volatility surface; the quality of the smoothing determines the quality of the recovered LV.

What Does Local Volatility Capture?

Every quoted vanilla price exactly: by construction the calibration error is zero.
The state-dependent nature of equity volatility: vol rises when spot falls, captured by σ(S, t).
Term-structure of volatility: σ(S, t) varies with t to match the term structure.

What Doesn't Local Volatility Capture?

Stochastic volatility dynamics: LV is deterministic; vol-of-vol is zero. For exotics that depend on the variance process (cliquets, forward-start, vol swaps), LV materially under-prices the embedded vol-of-vol risk.
The forward-skew dynamics: LV mis-prices barrier options because the forward-skew implied by LV flattens too quickly.
Jumps: like Black-Scholes and Heston, LV has continuous paths.

When Should You Use Local Volatility?

As the calibration baseline that gets the listed surface right by construction: the reference for cross-model divergence checks where LV provides the fixed-point baseline.
For pricing path-dependent products where the calibration-perfect spot-strike relationship is more important than the dynamics (some American puts, certain barrier products with European-style settlement of the barrier event).
In hybrid models (Local Stochastic Vol, or LSV) where LV provides the surface-matching backbone and a stochastic-vol perturbation adds dynamics. LSV combines the strengths of both: surface accuracy from LV and forward-dynamics realism from stochastic vol.
For exotic options whose payoffs depend most on the terminal distribution rather than the path through that distribution. LV's terminal distribution matches the listed market exactly.
As a sanity check on whether a model alternative (Heston, SABR) is sacrificing too much surface accuracy for its dynamics; comparing the alternative's prices to LV reveals the size of the tradeoff being made.

When Should You Not Use Local Volatility?

Pricing barriers, lookbacks, and other path-dependent exotics: LV under-prices the forward-skew dependence these products carry because the deterministic vol function flattens forward-vol dynamics that the products are sensitive to.
Pricing forward-start or cliquet-style options whose value depends on future smile shape: LV's deterministic vol function produces an unrealistic forward-smile that flattens too quickly compared to what stochastic vol predicts.
Anywhere stochastic-vol dynamics matter: Heston, Bates, or LSV hybrids are better tools when the volatility process needs its own randomness rather than being a deterministic function of spot and time.
Real-time intraday repricing where the calibration overhead of fitting a smooth IV surface is too expensive; Black-Scholes or Heston with cached parameters are cheaper.

Numerical Implementation of Local Volatility

Implementing LV in production is a numerical-analysis exercise. The Dupire formula's derivatives are estimated from a smoothed implied-volatility surface fit to the listed option prices. The smoothing scheme (cubic splines, SVI parameterization, eSSVI, or arbitrage-free interpolation) determines the quality of the recovered LV. Naive interpolation produces noisy LV estimates, especially in low-data regions of the surface. Production implementations use arbitrage-free fitting (where butterfly-spread no-arbitrage conditions are enforced as constraints) followed by analytical differentiation of the fitted surface. The platform's LV implementation uses an arbitrage-free SVI-style fit for stability across strikes and tenors.

The Local-Stochastic-Volatility Hybrid

Local volatility's calibration-perfect property combined with stochastic volatility's forward-skew-correct dynamics gives rise to LSV (Local-Stochastic-Volatility) hybrids. LSV models multiply the LV instantaneous volatility by a stochastic factor: the LV surface provides the surface-matching backbone, while the stochastic factor adds vol-of-vol dynamics. LSV is the production-grade choice for path-dependent exotic pricing where both surface accuracy and forward dynamics matter, a class of products LV alone or Heston alone don't handle satisfactorily. The platform doesn't expose LSV as a standalone model surface, but the LV and Heston calibrations together provide the building blocks for users who want to compose their own LSV pricing externally.

How OAS Uses Local Volatility

The platform offers Local Volatility as a calibration-perfect reference surface: prices from LV match the listed surface exactly, providing the baseline that other models (Black-Scholes, Heston, SABR) are compared against. The model divergence view uses LV as the fixed-point reference; deviations from LV are the model-implied tradeoffs each alternative model is making: Heston giving up some surface accuracy for cleaner forward dynamics, SABR fitting per-expiration smiles cleanly but losing term-structure consistency, Black-Scholes flattening the surface entirely. LV anchors the model-comparison framework because it's the only model that, by construction, prices every listed vanilla correctly.

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Related Concepts

Black-Scholes (vs) · Heston (vs) · SABR · Jump Diffusion · Variance Gamma · Stochastic Volatility · Volatility Skew · Volatility Smile · Leverage Effect · Calibration · Model Divergence · Implied Volatility · IV Crush · Dealer Gamma · Tail Risk · Vanna / Charm / Vomma Exposure · eSSVI Parameterization · Butterfly Arbitrage · PDE Methods · Local Volatility vs Stochastic Volatility · Black-Scholes vs Local Volatility · Model Landscape

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This section is part of the Options Analysis Suite Documentation. Browse the full model index or compare alternatives in the pricing calculator.

As of the latest snapshot, AAPL has an ATM implied volatility of 21.6%, IV rank 28% (percentile 11%); 20-day realized vol 16.3%. 25-delta skew is +1.1%, meaning OTM puts trade richer than OTM calls. The IV here is the input that pricing-model walkthroughs (Black-Scholes, Heston, SABR, local-vol) take as their starting point: each model decomposes the same observed quote into different latent dynamics (constant vol, stochastic vol, surface-fitted vol, etc.) which is why two models can agree on price but disagree on Greeks and on how vol will evolve.