Local Volatility

Local volatility σ(S, t) is a single, deterministic function that maps the current underlying price S and time t to an instantaneous volatility value. Unlike stochastic volatility models, there is no separate random process driving volatility—the entire evolution of volatility is pre-determined by the asset's price path. This makes the model Markovian: the future depends only on the current state (S, t), not on historical volatility paths.

• σ(S, t) is fully specified once calibrated
• No additional source of randomness beyond the Brownian motion driving S
• Simplifies numerical methods like finite difference PDE solvers

The defining advantage of local volatility is its ability to perfectly reproduce the market prices of all liquid vanilla options across strikes and maturities. Given a complete implied volatility surface, the Dupire Equation uniquely determines the local volatility function that makes the model's theoretical prices match every observed market quote exactly.

• Eliminates calibration error for vanilla instruments
• Serves as the foundational pricing map for exotic derivatives
• Requires a continuum of option prices; interpolation is needed in practice

The Dupire Equation is a forward partial differential equation that inverts the Black-Scholes formula. It expresses local volatility in terms of observable market prices:

σ²(K, T) = (∂C/∂T) / (½ K² ∂²C/∂K²)

This formula reveals that local volatility at strike K and maturity T is proportional to the calendar spread (∂C/∂T) divided by the butterfly spread (∂²C/∂K²). The denominator represents the risk-neutral density, meaning local volatility is high where the market assigns significant probability mass.

Local volatility models inherently exhibit sticky strike dynamics: when the underlying price moves, the implied volatility for a fixed strike remains unchanged. This contrasts with sticky delta dynamics, where implied volatility follows moneyness.

• Sticky strike: σ_imp(K) stays constant as S moves; the smile shifts relative to the forward
• Sticky delta: σ_imp(Δ) stays constant; the smile moves with the spot
• Local volatility's sticky strike behavior often underestimates the movement of the smile, leading to underpricing of cliquet and forward-starting options

A well-known deficiency of local volatility is the flattening of the forward smile. While the model exactly calibrates to today's implied volatility surface, the future implied volatility surfaces it predicts are significantly flatter than those observed historically.

• Forward-starting options are systematically underpriced
• The model lacks a volatility of volatility parameter to control smile convexity through time
• This limitation motivated the development of stochastic volatility models like Heston and SABR, which preserve smile dynamics forward
• In practice, local volatility is often combined with stochastic volatility in LSV (Local Stochastic Volatility) hybrid models

Local volatility models are typically solved using finite difference methods on a grid of spot prices and time steps. The deterministic nature of σ(S, t) allows efficient backward induction for pricing path-dependent options.

• Crank-Nicolson or ADI (Alternating Direction Implicit) schemes for 2D PDEs
• Monte Carlo simulation with Euler-Maruyama discretization: S(t+Δt) = S(t) + rS(t)Δt + σ(S(t), t)S(t)√Δt Z
• Requires careful handling of the volatility interpolation to avoid arbitrage and ensure numerical stability
• Popular in production systems for barrier options, accumulators, and TARF structures