Local Volatility Model: From Option Surface to State-Dependent Volatility

For educational purposes only; not investment advice.

Direct answer

A local volatility model makes instantaneous volatility a deterministic function of the current underlying level and time: dS_t=(r-q)S_tdt+σ_loc(S_t,t)S_tdW_t under a simple risk-neutral specification. Unlike Black-Scholes with one constant volatility, it can reproduce an entire cross-section of European option prices across strikes and expirations when calibrated to a sufficiently smooth, arbitrage-consistent surface.

The model is complete in its idealized diffusion setting and is useful for pricing path-dependent claims and computing hedges consistent with today’s vanilla surface. Exact fit today does not mean that it predicts tomorrow’s implied-volatility surface. Its future smile dynamics are imposed by the deterministic function and can differ materially from observed markets.

Dupire calibration mechanics

Let C(K,T) be a European call price as a function of strike K and expiration T, with constant rate r and dividend yield q for illustration. Dupire’s forward equation gives:

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

The strike curvature ∂²_KK C is connected to the risk-neutral terminal density. Therefore call prices must decrease and remain convex in strike, and calendar relationships must be consistent, before the formula is meaningful. In practice, desks first clean quotes, convert rates and dividends to forwards and discount factors, fit a smooth arbitrage-controlled surface, differentiate it, and then interpolate or extrapolate the local-volatility grid.

A calibrated σ_loc(K,T) is not simply the quoted implied volatility at that strike and maturity. Implied volatility is one constant Black-Scholes parameter that reproduces a particular option price; local volatility is the instantaneous state-dependent diffusion coefficient whose distribution reproduces the broader surface.

Illustrative Dupire calculation

At K=$100 and T=1 year, suppose a smoothed call surface gives C=$8, ∂_T C=10 dollars per year, ∂_K C=-0.45, and ∂²_KK C=0.018 per dollar. Let r=5% and q=2%. The numerator is:

10+(0.05-0.02)×100×(-0.45)+0.02×8=8.81.

The denominator is 0.5×100²×0.018=90. Local variance is therefore 8.81/90=0.0979 and local volatility is √0.0979=31.3%.

This is a numerical illustration, not a market estimate. If noisy fitting changed curvature from 0.018 to 0.012 while other inputs stayed fixed, the inferred volatility would rise to √(8.81/60)=38.3%. The large change demonstrates why quote filtering, no-arbitrage smoothing, grid stability, and out-of-sample checks matter.

Calibration and model-risk checklist

Use synchronized executable option quotes, forward prices, discount factors, dividends, contract terms, and settlement conventions.
Remove crossed, stale, zero-bid, and economically inconsistent observations under documented rules.
Enforce strike convexity and appropriate calendar monotonicity before taking numerical derivatives.
Test sensitivity to interpolation, smoothing penalties, quote weights, grid spacing, boundary conditions, and extrapolation.
Reprice all calibration vanillas and report errors in executable price units, not only implied-volatility points.
Compare local-volatility hedge behavior with Black-Scholes and stochastic-volatility alternatives under spot and smile shocks.
Stress gaps and jumps. A continuous diffusion cannot hedge an instantaneous discontinuity.
Treat the future surface dynamics as a model assumption; local volatility often produces smile movement different from the market.
Be cautious with barriers and other path-dependent claims whose value depends on the modeled path distribution, not only terminal vanilla prices.
Recalibrate when the surface changes and distinguish market P&L, carry, hedge error, and model recalibration effects.
Avoid extrapolating sparse wings or long maturities without explicit limits; derivatives can magnify small input errors.
Include liquidity, transaction costs, discrete hedging, early exercise, and counterparty or settlement terms outside the idealized equation.

Common misconceptions

“Local volatility equals implied volatility at each point.” The two objects have different definitions and uses.
“Exact calibration proves the model is true.” Many models can fit today’s vanilla prices while implying different paths and hedges.
“The Dupire formula can be applied directly to raw quotes.” It requires smooth derivatives and arbitrage-consistent inputs.
“A complete diffusion removes all risk.” Gaps, discrete hedging, liquidity, parameter changes, and model error remain.
“More surface detail always improves accuracy.” Differentiation can amplify noisy and illiquid quotes.
“The model forecasts volatility.” Calibration extracts a risk-neutral pricing function; it is not a physical forecast.
“Two models matching all vanillas must price every exotic equally.” Path distributions and future smile dynamics can differ.
“Negative local variance is a valid signal.” It usually indicates inconsistent data, smoothing, derivatives, or assumptions that require investigation.