Local volatility
Quantitative Finance glossary · Derivatives Pricing & Volatility Surfaces
Definition
Model in which instantaneous volatility is a deterministic function σ_loc(S,t) of spot and time, calibrated to perfectly reproduce the observed implied-vol surface via Dupire's formula: σ_loc^2(K,T) = dfracpartial C/partial T + r K partial C/partial K1/2 K^2 partial^2 C/partial K^2. The unique 1-factor Markovian completion of an arbitrage-free option surface. Mis-prices forward-skew and barrier options because the model's future skew flattens (counterfactually). Stochastic-local-volatility hybrids (SLV) are the dealer-grade fix.
Local volatility in plain English
"Volatility" means how jumpy a stock's price is, and option prices depend heavily on it. The catch: the market quotes different volatilities for options at different strike prices (the famous "smile" or "skew"), which a single fixed number can't explain. Local volatility solves this by making volatility a known function of just two things — the current price and the date — tuned so the model reproduces every option price on the board exactly (that's what Dupire's formula does). Think of it as a custom weather map where the expected turbulence depends on where the stock is and when. The honest catch professionals know: it nails today's prices but predicts that the smile will unrealistically flatten out in the future, so it mis-prices some path-sensitive options — which is why banks bolt on a randomness upgrade (stochastic-local vol).
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