Dupire's formula
Quantitative Finance glossary · Quantitative Finance
Definition
Closed-form inversion of the local-volatility function from observed call prices: σ_loc^2(K, T) = dfracpartial C/partial T + r K,partial C/partial K1/2 K^2,partial^2 C/partial K^2. Equivalently restated in implied-vol space (Gatheral). The 1-factor Markovian completion of an arbitrage-free option surface, exact by construction but a poor forward-vol predictor — local-vol future skew flattens, opposite to empirical behaviour.
Dupire's formula in plain English
Traders can SEE the market prices of options at many strike prices and expiry dates, but they'd love to know the hidden 'local volatility' — how jumpy the stock is expected to be at each specific price level and future time. Dupire's formula is a clever piece of algebra that flips the problem around: feed in all those observed option prices and it hands you back that hidden volatility map, exactly, no guessing. It's like reverse-engineering a recipe's oven temperature at every moment just from tasting how the cake turned out at different baking times. The catch: it perfectly matches today's prices but is a lousy fortune-teller — it predicts the market's future 'smile' (skew) will flatten out, when in reality it tends to stay bumpy.
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