Local Volatility

Local volatility, introduced by Dupire, lets instantaneous volatility be a deterministic function of the underlying price and time. Its centerpiece is a formula that recovers that function from quoted call prices — equivalently from the implied volatility smile — so the model reproduces the market’s vanilla prices exactly. That property made local volatility the industry workhorse for pricing path-dependent exotics consistently with the smile. Its weakness is dynamics: the calibrated model predicts future smiles that flatten unrealistically and produces hedge ratios that disagree with how real markets move. Stochastic volatility models, the next node, were developed largely to repair exactly that failure.