A local volatility model and a stochastic volatility model are both calibrated to fit every listed European price on a name exactly. Name a product they price differently and explain which one is closer to right.

A local volatility model and a stochastic volatility model are both calibrated to fit every listed European price on a name exactly. Name a product they price differently and explain which one is closer to right.

Approach: Fitting today's vanillas pins the marginal distribution at each expiry. Ask what a product needs beyond those marginals, and look at what each model implies about the smile seen from a future date.

They agree on every European payoff and disagree on anything depending on the forward skew, with a forward-starting cliquet the clearest case, and the stochastic volatility answer is usually closer to right. Fitting the listed surface pins only the marginal distributions at each expiry, and a cliquet or a barrier depends on the joint behaviour across dates, which the marginals leave free. Local volatility makes volatility a deterministic function of spot and time, so as time passes the smile flattens dramatically and the model prices a one-year forward-starting option off almost no skew. Stochastic volatility keeps a random variance with its own correlation to spot, so the forward skew persists at roughly today's level, which is what the market has historically shown. A desk that hedges a cliquet book on local volatility is short forward skew at a price nobody would quote, and the loss appears when the structure resets rather than when it is written.

Follow-up: Where does a mixed local stochastic volatility model put the leverage function, and what does its calibration cost you in stability?

Key concepts: local volatility, stochastic volatility, forward skew, cliquet, marginal distributions.