Probabilistic Interpretation of an Implied Volatility Smile

When the variance rates implied from option prices differ across strike prices, at most one of them can be interpreted as the variance rate of the underlying security price. We develop an arbitrage-free option pricing model... [ view full abstract ]

When the variance rates implied from option prices differ across strike prices, at most one of them can be interpreted as the variance rate of the underlying security price. We develop an arbitrage-free option pricing model with four stochastic state variables, one of which is the underlying security price. We show how to successively explicitly determine the other three state variables from three given co-terminal arbitrage-free implied variance rates. The resulting calibrated implied variance rate smile is given a simple probabilistic representation. To our knowledge, this is the first non-flat implied variance rate smile enjoying any probabilistic interpretation.