We analyze the specifications of option pricing models based on time-changed Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process...

We develop a simple robust method to distinguish the presence of continuous and discontinuous components in the price of an asset underlying options. Our method examines the prices of at-the-money and out-of-the-money options as the option's time-to-maturity approaches zero. We show that these...

We document a surprising pattern in S&P 500 option prices. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the...