In a tractable stochastic volatility model, we identify the price of the smile as the price of the unspanned risks traded in SPX option markets. The price of the smile reflects two persistent volatility and skewness risks, which imply a downward sloping term structure of low-frequency variance...

We provide a theoretical framework to uncover in a model-free way the relationship between international stochastic discount factors (SDFs), stochastic wedges, and financial market structures. Exchange rates are in general different from the ratio of international SDFs in incomplete markets, as...

We develop a novel class of time-changed Lévy models which are tractable and readily applicable, capture the leverage effect, and exhibit pure jump processes with finite or infinite activity. Our models feature four nested processes reflecting market, volatility and jump risks, and observation...