Option Prices in a Model with Stochastic Disaster Risk

Contrary to the Black-Scholes model, volatilities implied by index option prices depend on the exercise price of the option and are often higher than realized volatilities. We explain both facts in the context of a model that can also explain the mean and volatility of equity returns. Our model assumes a small risk of a rare disaster that is calibrated based on the international data on large consumption declines. We allow the risk of this rare disaster to be stochastic, which turns out to be crucial to the model's ability to explain both equity volatility and option prices. We explore different specifications for the stochastic rare disaster probability and show that the data favor a multifrequency process. Finally, we show that the model can simultaneously fit the time series of option prices and equities.