Multiscale exponential Levy-type models

We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential Levy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors-one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of Fouque et al. [Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, 2011] to models of the exponential Levy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility, as demonstrated in a calibration to S&P500 options data.