  We consider the problem of valuing a European option written on an asset
whose dynamics are described by an exponential L\'evy-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 \citet*{fpss} to models of the
exponential L\'evy type. From a financial perspective, the inclusion of jumps
and stochastic volatility allow us to capture the term-structure of implied
volatility. To illustrate the flexibility of our modeling framework we extend
five exponential L\'evy processes to include stochastic volatility and
jump-intensity. For each of the extended models, using a single fast-varying
factor of volatility and jump-intensity, we perform a calibration to the S&P500
implied volatility surface. Our results show decisively that the extended
framework provides a significantly better fit to implied volatility than both
the traditional exponential L\'evy models and the fast mean-reverting
stochastic volatility models of \citet{fpss}.

Pricing of Securities;Computational Finance;General Finance