In equity and foreign exchange markets the risk-neutral dynamics of the
underlying asset are commonly represented by a stochastic volatility model with
jumps. In this paper we consider a dense subclass of such models and develop
analytically tractable formulae for the prices of a range of first-generation
exotic derivatives. We provide closed form formulae for the Fourier transforms
of vanilla and forward starting options as well as the formula for the slope of
the implied volatility smile for large strikes. A simple explicit approximation
formula for the variance swap price is given. The prices of the volatility
swaps and other volatility derivatives are given as a one-dimensional integral
of an explicit function. Analytically tractable formulae for the Laplace
transform (in maturity) of the double-no-touch options and the Fourier-Laplace
transform (in strike and maturity) of the double knock-out call and put options
are obtained. The proof of the latter results is based on a novel complex
matrix Wiener-Hopf factorization.