In the setting of ‘‘affine’ ’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks ’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

I especially appreciate the extensive and helpful comments of an anonymous referee. I also appreciate the comments and suggestions of Bernard Dumas (the editor), Qiang Dai, and my dissertation committee members at Yale University, Don Andrews, George Hall, and especially Chris Sims. David Backus has generously provided part of the data used in this study. All of the