Gauge Transformations in the Dual Space, and Pricing and Estimation in the Long Run in Affine Jump-Diffusion Models

Abstract

We suggest a simple reduction of pricing European options in affine jump-diffusion models to pricing options with modified payoffs in diffusion models. The procedure is based on conjugation of infinitesimal generator of model with an operator of form $e^{i\Phi(-\sqrt{-1}\dd_x)}$ (gauge transformation in dual space). A general procedure for calculation of function $\Phi$ is given, with examples. As applications, we consider pricing in jump-diffusion models and their subordinated versions using eigenfunction expansion technique, and estimation of extremely rare jumps component. The beliefs of market about yet unobserved extreme jumps and pricing kernel can be recovered: market prices allow one to see the shape of things to come.