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

23 Pages Posted: 13 Jan 2020 Last revised: 26 Mar 2021

See all articles by Svetlana Boyarchenko

## Svetlana Boyarchenko

University of Texas at Austin - Department of Economics

## Sergei Levendorskii

Calico Science Consulting

Date Written: December 14, 2019

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

Keywords: affine jump-diffusions, eigenfunction expansion, long run, estimation, Ornstein-Uhlenbeck model, Vasicek model, square root model, CIR model

JEL Classification: C58, C63, C65, G12

Suggested Citation

Boyarchenko, Svetlana I. and Levendorskii, Sergei Z., Gauge Transformations in the Dual Space, and Pricing and Estimation in the Long Run in Affine Jump-Diffusion Models (December 14, 2019). Available at SSRN: https://ssrn.com/abstract=3504029 or http://dx.doi.org/10.2139/ssrn.3504029