Gauge transformations in the dual space, and pricing and estimation in the long run in affine jump-diffusion models

TL;DR

This paper introduces a gauge transformation approach to simplify option pricing in affine jump-diffusion models, enabling easier computation and estimation of rare jumps and market expectations about future extreme events.

Contribution

It proposes a novel gauge transformation method to reduce complex jump-diffusion pricing to diffusion models with modified payoffs, facilitating analysis and estimation.

Findings

Reduction of jump-diffusion pricing to diffusion models

Application of eigenfunction expansion for pricing and estimation

Ability to infer market expectations about rare jumps

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 $e^{i\Phi(-i\dd_x)}$ (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".