A First Option Calibration of the GARCH Diffusion Model by a PDE Method

TL;DR

This paper introduces a PDE-based numerical method for fast and accurate option calibration of the GARCH diffusion model, overcoming the lack of semi-analytic solutions and enabling practical use in financial modeling.

Contribution

It presents the first PDE calibration approach for the GARCH diffusion model, providing an efficient finite difference solver that replaces analytical solutions.

Findings

Calibration can be done in less than a minute.

The PDE solver is robust across various parameters.

The method produces benchmark results for future reference.

Abstract

Time-series calibrations often suggest that the GARCH diffusion model could also be a suitable candidate for option (risk-neutral) calibration. But unlike the popular Heston model, it lacks a fast, semi-analytic solution for the pricing of vanilla options, perhaps the main reason why it is not used in this way. In this paper we show how an efficient finite difference-based PDE solver can effectively replace analytical solutions, enabling accurate option calibrations in less than a minute. The proposed pricing engine is shown to be robust under a wide range of model parameters and combines smoothly with black-box optimizers. We use this approach to produce a first PDE calibration of the GARCH diffusion model to SPX options and present some benchmark results for future reference.