GARCH options via local risk minimization

TL;DR

This paper develops a quadratic hedging approach for GARCH-based options, introducing Girsanov-like measures to simplify strategies and recover classical pricing formulas with improved hedging performance.

Contribution

It presents a novel application of local risk minimization to GARCH models, introducing tractable risk-neutral measures and extending classical option pricing and hedging methods.

Findings

Existence of local risk-minimizing strategies under GARCH models.

Introduction of Girsanov-like measures for tractable hedging.

Recovery of classical pricing formulas with improved hedging schemes.

Abstract

We apply a quadratic hedging scheme developed by Foellmer, Schweizer, and Sondermann to European contingent products whose underlying asset is modeled using a GARCH process and show that local risk-minimizing strategies with respect to the physical measure do exist, even though an associated minimal martingale measure is only available in the presence of bounded innovations. More importantly, since those local risk-minimizing strategies are in general convoluted and difficult to evaluate, we introduce Girsanov-like risk-neutral measures for the log-prices that yield more tractable and useful results. Regarding this subject, we focus on GARCH time series models with Gaussian innovations and we provide specific sufficient conditions that have to do with the finiteness of the kurtosis, under which those martingale measures are appropriate in the context of quadratic hedging. When this equivalent martingale measure is adapted to the price representation we are able to recover out of it the classical pricing formulas of Duan and Heston-Nandi, as well as hedging schemes that improve the performance of those proposed in the literature.