Closed-Form Risk-Minimizing Hedge Ratios for Affine GARCH Models

Abstract

This paper presents closed-form expressions for risk-minimizing hedging strategies under affine GARCH models driven by Gaussian innovations. Our expressions are applicable to European derivatives with payoff functions that admit an inverse Laplace transform representation, such as calls and puts. The solutions are derived under risk-neutral dynamics based on a variance-dependent pricing kernel that incorporates market prices of both equity and variance risks. The hedging strategy that is obtained is locally risk-minimizing (LRM) under the risk-neutral measure and also variance-optimal in the sense that it minimizes the expected value of the squared terminal hedging error. In addition, a first-order approximation of the hedge ratio is provided, and the continuous-time LRM strategy based on the weak limit of the underlying GARCH model is derived. Several numerical experiments are conducted to compare our analytic formulas to Monte Carlo estimators, to investigate the accuracy of the proposed LRM approximations, and to test the numerical convergence of the GARCH-based LRM hedge ratio to its continuous-time counterpart.

Keywords: affine GARCH models, mean-variance hedging, local risk-minimization, weak convergence

JEL Classification: C58, G12, G13