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

34 Pages Posted: 5 Nov 2019

See all articles by Maciej Augustyniak

Maciej Augustyniak

University of Montreal - Department of Mathematics and Statistics

Alex Badescu

University of Calgary

Date Written: October 24, 2019

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

Suggested Citation

Augustyniak, Maciej and Badescu, Alex, Closed-Form Risk-Minimizing Hedge Ratios for Affine GARCH Models (October 24, 2019). Available at SSRN: https://ssrn.com/abstract=3475245 or http://dx.doi.org/10.2139/ssrn.3475245

Maciej Augustyniak (Contact Author)

University of Montreal - Department of Mathematics and Statistics ( email )

C.P. 6128, succursale Centre-ville
Montreal, Quebec H3C 3J7
Canada

HOME PAGE: http://dms.umontreal.ca/

Alex Badescu

University of Calgary ( email )

University of Calgary
Calgary, Alberta
Canada

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