Quadratic hedging schemes for non-Gaussian GARCH models |
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| Institution: | 1. Department of Mathematics and Statistics, University of Calgary, Calgary, Canada;2. School of Mathematical Sciences, University of Adelaide, SA 5005, Australia;3. Centre National de la Recherche Scientifique, Département de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16, route de Gray, F-25030 Besançon cedex, France;1. Centro de Investigación en Matemáticas (CIMAT), Apdo. Postal 402, 36000 Guanajuato, Gto., Mexico;2. Departamento de Matemáticas, Universidad de Guanajuato, CP 36240, Guanajuato, Gto., Mexico;1. Department of Meteorology, University of Reading, Reading RG6 6BB, UK;2. Energy and Resources Group, University of California, Berkeley, Berkeley, CA 94720, USA;3. Skeptical Science, Brisbane, QLD 4072, Australia;4. Institute for Astronomy, Royal Observatory, University of Edinburgh, Edinburgh EH93HJ, UK;5. School of Engineering, University of St. Thomas, St. Paul, MN 55105-1079, USA;1. The Oxford-Man Institute, University of Oxford, Eagle House, Walton Well Road, Oxford OX2 6ED, United Kingdom;2. Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom;1. Department of Mathematics, Kunsan National University, Kunsan 54150, Republic of Korea;2. Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea;1. Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain;2. Departamento de Análisis Matemático, Universidad de la Laguna, 38271, La Laguna (Tenerife), Spain |
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| Abstract: | We propose different schemes for option hedging when asset returns are modeled using a general class of GARCH models. More specifically, we implement local risk minimization and a minimum variance hedge approximation based on an extended Girsanov principle that generalizes Duan׳s (1995) delta hedge. Since the minimal martingale measure fails to produce a probability measure in this setting, we construct local risk minimization hedging strategies with respect to a pricing kernel. These approaches are investigated in the context of non-Gaussian driven models. Furthermore, we analyze these methods for non-Gaussian GARCH diffusion limit processes and link them to the corresponding discrete time counterparts. A detailed numerical analysis based on S&P 500 European call options is provided to assess the empirical performance of the proposed schemes. We also test the sensitivity of the hedging strategies with respect to the risk neutral measure used by recomputing some of our results with an exponential affine pricing kernel. |
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| Keywords: | GARCH models Local risk minimization Martingale measure Bivariate diffusion limit Minimum variance hedge |
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