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A Note on Hedging in ARCH and Stochastic Volatility Option Pricing Models

Author

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  • René Garcia
  • Eric Renault
Abstract
Recently, Duan (1995) proposed a GARCH option pricing formula and a corresponding hedging formula. In a similar ARCH-type model for the underlying asset, Kallsen and Taqqu (1994) arrive at a hedging formula different from Duan's , although they concur on the pricing formula. In this note, we explain the difference by pointing out that the formula developped by Kallsen and Taqqu corresponds to the usual concept of hedging in the context of ARCH-type models. We argue however that Duan's formula has some appeal and propose a stochastic volatility model which ensures its validity. We conclude by a comparison of ARCH-type and stochastic volatility option pricing models. Duan (1995) a proposé récemment une formule de valorisation d'option fondée sur un modèle GARCH ainsi que la formule de couverture correspondante. Dans un modèle similaire de type ARCH pour l'actif sous-jacent conduisant à la même formule de valorisation, Kallsen et Taqqu (1994) arrivent à une formule de couverture différente. Dans cette note, nous expliquons cette différence en soulignant que la formule de Kallsen et Taqqu correspond au concept usuel de couverture dans le cadre des modèles de type ARCH. Nous trouvons toutefois que la formule de couverture de Duan a un certain attrait et proposons un modèle de volatilité stochastique qui en assure la validité. Nous concluons par une comparaison des modèles ARCH et de volatilité stochastique pour la valorisation d'options.

Suggested Citation

  • René Garcia & Eric Renault, 1997. "A Note on Hedging in ARCH and Stochastic Volatility Option Pricing Models," CIRANO Working Papers 97s-13, CIRANO.
    • Handle: RePEc:cir:cirwor:97s-13
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    References listed on IDEAS

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    1. Marcel Boyer, 1997. "Competition and Access in Telecoms: ECPR, Global Price Cap, and Auctions," CIRANO Working Papers 97s-03, CIRANO.
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    Cited by:

    1. GARCIA, René & RENAULT, Éric, 1998. "Risk Aversion, Intertemporal Substitution, and Option Pricing," Cahiers de recherche 9801, Universite de Montreal, Departement de sciences economiques.
    2. Peter Christoffersen & Kris Jacobs, 2002. "Which Volatility Model for Option Valuation?," CIRANO Working Papers 2002s-33, CIRANO.
    3. Jean Pierre Fernández Prada Saucedo & Gabriel Rodríguez, 2020. "Modeling the Volatility of Returns on Commodities: An Application and Empirical Comparison of GARCH and SV Models," Documentos de Trabajo / Working Papers 2020-484, Departamento de Economía - Pontificia Universidad Católica del Perú.
    4. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    5. Matthias R. Fengler & Helmut Herwartz & Christian Werner, 2012. "A Dynamic Copula Approach to Recovering the Index Implied Volatility Skew," Journal of Financial Econometrics, Oxford University Press, vol. 10(3), pages 457-493, June.
    6. Ryszard Kokoszczyński & Paweł Sakowski & Robert Ślepaczuk, 2017. "Which Option Pricing Model Is the Best? HF Data for Nikkei 225 Index Options," Central European Economic Journal, Sciendo, vol. 4(51), pages 18-39, December.
    7. Bauwens, Luc & Lubrano, Michel, 2002. "Bayesian option pricing using asymmetric GARCH models," Journal of Empirical Finance, Elsevier, vol. 9(3), pages 321-342, August.
    8. Maciej Augustyniak & Frédéric Godin & Clarence Simard, 2017. "Assessing the effectiveness of local and global quadratic hedging under GARCH models," Quantitative Finance, Taylor & Francis Journals, vol. 17(9), pages 1305-1318, September.
    9. Bruno R'emillard & Sylvain Rubenthaler, 2012. "Optimal hedging in discrete time," Papers 1211.5035, arXiv.org.
    10. Xu Cheng & Eric Renault & Paul Sangrey, 2024. "Identifying the Volatility Risk Price Through the Leverage Effect," PIER Working Paper Archive 24-013, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    11. Fengler, Matthias R. & Hin, Lin-Yee, 2015. "Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints," Journal of Econometrics, Elsevier, vol. 184(2), pages 242-261.
    12. Jin-Chuan Duan & Peter H. Ritchken & Zhiqiang Sun, 2006. "Jump starting GARCH: pricing and hedging options with jumps in returns and volatilities," Working Papers (Old Series) 0619, Federal Reserve Bank of Cleveland.
    13. Maciej Augustyniak & Alexandru Badescu, 2021. "On the computation of hedging strategies in affine GARCH models," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(5), pages 710-735, May.

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