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

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  • René Garcia
  • Éric 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 & Éric 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. Bruno R'emillard & Sylvain Rubenthaler, 2012. "Optimal hedging in discrete time," Papers 1211.5035, arXiv.org.
    3. Peter Christoffersen & Kris Jacobs, 2002. "Which Volatility Model for Option Valuation?," CIRANO Working Papers 2002s-33, CIRANO.
    4. Bauwens, Luc & Lubrano, Michel, 2002. "Bayesian option pricing using asymmetric GARCH models," Journal of Empirical Finance, Elsevier, vol. 9(3), pages 321-342, August.
    5. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    6. 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.
    7. Jin-Chuan Duan & Peter H. Ritchken & Zhiqiang Sun, 2006. "Jump starting GARCH: pricing and hedging options with jumps in returns and volatilities," Working Paper 0619, Federal Reserve Bank of Cleveland.
    8. Matthias R. Fengler & Helmut Herwartz & Christian Werner, 2012. "A Dynamic Copula Approach to Recovering the Index Implied Volatility Skew," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 10(3), pages 457-493, June.

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