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

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  • René Garcia
  • Èric 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) arrived at a hedging formula different from Duan's although they concur on the pricing formula. In this note, we explain this difference by pointing out that the formula developed 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 we propose a stochastic volatility model that ensures its validity. We conclude by a comparison of ARCH‐type and stochastic volatility option pricing models.

Suggested Citation

  • René Garcia & Èric Renault, 1998. "A Note on Hedging in ARCH and Stochastic Volatility Option Pricing Models," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 153-161, April.
    • Handle: RePEc:bla:mathfi:v:8:y:1998:i:2:p:153-161
    DOI: 10.1111/1467-9965.00049
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    Cited by:

    1. Ma, Junmei & Wang, Chen & Xu, Wei, 2025. "A new lattice approach for risk-minimization hedging under generalized autoregressive conditional heteroskedasticity models," European Journal of Operational Research, Elsevier, vol. 321(3), pages 1021-1035.
    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. GARCIA, René & RENAULT, Éric, 1998. "Risk Aversion, Intertemporal Substitution, and Option Pricing," Cahiers de recherche 9801, Universite de Montreal, Departement de sciences economiques.
    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. Cheng, Xu & Renault, Eric & Sangrey, Paul, 2025. "Identifying the volatility risk price through the leverage effect," Journal of Econometrics, Elsevier, vol. 248(C).
    12. 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.
    13. 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.
    14. 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.
    15. 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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    JEL classification:

    • G1 - Financial Economics - - General Financial Markets

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