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A closed-form GARCH option pricing model

Author

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  • Steven L. Heston
  • Saikat Nandi

Abstract

This paper develops a closed-form option pricing formula for a spot asset whose variance follows a GARCH process. The model allows for correlation between returns of the spot asset and variance and also admits multiple lags in the dynamics of the GARCH process. The single-factor (one-lag) version of this model contains Heston's (1993) stochastic volatility model as a diffusion limit and therefore unifies the discrete-time GARCH and continuous-time stochastic volatility literature of option pricing. The new model provides the first readily computed option formula for a random volatility model in which current volatility is easily estimated from historical asset prices observed at discrete intervals. Empirical analysis on S&P 500 index options shows the single-factor version of the GARCH model to be a substantial improvement over the Black-Scholes (1973) model. The GARCH model continues to substantially outperform the Black-Scholes model even when the Black-Scholes model is updated every period and uses implied volatilities from option prices, while the parameters of the GARCH model are held constant and volatility is filtered from the history of asset prices. The improvement is due largely to the ability of the GARCH model to describe the correlation of volatility with spot returns. This allows the GARCH model to capture strike-price biases in the Black-Scholes model that give rise to the skew in implied volatilities in the index options market.

Suggested Citation

  • Steven L. Heston & Saikat Nandi, 1997. "A closed-form GARCH option pricing model," FRB Atlanta Working Paper 97-9, Federal Reserve Bank of Atlanta.
  • Handle: RePEc:fip:fedawp:97-9
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    References listed on IDEAS

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    Cited by:

    1. Gondzio, Jacek & Kouwenberg, Roy & Vorst, Ton, 2003. "Hedging options under transaction costs and stochastic volatility," Journal of Economic Dynamics and Control, Elsevier, vol. 27(6), pages 1045-1068, April.
    2. Duan, Jin-Chuan & Yu, Min-Teh, 1999. "Capital standard, forbearance and deposit insurance pricing under GARCH," Journal of Banking & Finance, Elsevier, vol. 23(11), pages 1691-1706, November.
    3. Duan, Jin-Chuan & Simonato, Jean-Guy, 2001. "American option pricing under GARCH by a Markov chain approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 25(11), pages 1689-1718, November.
    4. S. K. Bhaumik & D. Coondoo, 2003. "Econometrics of yield spreads in the money market: a note," Applied Financial Economics, Taylor & Francis Journals, vol. 13(9), pages 645-653.
    5. Capelle-Blancard, Gunther & Jurczenko, Emmanuel & Maillet, Bertrand, 2001. "The approximate option pricing model: performances and dynamic properties," Journal of Multinational Financial Management, Elsevier, vol. 11(4-5), pages 427-443, December.
    6. Yalincak, Orhun Hakan, 2005. "Criticism of the Black-Scholes Model: But Why Is It Still Used? (The Answer Is Simpler than the Formula)," MPRA Paper 63208, University Library of Munich, Germany.
    7. Vázquez, Miguel & Sánchez-Úbeda, Eugenio F. & Berzosa, Ana & Barquín, Julián, 2008. "Short-term evolution of forward curves and volatility in illiquid power market," MPRA Paper 8932, University Library of Munich, Germany, revised May 2008.

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