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Option pricing and ARCH processes

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  • Zumbach, Gilles

Abstract

Recent progresses in option pricing using ARCH processes for the underlying are summarized. The stylized facts are multiscale heteroscedasticity, fat-tailed distributions, time reversal asymmetry, and leverage. The process equations are based on a finite time increment, relative returns, fat-tailed innovations, and multiscale ARCH volatility. The European option price is the expected payoff in the physical measure P weighted by the change of measure dQ/dP, and an expansion in the process increment δt allows for numerical evaluations. A cross-product decomposition of the implied volatility surface allows to compute efficiently option prices, Greeks, replication cost, replication risk, and real option prices. The theoretical implied volatility surface and the empirical mean surface for options on the SP500 index are in excellent agreement.

Suggested Citation

  • Zumbach, Gilles, 2012. "Option pricing and ARCH processes," Finance Research Letters, Elsevier, vol. 9(3), pages 144-156.
  • Handle: RePEc:eee:finlet:v:9:y:2012:i:3:p:144-156
    DOI: 10.1016/j.frl.2012.01.002
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    References listed on IDEAS

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    1. Duan, Jin-Chuan & Zhang, Hua, 2001. "Pricing Hang Seng Index options around the Asian financial crisis - A GARCH approach," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1989-2014, November.
    2. Gilles Zumbach, 2009. "Time reversal invariance in finance," Quantitative Finance, Taylor & Francis Journals, vol. 9(5), pages 505-515.
    3. Peter Christoffersen & Redouane Elkamhi & Bruno Feunou & Kris Jacobs, 2010. "Option Valuation with Conditional Heteroskedasticity and Nonnormality," Review of Financial Studies, Society for Financial Studies, vol. 23(5), pages 2139-2183.
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    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    6. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
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    8. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 70-86.
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    Cited by:

    1. Hu, Jun & Kanniainen, Juho, 2015. "Asymptotic expansion of European options with mean-reverting stochastic volatility dynamics," Finance Research Letters, Elsevier, vol. 14(C), pages 1-10.
    2. Chen, Son-Nan & Chiang, Mi-Hsiu & Hsu, Pao-Peng & Li, Chang-Yi, 2014. "Valuation of quanto options in a Markovian regime-switching market: A Markov-modulated Gaussian HJM model," Finance Research Letters, Elsevier, vol. 11(2), pages 161-172.

    More about this item

    Keywords

    Option pricing; ARCH process; Implied volatility; Student innovations; Long memory volatility; Hedging cost and risk;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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