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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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    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.

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    More about this item

    Keywords

    Option pricing; ARCH process; Implied volatility; Student innovations; Long memory volatility; Hedging cost and risk;
    All these keywords.

    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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