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Option pricing under fast-varying and rough stochastic volatility

Author

Listed:
  • Josselin Garnier

    (Ecole Polytechnique)

  • Knut Sølna

    (University of California)

Abstract

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.

Suggested Citation

  • Josselin Garnier & Knut Sølna, 2018. "Option pricing under fast-varying and rough stochastic volatility," Annals of Finance, Springer, vol. 14(4), pages 489-516, November.
    • Handle: RePEc:kap:annfin:v:14:y:2018:i:4:d:10.1007_s10436-018-0325-4
    DOI: 10.1007/s10436-018-0325-4
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    References listed on IDEAS

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    Cited by:

    1. Cao, Jiling & Kim, Jeong-Hoon & Kim, See-Woo & Zhang, Wenjun, 2020. "Rough stochastic elasticity of variance and option pricing," Finance Research Letters, Elsevier, vol. 37(C).
    2. Bolko, Anine E. & Christensen, Kim & Pakkanen, Mikko S. & Veliyev, Bezirgen, 2023. "A GMM approach to estimate the roughness of stochastic volatility," Journal of Econometrics, Elsevier, vol. 235(2), pages 745-778.
    3. Ranieri Dugo & Giacomo Giorgio & Paolo Pigato, 2024. "The Multivariate Fractional Ornstein-Uhlenbeck Process," CEIS Research Paper 581, Tor Vergata University, CEIS, revised 28 Aug 2024.
    4. Benjamin James Duthie, 2019. "Portfolio optimisation under rough Heston models," Papers 1909.02972, arXiv.org.
    5. Cao, Jiling & Kim, Jeong-Hoon & Liu, Wenqiang & Zhang, Wenjun, 2023. "Rescaling the double-mean-reverting 4/2 stochastic volatility model for derivative pricing," Finance Research Letters, Elsevier, vol. 58(PB).

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    More about this item

    Keywords

    Stochastic volatility; Short-range correlation; Fractional Ornstein–Uhlenbeck process; Hurst exponent; Mean reversion;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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