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Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach

Author

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  • Elisa Alòs

    (Universitat Pompeu Fabra
    Barcelona Graduate School of Economics)

  • Kenichiro Shiraya

    (The University of Tokyo)

Abstract

This paper is devoted to studying the difference between the fair strike of a volatility swap and the at-the-money implied volatility (ATMI) of a European call option. It is well known that the difference between these two quantities converges to zero as the time to maturity decreases. In this paper, we make use of a Malliavin calculus approach to derive an exact expression for this difference. This representation allows us to establish that the order of convergence is different in the correlated and uncorrelated cases, and that it depends on the behavior of the Malliavin derivative of the volatility process. In particular, we see that for volatilities driven by a fractional Brownian motion, this order depends on the corresponding Hurst parameter H $H$ . Moreover, in the case H ≥ 1 / 2 $H\geq 1/2$ , we develop a model-free approximation formula for the volatility swap in terms of the ATMI and its skew.

Suggested Citation

  • Elisa Alòs & Kenichiro Shiraya, 2019. "Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach," Finance and Stochastics, Springer, vol. 23(2), pages 423-447, April.
    • Handle: RePEc:spr:finsto:v:23:y:2019:i:2:d:10.1007_s00780-019-00384-5
    DOI: 10.1007/s00780-019-00384-5
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Working Papers hal-03827332, HAL.
    2. Hyungbin Park, 2021. "Influence of risk tolerance on long-term investments: A Malliavin calculus approach," Papers 2104.00911, arXiv.org.
    3. Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2022. "On the implied volatility of Asian options under stochastic volatility models," Papers 2208.01353, arXiv.org, revised Mar 2024.
    4. Antoine Jacquier & Aitor Muguruza & Alexandre Pannier, 2021. "Rough multifactor volatility for SPX and VIX options," Papers 2112.14310, arXiv.org, revised Nov 2023.
    5. E. Al`os & F. Rolloos & K. Shiraya, 2023. "A lower bound for the volatility swap in the lognormal SABR model," Papers 2306.14602, arXiv.org, revised Aug 2023.
    6. Kim, Hyun-Gyoon & Kim, See-Woo & Kim, Jeong-Hoon, 2024. "Variance and volatility swaps and options under the exponential fractional Ornstein–Uhlenbeck model," The North American Journal of Economics and Finance, Elsevier, vol. 72(C).
    7. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Papers 2210.12393, arXiv.org.
    8. Elisa Al`os & Fabio Antonelli & Alessandro Ramponi & Sergio Scarlatti, 2022. "CVA in fractional and rough volatility models," Papers 2204.11554, arXiv.org.
    9. Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2023. "On the implied volatility of Inverse options under stochastic volatility models," Papers 2401.00539, arXiv.org, revised Sep 2024.
    10. Masaaki Fukasawa, 2020. "Volatility has to be rough," Papers 2002.09215, arXiv.org.
    11. Elisa Alos & Frido Rolloos & Kenichiro Shiraya, 2019. "On the difference between the volatility swap strike and the zero vanna implied volatility," Papers 1912.05383, arXiv.org, revised Dec 2020.

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    More about this item

    Keywords

    Malliavin calculus; Fractional volatility models; Volatility swaps;
    All these keywords.

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling

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