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

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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 \(H\). Moreover, in the case \(H\geq 1/2\), we develop a model-free approximation formula for the volatility swap in terms of the ATMI and its skew.

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Authors and Affiliations

  1. Dpt. Economia i Empresa, Universitat Pompeu Fabra, c/Ramón Trias Fargas, 25-27, Barcelona, Spain

    Elisa Alòs

  2. Barcelona Graduate School of Economics, c/Ramón Trias Fargas, 25-27, Barcelona, Spain

    Elisa Alòs

  3. Graduate School of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Kenichiro Shiraya

Authors
  1. Elisa Alòs
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  2. Kenichiro Shiraya
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Correspondence to Elisa Alòs.

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E.A. supported by grants ECO2014-59885-P and MTM2016-76420-P.

K.S. supported by CARF (Center for Advanced Research in Finance).

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Alòs, E., Shiraya, K. Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach. Finance Stoch 23, 423–447 (2019). https://doi.org/10.1007/s00780-019-00384-5

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  • DOI: https://doi.org/10.1007/s00780-019-00384-5

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