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Estimation of Integrated Volatility

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Fourier-Malliavin Volatility Estimation

Part of the book series: SpringerBriefs in Quantitative Finance ((BRIEFFINANCE))

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Abstract

The financial econometrics literature mainly focuses on the integrated volatility and cross-volatility on a fixed time horizon. Therefore, this chapter is devoted to the estimation of these quantities. In the context of the Fourier estimation method, the integrated volatilities are computed by simply taking the 0-th Fourier coefficient in formula (2.13). We begin with the study of the univariate estimator, for the ease of notation; nevertheless, the results holding for this case can be easily extended to the multivariate estimator that will be studied in Section 3.3, with special care to be paid for the asynchronous data case. Then, the issue of feasibility for these results is discussed by providing an estimator of the error asymptotic variance, called quarticity. Finally, the properties of the Fourier estimator versus different integrated volatility estimators proposed in the literature are outlined.

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Notes

  1. 1.

    Interested readers can find a deep study of semimartingale theory in Protter (1992).

  2. 2.

    The optimal rate of convergence for a non-parametric estimator of volatility is O(n 1∕2).

  3. 3.

    For an introduction of the concept of stable convergence in law see, e.g., Aldous and Eagleson (1978) and Jacod and Shiryaev (2003).

  4. 4.

    Section A.2.3 in the Appendix A contains a quick review of the Nyquist frequency.

  5. 5.

    Note that the definition of the Realized Volatility is the same as (3.8), but we prefer here to point out the time step size Δ t instead of the number of observations.

  6. 6.

    A more detailed discussion of the convergence of the Realized Volatility-type estimators can be found in Aït-Sahalia and Jacod (2014) Section 6.

  7. 7.

    Asymptotic conditions required for the irregular/asynchronous time grids and detailed proof can be found in Malliavin and Mancino (2009) Theorem 4.4.

  8. 8.

    The notion of Nyquist frequency is discussed in Section A.2.3

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Authors and Affiliations

  1. Department of Economics and Management, University of Firenze, Firenze, Italy

    Maria Elvira Mancino

  2. Department of Management, University Politecnica delle Marche, Ancona, Italy

    Maria Cristina Recchioni

  3. Department of Economics, University of Parma, Parma, Italy

    Simona Sanfelici

Authors
  1. Maria Elvira Mancino
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  2. Maria Cristina Recchioni
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  3. Simona Sanfelici
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Mancino, M.E., Recchioni, M.C., Sanfelici, S. (2017). Estimation of Integrated Volatility. In: Fourier-Malliavin Volatility Estimation. SpringerBriefs in Quantitative Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-50969-3_3

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