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Getting Inside the Latent Volatility

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Fourier-Malliavin Volatility Estimation

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

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Abstract

This chapter introduces the reader into some recent financial applications of the Fourier estimator. We exploit here the ability of the method to reconstruct the volatility as a stochastic function of time in the univariate and multivariate case; in other words, we can handle the volatility function as an observable variable. This property makes it possible to have insights into various volatility related financial quantities, such as volatility of volatility and leverage. The chapter begins with an empirical exercise in which the latent volatility is estimated; we discuss in some extent the issue of the presence of jumps in the financial data. Then, in Sections 6.2 and 6.3 it is shown how to iterate the procedure for the purpose of parameter identification and calibration of stochastic volatility models and how to estimate in a model-free fashion a second order effect, known as price-volatility feedback rate. Finally, in Section 6.4 we analyze the forecasting power of the Fourier estimator of integrated volatility by a simple Monte Carlo experiment and an empirical application. Further directions for additional applications are given in Section 6.5.

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Notes

  1. 1.

    In all the empirical experiments only one of the possible procedures to test for the presence of jumps in the data is considered, as the study of jumps goes beyond the scope of the present book.

  2. 2.

    We use this definition of volatility measure because TBV is less efficient than the Realized Volatility if no jumps occur.

  3. 3.

    An early attempt to use the Fourier method to identify the parameters of stochastic volatility models is present in Malliavin and Mancino (2002b), Barucci and Mancino (2010), Renò (2008), while a deep study is done in Sanfelici et al. (2015), Curato and Sanfelici (2015).

  4. 4.

    In the convolution (6.5) a Barlett kernel has been added, which improves the behavior of the estimator for very high observation frequencies.

  5. 5.

    The proof by Curato and Sanfelici (2015) require the differentiability in the sense of distributions of the drift and diffusion coefficients. We refer the interested reader to the paper for further details.

  6. 6.

    The extension to a finite number of assets can be found in Barucci et al. (2003). Further, the computation with non-zero drift is in Malliavin and Thalmaier (2006) Chapter 3

  7. 7.

    The use of prime stands here for the first derivative with respect to the level p(t).

  8. 8.

    The proof is in Malliavin and Thalmaier (2006).

  9. 9.

    The ESV models introduced by Meddahi (2001) include most continuous-time stochastic volatility models. Roughly speaking, under these models the volatility process depends only on a single (latent) state variable and can be expressed as a linear combination of the eigenfunctions of the infinitesimal generator associated with this latent variable.

  10. 10.

    A comparison with methods specifically designed to handle market microstructure noise can be found in Barucci et al. (2012).

  11. 11.

    The cutting frequency in this experiment differs from the one considered in Barucci et al. (2012), where N is selected in order to maximize the R 2. Moreover, the results in Table 6.4 are slightly less performing than those in Barucci et al. (2012) for low values of n due to the feasible minimization adopted here.

  12. 12.

    Even if a simple AR(1) model cannot perfectly capture the dynamic of the integrated volatility, this model has been mainly chosen to make the empirical analysis comparable with the Monte Carlo analysis.

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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). Getting Inside the Latent Volatility. In: Fourier-Malliavin Volatility Estimation. SpringerBriefs in Quantitative Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-50969-3_6

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