Download Fourier-Malliavin Volatility Estimation: Theory and Practice by Maria Elvira Mancino, Maria Cristina Recchioni, Simona PDF

By Maria Elvira Mancino, Maria Cristina Recchioni, Simona Sanfelici

This quantity is a basic presentation of the most theoretical houses of the Fourier-Malliavin volatility estimation, permitting the readers to event the potential for the technique and its software in a variety of monetary settings. Readers are given examples and tools to enforce this technique in a number of monetary settings and functions of real-life facts. a close bibliographic reference is integrated to allow an in-depth examine.

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Extra resources for Fourier-Malliavin Volatility Estimation: Theory and Practice

Example text

2 Consistency. 1). The following uniform convergence in probability holds lim 2 sup |σn,N,M (t) − σ 2 (t)| = 0. 1 Univariate Estimator 33 The proof can be found in Malliavin and Mancino (2009). 5) highlights the fact that the Fourier estimator is a global estimator, in the sense that it estimates the whole path t → σ 2 (t) in the interval of interest. Apart from results on the uniform convergence of the Fourier estimator, very few results are available for the global estimators, such those for the kernel-based estimators by Fan and Wang (2008) and the wavelet-based estimator by Hoffmann (1999) .

Tn = 2π } with step size ρ (n) = 2π /n. 14) ∑ n e−ikt j−1 g( nδ j (p)), j=1 where the function g can assume different specifications. 1) to reconstruct the path of the process ρ (σ 2 ). This can thus be translated into an estimator of the volatility by inverting the function ρ (·). The estimator of instantaneous volatility obtained so far is consistent and the Central Limit Theorem holds with rate of convergence equal to n(1−ν )/2 , where ν 7 A precise statement of the Itˆ o formula for general semimartingale is beyond the scope of this book; we refer the interested reader to the book by Protter (1992).

27). The localizing bandwidth must be carefully chosen in order to reduce oscillations and minimize the discretization errors due to last tick interpolation. Due to the particular asynchronicity structure at hands, here the bandwidth is 7 minute large. It is evident that the Fourier estimates are much smoother and hence reliable than the ones obtained by the Realized Covariance. 3. A preliminary attempt to investigate the impact of the cutting frequencies on the time scale used to reconstruct the volatility path is due to Mattiussi and Iori (2010).

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