Spectral estimation of the fractional order of a Lévy process

By Denis Belomestny. 2010

56 citations recorded by the source corpus.

We consider the problem of estimating the fractional order of a L´evy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate are derived, the asymptotic normality is proved and a data-driven algorithm based on aggregation is proposed. The performance of the estimator in both estimation and calibration setups is illustrated by a simulation study.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2fab50ab972dd8917f933a489baf2282bcd52fd7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Jean Jacod
2. Yacine Aït-Sahalia
3. Peter Tankov
4. Joan Bruna
5. Emmanuel Bacry
6. Jean-François Muzy
7. Stéphane Mallat
8. Aleksandar Mijatovic

In Statistics and Probability

1. Yacine Aït-Sahalia
2. Jean Jacod
3. Denis Belomestny
4. Aleksandar Mijatovic
5. Peter Tankov
6. Emmanuel Bacry
7. Jean-François Muzy
8. Joan Bruna

Citations in the corpus

• A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS (2012, methodology): “In contrast to the high-frequency setting, a spectral estimation algorithm for the Blumenthal–Getoor index of a Lévy process based on low-frequency historical and options data was proposed in Belomestny (2010).”
• Statistical inference for time-changed Lévy processes via composite characteristic function estimation (2010, methodology): “3 can be found in the preprint version of our paper Belomestny (2010a). Combining Lemma 6.”
• Intermittent process analysis with scattering moments (2013, methodology): “This index can be estimated using spectral methods in [8], which require an estimation of the characteristic function.”
• Testing whether jumps have finite or infinite activity (2011, result): “This is in line with the empirical results of a companion paper, Aı̈t-Sahalia and Jacod (2009a), which contains an extension to Itô semimartingales of the classical Blumenthal–Getoor index β for Lévy processes and estimators for β; see also Belomestny (2010) for different estimators.”
• Estimating the degree of activity of jumps in high frequency data (2009, background): “Also, Belomestny (2008) estimates the same index when there is no Brownian part and when, together with the prices, some option prices also are recorded. Another related problem is the estimation of the index β of a stable process [see, e.g., DuMouchel (1983)]. However, our situation here is fundame”
• Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data (2010, methodology): “Delattre, Sylvain, and Jean Jacod.”