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Spectral estimation of the fractional order of a Lévy process
2010, Belomestny, Denis
We consider the problem of estimating the fractional order of a Levy 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
Abelian theorems for stochastic volatility models with application to the estimation of jump activity of volatility
2011, Belomestny, Denis, Panov, Vladimir
In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models $(X, V)$, where both the state process $X$ and the volatility process $V$ may have jumps. Our results relate the asymptotic behavior of the characteristic function of $X_Delta$ for some $Delta > 0$ in a stationary regime to the Blumenthal-Getoor indexes of the Lévy processes driving the jumps in $X$ and $V$ . The results obtained are used to construct consistent estimators for the above Blumenthal-Getoor indexes based on low-frequency observations of the state process $X$. We derive the convergence rates for the corresponding estimator and prove that these rates can not be improved in general.