Estimating the Volatility Occupation Time via Regularized Laplace Inversion

Abstract

We propose a consistent functional estimator for the occupation time of the spot variance of an asset price observed at discrete times on a finite interval with the mesh of the observation grid shrinking to zero. The asset price is modeled nonparametrically as a continuous-time Itô semimartingale with nonvanishing diffusion coefficient. The estimation procedure contains two steps. In the first step we estimate the Laplace transform of the volatility occupation time and, in the second step, we conduct a regularized Laplace inversion. Monte Carlo evidence suggests that the proposed estimator has good small-sample performance and in particular it is far better at estimating lower volatility quantiles and the volatility median than a direct estimator formed from the empirical cumulative distribution function of local spot volatility estimates. An empirical application shows the use of the developed techniques for nonparametric analysis of variation of volatility.

The Realized Empirical Distribution Function of Stochastic Variance with Application to Goodness-of-Fit Testing

“…→ 0 with H = 1/2, as is consistent with the fixed T asymptotic theory in Li, Todorov, and Tauchen (2013). The deterioration of the speed condition is due to the added complexity from retrieving volatility robustly, which cuts the rate of convergence in half.…”

“…, n, where ∆ n is the time gap between consecutive observations and 1 In probability theory, F T (x) = T 0 1 {Yt≤x} dt is called the occupation-or local-time of the stochastic process Y (e.g., Geman and Horowitz, 1980). This convention was adopted by Li, Todorov, andTauchen (2013, 2016), who applied it to high-frequency volatility estimation. We normalize F T by T here, as we are heading toward a setting with stationary volatility and an asymptotic theory with T → ∞.…”

“…We estimate F T with F n,T over 1,000 simulations and plot the average (dashed line). The version proposed in Li, Todorov, andTauchen (2013, 2016) based on 1-and 5-minute sparse sampling is shown as a comparison ('×' and '•'). The dotted line is an infeasible estimate that has access to the whole record of X, (X i∆n ) n i=0 .…”

“…The sample path of σ t is shown in Panel A, while the associated EDF appears in Panel B. We report our estimator and compare it to Li, Todorov, and Tauchen (2013). We construct the latter statistic from 1-and 5-minute low-frequency returns.…”

“…As an aside, we note Li, Todorov, and Tauchen (2013) only prove consistency of the jump-truncated statistic in the presence of discontinuous X and no microstructure noise, but this can also be extended to the non-truncated estimator. Nonetheless, it may be preferable to truncate in practice, as it can act as a safeguard against finite sample distortions induced by jumps.…”