Inverse Realized Laplace Transforms for Nonparametric Volatility Density Estimation in Jump-Diffusions

Abstract

This article develops a nonparametric estimator of the stochastic volatility density of a discretely observed Itô semimartingale in the setting of an increasing time span and finer mesh of the observation grid. There are two basic steps involved. The first step is aggregating the high-frequency increments into the realized Laplace transform, which is a robust nonparametric estimate of the underlying volatility Laplace transform. The second step is using a regularized kernel to invert the realized Laplace transform. These two steps are relatively quick and easy to compute, so the nonparametric estimator is practicable. The article also derives bounds for the mean squared error of the estimator. The regularity conditions are sufficiently general to cover empirically important cases such as level jumps and possible dependencies between volatility moves and either diffusive or jump moves in the semimartingale. The Monte Carlo analysis in this study indicates that the nonparametric estimator is reliable and reasonably accurate in realistic estimation contexts. An empirical application to 5-min data for three large-cap stocks, 1997-2010, reveals the importance of big short-term volatility spikes in generating high levels of stock price variability over and above those induced by price jumps. The application also shows how to trace out the dynamic response of the volatility density to both positive and negative jumps in the stock price.

Estimating the Volatility Occupation Time via Regularized Laplace Inversion

“…However, the problem here is more complicated, since the estimand F T (·) itself is a random function, which in particular renders the regularization bias random, whereas in Todorov and Tauchen (2012a) and Kryzhniy (2003a,b), the object of interest is deterministic. We now turn to the asymptotic properties of the estimator F T,n,R n (x), where R n is a sequence of (strictly positive) regularization parameters that grows to +∞ asymptotically.…”

“…We can further compare our analysis here with Todorov and Tauchen (2012a), where somewhat analogous steps were followed to estimate the invariant probability density of the volatility process, but there are fundamental differences between the current paper and Todorov and Tauchen (2012a). First, unlike Todorov and Tauchen (2012a), the time span of the data is fixed and hence we are interested in pathwise properties of the latent volatility process over the fixed time interval.…”

“…As a technical by-product of our analysis, we provide primitive conditions for the smoothness of the volatility occupation density for a popular class of jump-diffusion stochastic volatility models. Overall, the current paper can be viewed as an extension of the results in Todorov and Tauchen (2012a) to the theoretically different setting of fixed time span and provides the theoretical justification for applying the method in Todorov and Tauchen (2012a) over different time horizons.…”

“…It is well known that smoothness conditions are important in the analysis of ill-posed problems (Carrasco et al (2007)). Indeed, our analysis of the stochastic regularization bias demands technical arguments that are very different from Todorov and Tauchen (2012a), where the invariant distribution is deterministic and smoother. As a technical by-product of our analysis, we provide primitive conditions for the smoothness of the volatility occupation density for a popular class of jump-diffusion stochastic volatility models.…”

“…First, unlike Todorov and Tauchen (2012a), the time span of the data is fixed and hence we are interested in pathwise properties of the latent volatility process over the fixed time interval. This is further illustrated by our empirical application which studies the randomness of the (occupational) interquartile range of various transforms of volatility.…”