Estimation of the instantaneous volatility

Abstract

This paper is concerned with the estimation of the volatility process in a stochastic volatility model of the following form: $d{X}_t=a_{t}dt+{\sigma }_{t}d{W}_t$, where $X$ denotes the log-price and $\sigma$ is a càdlàg semi-martingale. In the spirit of a series of recent works on the estimation of the cumulated volatility, we here focus on the instantaneous volatility for which we study estimators built as finite differences of the \textit{power variations} of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of $\sigma$. In particular, these theorems yield some confidence intervals for ${\sigma }_t$.

Submitted to Statistical Inference for Stochastic Processes. 28 pages, 4 figures

References in corpus (2)

• Testing for jumps in a discretely observed process
• Asymptotic properties of realized power variations and related functionals of semimartingales