Estimation of the instantaneous volatility

2008/12/18 by Alexánder Álvarez, Fabien Panloup, Alvarez, A. +5
Economics, Econometrics and Finance · #60F05 (Primary) #91B70 (Secondary) #91B82 #Complex Systems and Time Series Analysis #FOS: Economics and business #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistical Finance (q-fin.ST) #Statistics Theory (math.ST) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.0812.3538

openalex publication_date 2008/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is concerned with the estimation of the volatility process in a stochastic volatility model of the following form: dX_t=a_tdt+σ_tdW_t, where X denotes the log-price and σ 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 power variations of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of σ. In particular, these theorems yield some confidence intervals for σ_t.

Citations

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

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