## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 4 (2017), 1542-1578.

### Nonparametric change-point analysis of volatility

Markus Bibinger, Moritz Jirak, and Mathias Vetter

#### Abstract

In this work, we develop change-point methods for statistics of high-frequency data. The main interest is in the volatility of an Itô semimartingale, the latter being discretely observed over a fixed time horizon. We construct a minimax-optimal test to discriminate continuous paths from paths with volatility jumps, and it is shown that the test can be embedded into a more general theory to infer the smoothness of volatilities. In a high-frequency setting, we prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we develop methods to infer changes in the Hurst parameters of fractional volatility processes. A simulation study is conducted to demonstrate the performance of our methods in finite-sample applications.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 4 (2017), 1542-1578.

**Dates**

Received: February 2016

Revised: June 2016

First available in Project Euclid: 28 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1498636866

**Digital Object Identifier**

doi:10.1214/16-AOS1499

**Mathematical Reviews number (MathSciNet)**

MR3670188

**Zentralblatt MATH identifier**

06773283

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 62G10: Hypothesis testing

**Keywords**

High-frequency data nonparametric change-point test minimax-optimal test stochastic volatility volatility jumps

#### Citation

Bibinger, Markus; Jirak, Moritz; Vetter, Mathias. Nonparametric change-point analysis of volatility. Ann. Statist. 45 (2017), no. 4, 1542--1578. doi:10.1214/16-AOS1499. https://projecteuclid.org/euclid.aos/1498636866

#### Supplemental materials

- Complete proofs. We provide all remaining proofs for the results from Sections 3, 4 and 5.Digital Object Identifier: doi:10.1214/16-AOS1499SUPPASupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.
- Application and simulations. We present complementary simulations for different sample sizes accompanied by a sensitivity analysis of the dependence on $k_{n}$ and a discussion of data applications.Digital Object Identifier: doi:10.1214/16-AOS1499SUPPBSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.