Probabilistic models and statistics for electronic financial markets in the digital age

TL;DR

This paper reviews recent statistical methods for analyzing discretely observed semimartingales in electronic financial markets, focusing on jump detection, rough volatility, and microstructure noise, highlighting classical concepts and new developments.

Contribution

It offers a comprehensive review of recent advances in statistical techniques for financial market models, including new jump tests, analysis of rough volatility, and microstructure noise modeling.

Findings

Jump tests based on extreme value theory and order statistics.

Establishment of a minimax lower bound for volatility regularity.

Probabilistic foundation for a stochastic boundary model with microstructure noise.

Abstract

The scope of this manuscript is to review some recent developments in statistics for discretely observed semimartingales which are motivated by applications for financial markets. Our journey through this area stops to take closer looks at a few selected topics discussing recent literature. We moreover highlight and explain the important role played by some classical concepts of probability and statistics. We focus on three main aspects: Testing for jumps; rough fractional stochastic volatility; and limit order microstructure noise. We review jump tests based on extreme value theory and complement the literature proposing new statistical methods. They are based on asymptotic theory of order statistics and the R\'{e}nyi representation. The second stage of our journey visits a recent strand of research showing that volatility is rough. We further investigate this and establish a minimax lower bound exploring frontiers to what extent the regularity of latent volatility can be recovered in a more general framework. Finally, we discuss a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices and its probabilistic and statistical foundation.