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

Markus Bibinger

TL;DR

This paper surveys recent advances in statistics for discretely observed semimartingales in electronic financial markets, focusing on jumps, rough volatility, and one-sided limit-order microstructure noise. It develops and compares jump-detection tools based on extreme value theory, including a Gumbel-based test and a Rényi-order-statistics test with Deheuvels-type limits, and analyzes the identifiability and estimation limits of latent volatility regularity in rough volatility models. It also introduces a stochastic boundary framework (LOMN) for limit-order data, leveraging local minima to yield rate-optimal volatility estimators under one-sided noise and linking taxi-problem ideas to boundary estimation. The outlook highlights multivariate extensions, non-synchronous observations, co-jumps, and high-dimensional risk forecasting, outlining practical avenues for improved inference and risk management in ultra-high-frequency finance.

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 Rényi 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.

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

TL;DR

This paper surveys recent advances in statistics for discretely observed semimartingales in electronic financial markets, focusing on jumps, rough volatility, and one-sided limit-order microstructure noise. It develops and compares jump-detection tools based on extreme value theory, including a Gumbel-based test and a Rényi-order-statistics test with Deheuvels-type limits, and analyzes the identifiability and estimation limits of latent volatility regularity in rough volatility models. It also introduces a stochastic boundary framework (LOMN) for limit-order data, leveraging local minima to yield rate-optimal volatility estimators under one-sided noise and linking taxi-problem ideas to boundary estimation. The outlook highlights multivariate extensions, non-synchronous observations, co-jumps, and high-dimensional risk forecasting, outlining practical avenues for improved inference and risk management in ultra-high-frequency finance.

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 Rényi 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.
Paper Structure (7 sections, 7 theorems, 92 equations, 7 figures)

This paper contains 7 sections, 7 theorems, 92 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Elements of high-frequency statistics for semimartingales
  3. Jump detection in high-frequency data based on extreme value theory
  4. Is volatility rough?
  5. Limit order microstructure noise
  6. Outlook
  7. Proof of Theorem 2

Key Result

Lemma 1

Let $(E_j)_{1\le j\le n}$ be i.i.d. $\text{Exp}(1)$. The equality in distribution holds for $(\tilde{E}_j)$ i.i.d. $\text{Exp}(1)$.

Figures (7)

  • Figure 1: Log-price (left) in Heston model with 5 jumps and its increments (right).
  • Figure 2: Histograms of the statistics for $n=3{,}600$ from 1,000,000 Monte Carlo iterations and the densities of their asymptotic Gumbel, exponential and Deheuvels distributions.
  • Figure 3: Q-q plots with $(90+j)$% percentiles, $0\le j\le 9$, of the statistics for $n=3{,}600$ from 1,000,000 Monte Carlo iterations compared to the asymptotic Gumbel, exponential and Deheuvels distributions.
  • Figure 4: Left $m(q,\Delta)$ from \ref{['mq']} as function of $\log(\Delta)$, right: $\zeta_q$ as a function of $q$.
  • Figure 5: Bid and ask quotes and trade prices (black dots) for the Apple asset over a 10 minutes time interval.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Lemma 1
  • Proposition 3.1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • Theorem 2