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Statistical inference for rough volatility: Minimax Theory

Carsten Chong, Marc Hoffmann, Yanghui Liu, Mathieu Rosenbaum, Grégoire Szymanski

TL;DR

This work establishes that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes, and establishes minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them.

Abstract

Rough volatility models have gained considerable interest in the quantitative finance community in recent years. In this paradigm, the volatility of the asset price is driven by a fractional Brownian motion with a small value for the Hurst parameter $H$. In this work, we provide a rigorous statistical analysis of these models. To do so, we establish minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them. We notably obtain an optimal speed of convergence of $n^{-1/(4H+2)}$ for estimating $H$ based on n sampled data, extending results known only for the easier case $H>1/2$ so far. We therefore establish that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes.

Statistical inference for rough volatility: Minimax Theory

TL;DR

This work establishes that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes, and establishes minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them.

Abstract

Rough volatility models have gained considerable interest in the quantitative finance community in recent years. In this paradigm, the volatility of the asset price is driven by a fractional Brownian motion with a small value for the Hurst parameter . In this work, we provide a rigorous statistical analysis of these models. To do so, we establish minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them. We notably obtain an optimal speed of convergence of for estimating based on n sampled data, extending results known only for the easier case so far. We therefore establish that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes.
Paper Structure (42 sections, 36 theorems, 348 equations)

This paper contains 42 sections, 36 theorems, 348 equations.

Table of Contents

  1. Introduction
  2. The rough volatility paradigm
  3. Rough volatility in the literature
  4. Results of the paper
  5. The piecewise constant rough volatility model
  6. Model and notation
  7. Minimax lower bound
  8. Construction of an asymptotically minimax estimator
  9. The general rough volatility model
  10. Model and main results
  11. Minimax optimality
  12. Energy levels of the log integrated volatility
  13. Construction of the estimator
  14. Discussion
  15. Proof of Theorem \ref{['thm:lower_multiscale']}
  16. ...and 27 more sections

Key Result

Theorem 2

The rate is a lower rate of convergence for estimating $H$ in $\mathcal{E}^n$. Moreover, is a lower rate of convergence for estimating $\eta$ over $\mathcal{D}$ in $\mathcal{E}^n$ (with obvious modifications in the definition).

Theorems & Definitions (51)

  • Remark 1: About the asymptotic regime
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Lemma 6
  • Lemma 7
  • Proposition 8
  • Proposition 9
  • Lemma 10
  • ...and 41 more