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The Fourier estimator of spot volatility: Unbounded coefficients and jumps in the price process

L. J. Espinosa González, Erick Treviño Aguilar

TL;DR

The paper extends the Malliavin–Mancino Fourier estimator for spot volatility to settings with unbounded volatility and cadlag price dynamics. It develops a Bohr-convolution-based coefficient estimation framework and proves convergence of the estimator, including almost-sure results and uniform path convergence, in both continuous and discretely observed regimes. In the presence of jumps, the authors show that a rescaled trigonometric polynomial converges to the quadratic jump process $\Delta J^2$, enabling direct capture of jump activity via Fourier methods. The results are complemented by a case study with compensated Poisson jumps and numerical illustrations, highlighting practical applicability to models with jumps and irregular sampling.

Abstract

In this paper we study the Fourier estimator of Malliavin and Mancino for the spot volatility. We establish the convergence of the trigonometric polynomial to the volatility's path in a setting that includes the following aspects. First, the volatility is required to satisfy a mild integrability condition, but otherwise allowed to be unbounded. Second, the price process is assumed to have cadlag paths, not necessarily continuous. We obtain convergence rates for the probability of a bad approximation in estimated coefficients, with a speed that allow to obtain an almost sure convergence and not just in probability in the estimated reconstruction of the volatility's path. This is a new result even in the setting of continuous paths. We prove that a rescaled trigonometric polynomial approximate the quadratic jump process.

The Fourier estimator of spot volatility: Unbounded coefficients and jumps in the price process

TL;DR

The paper extends the Malliavin–Mancino Fourier estimator for spot volatility to settings with unbounded volatility and cadlag price dynamics. It develops a Bohr-convolution-based coefficient estimation framework and proves convergence of the estimator, including almost-sure results and uniform path convergence, in both continuous and discretely observed regimes. In the presence of jumps, the authors show that a rescaled trigonometric polynomial converges to the quadratic jump process , enabling direct capture of jump activity via Fourier methods. The results are complemented by a case study with compensated Poisson jumps and numerical illustrations, highlighting practical applicability to models with jumps and irregular sampling.

Abstract

In this paper we study the Fourier estimator of Malliavin and Mancino for the spot volatility. We establish the convergence of the trigonometric polynomial to the volatility's path in a setting that includes the following aspects. First, the volatility is required to satisfy a mild integrability condition, but otherwise allowed to be unbounded. Second, the price process is assumed to have cadlag paths, not necessarily continuous. We obtain convergence rates for the probability of a bad approximation in estimated coefficients, with a speed that allow to obtain an almost sure convergence and not just in probability in the estimated reconstruction of the volatility's path. This is a new result even in the setting of continuous paths. We prove that a rescaled trigonometric polynomial approximate the quadratic jump process.
Paper Structure (30 sections, 34 theorems, 180 equations, 3 figures)

This paper contains 30 sections, 34 theorems, 180 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The price process
  3. Preliminaries
  4. The Fourier coefficients of volatility
  5. Basic definitions on Fourier analysis
  6. The Bohr convolution
  7. Continuous observation of the price's path
  8. Error's representation
  9. A key estimation: The remainder's $\mathbb{L}_p$-norm
  10. Coefficients' convergence
  11. Uniform convergence of trigonometric polynomials
  12. Discrete observation
  13. Error representation
  14. A key estimation: The remainder's $\mathbb{L}_p$-norm
  15. Coefficients' convergence
  16. ...and 15 more sections

Key Result

Proposition 1

where

Figures (3)

  • Figure 1: Simulation of the logarithmic price process.
  • Figure 2: The rescaled trigonometric polynomial $\frac{2 \pi}{M}\mathcal{T}^{\circledast}_{N,M}$, degrees 10 and 50, respectively.
  • Figure 3: The rescaled trigonometric polynomial $\frac{2 \pi}{M}\mathcal{T}^{\circledast}_{N,M}$, degrees 100 and 700, respectively.

Theorems & Definitions (68)

  • Remark 1
  • Remark 2
  • Definition 1
  • Proposition 1
  • Lemma 2
  • Theorem 3
  • proof
  • Remark 3
  • Lemma 4
  • proof
  • ...and 58 more