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On the Realized Joint Laplace Transform of Volatilities with Application to Test the Volatility Dependence

XinWei Feng, Yu Jiang, Zhi Liu, Zhe Meng

TL;DR

This paper addresses inference on the joint distribution of two volatilities by estimating the empirical joint Laplace transform over $[0,T]$ from high-frequency data. It introduces overlapped and non-overlapped estimators, proves functional CLTs and a long-span central limit theorem under stationarity and mixing, and provides a feasible test for volatility dependence via a Delta-method construction and kernel-based covariance estimation. The methods are validated through extensive simulations showing efficiency gains from overlapped increments and robustness to jumps, and are complemented by an empirical study on S&P 500 stocks revealing prevalent volatility dependence. The work advances volatility risk assessment by enabling joint inference on volatilities with practical applicability to asset pricing and risk management using high-frequency data.

Abstract

In this paper, we first investigate the estimation of the empirical joint Laplace transform of volatilities of two semi-martingales within a fixed time interval [0, T] by using overlapped increments of high-frequency data. The proposed estimator is robust to the presence of finite variation jumps in price processes. The related functional central limit theorem for the proposed estimator has been established. Compared with the estimator with non-overlapped increments, the estimator with overlapped increments improves the asymptotic estimation efficiency. Moreover, we study the asymptotic theory of estimator under a long-span setting and employ it to create a feasible test for the dependence between volatilities. Finally, simulation and empirical studies demonstrate the performance of proposed estimators.

On the Realized Joint Laplace Transform of Volatilities with Application to Test the Volatility Dependence

TL;DR

This paper addresses inference on the joint distribution of two volatilities by estimating the empirical joint Laplace transform over from high-frequency data. It introduces overlapped and non-overlapped estimators, proves functional CLTs and a long-span central limit theorem under stationarity and mixing, and provides a feasible test for volatility dependence via a Delta-method construction and kernel-based covariance estimation. The methods are validated through extensive simulations showing efficiency gains from overlapped increments and robustness to jumps, and are complemented by an empirical study on S&P 500 stocks revealing prevalent volatility dependence. The work advances volatility risk assessment by enabling joint inference on volatilities with practical applicability to asset pricing and risk management using high-frequency data.

Abstract

In this paper, we first investigate the estimation of the empirical joint Laplace transform of volatilities of two semi-martingales within a fixed time interval [0, T] by using overlapped increments of high-frequency data. The proposed estimator is robust to the presence of finite variation jumps in price processes. The related functional central limit theorem for the proposed estimator has been established. Compared with the estimator with non-overlapped increments, the estimator with overlapped increments improves the asymptotic estimation efficiency. Moreover, we study the asymptotic theory of estimator under a long-span setting and employ it to create a feasible test for the dependence between volatilities. Finally, simulation and empirical studies demonstrate the performance of proposed estimators.

Paper Structure

This paper contains 11 sections, 69 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Asymptotic results
  4. Joint Infill and Long-time Span Asymptotics
  5. Simulation studies
  6. Fixed $T$
  7. Non-synchronous observation
  8. Test for the dependence of volatility processes
  9. Empirical study
  10. Conclusion
  11. Appendix: proofs

Figures (5)

  • Figure 1: The asymptotic variance of $V_n(u,v)$ and $U_n(u,v)$
  • Figure 2: Left panel: The histograms of $V_n(u,v)-\int_0^1\hbox{e}^{-\langle (u, v), ((\sigma_{s}^{X})^2, (\sigma_{s}^{Y})^2)\rangle}\mathrm{d}s$; Right panel: The histograms of $U_n(u,v)-\int_0^1\hbox{e}^{-\langle (u, v), ((\sigma_{s}^{X})^2, (\sigma_{s}^{Y})^2)\rangle}\mathrm{d}s$.
  • Figure 3: Left panel: The histograms of $\frac{1}{\sqrt{\Delta_n\tilde{\Gamma}_{n}}}\left(V_n(u,v)-\int_0^1\hbox{e}^{-\langle (u, v), ((\sigma_{s}^{X})^2, (\sigma_{s}^{Y})^2)\rangle}\mathrm{d}s\right)$; Right panel: The histograms of $\frac{1}{\sqrt{\Delta_n\tilde{\Gamma}_{n}}}\left(U_n(u,v)-\int_0^1\hbox{e}^{-\langle (u, v), ((\sigma_{s}^{X})^2, (\sigma_{s}^{Y})^2)\rangle}\mathrm{d}s\right)$.
  • Figure 4: The Non-synchronous observations of price processes $X$ and $Y$.
  • Figure 5: The histograms of $U_{n}'(u,v)-\int_0^1\hbox{e}^{-\langle (u, v), ((\sigma_{s}^{X})^2, (\sigma_{s}^{Y})^2)\rangle}\mathrm{d}s$.