[Paper Review] Common Underlying Dynamics in an Emerging Market: From Minutes to Months

Renato Vicente, Charles M. de Toledo+2 more|ArXiv.org|Feb 6, 2004

Complex Systems and Time Series AnalysisEconomics, Econometrics and Finance8 references3 citations

TL;DR

This paper demonstrates that the Heston stochastic volatility model with a single parameter set can accurately describe price fluctuations in Brazil's IBOVESPA index across time scales from minutes to months over a 35-year period of economic and political turmoil. To resolve the model's failure in capturing the empirical volatility autocorrelation function, the authors introduce a two-time-scale extension with a slow mean-reverting volatility process, revealing a universal underlying market microstructure dynamics despite macroeconomic instability.

ABSTRACT

We analyse a period spanning 35 years of activity in the Sao Paulo Stock Exchange Index (IBOVESPA) and show that the Heston model with stochastic volatility is capable of explaining price fluctuations for time scales ranging from 5 minutes to 100 days with a single set of parameters. We also show that the Heston model is inconsistent with the observed behavior of the volatility autocorrelation function. We deal with the latter inconsistency by introducing a slow time scale to the model. The fact that the price dynamics in a period of 35 years of macroeconomical unrest may be modeled by the same stochastic process is evidence for a general underlying microscopic market dynamics.

Motivation & Objective

• To investigate whether a single stochastic volatility model can describe price dynamics across multiple time scales in an emerging market.
• To assess the Heston model’s ability to reproduce empirical return distributions and volatility clustering in the IBOVESPA index over 35 years of economic and political instability.
• To identify and resolve inconsistencies between the Heston model and observed volatility autocorrelation functions in financial time series.
• To propose and validate an extended Heston model with a slow time scale to better capture long-memory volatility behavior.
• To provide evidence for a common, underlying microscopic market mechanism that persists despite structural macroeconomic changes.

Proposed method

• The Heston model is applied to IBOVESPA data, using geometric Brownian motion with stochastic volatility governed by a Cox-Ingersoll-Ross (CIR) process.
• The Fokker-Planck equation for log-returns is solved semi-analytically using Fourier and Laplace transforms, enabling derivation of return distribution dynamics.
• A two-time-scale extension is introduced, where volatility reverts to a second, slowly evolving mean-reverting process with a much longer relaxation time.
• The extended model incorporates three Wiener processes with specified cross-correlations, and the volatility autocorrelation function is analytically derived as a sum of two exponential decays.
• The model parameters are fitted to empirical data using a moving window analysis to assess non-stationarity in the mean volatility over time.
• The solution is validated by comparing simulated return distributions and autocorrelation functions with empirical data across time scales from minutes to 100 days.

Experimental results

Research questions

• RQ1 Can the Heston model with a single parameter set explain return dynamics in an emerging market like Brazil across time scales from minutes to months?
• RQ2 Why does the standard Heston model fail to reproduce the empirically observed volatility autocorrelation function in the IBOVESPA index?
• RQ3 How can a stochastic volatility model be extended to account for long-memory volatility behavior observed in financial time series?
• RQ4 To what extent is the statistical behavior of financial returns in an emerging market robust to macroeconomic instability and structural changes?
• RQ5 What does the persistence of a single stochastic process across decades imply about the underlying mechanisms of market microstructure?

Key findings

• The Heston model successfully reproduces the diffusion process and return distribution of IBOVESPA across time scales from 5 minutes to 100 days using a single set of parameters.
• The model's volatility autocorrelation function does not match empirical data, indicating a fundamental inconsistency in the single relaxation time assumption.
• The extended two-time-scale model with a slow mean-reverting process for the long-term volatility level produces an autocorrelation function that matches empirical observations.
• The slow relaxation time in the extended model is found to be γ₂⁻¹ = 144.9 days, with a volatility volatility parameter κ₂ = 1.0 × 10⁻⁴ days⁻¹.
• Despite 35 years of hyperinflation, currency crises, and political upheavals, the statistical behavior of IBOVESPA remains consistent with a single underlying stochastic process.
• The robustness of the model across macroeconomic instability suggests the existence of universal, microscopic market dynamics independent of macroeconomic conditions.