Quantitative Investigation of Univariate Time Series Behavior in Financial Markets

Abstract

This thesis contributes to a growing body of work in statistical analysis of univariate time series in financial markets, bridging the conventional markets, such as the S&P500 index, with emerging digital assets like Bitcoin cryptocurrency. The research focuses on understanding the effects of deterministic factors and market volatility.

We first develop governing equations to describe the S&P500 price return, utilizing a q-Gaussian diffusion process that exhibits fractional, non-linear diffusion characteristics. This approach decomposes time series into deterministic trends and stochastic fluctuations. The application of this q-Gaussian model provides a better fit for describing return fluctuations.

The comparative analysis extends to Bitcoin price returns, revealing stylized facts similar to conventional markets, including heavy tails, self-similarity, volatility clustering, and short-time autocorrelation. The q-Gaussian process effectively describes the anomalous diffusion in Bitcoin price returns, presenting a power-law decay in terms of volatility against diffusion time.

The temporal evolution of exponents characterizing anomalous characteristics is explored, deriving a specific category of solutions for variable order nonlinear Fokker-Planck equations, formulated as variable order q-Gaussian functions. The variable order diffusion process is governed by the Central Limit Theorem, transitioning from a q-Gaussian to a Gaussian distribution over time, providing a more robust model for real-world financial systems.

Finally, this thesis investigates non-stationary effects in price return for both markets. Results highlight that localized stationarity can be achieved by truncating the time series into segments, and for each segment, removing the trend and volatility of the price return is required. Moreover, we demonstrate that external events can significantly impact market statistics.