Stylized Facts

Stylized Facts and Their Microscopic Origins: Clustering, Persistence, and Stability in a 2D Ising Framework ArXiv ID: 2512.17925 “View on arXiv” Authors: Hernán Ezequiel Benítez, Claudio Oscar Dorso Abstract The analysis of financial markets using models inspired by statistical physics offers a fruitful approach to understand collective and extreme phenomena [“3, 14, 15”] In this paper, we present a study based on a 2D Ising network model where each spin represents an agent that interacts only with its immediate neighbors plus a term reated to the mean field [“1, 2”]. From this simple formulation, we analyze the formation of spin clusters, their temporal persistence, and the morphological evolution of the system as a function of temperature [“5, 19”]. Furthermore, we introduce the study of the quantity $1/2P\sum_{“i”}|S_{“i”}(t)+S_{“i”}(t+Δt)|$, which measures the absolute overlap between consecutive configurations and quantifies the degree of instantaneous correlation between system states. The results show that both the morphology and persistence of the clusters and the dynamics of the absolute sum can explain universal statistical properties observed in financial markets, known as stylized facts [“2, 12, 18”]: sharp peaks in returns, distributions with heavy tails, and zero autocorrelation. The critical structure of clusters and their reorganization over time thus provide a microscopic mechanism that gives rise to the intermittency and clustered volatility observed in prices [“2, 15”]. ...

Phase Transitions in Financial Markets Using the Ising Model: A Statistical Mechanics Perspective ArXiv ID: 2504.19050 “View on arXiv” Authors: Bruno Giorgio Abstract This dissertation investigates the ability of the Ising model to replicate statistical characteristics, or stylized facts, commonly observed in financial assets. The study specifically examines in the S&P500 index the following features: volatility clustering, negative skewness, heavy tails, the absence of autocorrelation in returns, and the presence of autocorrelation in absolute returns. A significant portion of the dissertation is dedicated to Ising model-based simulations. Due to the lack of an analytical or deterministic solution, the Monte Carlo method was employed to explore the model’s statistical properties. The results demonstrate that the Ising model is capable of replicating the majority of the statistical features analyzed. ...

International Financial Markets Through 150 Years: Evaluating Stylized Facts ArXiv ID: 2504.08611 “View on arXiv” Authors: Unknown Abstract In the theory of financial markets, a stylized fact is a qualitative summary of a pattern in financial market data that is observed across multiple assets, asset classes and time horizons. In this article, we test a set of eleven stylized facts for financial market data. Our main contribution is to consider a broad range of geographical regions across Asia, continental Europe, and the US over a time period of 150 years, as well as two of the most traded cryptocurrencies, thus providing insights into the robustness and generalizability of commonly known stylized facts. ...

CoFinDiff: Controllable Financial Diffusion Model for Time Series Generation ArXiv ID: 2503.04164 “View on arXiv” Authors: Unknown Abstract The generation of synthetic financial data is a critical technology in the financial domain, addressing challenges posed by limited data availability. Traditionally, statistical models have been employed to generate synthetic data. However, these models fail to capture the stylized facts commonly observed in financial data, limiting their practical applicability. Recently, machine learning models have been introduced to address the limitations of statistical models; however, controlling synthetic data generation remains challenging. We propose CoFinDiff (Controllable Financial Diffusion model), a synthetic financial data generation model based on conditional diffusion models that accept conditions about the synthetic time series. By incorporating conditions derived from price data into the conditional diffusion model via cross-attention, CoFinDiff learns the relationships between the conditions and the data, generating synthetic data that align with arbitrary conditions. Experimental results demonstrate that: (i) synthetic data generated by CoFinDiff capture stylized facts; (ii) the generated data accurately meet specified conditions for trends and volatility; (iii) the diversity of the generated data surpasses that of the baseline models; and (iv) models trained on CoFinDiff-generated data achieve improved performance in deep hedging task. ...

Generation of synthetic financial time series by diffusion models ArXiv ID: 2410.18897 “View on arXiv” Authors: Unknown Abstract Despite its practical significance, generating realistic synthetic financial time series is challenging due to statistical properties known as stylized facts, such as fat tails, volatility clustering, and seasonality patterns. Various generative models, including generative adversarial networks (GANs) and variational autoencoders (VAEs), have been employed to address this challenge, although no model yet satisfies all the stylized facts. We alternatively propose utilizing diffusion models, specifically denoising diffusion probabilistic models (DDPMs), to generate synthetic financial time series. This approach employs wavelet transformation to convert multiple time series (into images), such as stock prices, trading volumes, and spreads. Given these converted images, the model gains the ability to generate images that can be transformed back into realistic time series by inverse wavelet transformation. We demonstrate that our proposed approach satisfies stylized facts. ...

Can GANs Learn the Stylized Facts of Financial Time Series? ArXiv ID: 2410.09850 “View on arXiv” Authors: Unknown Abstract In the financial sector, a sophisticated financial time series simulator is essential for evaluating financial products and investment strategies. Traditional back-testing methods have mainly relied on historical data-driven approaches or mathematical model-driven approaches, such as various stochastic processes. However, in the current era of AI, data-driven approaches, where models learn the intrinsic characteristics of data directly, have emerged as promising techniques. Generative Adversarial Networks (GANs) have surfaced as promising generative models, capturing data distributions through adversarial learning. Financial time series, characterized ‘stylized facts’ such as random walks, mean-reverting patterns, unexpected jumps, and time-varying volatility, present significant challenges for deep neural networks to learn their intrinsic characteristics. This study examines the ability of GANs to learn diverse and complex temporal patterns (i.e., stylized facts) of both univariate and multivariate financial time series. Our extensive experiments revealed that GANs can capture various stylized facts of financial time series, but their performance varies significantly depending on the choice of generator architecture. This suggests that naively applying GANs might not effectively capture the intricate characteristics inherent in financial time series, highlighting the importance of carefully considering and validating the modeling choices. ...

No Tick-Size Too Small: A General Method for Modelling Small Tick Limit Order Books ArXiv ID: 2410.08744 “View on arXiv” Authors: Unknown Abstract Tick-sizes not only influence the granularity of the price formation process but also affect market agents’ behavior. We investigate the disparity in the microstructural properties of the Limit Order Book (LOB) across a basket of assets with different relative tick-sizes. A key contribution of this study is the identification of several stylized facts, which are used to differentiate between large, medium, and small-tick assets, along with clear metrics for their measurement. We provide cross-asset visualizations to illustrate how these attributes vary with relative tick-size. Further, we propose a Hawkes Process model that {"\color{black"}not only fits well for large-tick assets, but also accounts for }sparsity, multi-tick level price moves, and the shape of the LOB in small-tick assets. Through simulation studies, we demonstrate the {"\color{black"} versatility} of the model and identify key variables that determine whether a simulated LOB resembles a large-tick or small-tick asset. Our tests show that stylized facts like sparsity, shape, and relative returns distribution can be smoothly transitioned from a large-tick to a small-tick asset using our model. We test this model’s assumptions, showcase its challenges and propose questions for further directions in this area of research. ...

Reinforcement Learning in Agent-Based Market Simulation: Unveiling Realistic Stylized Facts and Behavior ArXiv ID: 2403.19781 “View on arXiv” Authors: Unknown Abstract Investors and regulators can greatly benefit from a realistic market simulator that enables them to anticipate the consequences of their decisions in real markets. However, traditional rule-based market simulators often fall short in accurately capturing the dynamic behavior of market participants, particularly in response to external market impact events or changes in the behavior of other participants. In this study, we explore an agent-based simulation framework employing reinforcement learning (RL) agents. We present the implementation details of these RL agents and demonstrate that the simulated market exhibits realistic stylized facts observed in real-world markets. Furthermore, we investigate the behavior of RL agents when confronted with external market impacts, such as a flash crash. Our findings shed light on the effectiveness and adaptability of RL-based agents within the simulation, offering insights into their response to significant market events. ...

Stylized Facts and Market Microstructure: An In-Depth Exploration of German Bond Futures Market ArXiv ID: 2401.10722 “View on arXiv” Authors: Unknown Abstract This paper presents an in-depth analysis of stylized facts in the context of futures on German bonds. The study examines four futures contracts on German bonds: Schatz, Bobl, Bund and Buxl, using tick-by-tick limit order book datasets. It uncovers a range of stylized facts and empirical observations, including the distribution of order sizes, patterns of order flow, and inter-arrival times of orders. The findings reveal both commonalities and unique characteristics across the different futures, thereby enriching our understanding of these markets. Furthermore, the paper introduces insightful realism metrics that can be used to benchmark market simulators. The study contributes to the literature on financial stylized facts by extending empirical observations to this class of assets, which has been relatively underexplored in existing research. This work provides valuable guidance for the development of more accurate and realistic market simulators. ...

Revisiting Cont’s Stylized Facts for Modern Stock Markets ArXiv ID: 2311.07738 “View on arXiv” Authors: Unknown Abstract In 2001, Rama Cont introduced a now-widely used set of ‘stylized facts’ to synthesize empirical studies of financial price changes (returns), resulting in 11 statistical properties common to a large set of assets and markets. These properties are viewed as constraints a model should be able to reproduce in order to accurately represent returns in a market. It has not been established whether the characteristics Cont noted in 2001 still hold for modern markets following significant regulatory shifts and technological advances. It is also not clear whether a given time series of financial returns for an asset will express all 11 stylized facts. We test both of these propositions by attempting to replicate each of Cont’s 11 stylized facts for intraday returns of the individual stocks in the Dow 30, using the same authoritative data as that used by the U.S. regulator from October 2018 - March 2019. We find conclusive evidence for eight of Cont’s original facts and no support for the remaining three. Our study represents the first test of Cont’s 11 stylized facts against a consistent set of stocks, therefore providing insight into how these stylized facts should be viewed in the context of modern stock markets. ...