Econophysics & Complex Systems

We assess the applicability of rough volatility models to Bitcoin realized volatility using the normalised p-variation framework of Cont and Das (2024). Applying this model-free estimator to high-frequency Bitcoin data from 2017 to 2024 across multiple sampling resolutions, we find that the normalis

Universal power laws have been scrutinised in physics and beyond, and a long-standing debate exists in econophysics regarding the strict universality of the nonlinear price impact, commonly referred to as the square-root law (SRL). The SRL posits that the average price impact $I$ follows a power law

In this work, we aim to reconcile several apparently contradictory observations in market microstructure: is the famous “square-root law” of metaorder impact, which decays with time, compatible with the random-walk nature of prices and the linear impact of order imbalances? Can one entirely explain

We propose a microstructural model for the order flow in financial markets that distinguishes between {\it core orders} and {\it reaction flow}, both modeled as Hawkes processes. This model has a natural scaling limit that reconciles a number of salient empirical properties: persistent signed order

Based on the cryptocurrency market dynamics, this study presents a general methodology for analyzing evolving correlation structures in complex systems using the $q$-dependent detrended cross-correlation coefficient ρ(q,s). By extending traditional metrics, this approach captures correlations at var

Portfolio optimization is a routine asset management operation conducted in financial institutions around the world. However, under real-world constraints such as turnover limits and transaction costs, its formulation becomes a mixed-integer nonlinear program that current mixed-integer optimizers of

We develop a rotation-invariant neural network that provides the global minimum-variance portfolio by jointly learning how to lag-transform historical returns and how to regularise both the eigenvalues and the marginal volatilities of large equity covariance matrices. This explicit mathematical mapp

This paper investigates the impact of posterior drift on out-of-sample forecasting accuracy in overparametrized machine learning models. We document the loss in performance when the loadings of the data generating process change between the training and testing samples. This matters crucially in set

This work extends and complements our previous theoretical paper on the subtle interplay between impact, order flow and volatility. In the present paper, we generate synthetic market data following the specification of that paper and show that the approximations made there are actually justified, wh

A multivariate fractional Brownian motion (mfBm) with component-wise Hurst exponents is used to model and forecast realized volatility. We investigate the interplay between correlation coefficients and Hurst exponents and propose a novel estimation method for all model parameters, establishing consi

We introduce the multivariate Log S-fBM model (mLog S-fBM), extending the univariate framework proposed by Wu \textit{“et al.”} to the multidimensional setting. We define the multidimensional Stationary fractional Brownian motion (mS-fBM), characterized by marginals following S-fBM dynamics and a sp

Financial returns are known to exhibit heavy tails, volatility clustering and abrupt jumps that are poorly captured by classical diffusion models. Advances in machine learning have enabled highly flexible functional forms for conditional means and volatilities, yet few models deliver interpretable s

The non-fungible token (NFT) market emerges as a recent trading innovation leveraging blockchain technology, mirroring the dynamics of the cryptocurrency market. The current study is based on the capitalization changes and transaction volumes across a large number of token collections on the Ethereu

In financial markets, the market order sign exhibits strong persistence, widely known as the long-range correlation (LRC) of order flow; specifically, the sign correlation function displays long memory with power-law exponent $γ$, such that $C(τ) \propto τ^{-γ}$ for large time-lag $τ$. One of the mo

We report the first application of a tailored Complexity-Entropy Plane designed for binary sequences and structures. We do so by considering the daily up/down price fluctuations of the largest cryptocurrencies in terms of capitalization (stable-coins excluded) that are worth $circa ,, 90 %$ of the t

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 ob

Understanding the impact of trades on prices is a crucial question for both academic research and industry practice. It is well established that impact follows a square-root impact as a function of traded volume. However, the microscopic origin of such a law remains elusive: empirical studies are pa

In this paper, we introduce the Dynamic Modularity-Spectral Algorithm (DynMSA), a novel approach to identify clusters of stocks with high intra-cluster correlations and low inter-cluster correlations by combining Random Matrix Theory with modularity optimisation and spectral clustering. The primary

We introduce a novel large-scale deep learning model for Limit Order Book mid-price changes forecasting, and we name it `HLOB’. This architecture (i) exploits the information encoded by an Information Filtering Network, namely the Triangulated Maximally Filtered Graph, to unveil deeper and non-trivi

Correlations in complex systems are often obscured by nonstationarity, long-range memory, and heavy-tailed fluctuations, which limit the usefulness of traditional covariance-based analyses. To address these challenges, we construct scale and fluctuation-dependent correlation matrices using the multi

Financial markets exhibit an apparent paradox: while directional price movements remain largely unpredictable–consistent with weak-form efficiency–the magnitude of price changes displays systematic structure. Here we demonstrate that real-time order-flow entropy, computed from a 15-state Markov tran

This paper proposes a reinforcement learning–based framework for cryptocurrency portfolio management using the Soft Actor–Critic (SAC) and Deep Deterministic Policy Gradient (DDPG) algorithms. Traditional portfolio optimization methods often struggle to adapt to the highly volatile and nonlinear dyn

The finite sample effect on the Hurst exponent (HE) of realized volatility time series is examined using Bitcoin data. This study finds that the HE decreases as the sampling period $Δ$ increases and a simple finite sample ansatz closely fits the HE data. We obtain values of the HE as $Δ\rightarrow 0

This study presents a unified, distribution-aware, and complexity-informed framework for understanding equity return dynamics in the Indian market, using 34 years (1990 to 2024) of Nifty 50 index data. Addressing a key gap in the literature, we demonstrate that the price to earnings ratio, as a valu

Multifractality is a concept that helps compactly grasping the most essential features of the financial dynamics. In its fully developed form, this concept applies to essentially all mature financial markets and even to more liquid cryptocurrencies traded on the centralized exchanges. A new element

We investigate the impact of several critical events associated with the Russo Ukrainian war, started officially on 24 February 2022 with the Russian invasion of Ukraine, on ten European electricity markets, two natural gas markets (the European reference trading hub TTF and N.Y. NGNMX market) and h

This study investigates the relationships between agricultural spot markets and external uncertainties via the multifractal detrending moving-average cross-correlation analysis (MF-X-DMA). The dataset contains the Grains & Oilseeds Index (GOI) and its five sub-indices of wheat, maize, soyabeans, ric

Networks of financial exposures are the key propagators of risk and distress among banks, but their empirical structure is not publicly available because of confidentiality. This limitation has triggered the development of methods of network reconstruction from partial, aggregate information. Unfort

We study the temporal evolution of the holding-time distribution of bitcoins and find that the average distribution of holding-time is a heavy-tailed power law extending from one day to over at least $200$ weeks with an exponent approximately equal to $0.9$, indicating very long memory effects. We a

Drawing inspiration from the significant impact of the ongoing Russia-Ukraine conflict and the recent COVID-19 pandemic on global financial markets, this study conducts a thorough analysis of three key crude oil futures markets: WTI, Brent, and Shanghai (SC). Employing the visibility graph (VG) meth

Trade prices of about 1000 New York Stock Exchange-listed stocks are studied at one-minute time resolution over the continuous five year period 2018–2022. For each stock, in dollar-volume-weighted transaction time, the discrepancy from a Brownian-motion martingale is measured on timescales of minute

This paper studies the q-learning, recently coined as the continuous time counterpart of Q-learning by Jia and Zhou (2023), for continuous time Mckean-Vlasov control problems in the setting of entropy-regularized reinforcement learning. In contrast to the single agent’s control problem in Jia and Zh

We construct an ensemble of correlation matrices from high-frequency foreign exchange market data, with one matrix for every day for 446 days. The matrices are symmetric and have vanishing diagonal elements after subtracting the identity matrix. For this case, we construct the general permutation in

We present a scalable, hardware-aware methodology for extending the Variational Quantum Eigensolver (VQE) to large, realistic Dynamic Portfolio Optimization (DPO) problems. Building on the scaling strategy from our previous work, where we tailored a VQE workflow to both the DPO formulation and the t

We address the attribution problem for apparent slow collective dynamics: is the observed persistence intrinsic, or inherited from a persistent driver? For the leading eigenvalue fraction $ψ_1=λ_{\max}/N$ of S&P 500 60-day rolling correlation matrices ($237$ stocks, 2004–2023), a VIX-coupled Ornst

This paper investigates the impact of environmental, social, and governance (ESG) constraint on a regularized mean-variance (MV) portfolio optimization problem in a large-dimensional setting, in which a positive definite regularization matrix is imposed on the sample covariance matrix. We first deri

Recent crises like the COVID-19 pandemic and geopolitical tensions have exposed vulnerabilities and caused disruptions of supply chains, leading to product shortages, increased costs, and economic instability. This has prompted increasing efforts to assess systemic risk, namely the effects of firm d

Multivariate Distributions are needed to capture the correlation structure of complex systems. In previous works, we developed a Random Matrix Model for such correlated multivariate joint probability density functions that accounts for the non-stationarity typically found in complex systems. Here, w

The distribution of income and wealth in developed economies exhibits a robust two-class structure: an exponential (Boltzmann–Gibbs) bulk covering $\sim!97%$ of the population, and a power-law (Pareto) tail in the upper $\sim!3%$. We derive this structure from first principles via an explicit m

Portfolio optimization under cardinality constraints transforms the classical Markowitz mean-variance problem from a convex quadratic problem into an NP-hard combinatorial optimization problem. This paper introduces a novel approach using THRML (Thermodynamic HypergRaphical Model Library), a JAX-bas

Speculative bubbles exhibit common statistical signatures across many financial markets, suggesting the presence of universal underlying mechanisms. We test this hypothesis in the Iranian stock market, an economy that is highly isolated, subject to capital controls, and largely inaccessible to forei

This paper introduces a comprehensive framework for Financial Information Theory by applying information-theoretic concepts such as entropy, Kullback-Leibler divergence, mutual information, normalized mutual information, and transfer entropy to financial time series. We systematically derive these m

We formalize the paradox of an omniscient yet lazy investor - a perfectly informed agent who trades infrequently due to execution or computational frictions. Starting from a deterministic geometric construction, we derive a closed-form expected profit function linking trading frequency, execution co

Covariance matrices estimated from short, noisy, and non-Gaussian financial time series are notoriously unstable. Empirical evidence suggests that such covariance structures often exhibit power-law scaling, reflecting complex, hierarchical interactions among assets. Motivated by this observation, we

We introduce a novel distribution-based estimator for the Hurst parameter of log-volatility, leveraging the Kolmogorov-Smirnov statistic to assess the scaling behavior of entire distributions rather than individual moments. To address the temporal dependence of financial volatility, we propose a ran

Purpose: This study introduces a novel framework for identifying and exploiting predictive lead-lag relationships in financial markets. We propose an integrated approach that combines advanced statistical methodologies with machine learning models to enhance the identification and exploitation of pr

We show that assuming that the returns are independent when conditioned on the value of their variance (volatility), which itself varies in time randomly, then the distribution of returns is well described by the statistics of the sum of conditionally independent random variables. In particular, we

This paper investigates the structural dynamics of stock market volatility through the Financial Chaos Index, a tensor- and eigenvalue-based measure designed to capture realized volatility via mutual fluctuations among asset prices. Motivated by empirical evidence of regime-dependent volatility beha

Implied volatility IV is a key metric in financial markets, reflecting market expectations of future price fluctuations. Research has explored IV’s relationship with moneyness, focusing on its connection to the implied Hurst exponent H. Our study reveals that H approaches 1/2 when moneyness equals 1

The Kullback-Leibler cluster entropy $\mathcal{“D_{C”}}[“P | Q”] $ is evaluated for the empirical and model probability distributions $P$ and $Q$ of the clusters formed in the realized volatility time series of five assets (SP&500, NASDAQ, DJIA, DAX, FTSEMIB). The Kullback-Leibler functional $\mathc

Granger causality can uncover the cause and effect relationships in financial networks. However, such networks can be convoluted and difficult to interpret, but the Helmholtz-Hodge-Kodaira decomposition can split them into a rotational and gradient component which reveals the hierarchy of Granger ca

A non-fungible token (NFT) market is a new trading invention based on the blockchain technology which parallels the cryptocurrency market. In the present work we study capitalization, floor price, the number of transactions, the inter-transaction times, and the transaction volume value of a few sele

In this study, we introduce a physical model inspired by statistical physics for predicting price volatility and expected returns by leveraging Level 3 order book data. By drawing parallels between orders in the limit order book and particles in a physical system, we establish unique measures for th

We introduce a Path Shadowing Monte-Carlo method, which provides prediction of future paths, given any generative model. At any given date, it averages future quantities over generated price paths whose past history matches, or shadows’, the actual (observed) history. We test our approach using path

Considering that both the entropy-based market information and the Hurst exponent are useful tools for determining whether the efficient market hypothesis holds for a given asset, we study the link between the two approaches. We thus provide a theoretical expression for the market information when l

Portfolio replication, or the construction of a tradable basket of assets to match the risk-return profile of a target benchmark, is fundamentally an ill-posed inverse problem. When restricted to a subset of available assets, classical variance-minimizing models often yield unstable, over-leveraged

Scaling laws in modern deep learning describe how held-out loss improves as model capacity, training data, and compute increase, often following power-law trends. We investigate whether analogous scaling regularities arise in actuarial ratemaking, where data are tabular, heterogeneous, and noisy, an

Whether heterogeneous investor flows transmit private information across stocks or merely reflect coordinated responses to public signals remains an open question in market microstructure. We construct Transfer Entropy (TE) networks from investor-type flows – foreign, institutional, and individual

The rough Bergomi (rBergomi) model, characterised by its roughness parameter $H$, has been shown to exhibit multiscaling behaviour as $H$ approaches zero. Multiscaling has profound implications for financial modelling: it affects extreme risk estimation, influences optimal portfolio allocation acros

We use the discrete Ollivier-Ricci graph curvature with Ricci flow to examine the intrinsic geometry of financial markets through the empirical correlation graph of the NASDAQ 100 index. Our main result is the development of a technique to perform surgery on the neckpinch singularities that form dur

Financial markets are dynamic, interconnected systems where local shocks can trigger widespread instability, challenging portfolio managers and policymakers. Traditional correlation analysis often miss the directionality and temporal dynamics of information flow. To address this, we present a unifie

We empirically analyze the reversion of financial market trends with time horizons ranging from minutes to decades. The analysis covers equities, interest rates, currencies and commodities and combines 14 years of futures tick data, 30 years of daily futures prices, 330 years of monthly asset prices

Macroscopic properties of equity markets affect the performance of active equity strategies but many are not adequately captured by conventional models of financial mathematics and econometrics. Using the CRSP Database of the US equity market, we study empirically several macroscopic properties defi

Production networks constitute the backbone of every economic system. They are inherently fragile as several recent crises clearly highlighted. Estimating the system-wide consequences of local disruptions (systemic risk) requires detailed information on the supply chain networks (SCN) at the firm-le

An increase in the novelty of news predicts negative stock market returns and negative macroeconomic outcomes over the next year. We quantify news novelty - changes in the distribution of news text - through an entropy measure, calculated using a recurrent neural network applied to a large news corp

We identify a robust structural signature of stock markets during exogenous shock events by analyzing collective return dynamics across G5 countries. Using Random Matrix Theory, we introduce the complexity gap, defined as the difference between the normalized largest eigenvalue and the average pairw

A new wave of work on covariance cleaning and nonlinear shrinkage has delivered asymptotically optimal analytical solutions for large covariance matrices. Building on this progress, these ideas have been generalized to empirical cross-covariance matrices, whose singular-value shrinkage characterizes

This paper introduces the Fractal-Chaotic Oscillation Co-driven (FCOC) framework, a novel paradigm for financial volatility forecasting that systematically resolves the dual challenges of feature fidelity and model responsiveness. FCOC synergizes two core innovations: our novel Fractal Feature Corre

We study the eigenvalue of the Wigner random matrix, which is created from a time series with temporal correlation. We observe the deformation of the semi-circle law which is similar to the eigenvalue distribution of the Wigner-Lèvy matrix. The distribution has a longer tail and a higher peak than t

We model the time evolution of single win odds in Japanese horse racing as a stochastic process, deriving an Ornstein–Uhlenbeck process by analyzing the probability dynamics of vote shares and the empirical time series of odds movements. Our framework incorporates two types of bettors: herders, who

We study optimal stopping for diffusion processes with unknown model primitives within the continuous-time reinforcement learning (RL) framework developed by Wang et al. (2020), and present applications to option pricing and portfolio choice. By penalizing the corresponding variational inequality fo

Modeling the dynamics of non-stationary stochastic systems requires balancing the representational power of deep learning with the mathematical transparency of classical models. While classical Markov transition operators provide explicit, theoretically grounded rules for system evolution, their emp

Volatility is the canonical measure of financial risk, a role largely inherited from Modern Portfolio Theory. Yet, its universality rests on restrictive efficiency assumptions that render volatility, at best, an incomplete proxy for true risk. This paper identifies three fundamental inconsistencies:

We introduce a method for describing eigenvalue distributions of correlation matrices from multidimensional time series. Using our newly developed matrix H theory, we improve the description of eigenvalue spectra for empirical correlation matrices in multivariate financial data by considering an inf

The analysis of logarithmic return distributions defined over large time scales is crucial for understanding the long-term dynamics of asset price movements. For large time scales of the order of two trading years, the anticipated Gaussian behavior of the returns often does not emerge, and their dis

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

This paper offers a thorough examination of the univariate predictability in cryptocurrency time-series. By exploiting a combination of complexity measure and model predictions we explore the cryptocurrencies time-series forecasting task focusing on the exchange rate in USD of Litecoin, Binance Coin

Starting from a basic model in which the dynamic of the transaction prices is a geometric Brownian motion disrupted by a microstructure white noise, corresponding to the random alternation of bids and asks, we propose moment-based estimators along with their statistical properties. We then make the

Limit Order Books (LOBs) serve as a mechanism for buyers and sellers to interact with each other in the financial markets. Modelling and simulating LOBs is quite often necessary for calibrating and fine-tuning the automated trading strategies developed in algorithmic trading research. The recent AI

We propose a formulation to construct new classes of financial price processes based on the insight that the key variable driving prices $P$ is the earning-over-price ratio $γ\simeq 1/P$, which we refer to as the earning yield and is analogous to the yield-to-maturity of an equivalent perpetual bond

The Nested factor model was introduced by Chicheportiche et al. to represent non-linear correlations between stocks. Stock returns are explained by a standard factor model, but the (log)-volatilities of factors and residuals are themselves decomposed into factor modes, with a common dominant volatil

Markowitz’s optimal portfolio relies on the accurate estimation of correlations between asset returns, a difficult problem when the number of observations is not much larger than the number of assets. Using powerful results from random matrix theory, several schemes have been developed to “clean” th

The increasing use of cryptoassets for international remittances has proven to be faster and more cost-effective, particularly for migrants without access to traditional banking. However, the inherent volatility of cryptoasset prices, independent of blockchain-based remittance mechanisms, introduces

This paper studies a discrete-time mean-variance model based on reinforcement learning. Compared with its continuous-time counterpart in \cite{“zhou2020mv”}, the discrete-time model makes more general assumptions about the asset’s return distribution. Using entropy to measure the cost of exploration

In this paper, we perform a comprehensive study of different covariance and precision matrix estimation methods in the context of minimum variance portfolio allocation. The set of models studied by us can be broadly categorized as: Gaussian Graphical Model (GGM) based methods, Shrinkage Methods, Thr

Fractional Brownian motion has become a standard tool to address long-range dependence in financial time series. However, a constant memory parameter is too restrictive to address different market conditions. Here we model the price fluctuations using a multifractional Brownian motion assuming that

In this study the cross-correlations between the cryptocurrency market represented by the two most liquid and highest-capitalized cryptocurrencies: bitcoin and ethereum, on the one side, and the instruments representing the traditional financial markets: stock indices, Forex, commodities, on the oth

Uniswap is a Constant Product Market Maker built around liquidity pools, where pairs of tokens are exchanged subject to a fee that is proportional to the size of transactions. At the time of writing, there exist more than 6,000 pools associated with Uniswap v3, implying that empirical investigations

Multifractal analysis is a forecasting technique used to study the scaling regularity properties of financial returns, to analyze the long-term memory and predictability of financial markets. In this paper, we propose a novel structural detrended multifractal fluctuation analysis (S-MF-DFA) to inves

Relative entropy, as a divergence metric between two distributions, can be used for offline change-point detection and extends classical methods that mainly rely on moment-based discrepancies. To build a statistical test suitable for this context, we study the distribution of empirical relative entr

The linear fractional stable motion (LFSM) extends the fractional Brownian motion (fBm) by considering $α$-stable increments. We propose a method to forecast future increments of the LFSM from past discrete-time observations, using the conditional expectation when $α>1$ or a semimetric projection ot

In 1955 Fermi, Pasta, Ulam and Tsingou performed first numerical studies with the aim to obtain the thermalization in a chain of nonlinear oscillators from dynamical equations of motion. This model happend to have several specific features and the dynamical thermalization was established only later

We study the allocation of synthetic portfolios under hierarchical nested, one-factor, and diagonal structures of the population covariance matrix in a high-dimensional scenario. The noise reduction approaches for the sample realizations are based on random matrices, free probability, deterministic

In this paper, we analyze the asymptotic behavior of the main characteristics of the mean-variance efficient frontier employing random matrix theory. Our particular interest covers the case when the dimension $p$ and the sample size $n$ tend to infinity simultaneously and their ratio $p/n$ tends to

We take inspiration from statistical physics to develop a novel conceptual framework for the analysis of financial markets. We model the order book dynamics as a motion of particles and define the momentum measure of the system as a way to summarise and assess the state of the market. Our approach p

Agent-based models (ABMs) are a promising approach to modelling and reasoning about complex systems, yet their application in practice is impeded by their complexity, discrete nature, and the difficulty of performing parameter inference and optimisation tasks. This in turn has sparked interest in th

This work aims to deal with the optimal allocation instability problem of Markowitz’s modern portfolio theory in high dimensionality. We propose a combined strategy that considers covariance matrix estimators from Random Matrix Theory~(RMT) and the machine learning allocation methodology known as Ne

Cryptocurrencies are considered the latest innovation in finance with considerable impact across social, technological, and economic dimensions. This new class of financial assets has also motivated a myriad of scientific investigations focused on understanding their statistical properties, such as

This paper introduces a novel algorithm for generating realistic metaorders from public trade data, addressing a longstanding challenge in price impact research that has traditionally relied on proprietary datasets. Our method effectively recovers all established stylized facts of metaorders impact,

Twenty five years ago, several authors proposed to describe the forward interest rate curve (FRC) as an elastic string along which idiosyncratic shocks propagate, accounting for the peculiar structure of the return correlation across different maturities. In this paper, we revisit the specific “stif