Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders

A Non-Stochastic Special Model of Risk Based on Radon Transform

The concept of risk is fundamental in various scientific fields, including physics, biology and engineering, and is crucial for the study of complex systems, especially financial markets. In our research, we introduce a novel risk model that has a natural transactional-financial interpretation. In our approach, the risk of holding a financial instrument is related to the measure of the possibility of its loss. In this context, a financial instrument is riskier the more opportunities there are to dispose of it, i.e., to sell it. We present a model of risk understood in this way, introducing, in particular, the concept of financial time and a financial frame of reference, which allows for associating risk with the subjective perception of the observer. The presented approach does not rely on statistical assumptions and is based on the transactional interpretation of models. To measure risk, we propose using the Radon transform. The financial concept of risk is closely related to the concepts of uncertainty, entropy, information, and error in physics. Therefore, the well-established algorithmic aspects of the computed tomography method can be effectively applied to the broader field of uncertainty analysis, which is one of the foundational elements of experimental physics.

Thermodynamic Analysis of Financial Markets: Measuring Order Book Dynamics with Temperature and Entropy

This study bridges finance and physics by applying thermodynamic concepts to model the limit order book (LOB) with high-frequency trading data on the Bitcoin spot. We derive the measures of Market Temperature and Market Entropy from the kinetic and potential energies in the LOB to provide a deeper understanding of order activities and market participant behavior. Market Temperature emerges as a robust indicator of market liquidity, correlating with liquidity measures such as Active Quote Volume, bid-ask spread and match volume. Market Entropy, on the other hand, quantifies the degree of disorder or randomness in the LOB, providing insights into the instantaneous volatility of price in the high-frequency trading market. Our empirical findings not only broaden the theoretical framework of econophysics but also enhance comprehensive understanding of the market microstructure and order book dynamics.

Inferring Microscopic Financial Information from the Long Memory in Market-Order Flow: A Quantitative Test of the Lillo-Mike-Farmer Model

In financial markets, the market-order sign exhibits strong persistence, widely known as the long-range correlation (LRC) of order flow; specifically, the sign autocorrelation function (ACF) displays long memory with power-law exponent γ, such that C(τ)∝τ^{-γ} for large time-lag τ. One of the most promising microscopic hypotheses is the order-splitting behavior at the level of individual traders. Indeed, Lillo, Mike, and Farmer (LMF) introduced in 2005 a simple microscopic model of order-splitting behavior, which predicts that the macroscopic sign correlation is quantitatively associated with the microscopic distribution of metaorders. While this hypothesis has been a central issue of debate in econophysics, its direct quantitative validation has been missing because it requires large microscopic datasets with high resolution to observe the order-splitting behavior of all individual traders. Here we present the first quantitative validation of this LMF prediction by analyzing a large microscopic dataset in the Tokyo Stock Exchange market for more than nine years. On classifying all traders as either order-splitting traders or random traders as a statistical clustering, we directly measured the metaorder-length distributions P(L)∝L^{-α-1} as the microscopic parameter of the LMF model and examined the theoretical prediction on the macroscopic order correlation γ≈α-1. We discover that the LMF prediction agrees with the actual data even at the quantitative level. We also discuss the estimation of the total number of the order-splitting traders from the ACF prefactor, showing that microscopic financial information can be inferred from the LRC in the ACF. Our Letter provides the first solid support of the microscopic model and solves directly a long-standing problem in the field of econophysics and market microstructure.

Potential fields and fluctuation-dissipation relations derived from human flow in urban areas modeled by a network of electric circuits

Owing to the big data the extension of physical laws on nonmaterial has seen numerous successes, and human mobility is one of the scientific frontier topics. Recent GPS technology has made it possible to trace detailed trajectories of millions of people, macroscopic approaches such as the gravity law for human flow between cities and microscopic approaches of individual origin-destination distributions are attracting much attention. However, we need a more general basic model with wide applicability to realize traffic forecasting and urban planning of metropolis fully utilizing the GPS data. Here, based on a novel idea of treating moving people as charged particles, we introduce a method to map macroscopic human flows into currents on an imaginary electric circuit defined over a metropolitan area. Conductance is found to be nearly proportional to the maximum current in each location and synchronized human flows in the morning and evening are well described by the temporal changes of electric potential. Surprisingly, the famous fluctuation-dissipation theorem holds, namely, the variances of currents are proportional to the conductivities akin to an ordinary material.

Analysis of Individual High-Frequency Traders’ Buy–Sell Order Strategy Based on Multivariate Hawkes Process

Traders who instantly react to changes in the financial market and place orders in milliseconds are called high-frequency traders (HFTs). HFTs have recently become more prevalent and attracting attention in the study of market microstructures. In this study, we used data to track the order history of individual HFTs in the USD/JPY forex market to reveal how individual HFTs interact with the order book and what strategies they use to place their limit orders. Specifically, we introduced an 8-dimensional multivariate Hawkes process that included the excitations due to the occurrence of limit orders, cancel orders, and executions in the order book change, and performed maximum likelihood estimations of the limit order processes for 134 HFTs. As a result, we found that the limit order generation processes of 104 of the 134 HFTs were modeled by a multivariate Hawkes process. In this analysis of the EBS market, the HFTs whose strategies were modeled by the Hawkes process were categorized into three groups according to their excitation mechanisms: (1) those excited by executions; (2) those that were excited by the occurrences or cancellations of limit orders; and (3) those that were excited by their own orders.

Linear response theory in stock markets

Linear response theory relates the response of a system to a weak external force with its dynamics in equilibrium, subjected to fluctuations. Here, this framework is applied to financial markets; in particular we study the dynamics of a set of stocks from the NASDAQ during the last 20 years. Because unambiguous identification of external forces is not possible, critical events are identified in the series of stock prices as sudden changes, and the stock dynamics following an event is taken as the response to the external force. Linear response theory is applied with the log-return as the conjugate variable of the force, providing predictions for the average response of the price and return, which agree with observations, but fails to describe the volatility because this is expected to be beyond linear response. The identification of the conjugate variable allows us to define the perturbation energy for a system of stocks, and observe its relaxation after an event.

The prediction of fluctuation in the order-driven financial market

Risk prediction is one of the important issues that draws much attention from academia and industry. And the fluctuation-absolute value of the change of price, is one of the indexes of risk. In this paper, we focus on the relationship between fluctuation and order volume. Based on the observation that the price would move when the volume of order changes, the prediction of price fluctuation can be converted into the prediction of order volume. Modelling the trader's behaviours-order placement and order cancellation, we propose an order-based fluctuation prediction model. And our model outperforms better than baseline in OKCoin and BTC-e datasets.

Effect of Savings on a Gas-Like Model Economy with Credit and Debt

In kinetic exchange models, agents make transactions based on well-established microscopic rules that give rise to macroscopic variables in analogy to statistical physics. These models have been applied to study processes such as income and wealth distribution, economic inequality sources, economic growth, etc., recovering well-known concepts in the economic literature. In this work, we apply ensemble formalism to a geometric agents model to study the effect of saving propensity in a system with money, credit, and debt. We calculate the partition function to obtain the total money of the system, with which we give an interpretation of the economic temperature in terms of the different payment methods available to the agents. We observe an interplay between the fraction of money that agents can save and their maximum debt. The system’s entropy increases as a function of the saved proportion, and increases even more when there is debt.

The microscopic relationships between triangular arbitrage and cross-currency correlations in a simple agent based model of foreign exchange markets

Foreign exchange rates movements exhibit significant cross-correlations even on very short time-scales. The effect of these statistical relationships become evident during extreme market events, such as flash crashes. Although a deep understanding of cross-currency correlations would be clearly beneficial for conceiving more stable and safer foreign exchange markets, the microscopic origins of these interdependencies have not been extensively investigated. This paper introduces an agent-based model which describes the emergence of cross-currency correlations from the interactions between market makers and an arbitrager. The model qualitatively replicates the time-scale vs. cross-correlation diagrams observed in real trading data, suggesting that triangular arbitrage plays a primary role in the entanglement of the dynamics of different foreign exchange rates. Furthermore, the model shows how the features of the cross-correlation function between two foreign exchange rates, such as its sign and value, emerge from the interplay between triangular arbitrage and trend-following strategies. In particular, the interaction of these trading strategies favors certain combinations of price trend signs across markets, thus altering the probability of observing two foreign exchange rates drifting in the same or opposite direction. Ultimately, this entangles the dynamics of foreign exchange rate pairs, leading to cross-correlation functions that resemble those observed in real trading data.

Stock markets: A view from soft matter

Different attempts to describe financial markets, and stock prices in particular, with the tools of statistical mechanics can be found in the literature, although a general framework has not been achieved yet. In this paper we use the physics of many-particle systems and the typical concepts of soft matter to study two sets of US and European stocks, comprising the biggest and most stable companies in terms of stock price and trading. Upon correcting for the center-of-mass motion, the structure and dynamics of the systems are studied (in the European set, the structure is studied for the UK subset only). The pair distribution of the stocks, corrected to account for the nonuniform distribution of prices, is close to 1, indicating that there is no direct interaction between stocks, similar to an ideal gas of particles. The dynamics is studied with the mean-squared price displacement (MSPD); the price correlation function, equivalent to the intermediate scattering function; the price fluctuation distribution; and two parameters for collective motions. The MSPD grows linearly and the velocity autocorrelation function is zero, as for isolated Brownian particles. However, the intermediate scattering function follows a stretched exponential decay, the fluctuation distributions deviate from the Gaussian shape, and strong collective motions are identified. These results indicate that the dynamics is much more complex than an ideal gas of Brownian particles, and similar, to some extent, to that of undercooled systems. Finally, two physical systems are discussed to aid in the understanding of these results: a low density colloidal gel, and a dense system of ideal, infinitely thin stars. The former reproduces the dynamical properties of stocks, linear mean-squared displacement (MSD), non-Gaussian fluctuation distribution, and collective motions, but also has strong structural correlations, whereas the latter undergoes a glass transition with the structure of an ideal gas, but the MSD has the typical two-step growth of undercooled systems.

Network Dynamics of a Financial Ecosystem

Global financial crises have led to the understanding that classical econometric models are limited in comprehending financial markets in extreme conditions, partially since they disregarded complex interactions within the system. Consequently, in recent years research efforts have been directed towards modeling the structure and dynamics of the underlying networks of financial ecosystems. However, difficulties in acquiring fine-grained empirical financial data, due to regulatory limitations, intellectual property and privacy control, still hinder the application of network analysis to financial markets. In this paper we study the trading of cryptocurrency tokens on top of the Ethereum Blockchain, which is the largest publicly available financial data source that has a granularity of individual trades and users, and which provides a rare opportunity to analyze and model financial behavior in an evolving market from its inception. This quickly developing economy is comprised of tens of thousands of different financial assets with an aggregated valuation of more than 500 Billion USD and typical daily volume of 30 Billion USD, and manifests highly volatile dynamics when viewed using classic market measures. However, by applying network theory methods we demonstrate clear structural properties and converging dynamics, indicating that this ecosystem functions as a single coherent financial market. These results suggest that a better understanding of traditional markets could become possible through the analysis of fine-grained, abundant and publicly available data of cryptomarkets.

Classification of position management strategies at the order-book level and their influences on future market-price formation

Financial prices fluctuate as a results of the market impact of the flow of transactions between traders. Reciprocally, several studies of market microstructure have shown how decisions of individual traders or banks, implemented in their trading strategies, are affected by historical market information. However, little is known about the detailed processes of how such trading strategies at the micro level recursively affect future market information at the macro level. Using a special fined-grained dataset that allows us to track the complete trading behavior of specific banks in a U.S. dollar (USD) versus Japanese yen (JPY) market, we find that position management methods, defined as the number of units of USD bought or sold by banks against JPY, can be classified into two strategies: (1) banks increase their positions by trading in the same direction repeatedly, or (2) banks attempt to reduce their inventories by rapidly shifting their positions toward zero. We then demonstrate that their systematic position management strategies strongly influence future market prices, as demonstrated by our ability using this information to predict market prices about fifteen minutes in advance. Further, by detecting outlier trades, we reveal that traders seem to switch their strategies when they become aware of outlier trades. The evidence obtained here suggests that positions, which are a consequence of historical trading decisions based on the position management strategies of each bank, strongly influence future market prices, and we unravel how market prices at the macro level evolve through an interactive process involving the interaction between well-defined trading strategies at the micro level.

Q-Gaussian diffusion in stock markets

We analyze the Standard & Poor's 500 stock market index from the past 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong superdiffusion together with short-time correlations and the second one corresponds to weak superdiffusion with weak time correlations. Both regimes are well described by q-Gaussian distributions. The porous media equation-a special case of the Tsallis-Bukman equation-is used to derive the governing equation for these regimes and the Black-Scholes diffusion coefficient is explicitly obtained from the governing equation.

Crossover from Linear to Square-Root Market Impact

Using a large database of 8 million institutional trades executed in the U.S. equity market, we establish a clear crossover between a linear market impact regime and a square-root regime as a function of the volume of the order. Our empirical results are remarkably well explained by a recently proposed dynamical theory of liquidity that makes specific predictions about the scaling function describing this crossover. Allowing at least two characteristic timescales for the liquidity ("fast" and "slow") enables one to reach quantitative agreement with the data.

Ecology of trading strategies in a forex market for limit and market orders

There is a growing interest to understand financial markets as ecological systems, where the variety of trading strategies correspond to that of biological species. For this purpose, transaction data for individual traders are studied recently as empirical analyses. However, there are few empirical studies addressing how traders submit limit and market order at the level of individual traders. Since limit and market orders are key ingredients finally leading to transactions, it would be necessary to understand what kind of strategies are actually employed among traders before making transactions. Here we demonstrate the variety of limit-order and market-order strategies and show their roles in the financial markets from an ecological perspective. We find these trading strategies can be well-characterized by their response pattern to historical price changes. By applying a clustering analysis, we provide an overall picture of trading strategies as an ecological matrix, illustrating that liquidity consumers are likely to exhibit high trading performances compared with liquidity providers. Furthermore, we reveal both high-frequency traders (HFTs) and low-frequency traders (LFTs) exhibit high trading performance, despite the difference in their trading styles; HFTs attempt to maximize their trading efficiency by reducing risk, whereas LFTs make their profit by taking risk.