Trading and Price Diffusion: Stock Market Modeling Using the Approach of Statistical Physics Ph.D. thesis statements. Supervisors: Dr.

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1 Trading and Price Diffusion: Stock Market Modeling Using the Approach of Statistical Physics Ph.D. thesis statements László Gillemot Supervisors: Dr. János Kertész Dr. Doyne Farmer BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS DEPARTMENT OF THEORETICAL PHYSICS 2006

2 1 Background and motivation The economy is an evolutionary system present in everyday life, which is devoted to serving people s everyday material needs as well as possible. Similarly to other evolutionary systems, efficiency is in the spotlight of concern. The most essential method to increase efficiency is specialization, that is, separation of tasks in the process of production of goods and services according to their type. In contrast, the needs are diverse on the consumption side. The only way to make this consistent if people are allowed swap their goods. Thus, increasing efficiency implies the necessity of trading. Indeed, trading is perhaps the most essential organizing mechanism in the economy. However, trading raises the problem of comparing the value of goods, that is, pricing, and consequently, price formation. Economics, just like many other social science fields, aims to describe a system with a large number of coupled constituents. In this case, the constituents are individuals and institutions, and the interaction is trading between them. As a consequence of the diversity of the elements and the compound disciplines (evolution, psychology, game theory, networks, technology, etc.), the economy is a complex system. Conventionally, economics is divided into two areas: microeconomics, that describes the individual constituents, and macroeconomics, that is focused on the global effects. However, there is a little emphasis on how the microscopic elements and mechanisms affect the collective behavior and what are their consequences at the global level. In order to understand this problem, statistical physics provide useful tools and framework. The breakthroughs in computer technology over the last couple of decades have created new opportunities in the research 1

3 of complex systems. Computer simulations of models play an increasingly important role, since the possibility of analytical approach quickly reaches its limits. Furthermore, improvements in computer technology boosted data gathering and analysis. Observations suggest to extend the discipline of economics with alternative points of view. According to the neoclassical economics, the price of a stock can be derived from the profitability of the respective company unambiguous. The price can change only if relevant news arrives in the market. In reality, the price of a stock changes much more frequently than relevant news arrives. On the other hand, the magnitudes of changes are not in agreement with the value based approach. Indeed, price often changes when a transaction completes, as one can expect from the trading mechanism. This causes non-negligible price fluctuations at high frequencies. Statistical regularities, the so-called stylized facts, affirmed the relevance of this approach. Price changes can satisfy the assumption of unpredictability only if they can be characterized as a random walk. Thus, price formation is a diffusive process. Based on observations in real data, the three most important stylized facts concerning the price diffusion are (1) the lack of autocorrelation in the return time series, (2) the fat tails of the return distributions, and (3) the long memory of the return size time series. The importance of the stylized facts is rooted in their robust presence in real data. The stylized facts can be observed (1) for different asset types of (2) the different issuers, (3) across different markets, (4) for different time periods, and (5) for different time scales. That is, the stylized facts are macroscopic qualitative properties which are independent of most of the microscopic details. (For example, the trading structure and rules can be significantly different in the different markets.) The origin of the 2

4 stylized facts is not understood yet, however, their implications are very important. The stylized facts are assumed to be a sign of the existence of principle underlying microscopic mechanisms in the economy, which is suspected of being rooted in the mechanism of trading. The nature of this problem and the adequate physical approach established a new interdisciplinary area which is often referred as econophysics. In this thesis, I focus on the formalization and analysis of the trading mechanism, as well as its effects on the price formation. The most convenient background is provided by stock exchange trading, since it is the most idealized existent form of trading and trading data is available in large quantities in electronic format. Most of the modern financial markets use a centralized, electronic, continuous double auction trading mechanism based on the socalled limit order books. The rules of this trading mechanism are simple, yet realistic, thus it provides a proper framework for studying trading. This mechanism is robust since it is in accordance with the interests of both trader parties. The aggregated behavior of traders present in the market can be described by the order flow, which uniquely defines the price formation and related quantities, such as market impact, liquidity, supply and demand, price diffusion coefficient, as well as the relationship between these quantities. 2 Goals The goal of my study was to introduce an appropriate formalism to describe the elementary trading process, and to apply this formalism in a simple stock trading model. The model was required to contain measurable parameters exclusively in order to keep the 3

5 predictions of the model falsifiable, and to ensure possibility to derive relationships and scaling properties of the important trading related quantities. Furthermore, my research also aimed to gain a deeper understanding of the origin of the stylized facts on different time scales, using the reconstructed limit order books of the London Stock Exchange (LSE). 3 Applied methods My study involves real data analysis, modeling and theoretical analysis of models, as well as computer simulations. In order to accomplish these tasks, a large amount of effort was taken to develope a programming library in C language, which provides a framework for the analysis of limit order based trading. The library supports (1) the simulation of limit order based trading, (2) the implementation of the limit order based models, (3) the theoretically relevant mappings of the order flow, (4) the analysis of the essential quantities with complex conditions, (5) the filtering of the LSE data, and (6) the reconstruction of the LSE data. I used a Sun Gridware computer cluster to run simulation that requires high computational power. In order to analyze simulation result and manage data I created further programs in C, MatLab, R, awk, shell scripts. 4 Thesis statements 1. I introduce a simple stochastic trading model, which assumes independently trading agents. I demonstrate how the most relevant quantities (market impact, liquidity, bid-ask spread, price diffusion coefficient, etc.) can be transformed into nondimensional forms using the scaling properties of the three 4

6 naturally occurring dimensions (time, price and the number of shares). Dimension analysis provides scaling rules for these quantities as a function of the first order parameters of the order-flow [1, 2, 3]. 2. Using the same model, I show how can be the dimension of the parameter space reduced by three. I demonstrate that even the simplest realistic trading models should incorporate an important parameter, the nondimensionalized order size, which describes the granularity of the trading process. Based on the model I also show that this granularity parameter has a stronger influence on the price formation characteristics (market impact, liquidity, bid-ask spread, diffusion coefficient, etc.) than the price discreteness parameter, the nondimensionalized tick size. The latter just basically makes the price scale discrete in a trivial manner [1, 2, 3]. 3. The model shows that the process of trading, despite its simplicity, leads to non-trivial behavior in the diffusion. Using computer simulation, I demonstrate that the return time series cannot be uncorrelated (required by market efficiency and the first stylized fact) in a liquid market if all the traders make decisions independently of each other [1]. 4. The understanding of the market impact, that is, the order size dependence of price changes is interesting both from practical and theoretical points of view. It is a common assumption that large price changes are caused by large orders. Based on analysis of data from the London Stock Exchange (LSE), I show that this is not the case. Indeed, the size of the finite price movements are independent of the size of the order causing the change. The order size dependence of market impact is driven by the order size dependence of the 5

7 probability that the order causes a non-zero price change. The sizes of the non-zero price changes are dominated by the instantaneous liquidity of the market, they are almost always the same as the difference between the first and second filled price level in the current state of the limit order book [4]. 5. Although the price variation is often modeled as a diffusive process which is continuous in time, the limit order book based trading mechanism clearly points out that it is discrete. The stylized facts observed on longer time scales are dependent on both the characteristics of the transaction level price time series and fluctuations in trading activity, including frequency of price changes and the fluctuation of traded volume. Using data from the London Stock Exchange and New York Stock Exchange, I show that the second and third stylized facts (fat tails of return distributions and long memory in volatility time series) observed on longer time scales are not primarily caused by the fluctuations in the trading activity or the long memory of that. This contradicts the view which is widely accepted in the literature. The stylized facts can be observed on elementary price change scale. The fat tails of return distributions in longer time scales are dominated by the size of individual price changes at the transaction time scale, and not by number of transaction or by the traded volume in the time intervals. The long time scale volatility clustering is dominated by the clustering of the size of the elementary price changes, rather than the long memory in the measures of the trading activity [5]. 6

8 References [1] E. Smith, J. D. Farmer, L. Gillemot and S. Krishnamurthy, Statistical theory of the continuous double auction, Quantitative Finance 3 (3) (2003) [2] Marcus G. Daniels, J. Doyne Farmer, László Gillemot, Giulia Iori and Eric Smith Quantitative model of price diffusion and market friction based on trading as a mechanistic random process, Physical Review Letters (2003) [3] G. Iori, M. G. Daniels, J. D. Farmer, L. Gillemot, S. Krishnamurthy and E. Smith, An analysis of price impact function in order driven markets, Physica A , (2003) [4] J. Doyne Farmer, László Gillemot, Fabrizio Lillo, Szabolcs Mike and Anindya Sen What really causes large price changes? Quantitative Finance 4 (4) (2004) [5] László Gillemot, J. Doyne Farmer and Fabrizio Lillo There s more to volatility than volume, Quantitative Finance 6 (5) (2006) 7

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