Liquid markets and market liquids Collective and single–asset dynamics in financial markets

We characterize the collective phenomena of a liquid market. By interpreting the behavior of a no–arbitrage N asset market in terms of a particle system scenario, (thermo)dynamical–like properties can be extracted from the asset kinetics. In this scheme the mechanisms of the particle interaction can be widely investigated. We test the verisimilitude of our construction on two–decade stock market daily data (DAX30) and show the result obtained for the interaction potential among asset pairs. PACS. 02.50.Sk Multivariate analysis – 89.90.+n Other areas of general interest to physicists Since the late 80s, with the introduction of electronic trading, huge quantities of financial data became available (or at least on sale) for both investment and research analysis. Quite unusually outside the natural science panorama, this novelty opened the way to test the reliability of theories and conjectures about the behavior of financial markets. One of them, a paradigm for financial mathematics, is the random character of markets [1], that is unpredictability. It has recently been proved, nevertheless, that a certain degree of correlation is still present on extremely short time scales [2]. Despite that, the intermediate scales are dominated by random behavior with Lévy stable statistics of asset returns [3]. The possibility to extract information on the future evolution of a single asset by knowing a big enough ensemble of its past values matters indeed institutional traders, who can generally intervene on the market in real time (with delays smaller than few seconds). Their presence reduces at minimum time correlations and consequently speculation possibilities. Time dependence is however only one possible domain for surveying similar patterns inside financial signals. The other domain for correlation detection, whose exploration was greatly facilitated by modern computation facilities is the ‘spatial’ one. In fact, albeit much efforts are spent in studying correlations in the time dynamics of a single asset (see [4] and [5] for a digest of the recent physicist and economist approach, respectively), there are many applicative and fundamental reasons for understanding deeply spatial, commonly referred as multivariate, correlations. A financial market is not simply a juxtaposition of different prices which are organised on an independent basis, but rather a complex system of interacting constituents [6]. The latter are then monitored by sampling a e-mail: [email protected] single prices with respect to an arbitrary currency. Hence the study of correlations among different asset time signals is of peculiar importance. By the way, this is also the case in many problems involved in the modern risk management theory, where the composition of a certain portfolio strongly depends on the movements of different underlying assets. On a more fundamental level, the interesting issue is the comprehension of how price changes can be separated, with a sufficient degree of confidence, in single asset– and collective– behavior. Since the Markowitz’s work on the theory of optimal portfolio [7], much effort has been spent to characterize correlation matrices of financial assets [8]. In recent contributions, different physics concepts have been adopted to endeavor this type of problem, mainly because the study of correlations represents a paradigm of a wide class of physical problems for which powerful tools have been developed. A bivariate analysis of the futures on the German and Italian bonds showed that despite the perfect uncorrelation of the single tracks, the crosscorrelation of the two signals was significantly non zero: the signals considered described two random, but similar, processes [9]. This behavior emerges quite generally in the stock market, where certain asset clusters ‘move’ in a particularly correlated way with respect to remaining titles. Using equal time cross–correlation matrices and several physics–borrowed tools such as the random matrix theory, these conjectures have been quantified [10]. In a recent study, the structure of a N stock market has been investigated as regarding the multivariate structure in a global window period [11]. In this paper, we propose a method to investigate asset correlations by interpreting asset growth rates as observables of a particle system scenario. This idea is carried out by introducing a formal map between the logarithmic returns and the distances among gas particles. The strength 2 Gianaurelio Cuniberti, Lorenzo Matassini: Liquid markets and market liquids of this analogy resides in the possibility to separate collective motion from the single asset dynamics through the investigation of mutual interactions among titles. Wielded by the theory of liquids, we can study the thermodynamics of the system and interprete its temperature as a measure of spatial volatility, as compared with the more familiar (temporal) volatility. The 2–asset interacting potential is then calculated on the isothermal (isovolatile) market. In the remainder of this paper a time dependent asset– distance and a moving frame model are introduces. The implementation of this scheme is performed on daily stock market data taken among the 30 most capitalized titles forming the Deutscher Aktien indeX (DAX30) in the period 30 Dec 1987 to 7 Mar 1995 (1800 trading days). To maintain a continuity of quotation, we have selected the maximal subset of 23 assets which, in the above mentioned period, remained in the DAX30 basket and did not split. Our discussion and comments conclude the paper. As a general starting point, we consider a collection of asset, which is a suitable subpart of titles in a stock market (better if one representative for every economic sector), a collection of currency prices, or any combination of them. The value of the asset Ωi at time t, is expressed in unity of asset Ωj by means of conversion factors Pij(t): Ωi(t) = Pij(t)Ωj(t). (1) The indices i and j span all N considered assets forming the market. By writing eq. (1) for another couple of indices, a no–arbitrage equation for a liquid market is obtained Pij = PikPkj . Its multiplicative symmetry is reflected in a corresponding additive symmetry of the logarithmic returns