Phase Transitions, Renormalization and Yang-Lee Zeros in Stock Markets
aa r X i v : . [ q -f i n . GN ] M a y Phase Transitions, Renormalizationand Yang-Lee Zeros in Stock Markets
J. L. Subias ∗ Departamento de Ingenieria de Diseno y Fabricacion, Universidad de Zaragoza,C/Maria de Luna 3, 50018-Zaragoza, Spain (Dated: April 18, 2015)
Abstract
The present paper analyses the formal parallelism existing between the laws of thermodynam-ics and some economic principles. Based on previous works, we shall show how the existencein Economics of principles analogous to those in thermodynamics involves the occurrence of eco-nomic events that remind of well-known phenomenological thermodynamic paradigms (i.e., themagnetocaloric effect and population inversion). We shall also show how the phase transition andrenormalization theory provides a natural framework to understand and predict trend changes instock markets. Finally, current negotiation strategies in financial markets are briefly reviewed.
PACS numbers: 89.65.Gh, 05.70.Fh, 05.70.-a, 87.23.Ge ∗ e-mail for correspondence to the author:[email protected] . INTRODUCTION One of the most interesting findings in Science is duality -i.e., the fact that there arelaws formally identical in two different fields of the same discipline and that their principlesare simply equivalent, thus requiring mere substitution of each concept by its correspond-ing pair. For instance, there are dual spaces in Geometry, so that -if dual concepts areinterchanged in a true sentence- the corresponding dual sentence shall be obtained, andshall also be true. For example, the sentence ”three non-aligned points determine a plane”find its dual sentence in ”three non-parallel planes determine a point”. Plane and pointare dual concepts. However, the most remarkable aspect is that duality can occur betweenwidely differing fields in a discipline. Let us consider L. Onsager’s widely-known example.Onsager perspicaciously realized that the Langevin equation could satisfactorily describe acompletely different phenomenon from that it had initially been conceived for -he realizedthat changing names in the Langevin equation leads to an equation that satisfactorily solvesan apparently unrelated problem in statistical physics. At this point, the question we poseis: can duality occur between completely different disciplines, so that it even transcends thereach itself of Physics and Mathematics? If there is duality between fundamental princi-ples, all laws, theorems, equations, etc. would then admit other formulations that wouldonly involve changing every concept in them by its corresponding pair. Can this dualityoccur between such different disciplines as Physics and Economics? A fundamental law ofPhysics usually taught in high school states that ”Energy can be neither created nor de-stroyed, but it can only change form” . In Economics there is a dual principle to the previousone: ”Money can be neither created nor destroyed, it can only change pocket” . On the otherhand, the second law of thermodynamics -which states that the entropy of an isolated sys-tem never decreases- is known to have universal validity and to apply in Biology, Ecology,Sociology and, of course, Economics. Perhaps the most difficult aspect to ”grasp” is that,in Economics, the ”no arbitrage” principle also has its counterpart. Indeed, consider thefollowing statement - ”if between markets A and B there is no arbitrage possibilities, norbetween markets B and C, therefore there cannot be arbitrage possibilities between marketsA and C either” . When analysed in depth, this is nothing but the zeroth law of thermody-namics applied to Economics. If the three fundamental laws of thermodynamics also applyin Economics, it is then logical to think that numerous economic phenomena will have their2orresponding thermodynamic equivalent. The reason why these dualisms have not beenobserved so far is that most part of the task is still to be done: namely, translating physicalvariables (temperature, pressure, etc.) into economic variables, as Onsager did with general-ized forces, generalized coordinates , etc. The present work summarises the findings obtainedin previous works on phenomenological thermodynamics paradigms that have counterpartsin Economics (namely, population inversion and magnetocaloric effect). Then, the latestfindings on phase transitions, renormalization and Yang-Lee zeros in the financial contextshall be described.
II. EFFECTIVE TEMPERATURES
As a starting point for the present study, we shall adopt the definitions -applied to thefield of financial markets- of the following variables: temperature, energy, magnetic fieldand entropy, as formulated in [1]. Let us imagine the following context: a mixture ofeconomic agents in perpetual interaction. A Brownian particle is suspended in this mixture,and randomly impacted by the numerous economic agents, describing a geometrical, one-dimensional Brownian motion. The foregoing is a possible conception of a financial market-i.e., the time series of stock prices is conceived as the movement of a hypothetical Brownianparticle hit by economic agents. This particle is then equivalent to a thermometer thatallows us to measure the effective temperature of the system it is suspended in. Such effectivetemperature is an estimation of market temperature and can be calculated by (5) in [1]. Somecases in which the temperature sequence imitated well-known processes in phenomenologicalthermodynamics are described in [1] and shall be summarised next.
A. Population inversion
Figure 1 (a) shows a type of graph that imitates the phenomenon known as populationinversion in statistical physics. In this phenomenon, the external field is inverted so brusquelythat spins cannot adapt to such a sudden commutation. This leaves the system in a highlyunstable state. For some time, spins shall try to adapt to a new stable equilibrium. Duringthis transition, temperature firstly rises and then falls, thus corresponding to a theoreticalstep from T = −∞ to T = + ∞ [2]. Note the dramatic fall of stock prices that takes place3 .605.405.205.00 q u o t e s +40+200-20 T t (b)(a) Sep 2003 Oct Novtime (trading days)time (trading days)+200-20-40-60 t D o w J o n e s A B AC time t e m p e r a t u r e (c)(d) ( € ) T FIG. 1. (a) Time series of temperatures T t corresponding to (b) the Dow Jones index. The inset C shows how a discrete register of a physical population inversion would be. Compare the inset withdetail A . (c) Time series of temperatures T t corresponding to (d) quotes of a particular stock.After a dividend payout, this stock suffers an extreme “cooling” near to T t = 0 (detail A ). in the following days.
1. Explanation of this phenomenon in economic terms
The emergence of some breaking bad news that go against the status at the time being(the magnetic field is brusquely inverted ) leads investors ( spins ) to instantly change theirmind (very brief spin-field interaction). However, selling their parcels of shares will takethem much longer (much longer spin-network interaction). The difference between thesetwo typical times is what favours the population inversion phenomenon. Some hurry to selltheir shares, and market temperature then turns negative and sharply falls towards −∞ ,and then comes back from + ∞ , decreasing and eventually becoming stable. Nevertheless,full balance has not been reached yet, since the sectors reacting first are those that own privileged information -i.e., those that know the bad news before they break into the media.In the following days, the remaining parts of the markets will react and stock prices willslump. 4 . Magnetocaloric effect In a different environment, Fig. 1 (c) shows the case of a stock security whose investorsare ’aligned’ awaiting the payment of abundant dividends. This expectation represents astrong magnetic field that suddenly vanishes the day after dividend payment. At this time,stock prices undergo ’cooling’ close to T = 0, imitating the magnetocaloric effect (detail A).
1. Explanation of this phenomenon in economic terms
Investors ( spins ) expect to be paid abundant dividends: almost nobody sells or buys(spins are mainly aligned under a powerful magnetic field ). On the scheduled date, paymentis made (the magnetic field suddenly disappears ) and investors randomly decide to keepor sell their shares after being paid their corresponding dividends. Thus, it shifts from avery ordered situation to a very disordered one with no money provision ( energy ), since theremaining agents in the market know that dividends are to be deducted from stock pricesand, accordingly, stay out of it. Therefore, disordered state can only be reached throughinteraction at internal level: some buy what others sell and ’temperature’ drops sharply(not market temperature, but that of these specific shareholders). The crux of the matteris that if shareholders are slightly ’ferromagnetic’, stock prices can then drop sharply, thusmeeting the following well-known financial rule: ”market needs money for shares to go up,yet it needs nothing for them to drop”. In magnetocaloric terms, we would say that ”forthe temperature of the magnetic material to rise, we need energy , yet we need nothing tocool it”.
C. Limitations of the temperature signing algorithm
Given the presence of random noise in the historic series of stock market prices, it wouldnot be reasonable for a temperature signing algorithm to be based on one only method.From a probabilistic viewpoint, sign determination would be more reasonable through twodifferent ways. If the conditions (15) and (16) in [1] based on equation (13) in [1] are takenas the first way, the second could be the rising trend in the absolute value of temperaturedue to the instability inherent to the states in which magnetic energy per spin E is over zero,(17) in [1]. Figure 2 represents a Monte Carlo simulation in which it can be observed how5he entropic curve deforms due to slight ferromagnetism. At the same time, the hysteresisloop is increased, thus partially losing magnetic energy (Fig. 3). For strong ferromagnetismlevels, deformation is so remarkable that the criterion of the entropic curve slope (15) and(16) in [1] is no longer valid to determine the sign of the temperature (although the growingvolatility criterion (17) in [1], based on the Black-Scholes equation, remains valid). FIG. 2. Monte Carlo simulation for J =0(paramagnetism) and J ≈ j = 1 /
2. Slightlyferromagnetic behaviour can be observedto move away from ideal paramagnetic be-haviour. -(cid:2)(cid:3)(cid:4) -0.2 (cid:5)(cid:6)(cid:7)(cid:8)(cid:9) (cid:10)(cid:11)(cid:12)(cid:13) (cid:14)(cid:15)(cid:16)(cid:17)(cid:18) 0(cid:19)(cid:20)(cid:21) (cid:22)(cid:23)(cid:24) (cid:25)(cid:26)(cid:27)(cid:28) (cid:29)(cid:30)(cid:31)
FIG. 3. Hysteresis loop Monte Carlo sim-ulation for J ≈ j = 1 /
2. A small part of themagnetic energy is lost.
III. SCALE INVARIANCE AND TEMPERATURE RENORMALIZATION
Numerous studies published in scientific journals support the thesis on the fractal natureof stock market prices. The description itself by R. N. Elliott of his own theory is a perfectexample of fractality. The most striking aspect is that this description had already beenmade from an empirical viewpoint many years before (1939) Mandelbrot got his works pub-lished and defined the concept of fractality. If we gave an expert financial analyst a stockmarket graph in which units from X and Y axes have been removed, this analyst could notdetermine whether this graph is intradiary, diary, weekly or monthly. This scale invariancein the field of time is therefore unquestionable. The subsequent issue is how this affects stockmarket dynamics. The spin system model proposed in [1] may allow two perfectly differen-tiated phases with a phase transition at a hypothetical Curie temperature. This happensto fit in with experience: financial analysts distinguish two clearly different phases (uptrend6nd downtrend), and transition periods characterised by high volatility and lateral move-ment in stock prices. On the other hand, scale invariance leads us to think of the possibilityof applying some kind of renormalization. Leaving aside profound considerations on thetopological dimension of the spin network, it is easy to understand that space interactionsbecome variations through time, thus existing some kind of correspondence between net-work topology and Brownian movement of stock prices. In other words, the observed scaleinvariance (in the time domain) of stock prices would be the consequence of some kind oftopological invariance in the spin network. Let use suppose we dispose of the following data(illustrated in Fig. 4): time series of intradiary stock prices, and frequency of interactions(transactions). This series would be associated with an intraday temperature given by (5) in
29 may 30 may (a)
29 may 30 may 2 jun 3 junVolume (b)
FIG. 4. (a)Graphic representation of the time series of stock intraday prices and (b) the same timeseries after renormalization. [1]. Relative frequency could be compared with probability density of interactions occurringat a specific time point. We would empirically ascertain that this probability is remarkablyhigher in the opening and closing auctions (beginning and end of the day), as shown in thisfigure. Stock-price renormalization (in the time domain) would then consist on substitutingthe original intradiary time series by another series in which the set of intradiary values aresubstituted by opening and closing prices values weighted by the aforementioned relativefrequency. Graphically, this would be equivalent to reducing the original series substitutingwhole sets of terms by their average value. If the reduced and weighted (renormalized) timeseries is applied the same temperature formula (5) in [1], a second renormalized interdaytemperature will be obtained. 7 . Yang-Lee Zeros
For spins organised in networks of simple topology, energy partition function Z N ( T, H )is a parametric function of temperature T and external field H , N being the number ofparticles. By means of renormalization, networks goes N → N ′ < N , the number of degreesof freedom thus being reduced. Repeating the process N < N ′ < N ′′ < N ′′′ ... would leadus to 1 or 2 degrees of freedom, which is the trivial case, and we would have a series ofsubsequently renormalized temperatures T < T ′ < T ′′ < T ′′′ ... , so that T ( i +1) = R ( T ( i ) ),where R stands for the renormalization transformation. On the other hand, the partitionfunction, transformed into polynomial form by means of a change of variable, has a set ofzeros that correspond to points in the complex plane. Years ago, Yang and Lee provedphase transitions to be related to the distribution of zeros in the partition function [3, 4].More precisely, as one gets closer to the thermodynamic limit, zeros begin to accumulate ina specific region of the complex plane and tend to ’mark’ the real axis on the point corre-sponding to Curie temperature T C (Fig. 5 a), at which ferro-paramagnetic phase transitionoccurs. Furthermore, the map of zeros in the thermodynamic limit is equal to the Julia
Im(z) (a) phasetransition
Re(z) Im(z) (b)
Re(z)
FIG. 5. (a) Accumulation of Yang-Lee zeros around the phase transition temperature; and (b)Map of zeros for some specific conditions. set of the renormalization transformation. Thus, curious fractals such as the one shown inFig. 5 b are obtained. However, the most interesting properties are that Curie temperature T C repels the theoretical renormalization process, and that this temperature is invariant relative to the renormalization transformation -i.e., T C = R ( T C ). As an econophysic corol-lary of the latter property, it may be guessed that, for a financial market, if two or three8 (c)(a)(b) -0.5 FIG. 6. (a) daily securities prices (End of Day) of a particular blue chip; (b) trading volumes; and(c) three successively-renormalized temperatures. Arrows indicate hypothetical phase transitionpoints, which anticipate changes in trends. successively-renormalized temperatures are calculated, downtrend-uptrend phase transitionwould then be found at those points with the same temperatures. In other words, phasetransition could be identified when intraday temperatures get significantly closer to interday(renormalized) temperatures. Figure 6 may confirm (although not statistically validated)this phase transition hypothesis, as it represents a well-known blue chip in Spanish IBEX 35.Arrows indicate hypothetical phase transition points in which temperatures values coincideand which also coincide with changes in price trends, thus marking trading opportunities.
Appendix A: Trading strategies
Nowadays important economic decisions are strongly influenced by powerful softwaretools. Thus, it can generally be said that a half-human, half-machine factor governs whatsome have called the modern cybernetic economy. Getting into further detail, economicagents (institutional or retail investors, general public, etc.) can be observed to follow dif-ferent trading strategies. Perhaps the most determining factor to choose a specific strategyis the frequency with which buying-selling cycles are to be completed, which in turns deter-mines the mean lifetime and volatility of the assets in the corresponding investment portfolio.Anyway, the objective always remains the same: maximizing the portfolio’s growth ratio.The main current strategies are: 9
High Frequency Trading (HFT) basically consists on managing an investment portfoliowhose assets’ mean lifetime range from fractions of a second to some seconds. Since ahuman trader cannot take and perform decisions in such brief lapses of time, buying-selling orders are given by completely-automatic specialized algorithms, thus limitingthe human factor to modulating or directing algorithm activity, as well as constantalgorithm modification and enhancement. Thus, algorithms behave like an automatonthat takes and perform decisions in fractions of a second. Low latency HFT -basedexclusively on tiny price arbitrage advantages that render benefits of only a fractionof a currency cent- is a class of its own. A low latency algorithm opens and closesportfolio positions in a matter of microseconds and formal parallelism with a Brownianmotor in physics is rather clear. In a more general context, automated or algorithmictrading is known as the use of software platforms that -preprogrammed with a set ofrules or algorithms- send buying-selling orders to the market automatically. The setof preprogrammed routines is equivalent to a robot colloquially known as robotrader. • Day trading: traders with technical analysis-based software tools complete buying-selling cycles on the same day -e.g., buying in the morning and selling in the evening.This strategy is exclusively based on technical analysis (Dow theory, Elliott waves,statistical indicators, etc.) • Low Frequency Trading: this concept may include institutional or private investorsthat make use of fundamental analysis tools (macro- and micro-economic indicators,financial ratios, etc.) and whose investment portfolios are much less volatile than inthe previous cases.Algorithmic trading is widely used by large banks and institutional investors. The set ofrules or algorithms can be based on any technique from technical or fundamental analysisto statistical physics. In 2006, 30% of the negotiation volume in EU and US stock marketswas estimated to be completed by algotraders or some other kind of automated trading.In 2009, several studies suggested that HFT completed at least 60% of the total tradingvolume in the US. The prevalence of algotraders in some markets has even reached 80%. Asa result, the massive use of these techniques is believed to have radically changed the market’smicrostructure and dynamics. Algorithmic trading in general was under debate until theUS Securities and Exchange Commission informed that these techniques contributed to the10ave of increased selling caused by 2010 Flash Crash (a sudden crash in which Dow Jonesplunged about 1000 points in only some minutes’ time, after which a large part of the losswas recovered). [1] J. L. Subias, “Negative kelvin temperatures in stock markets,” e-print arXiv:cond-mat/1206.1272 (2013).[2] R. K. Pathria and P. D. Beale,
Statistical Mechanics (Academic Press, New York, 2011).[3] C. N. Yang and T. D. Lee, Phys. Rev. , 404 (1952).[4] C. N. Yang and T. D. Lee, Phys. Rev. , 410 (1952)., 410 (1952).