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A brief overview of more than 25 years of Econophysics

Rosario Nunzio Mantegna Central European University, Budapest, Hungary

Palermo University, Palermo, Italy

1 27 July 2016 IFT Colloquium - Sao Paulo

27 July 2016 2

“Mantegna and Stanley (2000, pp. viii-ix) define “the multidisciplinary field of econophysics” as “a neologism that denotes the activities of physicists who are working on economics problems to test a variety of new conceptual approaches deriving from the physical sciences.”

ECONOPHYSICS The New Palgrave Dictionary of Economics, 2nd edition J. Barkley Rosser, Jr. James Madison University

A definition

IFT Colloquium - Sao Paulo

Mantegna, R.N. and Stanley, H.E., 1999. Introduction to econophysics: correlations and complexity in finance. Cambridge university press.

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The first printed record of the neologism


The first econophysics paper


Another earlier paper of econophysics


Another early work on econophysics

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Econophysics is a hybrid discipline


L. Walras H. Poincaré


Jan Tinbergen and the gravity model of international trade. First Nobel laureate in Economics (1969)


The approach of Econophysics �� Econophysics aims to complement the approach of

other traditional disciplines in the study of economic and social systems. In particular in Econophysics:

- There is a continuous feedback between empirical observations and development of models; - Models may consider agents with bounded rationality and heterogeneity; - Data mining of large datasets is considered a key aspects to develop stylized facts to be used to develop models.


The background of Econophysics: Concepts and methods of statistical physics and theoretical physics.

Several laws and theories in Physics have a statistical basis. Some examples are: - Friction; - Ohm's laws; -  Critical phenomena; - Random walks; - Chaos and the theory of dynamical systems; - Turbulence; - Network theory; -  Percolation theory; - .............................


Some disciplines which traditionally were characterized by a low rate of production of scientific data have rapidly moved to a status of disciplines producing a high rate of data and information.

Changes in the scientific practice

For example biology, medical sciences and social sciences today are characterized by a huge rate of production of scientific data.

Cover of the special issue of Nature (September 4th, 2008)


We therefore need new methodological approaches and new techniques. Often the development of these new methods and techniques emerges in an interdisciplinary (hybrid) environment involving physicists, computer scientists, mathematicians and social scientists.

It is not only a matter of the size of the information which is produced and available. The nature of data and the associated data mining and data interpreting procedures raise new challenges.

In most disciplines data produced are global and of observational nature. This is quite different from what was the standard in the XXth century when experimental recording was localized and experiments where highly controlled.


Ettore Majorana

A pioneering point of ��view:“The value ��

of statistical laws in ��physics and social

sciences” ��


Majorana’s main message:

“Indeed our final aim will be to illustrate the renovation that the traditional concept of statistical laws must undertake as a consequence of the new direction followed by contemporary physics.”

This lack of determinism at the level of an elementary physical system motivated him to suggest a formal analogy between statistical laws observed in physics and the social sciences. In his article, he concludes that there is an “essential analogy between physics and the social sciences, between which an identity of value and method has turned out”.


Finance

Economics

Agent-based models

Networks

Data mining

Research topics in econophysics: A personal selection.


Finance •  Univariate and multivariate stochastic processes (stylized facts); •  Fractal and multifractal description; •  Derivative pricing; •  Portfolio optimization; •  Random matrix theory (course of dimensionality); •  Market microstructure and high-frequency trading; •  Optimal execution; •  Trading decisions of individual investors; •  Rare events. Price dynamics after (and before) a rare event •  Crypto-currencies.

Economics •  Wealth distribution; •  Firm growth; •  Firm organization; •  International trade; •  Inequalities.


Agent based models •  El Farol bar and the Minority game; •  Agent based models of financial markets (price discovery, order book, network relationships, global financial system); •  Agent based models in macroeconomics.

Networks •  Proximity (similarity and dissimilarity) based networks; •  Interbank market •  Economic and financial networks (example International Trade); •  Role of social networks in economic problems; •  Viral marketing and word-of-mouth diffusion of innovation; •  Networked markets; •  Systemic risk: Resilience and vulnerability of a network.


Data mining

•  New data mining tools for the detection of stylized facts; •  Impact of news in trading and social decisions; •  Information demand, diffusion of information and price dynamics; •  Methods able to detect statistical regularities in heterogeneous systems.


daily volatility

The price (and return) dynamics of a financial asset

1/1984 1/1989


Financial markets act as institutions performing delocalized and impersonal information aggregation.


Z=r/σ

r(τ)=ln(p(t+τ))-ln(p(t))

Paul Lévy

R.N. Mantegna, H.E. Stanley, Scaling Behavior in the Dynamics of an Economic Index,Nature 376, 46-49 (1995)

There is a large evidence that the second moment of the unconditional pdf is FINITE!!

One-minute S&P500 index returns (1984-1988)

Gaussian Lévy α=1.4


Extreme events: tail behavior

Pioneering work in this field has been done by Casper G. de Vries and collaborators (1990) for the foreign exchange market. In 1996 T. Lux observed similar power-law behavior for daily stock return tails of the German Stock Exchange.

T. Lux observed that the stocks composing the DAX index were characterized by positive and negative tails of the cumulative return distribution of exponent α≅3

- Koedijk K.G., Schafgans M.M.A., dr Vries C.G., The tail index of exchange rate returns, J. of International Economics 29, 93-108 (1990);

- Lux T., The stable Paretian hypothesis and the frequency of large returns: an examination of major German stocks, Applied Financial Economics 6, 463-475 (1996);


The data were extracted from the TAQ database. They analyzed transactions of the 1000 most capitalized stocks traded in the NYSE in 1993-1994

A similar result was obtained in Boston ‡ by investigating high-frequency data of the NYSE

‡ by P Gopikrishnan, M Meyer, LAN Amaral, HE Stanley, Inverse cubic law for the distribution of stock price variations, EPJB 3, 139 (1998)


Empirical volatility autocorrelation The volatility autocorrelation is a slow-decaying function

The decay is compatible with a power-law decay

( )( ) ηττσ −∝ACF

S&P 500 sampled at 1 min time horizon 1984 - 1996

3.0≈η

In econophysics we find stochastic processes that are long-range correlated


Several groups have noticed that the degree of leptokurtosis varies at different time horizons. Specifically, there is a progressive convergence towards the geometric Brownian motion at longer time horizons.

This observation has motivated a certain number of multifractal models of price returns:


- Muzy JF, Delour J, Bacry E, Modelling cascade process to stochastic volatility model, Eur. Phys. J. B 17, 537-548 (2000) fluctuations of financial time series: from

Multifractal models in turbulence and in financial markets

- B. Mandelbrot, A. Fisher and L. Calvet, A Multifractal Model of Asset Returns, Cowles Foundation Discussion Paper #1164, (1997)

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0.600.700.800.901.001.101.201.301.40

The complexity of the correlation matrix of a portfolio of financial assets.

Random Matrix Theory of financial correlation


€

ρ λ( ) =T

2π λσ 2 λmax − λ( ) λ − λmin( ), where λminmax =σ 2 1+1/Q± 2 1/Q( ), σ 2 =1− λ1

N, and Q =

TN

.

L. Laloux, P. Cizeau, J.-P. Bouchaud & M. Potters, Phys. Rev. Lett. 83, 1468 (1999). V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, and H. E. Stanley, Phys. Rev. Lett. 83, 1471 (1999).


Characteristics of the eigenvalue spectrum

The spectrum of a typical portfolio can be divided in three classes of eigenvalues:

1) The largest eigenvalue describes the common behavior of stocks (what is called “the market”). It is incompatible with the random matrix theory of random variables.

2) A fraction of 5% of the eigenvalues is also incompatible with the random matrix theory because eigenvalues fall outside ] λmin, λmax[


The low part of the spectrum

3) The remaining eigenvalues assume values between λmin, λmax and therefore one cannot say whether the eigenspace, which is corresponding to these eigenvalues, contains information or not.


Which is the meaning of largest eigenvalues?

Stanley and collaborators relate some of them to some economic sectors V- Plerou et al, Phys. Rev. E65, 066126 (2002) P. Gopikrishnan et al, Phys. Rev. E64, 035106 (2001)

€

XSk = PSi

i=1

n

∑ uik[ ]2

€

PSi =1nSi0

" # $

% $

33

Similarity based networks: Minimum Spanning Tree

Mantegna, R.N., 1999. Hierarchical structure in financial markets. The European Physical Journal B-Condensed Matter and Complex Systems, 11(1), pp.193-197.

27 July 2016 IFT Colloquium - Sao Paulo

US markets daily returns 1989-1995 (Dow Jones 30)

Correlations and hierarchical structures

Define a similarity measure between the elements of the

system

Construct the list S by ordering similarities in decreasing order

Starting from the first element of S,

add the corresponding link if and only if

the graph is still a Forest or a Tree

Minimum Spannig Tree MST

34

Planar Maximally Filtered Graph

27 July 2016 IFT Colloquium - Sao Paulo

N=100 (NYSE) daily returns 1995-1998 T=1011

Tumminello, M., Aste, T., Di Matteo, T. and Mantegna, R.N., 2005. A tool for filtering information in complex systems. PNAS, 102(30), pp.10421-10426.

Define a similarity measure between the elements of the system

Construct the list S by ordering similarities in decreasing order

Starting from the first element of S,

add the corresponding link if and only if

the graph is still Planar (g=0)

Planar Maximally Filtered Graph

PMFG


Econophysics has contributed to option pricing


Path Integral Approach to Option Pricing


Econophysics is contributing to market microstructure - empirical investigation and modeling of market impact; - empirical investigation and modeling of order book stylized facts; - long memory of the symbolic sequences of order submissions; - explaining power of zero intelligence agent based models; - Optimal execution; - Algorithmic trading and High frequency trading.


sell limit orders

best ask a

buy limit orders

best bid b

cancellation sell market orders

new limit order submission

Schematic representation of the order book

bid-ask spread

price

midprice a+ b2


Schematic representation of the order book: AZN order book on September 4th, 2002

- sell limit orders

- buy limit orders

X sell market orders

¢ buy market orders


P. Bak, M. Paczuski, and M. Shubik, “Price variations in a stock market with many agents,” Physica A: Statistical Mechanics and its Applications, vol. 246, no. 3, pp. 430–453, 1997.

Order book dynamics as a reaction diffusion processes

Challet, D. and Stinchcombe, R., 2001. Analyzing and modeling 1+ 1d markets. Physica A: Statistical Mechanics and its Applications, 300(1), pp.285-299.

Maslov, S., 2000. Simple model of a limit order-driven market. Physica A: Statistical Mechanics and its Applications, 278(3), pp.571-578.


Price impact of a single transaction��

Price impact function is not the same for all the stocks but a scaling relation taking into account the liquidity of the stock can be observed

1996

Lillo, F., Farmer, J.D. and Mantegna, R.N., 2003. Econophysics: Master curve for price-impact function. Nature, 421(6919), pp.129-130.

Evidence of long memory in order submission��

Lillo and Farmer considered the symbolic time series obtained by replacing buyer initiated trades with +1 and seller initiated trades with -1

The time series of the sign of submitted market orders is a long memory process


Lillo, F. and Farmer, J.D., 2004. The long memory of the efficient market. Studies in nonlinear dynamics & econometrics, 8(3).


Average volume in the order book at a given time

Bouchaud, J. -P., M. Mezard, and M. Potters, Statistical properties of the stock order books: empirical results and models, Quantitative Finance, 2002, 2(4), 251–256.

Order book stylized facts


The minority game

In order to stylize the El Farol model, Challet and Zhang gave a simpler definition of the El Farol bar model which they called minority game

In their model:

N agents take an action ai(t) deciding either to go (ai(t) = 1) or to stay at home (ai(t) = -1)

The agents who take the minority action win, whereas the ones taking the majority action loose

- Challet D, Zhang YC, Emergence of cooperation and organization in an evolutionary game, Physica A 246, 407-418 (1997)

Agent based models

18/5/2016 Dip. Fisica e Chimica - Palermo 45

Time evolution of the attendance

- From E. Moro, The Minority Game: an introductory guide, 2004

s=2 , m=2

s=2 , m=7

s=2 , m=15

N=301

18/5/2016 Dip. Fisica e Chimica - Palermo 46

A control parameter exists

†Savit, R., Manuca, R. and Riolo, R., 1999. Adaptive competition, market efficiency, and phase transitions. Physical Review Letters, 82(10), p.2203.

α =2m

N

Less number of effective strategies than agents. Evolution in groups.

Higher number of effective strategies than agents

47 18/5/2016 Dip. Fisica e Chimica - Palermo

There is a non-ergodic phase α < αc and an ergodic phase α > αc

The variable H acts as an order parameter H =

12m

sign A t +1( )( ) |µ t( )2

µ=1

2m

∑

information can be extracted from the public outcome μ(t), i.e., the attendance sequence of m periods observed at time t

No information can be extracted from μ(t)

y(0) indicates so-called "flat" initial conditions


The economic background of the Lux-Marchesi model

American Economic Review 80, 181-185 (1990)

E. Zeeman, On the unstable behavior of stock exchange, J. of Math. Econ. 1, 39-49 (1974)

A. Beja and M. Goldman, On the dynamic behavior of prices in disequilibrium, J. of Finance 34, 235-247 (1980).

Lux – Marchesi (1999)

T. Lux and M. Marchesi. Scaling and criticality in a stochastic multi-agent model of a financial market. Nature, 397:498–500, 1999.


- Lux T, Marchesi M, Scaling and criticality in a - stochastic multi-agent model of a financial - market, Nature 397, 498-500 (1999)

The model shows leptokurtosis of returns, volatility clustering and power law behavior of volatility scaling.


The interbank market

M. Boss, H. Elsinger, M. Summer, and S. Thurner, The Network Topology of the Interbank Market, Quantitative Finance (2004).

De Masi, G., Iori, G. and Caldarelli, G., 2006. Fitness model for the Italian interbank money market. Physical Review E, 74(6), p.066112.

52 April 19, 2011 Swissquote & EPFL day on quantitative finance

Heterogeneity is a key ingredient of economic and social complex systems. Nordic Center Securities Data (NCSD) (today Euroclear) collects a database recording the daily ownership of portfolios and trading records from January 1, 1995 of different classes of investors domiciled in Finland (companies, institutional governmental investors, foreign investors, no profit organizations, financial institutions and households). Identity of the investors is coded for privacy reasons. This database has been extensively investigated at the aggregated level of classes of investors in the financial literature by Grinblatt and Keloharju (2000, 2001,2009).


The set of investors is quite heterogeneous in terms of volume and frequency of trading therefore a comprehensive analysis of variables monitoring their activity like, for example, inventory variation is challenging.

Ex: Cumulative probability density P(X > Nt) of the number of transactions Nt performed by 41250 Finnish legal entities trading the Nokia stock (typically at the Nordic Stock Exchange) during 2003.


In the presence of heterogeneity the use of categorical variables can help. We therefore performed investigations by using categorical variables. This choice allowed us to perform a comprehensive analysis of a very large set of investors in spite of the huge level of heterogeneity present in it.

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+,,) -,,) .,,) /,,) 0,,,) 0+,,)

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27 July 2016 IFT Colloquium - Sao Paulo 55 55

Trading activity of single investors investigated with the tool of Statistically Validated Networks: profile of the 4 largest clusters of the Bonferroni network

Buy

Sell

BuySell

M. Tumminello, F. Lillo, J. Piilo and R.N.M., Identification of clusters of investors from their real trading activity in a financial market, New Journal of Physics 14, (2012) 013041


Impact of news and impact of the demand of information Data mining and price discovery

Bollen, J., Mao, H. and Zeng, X., 2011. Twitter mood predicts the stock market. Journal of Computational Science, 2(1), pp.1-8.


Conclusions After more than twenty-five years econophysics has obtained a series of results in the analysis and modeling of several economic and financial systems.

Econophysics provides an approach to the investigation of social and economic complex systems which is complementary to the ones of economists, financial mathematicians, econometricians and computer scientists.

The key ingredients are the approach towards data analysis and modeling not necessarily restricted to micro-founded problems, the statistical physics background especially for non equilibrium processes and the analysis and modeling of large datasets with new data mining tools and techniques.


Thank you!!

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