Applying Physics Methods to Economics and Finance

Sergei MaslovBrookhaven National Laboratory

What is econophysics?

• Biophysics, geophysics study physical processes in biology or geology

• Econophysics tries to apply physics methods to economics

• Price fluctuations look like a particularly promising subject:– large and non-gaussian– long temporal correlations– scaling and power laws– universal

A

Abstracts of Working Papers in Economics

Administrative Science Quarterly

Africa Research Bulletin - Economic, Financial and Technical Series

Agribusiness - An International Journal

Agricultural and Resource Economics Review

Agricultural Economics

American Economic Review

American Journal of Agricultural Economics

American Journal of Economics and Sociology

Annales dEconomie et de Statistique

Annals of Public and Cooperative Economics

Annals of Regional Science

Applied Economics

Applied Economics Letters

Applied Financial Economics

Applied Mathematical Finance

Aquaculture Economics and Management

Asia-Pacific Economic Review

Asian Economic Journal

Asian Survey

Asian-Pacific Economic Literature

Asia Pacific Journal of Economics and Business

Atlantic Economic Journal

Australian Economic History Review

Australian Economic Papers

Australian Economic Review

Australian Journal of Agricultural and Resource Economics

B

Briefing Notes in Economics

Brazilian Electronic Journal of Economics

Brookings Papers on Economic Activity

Bulletin of Economic Research

Bulletin of Indonesian Economic Studies

Business History

Business History Review

C

Cambridge Journal of Economics

Canadian Journal of Economics

Carnegie-Rochester Conference Series on Public Policy

China Economic Review

Communist Economies and Economic Transformation

Comparative Economic Studies

Computational Economics

Computers in Higher Education Economics Review

Constitutional Political Economy

Contemporary Economic Policy

D

De Economist Derivatives, Use, Trading and Regulation Development and Change Development Policy Review Discrete Dynamics in Nature and Society

E

Eastern Economic Journal East European Politics and Societies Ecological Economics Econometrica Econometric Reviews Econometrics Journal Econometric Theory Economia Politica Rivista di Teoria e Analisi Economica Economic Affairs Economic and Industrial Democracy Economic Development and Cultural Change Economic Development Quarterly Economic History Review Economic Inquiry Economic Issues Economic Journal Economic Modelling Economic Outlook Economic Policy Economic Record Economic Systems Economics and Philosophy Economics and Politics Economics Letters Economics of Education Review Economics of Innovation and New Technology Economics of Planning Economic Systems Economic Systems Research Economic Theory Economie Appliquee Economies et Societes Economy and Society Education Economics Electronic Journal of Business and Organization Ethics Electronic Journal of Evolutionary Modelling and Economic Dynamics (e-JEMED) Empirica Empirical Economics Energy Economics Energy Journal Environmental and Resource Economics Environment and Development Economics Environment and Planning European Economic Review European Finance Review European Journal of Finance European Journal of Industrial Relations European Journal of Law and Economics European Journal of Political Economy European Journal of the History of Economic Thought European Review of Agricultural Economics European Review of Economic History Experimental Economics Explorations in Economic History

F

Federal Reserve Bank of Richmond Economic Quarterly Feminist Economics Finance and Development Finance and Stochastics Financial Counseling and Planning Financial History Review Finnish Journal of Business Economics Fiscal Studies G

Games and Economic Behavior Geneva Papers on Risk and Insurance Theory Global Business and Economics Review Global Economy Quarterly Growth and Change

H

Health Economics History of Economic Ideas

I

IMF Staff Papers Indian Economic Journal Indian Economic and Social History Review Industrial and Corporate Change Industrial and Labor Relations Review Industry and Innovation Information Economics and Policy Insurance Mathematics and Economics International Advances in Economic Research International Economic Review International Finance International Game Theory Review International Journal of Finance and Economics International Journal of Forecasting International Journal of Game Theory International Journal of Industrial Organization International Journal of Islamic Financial Services International Journal of Production Economics International Journal of Social Economics International Journal of the Economics of Business International Journal of Theoretical and Applied Finance International Organization International Regional Science Review International Review of Applied Economics International Review of Economics and Finance International Review of Finance International Review of Financial Analysis International Tax and Public Finance International Trade Journal Investigaciones Economicas

Alphabetic list of Economics Journals A-I

Theoretical economics is dominated by pure mathematics:

• Lemma/theorem style is required

• Little effort to compare theoretical predictions to “experimental data” - say, price record from real stock markets

• Bulk of papers are inaccessible and of no interest to “experimentalists” - practitioners of the field

George Soros on theoretical economics

“Existing theories about the behavior of stock prices are remarkably inadequate. They are of so little value to the practitioner that I am not even fully familiar with them. The fact that I could get by without them speaks for itself.”

G. Soros, “Alchemy of Finance” 1994

What pure mathematics has contributed to economics?

• Game theory approach– Concept of Nash equilibrium, where no player

can improve on his/her strategy

• Assumptions about stock price fluctuations– Gaussian– Uncorrelated. Justification: Efficient Market

Hypothesis, which states that all correlations are arbitraged away

Observations disagree:

• Short term fluctuations are strongly non-gaussian

• Price increments are correlated:– signs of price increments are uncorrelated in

agreement with Efficient Market Hypothesis– magnitude of price fluctuations has long

temporal correlations

Why physics can contribute to economics?

• We can work with real market data derive empirical laws and construct phenomenological theories

• Statistical physics has useful approaches to systems, composed of many interacting parts

Drawbacks:

• Interacting units in economics are thinking agents with adaptive strategies and not “mindless” particles obeying simple microscopic laws

Why bother?• Lots of high quality data:

– www.nyse.com has free daily volumes and closing values of DJIA for 1966-1998

– Trades and Quotes Database: a monthly CD-ROM with every transaction at NYSE, AMEX, and NASDAQ

• Journals to publish your results:– International Journal of Theoretical and Applied

Finance

– Physica A

– Nature (if you feel lucky)

– PRL, PRE (if your paper satisfies a set of rather restrictive conditions)

Short-term price fluctuations

• Basic object of study:historical record of price incrementsp(t)=p(t+t)-p(t)or returns r(t)= p(t)/p(t)= p(t+t)/p(t)-1 depend on time-lag t

• Short term (small t)increments do not dependon trends and politics but tell us something about the market mechanism itself.

9am 1/18/2000 4pm 1/19/2000

DJIA at 1min resolution

Mainstream theoretical economy on price fluctuations

• p(t) is a multiplicative random walk, i.e. ln[p(t+Δt)]=ln[p(t)]+ηΔt(t), where η(t) is a gaussian uncorrelated random variable

10%

-10%

Quick experiment: free data fromwww.nyse.com/marketinfo/nysestatistics.html

In a gaussian world the probability of the October 1987 crash would be 10-135!

10%

-10%

If not Gaussian then what?Small to intermediate p :

Pareto-Levy distribution with a power-law tail 1+2.4 (Mandelbrot, 1963).

• It is a stationary pdf for the sum of random variables with diverging second moment: p-(1+) p2 d p =

• non-random walk exponent in p vs. t <|p(t)-p(0)|>~t1/

From Mantegna,Stanley, Nature (1995)

SP500 in 1984-1989 at 1 min resolution

(1.5mln data points)

Crossover to 1+4 forSERIOUSLY large fluctuations

V. Plerou et al (BU group), 19991000 largest companies from TAQ

in 1994-1995 at 5min resolution

Is there universality in μ?

ForEx data by Dacorogna et al (1998)

Stock data by Plerou et al (1998)

Correlations in S&P500

< p(0) p(t) > < |p(0)| |p(t)| >

day yearmonth

from Y. Liu et al, 1999

The list of empirical facts about price fluctuations

• Probability distribution P(t p) is strongly non-gaussian especially when measured for t < several days• Crossover from Pareto-Levy distribution with 1+μ~2.4 to power law tail with 1 +μ~4.• Correlations in signs of price changes < p(0) p(t) > quickly decay to zero• Correlations in magnitudes of price changes persist from minutes to years (!) as manifested by by higher order correlators < |p(0)| |p(t)| > or < p(0)2 p(t)2 >

What models are used to explain these empirical laws?

• Econometrists use models with ugly acronyms ARCH(n), GARCH(n,m) (AutoRegressive Conditional Heteroscedasticity): Variance of a random walk has some history dependence on its previous values

— Don’t explain anything— Not very successful in reproducing real data

• Physicists crave for simple models keeping only the essential ingredients:

1. Models where players use and evolve strategies2. Models based just on the trading rules of the market (how supply/demand determines the price)

Limit order model

• No strategies (obvious oversimplification)• No market makers (limit & market orders placed by buyers sellers themselves through a computerized trading system• A buyer/seller has two options: - to place a limit order, i.e. the instruction to sell/buy a unit of stock if the price raises above/falls below a predetermined level - place a market order i.e. an instruction to buy/sell from the lowest bid/highest ask limit order• Limit orders are placed with a random offset Δ from the latest transaction price

Price signal in a limit order model

Limit order book

Price axis

bid-ask spread

limit orders to selllimit orders to buy

• Is there asymmetry between sell and buy sides of the book?• How density of limit orders depends on the distance from the current price?• How bid-ask spread scales with volume?

Minority Game (MG)• Introduced by Brian Arthur as “El Farol”

bar problem (1994)• Reformulated for stock prices by D. Chalet and Y.-

C. Zhang (1997)– at each time step N (odd) players (traders) choose

one of two sides B (buy stock ) or S (sell stock)

– minority side wins and “stock price” reflects it as(-) if Buyers won or (+) if Sellers won

– each player has a pool of S strategies based on outcomes (+ or -) of the last M steps of the game

– a virtual score is kept for each strategy

– each player always uses the best strategy from his pool

Testing the efficient market hypothesis in MGAll traders have 4 time-step memory:

i.e. all their strategies try to guess the next outcome based on 24=16 possible configurations of the last 4 outcomes

Prediction of the future based on 4

time-steps past

Y.Li, R. Riolo, & R. Savit, 1998 Y.Li, R. Riolo, & R. Savit, 1998

Prediction of the future based on 5

time-steps past

“I believe that microscopic market simulations have an important role to play in economics and finance. If it takes people from outside economics and finance - perhaps physicists -to demonstrate this role, it won't be for the first time that outsiders have made substantial contributions to these fields.”

From the letter by Nobel Laureate in Economics Harry M. Markowitz to “econophysicist” Dietrich Stauffer

Web sources on econophysics:

• Econophysics Forum: bimonthly articles and editorials, daily selection of relevant papers from cond-mat, list of conferences and books http://www.unifr.ch/econophysics/

• “Entry level” free data to play with http://www.nyse.com/marketinfo/nysestatistics.html