# Presentation: 042704

## Applying Physics Methods to Economics and Finance

- Sergei MaslovBrookhaven National Laboratory

**Notes:**
Thanks for the introduction. In this talk I will try to give you a brief overview of the new emerging branch of Statistical Physics, which became known by the name of “econophysics”. The first official conference on econophysics took place in Budapest in 1997 and since than this field experienced an explosion of activity mostly limited to Europe.
It is enough to say that from 36 econophysics  conferences announced for this year only 4 are held in US, while the rest is all over Europe. Perhaps it explains why my modest contribution to this field has a pronounced seasonal periodicity: it blooms during my annual summer visits to Switzerland, keeps going for a couple of months after I get back and goes into hibernation in winter. 

I want to use this opportunity to thank my host and collaborator Yi-Cheng Zhang from the Institute of Theoretical Physics in Fribourg, Switzerland who introduced me to this field. His enthusiasm for the future of econophysics proved to be contagious.

## 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

**Notes:**
Why physicists decided to start thinking about the economics? We are familiar with interdisciplinary branches of physics such as biophysics or geophysics studying physical processes in biology and geology. They do not pretend to know all the answers: in fact their field of interests constitute only a minor part of their adopted discipline. At a first place economics it is not a place for physicists to be: you cannot study the force-load diagram of molecular motors, like biophysicists do. You cannot describe the plate tectonics in terms of convective dynamics of magma like geophysicists do. There is no physical processes in economics.
What we hope to be able to do is to apply the statistical physics methods combined with plain common sense to some of the questions in economics.
One branch of economics looks particularly promising for the application of ideas of statistical physics.  It is the study of the market price fluctuations.It has all features which make the heart of every statistical physicists to beat faster:
they are large and non-gaussian, have long temporal correlations, exhibit scaling and power laws. Perhaps most importantly on short time scales fluctuations in the price of stocks, bonds, foreign currencies, commodities such as oil or cotton as well as any other risky asset look remarkably similar
It is clear that these fluctuations are due to a stochastic process of some kind which is driven by collective actions of many individual agents. 
And many physicists are now actively working to derive regular reproducible results for this process.

## 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

-           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

**Notes:**
But physicists are not moving into an empty field. Preparing this talk I decided to count how many economics-related journals there is. No matter how small was the font I managed to fit only A-J part of the index. SHOW NEXT SLIDE
It’s enough to open one journal to see that our position of the discipline, helping economics to quantitatively understand their data  is already taken by mathematics. 
What immediately  jumps to your eyes is lemma/theorem notation which seems totally out of place in a “soft” discipline such as economics. Economists are ready to sacrifice virtually everything to be able to “prove” certain statements. 
Readability goes first: I know papers where serious authors talk about the “continuum of agents” on an interval [0,1] only because they can prove certain things for continuum.
This emphasis on rigorous proofs means no or little comparison with “experiment”- data from real stock market. Sometimes they can “prove” things that are simply not true for real market
The consequence of this approach is a total disregard of the theory by the “experimentalists” - practitioners of the field, whose every day job is to predict 
and use market price fluctuations.
Their opinion is clearly formulated by, arguably, the best practitioner in the field - famous trader George Soros.

## 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

**Notes:**
In his book “Alchemy of Finance” he writes:

## 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

**Notes:**
I am far from saying that mathematics didn’t contribute anything to economics. In fact the original ideas based on the game theory constituted a reasonable “zero-order approximation”. It is not mathematicians fault that after this first step the mainstream theoretical economics got stuck polishing minor details of useless proofs. Many serious economists understand that their field is ripe for a fresh input and many look at physicists as likely candidates for this fresh blood. 
Let me attempt to give you a crash course in foundations of modern economics:
The cornerstone of mathematical economics is played by an elegant branch of mathematics known as the Game theory. 
The most useful concept is that of Nash equilibriumన (it was formulated by a young Princeton graduate student in 1951 who was awarded a Nobel prize in Economics in 1994). It promotes the view of the whole economics as a game played by perfectly rational infinitely smart players using deductive logic. The game is frozen in a static state where each player uses the best strategy at his disposal provided other players do not change their strategies. 
There is no obvious connection from this equilibrium picture to stock price fluctuations: in fact in a perfect equilibrium world there should be no fluctuations at all. Here the mainstream economics makes a big leap of faith by postulating that price fluctuations are perfectly Gaussian and uncorrelated.
There is one explicit prediction of the game theory about the behavior of stock prices, which is known as Effective Market Hypothesis: all correlation in stock prices that can be used for making money are arbitraged away.  Let’s quickly see how it works:suppose there is a logical way to predict the sign of stock price change between time t and t’. For the sake of argument let’s say one can deduce that the stock price would go up. The logical thing for a speculator to do is to buy stock be to buy stock right before the predicted change and to dump it immediately after. But buying stock creates an excess demand which drives the price at t up, while selling creates an excessive supply which drives price at t’ down.Both effects tend to reduce the price increase which was driving the whole process in the first place. Market has a selfcorrecting mechanism removing correlations in SIGNS of price fluctuations! This gives some support to the anzats that price fluctuations are uncorrelated.

## 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

**Notes:**
Observations disagree rather dramatically:

Fluctuations are not even approximately gaussian, especially when price increments are taken on short time scales, where there is no flow of information about the prospects of a given company. At such scale the price changes due to  trading process itself. 

Finally, the EMH that prices are uncorrelated seems to work pretty well for SIGNS of price changes, I.e. something that can be easily arbitraged away. The magnitude of price fluctuations has long temporal correlations, which is manifested in volatile and non-volatile periods on the market.

There is no clear understanding why volatility hassuch a long memory. One possibility is that strategies used by traders are not independent, and traders themselves are not particularly rational, which is manifested by the so-called herd effect.

## 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

**Notes:**
Why do we think that physics can do better than pure mathematics?
First of all, what I feel is the most important is that physicists are trained to work with real “experimental”data and construct phenomenological theories and derive empirical laws. This approach seems to me infinitely more useful than rigorous mathematical proofs based on unconfirmed postulates.

Second, physicists come with their own toolbox. Hopefully some of physics concepts may prove to be useful in describing economics. 
Of course in a dark and unexplored forest of economics physicists should not wander too far without experimental support. Otherwise they’ll become second mathematicians. They should not forget that economics is the “soft” science. Its basic units are thinking agent sand not mindless particles:

That is why very popular and powerful approach is in terms of evolutionary or adaptive models, where the behavior of agents is not fixed by rigid rules but evolves in response to the market itself.

## 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)

**Notes:**
Now that we see that there is hope of applying physics methods to economics we should ask ourselves: why bother? In other words, what is the advantage of economics compared to other “soft” sciences such as evolutionary biology or psychology? 
First great advantage of economics is plenty of high quality data readily available:
free daily NYSE
 if one needs a more detailed and accurate test one could buy TQD of 24 CD-ROMs with every transaction for 100 biggest companies made in 1994-95 on three major US stock markets such as NY Stock Exchange, NASDAQ, and AMEX. Taking these data at 1 minute interval gives roughly 40 million data points.
Secondly, there exist journals of established reputation where one can publish the results:
 this journal was created specifivally for econophysics community and its editors are almost all physicists (apart from advisor a last year Nobelist in economics Robert Merton). Unfortunately, it looks like it starts to get contaminated by second rate papers by mainstream economists. We’ll see if its future volumes will be worth looking through.
 Physica A  is trying to establish itself as a major journal for inerdisciplinary physics. It accepts reasonable papers on econophysics without delays.
 Finally if you feel particularly lucky you may try Nature: it has a record of publishing about half a dosen papers on econophysics

## Short-term price fluctuations

- Basic object of study:historical record of price incrementsp(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.

**Notes:**
Now, having finished with the introduction I can start a more concrete part of my talk.
The main subject which I  would discuss in this talk is statistics of price increments and various ways to model them.All you need to have in order to study price increments is a historical record of price of some risky asset or index. The longer and the better is its spatial resolution the better statistics we would have.The next step is to pick up the time lag Delta t and essentially to differentiate the price signal with step Delta t. It would be even better to normalize it by the starting price p(t) or for very long Delta t take the increment in the logarithm of price, which makes sense due to approximately multiplicative nature of price fluctuations. A common objection which I hear from people about physicists studying price fluctuations is: but they depend on so many factors such as political decisions, latest Greenspan announcements, wars, news from companies about their earnings etc. The answer to this is to consider as short time intervals as possible at which price fluctuations come from market mechanism, law of supply and demand itself. INTRIODUCE THE FIGURE HERE. On this figure there are all these wiggles which have nothing to do with breaking news. So there are two advantages to use small Dt: better statistics and cleaner data.

## 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%

**Notes:**
As I already mentioned the mainstream theoretical economics postulates that the logarithm of the price performs a gaussian random walk. Here is how  a sample of 8000 gaussian variables looks like.

## Quick experiment: free data fromwww.nyse.com/marketinfo/nysestatistics.html

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

- 10%

- -10%

**Notes:**
And here is the real market signal. For this quick experiment I used free daily data on closing value of DJIA from 1966 to 1998. So here Dt=1day. What immediately jumps to one’s eyes is the existence of regions of high and law volatility. To compare to a gaussian theory let me plot the histogram of these increments. What immediately strikes the eye is that it is approximately symmetric. You may ask:  we know that stock was consistently growing all this time. Where is this growth? The answer is: it is hidden in the tiny offset of the center.
Red gaussian has the same dispersion as measured in the signal itself, whereas black gaussian is an unsuccessful attempt to fit a parabola at least to the very center of the distribution.
Perhaps the most spectacular way to illustrate the non-gaussian nature of this distribution is to say that according to red gaussian the probability of 1987 crash would be 10^-135! We must have been extremely lucky (or unlucky) to witness such a rare event. 

## 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)

**Notes:**
Here I present a more professional attempt to measure the Pdf of price fluctuations. Here the underlying index is SP500 at 1min resolution.
The blue curve is gaussian which does not even come close to describe the distribution. The red curve is more interesting. No many people know that not all uncorrelated stochastic processes satisfy the CLT. Indeed, CLT requires a finite second moment. If one has a sequence of random variables drawn from a distribution with long power law tails with exponent 1+mu where 1<mu<2 we have a situation where average is defined by standard deviation is not. The sum of such variables converges to a non-gaussian limit knows as Pareto-Levy distribution. The price signal in such process has a non-random walk exponent 1/mu.
After observing cotton price fluctuations Mandelbrot in 1963 conjectured that price fluctuations have PL form. That was well before he came up with the concept of fractals. This was a nice empirical model but reality is more complicated:a careful study indicates that really large fluctuations are less likely than Pareto-Levy distribution predicts them to be. In particular the second moment is FINITE. How exactly the Pareto-Levy distribution is truncated is a subject of controversy among econophysicists: one group claims that there is an exponential cutoff, while the other observes the crossover to a steeper power law, such that the standard deviation is finite. The consensus seems to incline towards the latter.

## Crossover to 1+4 forSERIOUSLY large fluctuations

- V. Plerou et al (BU group), 1999
- 1000 largest companies from TAQ in 1994-1995 at 5min resolution

**Notes:**
To explore very far tails one needs to work with many stocks (just to collect sufficient statistics)
That is exactly what was done by BU group in this paper by Plerou and collaborators. They normalized fluctuations of 1000 individual companies 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)

**Notes:**
I would not take this very seriously if I haven’t had corroborating evidence from another group studying completely different object: currency exchange rates.

## Correlations in S&P500

- < p(0) p(t) >

- < |p(0)| |p(t)| >

- day

- year

- month

- from Y. Liu et al, 1999

**Notes:**
Of course, the knowledge of PdF of price fluctuation does not give us the whole information about price fluctuation. In order to be able to derive PdF of price fluctuations taken with 1 day interval from PdF of the price fluctuations taken at 30 minutes interval one needs to know how these price increments are correlated with each other. Here for once mainstream economics gives a clear and unambiguous answer, which is given by Efficient Market Hypothesis:
The answer is - prices should be uncorrelated. Any correlation is arbitraged away by collective actions of traders.
At the first glance statistical physics agrees: correlations between price increments fall off exponentially with a time constant 4 minutes. 
But the absence of the second moment does not mean there is no correlations: there higher order correlation functions.
It turns out that it is signs of price increments which lose their coherence very quickly, while the absolute value of price increments remains correlated over very long time intervals. This does not contradict the spirit of EMH: indeed, only price correlations that can be used for prediction should be eliminated.
The long range correlation in the absolute value of price increments would mean that if the market is very volatile today it will be volatile tomorrow and the day after tomorrow. But there is no way you can exploit this knowledge. In fact exactly this type of correlations was observed in real S&P500 data. The correlations in absolute price increments seem to fall off as a power law with extremely small exponent gamma=0.3. There is no obvious change in this behavior on scale of days, month and even years.

## 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 >

**Notes:**
This concludes the first part of my talk. I just want to summarize the details:

## 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 strategies
- 2.  Models based just on the trading rules of the       market (how supply/demand determines the price)

**Notes:**
The next step is: OK we have all these nice empirical laws. Maybe they are even universal. How can we explain them?
The existence (but not the exact functional form) of “fat tails” is not really new to economists. In fact there is the whole branch of economics called econometrics devoted to just measuring a signal like this. What models do they use?ARCH and GARCH.
Physicists want to understand. So as usual they start with toy models. There are two main directions in constructing these toy models: strategy based and market mechanism based.

I will start exploring this list from the end. Also, since I am giving this colloquium I can give a preferential treatment to my own model.

## 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

**Notes:**
Here I want to introduce a toy model. The main purpose of this model is not to give the right exponent but rather to draw researcher’s attention to the microscopic trading rules of the market and to show that one does not necessarily has to have evolving strategies and collective behavior of traders to get a nontrivial behavior of the price signal. 
Traders in my model are extremely dumb: they don’t have strategies at all! Instead they are given two options of how they can behave if they want to sell  or buy stock. And to introduce these options I need to get just a bit of technical definitions: there are two types of orders on any market: one is a market orders which is similar to a trader saying: “Rush, rush, quickly sell my stock at the best currently available price”. The second is limit order, which is when agent says: sell this stock only if its price raises above a predetermined level. Otherwise disregard my order.
So in my model agents just randomly place of of these types of orders and all orders are kept in limit order book. A new market order is satisfied with the best currently available limit order, while limit orders are kept indefinitely. 
 There are gross oversimplifications in my model (no strategies or market makers, all trades are of the same unit volume) but I wanted to keep this model as simple as possible to make my point

## Price signal in a limit order model

**Notes:**
Here is how the price signal in my model looks like: one can see correlations in volatility (with approximately right exponent describing their power law decay) and a pdf with a crossover between two power law regimes. The exponents are wrong, however! Obviously, to make it right one needs to mix in this model some extra secret ingredients. Which ones? I don’t know yet!

But the I will feel that I succeeded in introducing this model if after it physicists would start to pay more attention to limit orders as a potential source of nontrivial behavior.

## Limit order book

- Price axis

- bid-ask spread

- limit orders to sell

- limit 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?

**Notes:**
All nontrivial features of my modeland many secrets of real markets are hidden in a limit order book.
 Not readily available but will be in the future as market will change to the electronic format
 There are many interesting questions that are just starting to be asked by economists

## 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

**Notes:**
Now I want to introduce another toy model based on strategies and their evolutions: the Minority Game. 
In fact from the time this model found its way to physics community it created a whirlpool of activity: there are more than 20 papers devoted to this model. 
The original variant was introduced by famous Stanford/Santa Fe Institute economist Brian Arthur to illustrate concept of bounded rationality. He is one of the rebels coming from inside the economics and his main enemy is this concept of perfectly rational, deductively thinking agents. His own point of view which he relentlessly promotes is that in a complicated system such as real economy, compared to simple game such as tick-tack-toe, there is no place for rationality. Instead, the agents use inductive adaptive thinking, based on incomplete information small set of strategies. In 1994 he came up with the “bar problem”, where he studied the consequences of bounded rationality in decision making of members of the Santa Fe Institute who every Thursday decide whether to attend the Irish music concert in bar “El Farol” .
It took three years and two physicists to realize that this model can be readily reformulated as a very crude model of stock price fluctuations. In order not to confuse you I’ll forget about bar attendance and use this interpretation of the rules of the model, which was christened “Minority Game” by the authors.
The game is played by an odd number N of traders.
STRESS players base their decision on the same information

## Testing the efficient market hypothesis in MG

- All 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

**Notes:**
The beauty of this model is that in it one can study and (hopefully) understand better many of serious concepts of the mainstream economics such as Market Efficiency, herd effect, synergetic of producers and speculators, etc.
One of the things one could study in this model is the efficient market hypothesis: how strategies used by players remove correlations that can be used for price prediction? As we saw on an example of real price fluctuations all such correlations are removed, while correlations which cannot be used for sure personal gain are still present: see the right panel.
One should not take this observation at the face value, that it pays to average one’s observation of the market over longer time window. There are versions of Minority game where different players with different memory spans happily coexist.

## “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

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

**Notes:**
Hopefully I have convinced you that physics have lots of reasons to be interested in economics and have a lot to offer. This opinion is shared by a Nobel Prize winner in economics Harry Markowitz. In his recent letter to an econophysicist Dietrich Stauffer he wrote:

## 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

**Notes:**
Now I am ready to answer your questions. On my last slide I  just wanted to show two web sites you may find useful