UNIVERSITY OF TURKU

ECONOPHYSICS
UFYS3068

Lecture 1

Jyrki Piilo

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 Practical arrangements of the course

11 x 2h = 22h lectures, 6 exercise sessions, 5op/ECTS

Physics - advanced level. Can be taken as alternative-advanced level
course in theoretical physics.

Lectures: Thu 24.09 - Thu 03.12.

Thu 12-14
Lecture Hall Qh314 and from 26.10 onwards hall XVII
No lectures on Thu 22.10, Thu 29.10

Exercises: Place and time to be decided.

One session, options:
Wed 8 -10 hall XVII
Fri 10 -12 hall XVII
The time of the first session will be given later (~in 2 weeks).

Form of exam: written or oral exam

Lecturer: Jyrki Piilo, Quantum 158, jyrki.piilo@utu.fi,

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 Guest lectures

Antti Koura
Chief Risk Officer
Veritas Pension Insurance
Thursday 26th November @12.15 Hall XVII

Rosario N. Mantegna
Director
Observatory of Complex Systems
University of Palermo, Italy
Monday 30th November @14.15 Hall XX
(Luonnontieteiden talo 2)

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 Literature

This course follows directly:

Mantegna and Stanley: “An Introduction to Econophysics” (CUP 2000 ).

Wider presentation:
Bouchaud and Potters: “Financial Risk and Derivative Pricing - From Statistical
Physics to Risk Management” (CUP 2003 revised edition).

Other sources:
Johnson, Jefferies, Hui: “Financial Market Complexity - what physics tells us about
market behavior” (OUP 2003).
Lax, Cai, Xu: “Random Processes in Physics and Finance” (OUP 2006).
Roehner: “Patterns of Speculation: A Study in Observational Econophysics”
(CUP 2002).

(Be aware: quite a lot of literature available but quality varies)

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 What is econophysics ?

A definition by Mantegna and Stanley:

(they coined the term econophysics during 90s)

Activities of physicists who are working on economics problems to test a

variety of new conceptual approaches deriving from physical sciences.
(interdisciplinary aspect, no “sharp” border)

A classification of contributions of physicists to economics:

๏ Statistical characterization of price dynamics of assets (e.g. stocks). Key

concepts: random process, correlations, power-law distributions.
๏ Pricing of derivatives, e.g., option pricing.
๏ Risk management
๏ Correlations between stocks: analysis of correlation matrices.
๏ Agent-based modelling, minority games
๏ Order book dynamics.
๏ Income distributions of (i) individuals in stable society (ii) companies.
๏ Economic performance of complex organizations, e.g., entire countries or
universities [Lee et al. PRL 81, 3275 (1995)]

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 What is econophysics ?

Econophysics is a new interdisciplinary research field: physicists

contributing to economics.

Can econophysics give genuine contribution to economics?

Bouchaud:

Joe McCauley:
“Econophysics will
displace economics in
both the universities and
boardrooms”.

On the other hand

economists not yet very
convinced...
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What are you suppose to learn from this course ?

This course is not trying to sell you a well

established traditional research field...on the other
hand may turn to be very fruitful field with time

Things to learn:
๏ Basic idea of some of the ingredients in
econophysics
๏ Some current lines of research
๏ Contribution to your ability to make up your
own mind
+ it may help you find a job outside academia...

Note: you will NOT learn how to become rich.

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 Econophysicists in academia and physicists in financial institutions (quants)

Numerous research groups active in econophysics:

Santa Fe, MIT, Oxford, Palermo, Fribourg, Ecole Centrale Paris...
But there does not exist yet any chairs in econophysics in the world.

So far: 1 university where possible to obtain PhD in econophysics:

University of Houston.

The research community in Europe: >100 researchers.

In financial institutions (quantitative finance):
Large number of physicist hired by large financial companies on Wall Street,
London.
In Finland not yet in large numbers.
Examples of quant jobs: http://www.quantfinancejobs.com/

One of the most well-known physicist in financial world: Emanuel Derman:

Ph.D. in theoretical physics from Columbia University in 1973
Between 1973 and 1980 he did research in theoretical particle physics
From 1980 to 1985 he worked at AT&T Bell Laboratories.
1985 Dr Derman joined Goldman Sachs' fixed income division
Managing Director of Goldman Sachs in 1997.
In 2000 head of the firmʼs Quantitative Risk Strategies group
Currently professor at Columbia University (NYC)
Book: My Life as A Quant: Reflections on Physics and Finance

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 Physicists, econophysics, and quantitative finance

๏ Practioner of quantitative finance: quant

๏ Quant tasks: development of derivatives and other financial
instruments, pricing of derivatives, statistical arbitrage...
๏ Econophysics: academic research
๏ Quantitative finance is different to econophysics but sometimes no
sharp border.

An example of a job possible to

apply as a physicist:

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 Popular articles and scientific papers on econophysics

Some popular articles:

Physics Today, September 2005: “Is economics the next physical science ?”
Nature 441, 18 June 2006: News Feature: Culture Crash
Science 322, 21 November 2008: Science Careers (quants)
Nature 455, 30 October 2008: Economics Needs a Scientific Revolution
Nature 460, 6 August 2009: Meltdown Modelling
Nature 460, 6 August 2009: The Economy Needs Agent-Based Modelling

A sample of scientific papers:

R. Mantegna and H. Stanley, “Scaling behavior in the dynamics of an

economic index”, Nature 376, 46 (1995).
R. Mantegna and H. Stanley, “Turbulence and financial markets”, Nature
383, 587 (1996).
F. Lillo, J. D. Farmer, and R. Mantegna, “Single curve collapse of the price
impact function”, Nature 421, 129 (2003).
S. Thurner et al., “Leverage causes fat tails and volatility”,
preprint arXiv:0908.1555
A. Chakraborti et al.,review, “Econophysics: Empirical facts and agent-
based models”, preprint arXiv:0909.1974
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Contents of the course

I INTRODUCTION
1. Preface
2. Efficient market hypothesis and the unpredictability of the price changes

II PROBABILITY THEORY
3. Random walk
4. Lévy stochastic processes and limit theorems

III CHARACTERIZATION OF FINANCIAL DATA

5. Scales in financial data (price and time scales)
6. Stationarity and time correlation
7. Time correlation in financial time series

IV MODELING OF FINANCIAL DATA

8. Stochastic models of price dynamics
9. Scaling and its breakdown
10. ARCH and GARCH processes
11. Financial markets and turbulence

V CORRELATIONS BETWEEN STOCKS AND PORTFOLIO TAXONOMY

12. Correlation and anticorrelation between stocks
13. Taxonomy of a stock portfolio

VI OPTIONS
14. Options in idealized markets: Black & Scholes formula
(Economics “Nobel” prize 1997: Merton and Scholes:
for a new method to determine the value of derivatives")
15. Options in real markets

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 Contents of Lecture 1

I INTRODUCTION

1. Preface
Some historical developments
Central topics of contemporary research

2. Efficient market hypothesis and the unpredictability of the

price changes
Central concepts and paradigms
Arbitrage
Efficient market hypothesis

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 1. Preface
Physicist model ”complex systems” by using tools of statistical and theoretical physics.
E.g. fluid dynamics, turbulence etc.
Financial markets are well defined complex systems:
continuously monitored, transactions recorded – large amount of data.

Examples of financial markets:

Stock markets: NYSE, Nasdaq, OMX Nordic
Commodities markets: oil, metals, wheat etc:
New York Mercantile Exchange...
Derivatives markets: options, futures: Chigaco Board Options
Exchange...

In general, complex system may have all/some of the following features:

๏ Feedback (what to do next depends on the past state).
๏ Non-stationarity (statistical properties change with time).
๏ Many interacting agents (many components/parts with interact in time
dependent way).
๏ Adaptation (agent can change its behaviour depending how others
behave).
๏ Evolution (population evolves).
๏ Open system (system coupled to its environment).
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 1. Preface

Why are we interested in complex system like financial

markets:

๏ Fundamental issues – the urge to understand the

fundamental behaviour of financial markets.

๏ Practical issues – risk management

(simple practical motivation: maximize return with minimum
risk; the mechanisms affecting this very complex).

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 1.1 Motivation
1st revolution – expansion of financial markets

1973 – currencies began to be traded in financial markets (possible to

buy and sell currencies and the their values determined by foreign
exchange markets).

Volume of trading has increased rapidly (foreign exchange, derivative

products etc.).
๏This has created a suitable, well-defined and interesting system for
systematic studies.

2nd revolution – electronic trading

1980’s - electronic trading adapted to foreign exchange markets

(already part of stock exchanges).
๏Electronic storage of data of financial contracts (bid – ask quotes).
๏Consequence – huge amount of high- frequency financial data
available which can be exploited by researchers and financial
institutions for research and risk-management.
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 1.2 Pioneering approaches
Central concepts to be applied to financial markets:

๏ Power-law distributions
๏ Correlations
๏ Scaling
๏ Unpredictable time series and random processes.
(centralconcepts during the last 30 years in studying phase transitions, statistical mechanics,
nonlinear dynamics, and disordered systems).

The first use of power-law distribution and mathematical formalization of

random walk appeared in social sciences !

Pareto wealth distribution in a stable economy (1897):

y ∼ x−ν

y: number of people having income > x

v: exponent.

Pareto: v=1.5 and the law seemed to apply to England, Ireland, Germany, Peru,
some Italian cities like Napoli.

This is an example of power-law behavior of probability distribution:

The value of the probability distribution P(x) for large x depends on some power of x.
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 1.2 Pioneering approaches
A more recent example of Pareto
tails: gross income in US for
1983-2001(from the bulk
exponential to Pareto tail).
in: Chatterjee, A., S. Yarlagadda and B.
Chakraborti, eds., Econophysics of Wealth
Distribution.
Springer 2005.

The result indicates that there exists

different mechanisms for generating
income.

First formalization of random walk:

(this happened before Einstein and Brownian motion):
Bachelier (student of Poincare) 1900:
Pricing of options in speculative markets (today very important field).

Probability of price changes: Gaussian distribution

(nowadays changes in logarithm of price, cf. geometric Brownian motion).

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 1.2 Pioneering approaches
Black-Scholes (BS) option pricing model 1973:
Merton and Scholes – Price in Economic Sciences for the memory of Alfred Nobel
1997 for a new method to determine the value of derivatives.

BS needs correction:
๏ BS based on the assumption of geometric Brownian motion (GBM) of price
changes.
๏ Still open problem which stochastic process describes the changes in the
logarithm of prices.

• The need to know the stochastic process and its properties for risk management
(exact future price can not be known but one can have information on
probabilities of different prices)
Open problems:
1) tails of distributions are fatter than for GBM
2) the time fluctuations of the second moment of price changes.

Mandelbrot 1963: Price changes follow Levy stable distribution (obey generalized
central limit theorem).

Open problems: Is the variance of the distribution finite or infinite ?

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 1.3 The chaos approach

The time series of asset prices are unpredictable, which in principle can be
due 2 reasons:
1. Random process
2. Chaotic process.

Chaotic processes deal with non-linear dynamics which is unpredictable

but deterministic (sensitive to initial condition).

There has been some attempts to use chaos to describe financial markets.

Main argument against the use of chaos theory:

Seems unlikely that all the information affecting the asset prices can be
described by a small number of nonlinear deterministic equations.

Dominant part of the research focuses on random

process approach and we follow this in the course.

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 1.4 The present focus
The governing rules are rather stable and the time evolution of the system is
continously monitored.
Possible to develop models and test their accuracy and predictions.

What type of problems can be treatable ?

1. Statistical characterization of the stochastic process of price changes
Shape of the distribution, temporal memory, higher-order stastitical properties. E.g. long standing open
problem: is second moment of price changes finite ? Nowadays concensus says yes.

2. Agent-based modelling
Computer simulations: agents are given certain properties based on assumptions or real world observations,
agents interacting in markets.

3. Pricing of derivative instruments (quant oriented):

Price of the asset as a stochastic process, the derivative security is evaluated according to the statistical
properties of the asset price (fundamental aspect : nature of the random process. Applied: solution of the
option pricing problem). What happens when some of the assumptions of Black-Scholes formula are relaxed ?
How to use this information for the portfolio optimization ?

4. Modeling of real financial markets

Should reproduce the main properties of the stochastic dynamics of the price changes (Santa Fe artificial
stock market).

5. Study of correlations in financial time series.

Price correlations, volatility correlations, correlations between stocks etc.

6. The statistical properties of economic performances of companies, countries,individuals

Income distributions, growth rates. 20
 1.4 The present focus

Empirical vs. experimental science:

๏ Econophysics is based on empirical analysis

on financial or economic data.
๏ Experiments not performed. In more
traditional physics one finds fields with similar
situation.
๏ Assumption: one is able to verify/falsify
theories associated with the currently existing
data.

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 Contents of Lecture 1

I INTRODUCTION

1. Preface
Some historical developments
Central topics of contemporary research

2. Efficient market hypothesis and the unpredictability of the

price changes
Central concepts and paradigms
Arbitrage
Efficient market hypothesis

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 2. Efficient market hypothesis

Key point of the chapter: Why is the time series

of trading of financial product unpredictable/
stochastic ?

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 2.1 Concepts and variables
In financial markets large number of traders interact with each
other and react to external information to determine the best
price for a given item.

• E.g. In stock markets individual and institutional investors buy and

sell stocks. External information: information on the company,
how the the field of the company is developing, what is the ”large
scale” economic trend...

Traded: commodities (e.g. oil, wheat, metals), equities (stocks),

currencies ($, €), bonds, derivative products, carbon oxide quotas...

Localized markets: New York , Tokyo, London.

Specific stocks in specific stock exchanges (e.g. Nokia in Helsinki
and New York).
Delocalized markets: e.g. foreign exchange (accessible all over the
world).

Trading history gives time series on:

๏ Price
๏ Volume
๏ Number of transactions of a financial product.
Lets see an example of trading history:
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Nokia stock in Helsinki...
 2.1 Concepts and variables

An example of trading history:

Time Price Volume

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 2.2 Arbitrage

This concept is very closely related to efficient market hypothesis. Efficient

functioning of the market arises partly because of the traders exploit rapidly the
arbitrage opportunities.

Arbitrage:
Purchase and sale of same/similar asset in order to profit from price
discrepancies.

Example:
1 kg of oranges: Miami 0.40 EUR, Napoli 0.60 EUR, transport cost 0.10 EUR.
Buy 100 000 kg in Miami and sell them in Napoli.
Risk free profit: 100 000 x [0.60 – 0.40 – 0.10] EUR = 10 000 EUR.

Consequence when repeated:

the demand for oranges in Miami increase and decrease in Napoli,
the price goes up in Miami and goes down in Napoli.
The prices in both places become more ”rational”.

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 2.2 Arbitrage

New arbitrage opportunities appear continually and

are discovered in the markets. As soon as one begins
to exploit them, the system moves in a direction that
eliminates the arbitrage opportunity.

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 2.3 Efficient market hypothesis
This section explains one of the key concepts: efficient market hypothesis:
its formulation in terms of a Martingale, and how this explains the
unpredictable nature of the price changes in the short time scale.

Information about given asset incorporated in its price.

Or reversely: the time series of the price reflects the information.
The market is expected to be highly efficient to determine the most
rational price of the traded asset.
In some sense market is an information processor: information goes in
and the market “calculates” the price.

Question:
•If you “knew” that price of e.g. Nokia will go up tomorrow, what would you do today ?
• What do you think others traders with same info would do ?
•How would this affect the price today ?

How is the information processed in the markets ?

Example 1:

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 2.3 Efficient market hypothesis
How is the information processed in the markets ?
Example 2:
AP
Stocks Dive, Then Soar on Fed Decision
Tuesday August 7, 10:45 pm ET
By Joe Bel Bruno, AP Business Writer
Wall Street Rebounds After Fed Says Maintaining Inflation Fighting Is Top Priority

NEW YORK (AP) -- Wall Street overcame disappointment in the Federal Reserve's failure to move toward an easing of interest rates
Tuesday, and stocks made a late-day surge as the decision was seen as a sign the economy wasn't threatened by turmoil in the credit
markets. Investors were at first deeply disappointed that policymakers, who kept benchmark rates on hold at 5.25 percent, did not
provide any hints about a possible cut. But, after digesting the policy statement, they quickly gained solace the economy is likely to
withstand troubles in the mortgage industry. The Dow Jones industrials rose into positive territory from a 121 point deficit right after the
decision was announced.
 2.3 Efficient market hypothesis
How is the information processed in the markets ?
Example 3:

S&P500 yesterday:
 2.3 Efficient market hypothesis

Efficient market hypothesis: all the available information

is instantly processed when it reaches the market and it is
immediately reflected in a new value of the traded assets.

• Question: How does the market know what is the most

rational price of the asset, e.g. price of the stock of a certain
company given all the economical information related to the
company ?

Historical background of the efficient market hypothesis:

๏ Bachelier 1900: the price of asset in speculative market is a stochastic

process.
๏ During 1950s empirical results showed that correlations of rate of returns
on a short time scale are negligible. Behavior similar to uncorrelated random
walks.
๏ Explicit formulation by Samuelson in 1965 who showed mathematically
that prices fluctuate randomly by using the hypothesis of efficient markets
and rational behaviour. Lets look this in more detail...

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 2.3 Efficient market hypothesis

The consequence of efficient markets and rational behaviour:

๏ Expected value of the price of the asset at time t+1, when the
previous values of the asset are known, is equal to the asset value at
time t.
๏ However, price changes can be positive or negative.

E{Yt+1 |Y0 , Y1 , ..., Yt } = Yt

This type of stochastic process is called a Martingale
and corresponds to a fair game:
• Gains and losses cancel.
• Gambler’s expected future wealth coincides with the current wealth.
The point: the best expectation value of the future price is given by the current price.

For Martingale: see Mantegna&Stanley: Appendix B.

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 2.3 Efficient market hypothesis

๏ The information arriving to the market defines the future value of

the asset, *not* the past value. There is no way of making profit by
using the recorded history of the price fluctuations.

๏ The conclusion: the price changes are unpredictable from the

historical time series.

๏ Empirical investigations show that the time correlation between

price changes is negligible small.

๏ However: The information contained in additional time series

(not only the price of the asset) – earnings/price ratio, dividend
yields – it is possible to make predictions of the price on a long
time scale (longer than one month).

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2.3 Efficient market hypothesis

Weak form of efficient market hypothesis:

The past information on the price changes does not tell anything about
the future price (note on the small print in the fund advertisement: past
returns do not guarantee the future return).

Strict form of the efficient market hypothesis:

No information gives indications about the price changes in the future.

This corresponds to idealized system.

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2.3 Efficient market hypothesis

Studies show that weak form of efficient market hypothesis seem to be

satisfied rather well. The idealization in the strong form holds to certain extent
since ineffective behavior always present.

• Question: What does all this mean from the point of view of upward/downward long-term
trends ? (the cause for trends is the information which is not in the price history itself, in
the short time scale the price changes are stochastic)

The key point of the section:

The information arriving to the market affects the price of the assets.
This information is directly and instantaneously reflected in the asset
price (in the ideal case no arbitrage opportunities). The price changes
are unpredictable on the short time scale since the past time series of
the price changes does not contain information about the future
information.

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 2.4 Algorithmic complexity theory

This section explains how the algorithmic complexity theory can be used to
explain why the time series of returns seem to be random. Note: the
emphasis is on the word seem, doest not explain the origin of randomness
but indicates that this is the case.

Starting point:
Given object can be encoded in an n-digit binary sequence (e.g. word using ASCII
code to binary).

What is the complexity of the object:

Algorithmic complexity theory quantifies the complexity of this object in the following
way: the compexity is given by the bit lenght K(n) of the shortest computer program
that can print the given sequence.

For example: Compare algorithms needed to print

a) 125 000 decimals of the constant pi
b) the time series of daily values of the Dow-Jones Industrial Average (DJIA) from
1898 till the year that the message would contain 125 000 digits. DJIA 1997-2007

The value of pi is not random, and there exists very short

algorithm to produce the wanted digits.
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In contrast the algorithm with comparable efficiency
has not been found for the time series of Dow-Jones.
 2.4 Algorithmic complexity theory
Predictable series of symbols can be compressed whereas unpredictable
series can not be compressed

Time series of Dow Jones contains unpredictable elements.

Algorithmic complexity theory helps because

(i) gives connection between efficient market hypothesis and unpredictability

of stock returns (efficient market hypothesis requires that past price
statistics does not effect the future price changes and complexity theory
indicates that the time series can not be compressed)
(ii) Deviations from randomness allows to verify the validity and limitations of
the efficient market hypothesis
(iii) Makes no difference between a time series carrying large amount of
economic information and pure random process.

What about the information carried by the financial time series ?

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 2.5 Amount of information in a financial time series

The unpredictability of the price changes does not mean that financial time series
does not contain information. In fact the opposite is true.

Because the information contained in the time series is very large, it is very
difficult to extract a subset of economic information associated with some specific
aspect.

In principle one can know all the past information which affected the price but it is
not possible to figure out how each piece of information affected.
The consequence is a random price change without correlations in a short time
scale.

Market is not completely efficient if a given piece of information effects the price in
a specific way (e.g. positive news, it takes certain time before the price reflects in
a ”consensus” way the new price, information should affect the price
instantaneously and the consequence should be completely random price
statistics).

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 2.5 Idealized systems in physics and finance
In physics, idealized systems are used as a starting point for more advanced
studies. Also, idealized systems with appropriate assumptions may already tell
large amount of useful information about the system behavior.

In a similar manner, perfectly efficient market is an idealization whereas real

markets are only approximately efficient. If the right assumptions are used,
even this idealized case gives useful information about the market behavior.
The assumptions and the validity of the results can be empirically tested.

Once accepting this paradigm, an essential step is to characterize the

statistical properties of the random processes in financial markets. For this
we obviously need the concepts of probability theory...to be continued

END OF LECTURE 1

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 Contents of the course
D
I INTRODUCTION O
1. Preface
2. Efficient market hypothesis and the unpredictability of the price changes N
E
II PROBABILITY THEORY
3. Random walk
4. Lévy stochastic processes and limit theorems

III CHARACTERIZATION OF FINANCIAL DATA

5. Scales in financial data (price and time scales)
6. Stationarity and time correlation
7. Time correlation in financial time series

IV MODELING OF FINANCIAL DATA

8. Stochastic models of price dynamics
9. Scaling and its breakdown
10. ARCH and GARCH processes
11. Financial markets and turbulence

V CORRELATIONS BETWEEN STOCKS AND PORTFOLIO TAXONOMY

12. Correlation and anticorrelation between stocks
13. Taxonomy of a stock portfolio

VI OPTIONS
14. Options in idealized markets: Black & Scholes formula
(Economics “Nobel” prize 1997: Merton and Scholes:
for a new method to determine the value of derivatives")
15. Options in real markets
40