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Aspects of Financial Physics
Gianaurelio Cuniberti Max-Planck-Institut für
Physik komplexer Systeme
Collaborators M. Raberto Università di
Genova E. Scalas Università del Piemonte
Orientale G. Susinno Monis, London A.
Valleriani MPIKG, Berlin
www.infm.it/econophysics
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Why Physicists are interested in
Economics (mainly in Finance)? Is it really a
new trend?
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First you guess. Dont laugh, this is the most
important step. Then you compute the
consequences. Compare the consequences with
experience. If it disagrees with experience, the
guess is wrong. In this simple statement is the
key to science. Richard P. Feynman
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Why Physicists... ?
circumstantial reasons
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Why Physicists... ?
educational reasons
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Why Physicists... ?
scientific reasons

• Statistical Physics offers concepts and methods
  applied outside the ordinary physics domain
• evolution of biological species
• immunology
• neural computation
• image reconstruction
• optimization theory
• financial markets
• financial markets have the additional feature of
  a huge amount of empirical data, letting theories
  and conjectures being falsified in the spirit of
  the scientific method
• many agent interactions the realm of Statistical
  Mechanics

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• Antoine-Augustine Cournot
• (Gray, Haute-Saone, 1801 Paris, 1727)
• mathematician, economist
• the first brillant example of the application of
  mathematical ideas in economics Researches into
  the Mathematical Principle of the Theory of
  Wealth (1838)
• introduced theory of oligopolies in terms of
  profit maximization

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Vilfredo Pareto (Paris, 1848 Geneve,
1923) engineer His first book on the political
economy (Cours déconomie politique) included the
famous law of income distribution where N is
the number of people with income X and a,b are
constant (a power law).
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A. Cowles Can stock market forecasters
forecast? Econometrica 1, 309 (1933) M.G.
Kendall The analysis of economic time-series J.
Royal Statist. Soc. 96, 11 (1953) ...The series
looks like a wondering one, almost as the Demon
of Chance drew a random number ... and add it
to the current price to determine next weeks
price. 1953 circular letter of Savage (Yale)
about the work of Bachelier...
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we (and not only we) have forgotten about
Louis Bachelier (Le Havre, 1870 Saint Servan,
1946) mathematician 1900 in his Sorbonne PhD
thesis, introduced the random walk model for
asset prices traded in the Paris stock exchange.
Later in this century was acknowledged as the
father of modern mathematical finance. The
Fokker-Planck equation for diffusion of
probabilities is already in Bachelier 1900
work. L. Bachelier Théorie de la Speculation,
Gauthier-Villars, Paris (1900), reprinted in
1995, Editions Jaques Gabay, Paris. English
translation in P.H. Cootner, The Random Character
of Stock Market Prices, MIT press (1964)
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Elliot Montroll (1916 1983) statistical
physicist studied many complex problems such as
pollution control, traffic flow, population
dynamics, and development of countries from an
agricultural to an industrial society. He felt
that a physicit should be able to contribute to
these important problems of modern times. E.W.
Montroll and W.W. Badger, Quantitative Aspects of
Social Phenomena, Gordon and Breach, London
(1974)
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Benoit Mandelbrot (Warszawa, 1923) (applied)
mathematician In the sixties, was the first to
introduce stable distributions in finance and
economics, in order to explain fat tails of
empirical distributions. B. Mandelbrot, The
variation of Certain speculative Prices, Journal
of Business 36, 394 (1963) B. Mandelbrot,
Fractals and Scaling in Finance, Springer (1997)
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Fisher Black (1938 1995) physicist Myron S.
Scholes (1941) mathematician Robert C. Merton
(New York, 1944) electric engineer Merton and
Scholes have awarded of the Nobel Prize for
Economics in 1997 for a new method to determine
the value of derivatives In 1973, Black, Scholes
and Merton developed a formula for the valuation
of stock options their methodology paved the
way for economic valuation in many areas,
generated new types of financial instruments
and facilitated more efficient risk management
in society
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SDE a poor man approach
riskless curve
gaussian white noise
risky curve
log P(t) is (drifted) arithmetic Brownian Motion
P(t) is (drifted) geometric Brownian
Motion are prices geometric BMs?
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Failure of the Classical Theory
The Geometric Brownian motion is not appropriate
for certain security stocks prices
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Correlations in the Bond-Future Market

• future market
• bund and btp futures opened at different times
  during the period considered (October 91 -
  January 94)
• symbolic dynamics
• correlations
• bund and btp future overnight are
  crosscorrelated
• gambling
• automatic investors are introduced to study the
  possibility of arbitrage

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Futures and Returns
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Symbolic Dynamics
the bond walk (?0)
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Contingency Tables
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Disjoint Monte Carlo
Joint Monte Carlo
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Gambling the rules
1. day 0th before closure, open a short and a
long position on btp future 2. day nth after the
opening of bund future market, order the closure
of the convenient btp position 3. day nth
before closure, the closed position is opened
again 4. increment n, and go to step 2.

• Assumptions
• every operation is costless
• transactions happen exactly at the opening and
  closing prices
• the margin account can be always kept over the
  maintenance margin

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Profiles
Convenience
Yield profiles
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The lotto gambler strategy
use past information on the btp walk to forecast
(in the spirit of the technical analysis)
n
build the probabilities
and draw from them the convenient operation
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Conclusions

• motivations for econophysics
• future market
• the future prices considered are random
  variables crosscorrelated
• gambling
• there was possibility of arbitrage in the
  (frictionless) future market
• EPR
• bund and btp in the EPR scenario quantum
  entanglements and non-locality are, here, prior
  information

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Gambling the rules
1. day (n-1)th before closure, open a short and
a long position on btp future 2. day nth after
the opening of bund future market, order the
closure of the convenient btp position 3. day
nth before closure, close the position still
open 4. increment n, and go to step 1.

• Assumptions
• every operation is costless
• transactions happen exactly at the opening and
  closing prices
• the margin account can be always kept over the
  maintenance margin

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Useful Books

• Anderson, Arrow, and Pines. The Economy as an
  Evolving Complex System. (1988)
• Bouchaud and Potters. Théorie des Risques
  Financiers. (1997)
• Campbell, Lo, and MacKinaly. The Econometrics of
  Financial Markets. (1997)
• Embrecht, Klueppelberg, and Mikosch. Modelling
  Extemal Events for Insurance and Finance. (1997)
• Gouriéroux. ARCH Models and Financial
  Applications. (1997)
• Georgescu-Roegen. The Entropy Law and the
  Economic Process. (197)
• Hull. Options, Futures, and other Derivatives.
  3rd Ed (1996)
• Karatzas, and Shreve. Brownian Motion and
  Stochastic Calculus. (1998)
• Kloeden, and Platen. Numerical Solution of
  Stochastic Differential Equations. (1992)
• Mantegna, and Stanley. Scaling Concepts in
  Finance. (1998)
• Musiela, and Rutkowski. Martingale Methods in
  Financial Modelling. (1997)
• Wilmott. Option Pricing. (1997)
• ...ad libitum in

www.infm.it/econophysics
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Prior Information in the Bond Future Market a
Possibility for Arbitrage?
Gianaurelio Cuniberti Max-Planck-Institut für
Physik komplexer Systeme Marco Raberto INFM and
Università di Genova Enrico Scalas INFM and
Università del Piemonte Orientale
www.infm.it/econophysics