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Aspects of Financial Physics Gianaurelio Cuniberti Max-Planck-Institut für Physik komplexer Systeme www.infm.it/econophysics Collaborators: M. Raberto Università di Genova E. Scalas Università del Piemonte Orientale G. Susinno Monis, London A. Valleriani MPIKG, Berlin

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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. Don‘t 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 d‘économie 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 week‘s 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 log P(t) is (drifted) arithmetic Brownian Motion P(t) is (drifted) geometric Brownian Motion are prices geometric BMs? risky curve gaussian white noise

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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