Introduktion til Econophysics ----------------- Nykredit, 6. december 2007 Mogens Høgh Jensen, Niels Bohr Institutet 1. Econophysics: En ny disciplin indenfor fysikken gennem de sidste ca. 10 år. 2. Komplekse Systemer: Startede i 80 erne: Helt nye beregningsmetoder i fysikken: Kaos, fraktaler turbulens, tids-serie-analyse, korrelationer, osv.
3. Invers statistik (fra turbulens): forudbestemt gevinst-niveau. Hvornår nås det for første gang 4. For Dow Jones Indekset: Fordelingen af investeringstider: vel-defineret maksimum, optimal investerings-horisont. 5. For gevinst/tab niveauer af samme størrelse: maksimum for gevinst ligger på dobbelte tid af maksimum for tab: Asymmetri! 6. Asymmetri er ikke tilstede for enkelte aktier.
Samarbejde med: Anders Johansen, NBI; Ingve Simonsen, Trondheim Peter Ahlgren, Nykredit/NBI; Henrik Dahl, NBI/Nykredit Kim Sneppen, NBI, Raul Donangelo, Rio, Brazil Felippo Petroni, Rome Publikationer: MHJ, A. Johansen and I. Simonsen, Optimal Investment Horizons, Eur. Jour. Phys B 27, 583 (2002) MHJ, A. Johansen and I. Simonsen, Inverse Statistics in Economics: The gain-loss asymmetry, Physica A 324, 338 (2003). MHJ, A. Johansen, F. Petroni and I. Simonsen, Inverse Statistics in the Foreign Exchange Market, Physica A 340, 678 (2004). A. Johansen, MHJ and I. Simonsen, Inverse Statistics for Stocks and Markets, submitted (2005). R. Donangelo, MHJ, I. Simonsen and K. Sneppen, Synchronization and Asymmetry in Stock Markets: The Consequences of Fear, J. Stat. Mech. 11, L11001 (2006). P. Ahlgren, MHJ, I. Simonsen, R. Donangelo, K. Sneppen, Frustration driven stock market dynamics: Leverage effect and asymmetry, Physica A 383, 1-4 (2007).
Vi kommer fra Niels Bohr Institutet! Fysikkens Mekka.. Hvorfor econophysics?
Hvad er Econophysics? Startede for ca. 10-12 år siden Fysikere brugte metoder fra komplekse systemers fysik Store data mængder, lange tids-serier Kaos-terorier Fraktal teorier Turbulens teorier Statistik, korrelationer
Komplekse Systemer: Strange Attractors Ikke-periodisk bevægelse, stor afhængighed af begyndelses-betingerlserne Lorenz attractor
Kaotisk dynamik : To tids-serier: Afstand gror eksponentielt (begyndelsesbetingelser): Positiv Lyapunov eksponent λ definerer kaotisk dynamik Deterministisk støj!
Tids-serie fra turbulens model: Re u 5 Re u 14 Denne kan analyseres med embedding metode: Fraktaler
Fraktal Dimension: Ikke hel-tallig
Fraktal struktur Self-similær på alle skalaer!
Korrelationer: Vigtige i Komplekse Systemer: Hvem vekselvirker med hvem? Boids in Chamonix
Atmosfæren er kaotisk og turbulent :
Netværk Proteiner i Gær!
Direct Numerical Simulations (DNS): Not very large! Vincent-Meneguzzi
νu 16 2 Bump i en turbulent tids-serie
Kaotisk tids-serie Re u 14 Im u 11
Hvilken slags fluktuationer får man i turbulens? Ikke normal-fordelte!!
Hvilken slags fluktuationer får man i finans? Mantegna and Stanley
Også ikke normal-fordelte!!
Fluktuationer i turbulens er som i finans!
Turbulens Dow Jones Index With inflation Detrended (over 1000 days)
Børsens Investeringstillæg!
Estimate differences in the DJIA: Inverse statistics (initially defined for turbulence): When does it for first time exceed predescribed level :
Inverse statisctis for =0.05 Maximum: Optimal Investment horizon Power law tail Fit: generalized Gamma function:
Positive : Gains Negative : Losses DJIA Note: Asymmetry between gains and losses (maybe related to leverage effect.)
DJIA
SP500
NASDAQ
Inverse statistics for single stocks
Inverse statistics for single stocks
Averaged over many single stocks NOTE: No asymmetry
How to explain the asymmetry in the market? External events (wars, terror, earthquakes, hurricanes) introduces a fear factor in the market Psychology of society/market: When stocks begin to fall they do it synchronously Under up-trends stocks move more or less randomly
The Fear Factor Model (FFM) For the log-price of a stock: With prob. p : all stocks move downwards synchronously With prob. 1-p : they do independent biased random walks With prob. q : move upward With prob. 1-q : move downward q determined from: Requirement : s i (t) is drift-less p : fear factor N : # of stocks in the index
What to say about climate? Climate Models: Chaotic dynamical systems Small errors will grow
Is-kerne data fra Grønland
Klar asymmetri I klimaet over 60.000 år