Dynamics of Socio-Economic Systems: A Physics Perspective
Transcript of Dynamics of Socio-Economic Systems: A Physics Perspective
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Lecture I:Random Matrix Theory for Financial Markets
Bernd Rosenow Harvard University
References: V. Plerou, P. Gopikrishnan, B. Rosenow, L. Amaral, and
H.E. Stanley, PRL 83, 1473 (1999).
P. Gopikrishnan, B. Rosenow, V. Plerou, and H.E. Stanley,
Phys. Rev. E 64, R035106 (2001).
V. Plerou et al., Phys. Rev. E 65, 066126 (2002).
B. Rosenow et al, Europhys. Lett. 59, 500 (2002).
Dynamics of Socio-Economic Systems: A Physics Perspective,September 19, 2005
Related work: L. Laloux, P.Cizeau, J.-P. Bouchaud and M. Potters, PRL. 83, 1469 (1999)
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
statistical mechanics!
many constituents
complex interactions
stock prices as macroscopic variables
Stock Market
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Outline
• Description of cross-correlations in the stock market – analysis
with random matrix theory (RMT)
• Autocorrelations of correlated modes
• Time dependence of cross-correlations
• Application: optimal investments in the stock market
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Long Term Capital Management
past correlations = future correlations?
NOT ALWAYS !
LTCM loss: 2,5 Milliarden $
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Correlations and Interactions
stock pricechanges
stock market 1994-97
data points
1000 companies, i = 1,...,1000
t = 30 min
Correlations betweencompanies measured by
What to do with this huge (1,000,000 entries) matrix?
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Nuclear Physics
Model calculationsfail!
Wigner: single energy levels not so important, study statistical properties
assumptions: real symmetric random Hamiltonian
prediction: distribution ofnearest neighbor distances s
s
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Stock Market
s
eigenvalues of
no free parameter!
conclusion: up to now,
nuclear physics = economy
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Eigenvalue Distribution
Compare eigenvalues of C to those of a random matrix R constructedfrom i.i.d. time series
agreement randomness deviations information
length of time series
number of time series
Laloux et al., '99, Plerou et al., '99
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Computer Simulation with Octave
M = randn(400,200) generates 200 i.i.d. time series of length 400
C = corrcoef(M) calculates correlation coefficients
d = eig(C) calculates eigenvalues of C
x = linspace(0,5,500)
y = linspace(0,0,500)
for i=1:200, y= y + normal_pdf(x,d(i),0.001); end
y = y./200
plot(x,y)
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Distribution of Eigenvector Components
RMT-prediction for distribution
Laloux et al., '99,Plerou et al., '99
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Market Mode
Example: perfectly correlated returns
Construct portfolio
from eigenvector
and compare it to
S&P500 portfolio
Strong common component in stock prices: market
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Industry Sectors
Construct map of the industry
Eigenvector has contribution
from industry
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Interpretation of Deviating Eigenvectors
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Model for stock price fluctuations
imbalance of supply and demand
noise correlator
return on microscopic time scale
with
dynamics of supply and demand
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Model for ''Interacting'' Stocks
Langevin equation for
Slow dynamics for strongly correlated stocks!
coupling matrix, eigenvalues
relaxation time of eigenmode
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Autocorrelation function
Power law correlations reminiscent of critical spin systems
Detrended fluctuation analysis:
• detrended fluctuation function
• without autocorrelations
• implies power law
correlations
Power Law Correlations
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Are Correlations Stable?
period A period B
01/94 06/94 01/95
How to compare
eigenvectors and ?
(market) is very stable
stable for
980
990
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Dynamics of the Correlation Strength
daily returns of 422 CRSP stocks, 1962-96
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Optimal Investments
•One stock: return , risk
•Diversification into N stocks
•Optimal investment minimizes
interaction random field
Lagrange multipliers fix total return and total invested capital
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Practical Implementation
estimation period A testing period B
01/94 06/94 01/95
• actually realized values
• three different estimates for
i) historical matrix
ii) filtered correlation matrix
iii) standard assumption
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Accuracy of Risk Forecast
0.5 1 1.5 2 2.5 3
10
20
30
40
50
0.5 1 1.5 2 2.5 3
10
20
30
40
50
0.5 1 1.5 2 2.5 3
10
20
30
40
50
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Risk Reduction
1.4 1.5 1.6 1.7 1.8 1.9 2
10
20
30
40
50
(Filtered)C
//(Control)C
Does RMT prediction work best?
construct
and transform back to original basis 1 10 20 30 40 50 60
1.5
1.52
1.54
1.56
1
1
1
1
Number of eigenvalues
Ris
k %
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Random Ferromagnet
Futures Market: deposit coupled equations for signs
S. Galluccio et al., Physica A 257, 449 (1998)
• Spin glass problem for historical
• Random ferromagnet for filtered
only has large components, all other have
0
50
100
150
200
250
300
350
0 100 200 300 400 500 600 700 800 900 1000
line 1line 2
N s
0 500 1000
100
200
300
Rank of eigenvector
B. Rosenow, Bad Honnef, DPG-School on Dynamics of Socio-Economic Systems 2005
Conclusions
• meaningful correlations described by large eigenvalues of C
• portfolios defined by deviating eigenvectors have power law
autocorrelations
• correlations change in time
• Random matrix theory helps to reduce risk of optimal portfolios