U. U. LeidenLeiden, Feb. 22, 2010, Feb. 22, 2010Exercice Exercice #2#2

DFA: DFA: Detrended Detrended Fluctuation Fluctuation AnalysisAnalysis

Marcel Ausloos (*)7 rue des Chartreux, B-4122 Neupré

(*) previously at GRAPES(*) previously at GRAPES(Group for Research and Applications of Physics in Economy and Sociology)(Group for Research and Applications of Physics in Economy and Sociology)

A

P E

S

G R

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 22/xxx/xxx

MotivationMotivation

in in order order to to understand understand how how strong strong are Longare LongRange Range and and Short Range fluctuationsShort Range fluctuations

to to suggest some investment suggest some investment techniquetechnique

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 33/xxx/xxx

Get rich Get rich ? ? (à la Ausloos)(à la Ausloos)

obtain the obtain the power power law behavior describing the rmslaw behavior describing the rms..deviation deviation of of the the fluctuationsfluctuations

as a as a function function of of the window the window sizesize see see αα value value and and range of range of windowswindows

calculate the calculate the time time dependence dependence of of αα exponent exponent moving moving a a window with appropriate window with appropriate size size along the along the time axis,time axis,

supposing stationarity supposing stationarity of of the increments the increments !!!!!! invent some investment strategyinvent some investment strategy calculate the returns calculate the returns (gains, of course)(gains, of course)

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 44/xxx/xxx

DEFINITIONSDEFINITIONS

““VariabilityVariability””

““Log returnsLog returns””

““Normalized variabilityNormalized variability””

DΔt y(t) = y(t+Δt) - y(t)

Z(t,Δt) = [DΔt y(t) - <DΔt y>] /sΔt

Time lag effect ?

L Z(t ,Δt) = log[ y(t+Δt) /y(t) ]

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 55/xxx/xxx

Standard & Poor 500Standard & Poor 500

5056 data points

http://finance.yahoo.com

(normalized)

Jan. 01,1980- Dec. 31,1999

Daily data

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 66/xxx/xxx

S&P500 S&P500 PDF returnsPDF returns

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 77/xxx/xxx

Returns Returns 97-0297-02

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 88/xxx/xxx

Southern Oscillation IndexSouthern Oscillation Index

P (Darwin) -

P (Tahiti) =

1672 data points

http://www.cpc.ncep.noaa.gov/data/indices/

(normalized)

1886-2006

Monthly data

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 99/xxx/xxx

Detrended Detrended Fluctuation AnalysisFluctuation Analysis (integrated series or not?)(integrated series or not?)

x=0.56

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1010/xxx/xxx

DFA DFA webweb

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1111/xxx/xxx

Detrended Detrended Fluctuation AnalysisFluctuation Analysis

LongRangeCorrelations ?

on SP500increments

Brownianvalues

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1212/xxx/xxx

SP500 80-99 SP500 80-99 ββ

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1313/xxx/xxx

SGPSGP

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1414/xxx/xxx

Detrended Detrended Fluctuation AnalysisFluctuation Analysis

LongRangeCorrelations ?

on SOIincrements

NonBrownianvalues

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1515/xxx/xxx

Polish Polish group DFA for crashesgroup DFA for crashes

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1616/xxx/xxx

DAX DFA(q)DAX DFA(q)

+ DFAq

+ DFA q,m

Variants/generalizations

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1717/xxx/xxx

DFA(q)DFA(q)word length (L); word frequency (F)word length (L); word frequency (F)

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1818/xxx/xxx

DFA DFA forecasting strategyforecasting strategy If If αα < 0.5< 0.5 AntipersistenceAntipersistence Negative correlationNegative correlation «« very noisyvery noisy »» If If αα <<<< 0.5<<<< 0.5 up, down, up, down, up,up, down, up, down, up,

down, up, down,..down, up, down,.. .... .... probabilisticprobabilistic .. αα = 0 : = 0 : white white noisenoise

If If αα > 0.5> 0.5 PersistencePersistence Positive Positive correlationcorrelation «« very smoothvery smooth »» If If αα >>>> 0.5>>>> 0.5 Up, up, up, up, up, ..., down,Up, up, up, up, up, ..., down,

down, down, ..down, down, .. .... .... probabilisticprobabilistic αα = 1 : = 1 : smooth smooth lineline

αα = 0.5 (or = 0.5 (or ββ = 2) : = 2) : Brownian Brownian motionmotion

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 1919/xxx/xxx

Local DFA: DJIA Jan.84-Aug.97Local DFA: DJIA Jan.84-Aug.97

““locallocal”” (or temporal) (or temporal)DFADFA

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 2020/xxx/xxx

«« LocalLocal » » DFA DFA

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 2121/xxx/xxx

BEF - DEMBEF - DEM ((veryvery) ) antipersistentantipersistent α α = 0.23= 0.23 ...... N.B.N.B.

NLG/BEFNLG/BEF : 0.26: 0.26 DKK/BEF : 0.31DKK/BEF : 0.31 FRF/BEFFRF/BEF : 0.37: 0.37

N.B.N.B. JPY/USDJPY/USD : 0.55: 0.55 USD/CAD : 0.55USD/CAD : 0.55 GBP/USDGBP/USD :: 0.550.55 USD/DEM : 0.55USD/DEM : 0.55

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 2222/xxx/xxx

USD/DEM gainUSD/DEM gain

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 2323/xxx/xxx

GoldGold

Leiden,Leiden,Feb.2010Feb.2010

M. AusloosM. Ausloos 2424/xxx/xxx

Make Make moneymoney