Statistical Physics Approaches to Statistical Physics Approaches to Financial FluctuationsFinancial Fluctuations

Fengzhong WangFengzhong WangAdvisor: H. Eugene StanleyAdvisor: H. Eugene Stanley

Dec 13, 2007Dec 13, 2007

Collaborators: Philipp Weber, Woo-Sung Jung, Irena Vodenska, Kazuko Yamasaki and Shlomo Havlin

“Scaling and Memory of Intraday Volatility Return Intervals in Stock Markets”, Phys. Rev. E 73, 026117 (2006). “Statistical Regularities in the Return Intervals of Volatility”, Eur. Phys. J. B 55, 123 (2007).

OutlineOutline

• Questions: – What are financial fluctuations?– Why we study?

• Databases• Results:

– Scaling– Memory– Long-term correlations

• Take home message

What are financial fluctuations?What are financial fluctuations?

Help understand markets and control risk

Ex: Stock Price and Shares Traded

!!!Dollars Giga 4~

Shares! 10~ 8

!!!Dollars Giga 30 Lose

days 3in 3$ Drop

Why study financial fluctuations?Why study financial fluctuations?

Databases AnalyzedDatabases Analyzed• DAILY DATA

– U.S.A. Stocks, 1962-2007, total=107 records

– Foreign Exchange Rates, 1971-2007, total=105 records

– Crude Oil Futures, 1985-2007, total=104 records • INTRADAY DATA

– Trades And Quotes: 2001-2002, every U.S.A. transactions, total=109 records

30 stocks: Dow Jones Industrial Average (DJIA), sampling interval=1 min, total=107 records

– S&P 500 Index: 1984-1996, total=105 records, sampling interval=10 min

from Yahoo Finance; from Federal Reserve;from Energy Information Administration;from New York Stock Exchange.

How to Calculate Volatility?How to Calculate Volatility?

Step 1: Compute price change |log(p(t+1)/p(t))|

Step 2: Remove intraday pattern by dividing A(s)

Step 3: Normalize by standard deviation

Our Approach: Return Intervals Our Approach: Return Intervals qq of Volatility of Volatility

Step 2: Calculate all time intervals between volatilities above q

q=3

q=2

Step 1: CHOOSE a threshold q

Result #1: Scaling in Return IntervalsResult #1: Scaling in Return Intervals

xxf

fPq

exp~)( :Function Scaling

1 : (PDF)Function Density y Probabilit

Result #2: UniversalityResult #2: Universality

1 : ns)correlatio (no Shuffled

datasets 31 allfor 4.0 :ns)correlatio(with Original

)exp(~)(function Scaling

xxf

A) w.r.t. Sampling Intervals B) w.r.t. Stock Names

How to Analyze Memory?How to Analyze Memory?

S1

S8

Divide return intervals into 8 subsets: S1, S2, …, S8

Stock GE

Result #3: Conditional PDFResult #3: Conditional PDF

How to Measure Long-Term Correlation?How to Measure Long-Term Correlation?Method: Detrended Fluctuation AnalysisMethod: Detrended Fluctuation Analysis

tS Signal

ncorrelatio positive:5.0

ncorrelatio no:0.5

ncorrelatio negative:0.5

~/))(~

()(1

2

dNdSSdFN

ttt

(d)S~ Trend t

d

Result #4: Detrended Fluctuation AnalysisResult #4: Detrended Fluctuation Analysis

Surprise: Return interval correlations Volatility correlations

Result #5: Universality in CorrelationsResult #5: Universality in Correlations

Take Home MessageTake Home Message

• Return intervals scale.

• Scaling is universal for many markets and many time scales.

• Return intervals show memory.

• Scaling and memory are related to long-term correlations in volatility.