Statistical Physics Approaches to Financial Fluctuations Fengzhong Wang Advisor: H. Eugene Stanley Dec 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, (2006). “Statistical Regularities in the Return Intervals of Volatility”, Eur. Phys. J. B 55, 123 (2007).

Outline Questions: –What are financial fluctuations? –Why we study? Databases Results: –Scaling –Memory –Long-term correlations Take home message

What are financial fluctuations?  Help understand markets and control risk Ex: Stock Price and Shares Traded Why study financial fluctuations?

Databases Analyzed DAILY DATA –U.S.A. Stocks, , total=10 7 records  –Foreign Exchange Rates, , total=10 5 records  –Crude Oil Futures, , total=10 4 records  INTRADAY DATA –Trades And Quotes: , every U.S.A. transactions, total=10 9 records   30 stocks: Dow Jones Industrial Average (DJIA), sampling interval=1 min, total=10 7 records –S&P 500 Index: , total=10 5 records, sampling interval=10 min  from Yahoo Finance;  from Federal Reserve;  from Energy Information Administration;  from New York Stock Exchange.

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  q 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 Intervals

Result #2: Universality A) w.r.t. Sampling Intervals B) w.r.t. Stock Names

How to Analyze Memory? S1 S8 Divide return intervals into 8 subsets: S1, S2, …, S8 Stock GE

Result #3: Conditional PDF

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

Result #4: Detrended Fluctuation Analysis Surprise: Return interval correlations  Volatility correlations

Result #5: Universality in Correlations

Take 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.