1. 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, 026117 (2006). “Statistical Regularities in the Return Intervals of Volatility”, Eur. Phys. J. B 55, 123 (2007).

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

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

4. Databases Analyzed DAILY DATA –U.S.A. Stocks, 1962-2007, total=10 7 records  –Foreign Exchange Rates, 1971-2007, total=10 5 records  –Crude Oil Futures, 1985-2007, total=10 4 records  INTRADAY DATA –Trades And Quotes: 2001-2002, 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: 1984-1996, total=10 5 records, sampling interval=10 min  from Yahoo Finance;  from Federal Reserve;  from Energy Information Administration;  from New York Stock Exchange.

5. 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

6. 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

7. Result #1: Scaling in Return Intervals

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

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

10. Result #3: Conditional PDF

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

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

13. Result #5: Universality in Correlations

14. 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.