2. The Econophysics of the Brazilian Real-US Dollar Rate Sergio Da Silva Department of Economics, Federal University of Rio Grande Do Sul Raul Matsushita Department of Statistics, University of Brasilia Iram Gleria Department of Physics, Federal University of Alagoas Annibal Figueiredo Department of Physics, University of Brasilia

3. This presentation and the associated paper are available at SergioDaSilva.com

4. Data Daily and intraday Daily series 2 January 1995 to 31 December 2003 15-minute series 9:30AM of 19 July 2001 to 4:30PM of 14 January 2003

5. Raw Daily Series

6. Daily Returns

7. Raw Intraday Series

8. Intraday Returns

9. Discoveries Related to regularities found in the study of returns for increasing

10. Power Laws Log-Log Plots Newtonian law of motion governing free fall can be thought of as a power law Dropping an object from a tower

11. Power Laws Drop Time versus Height of Free Fall The relation between height and drop time is no linear

12. Power Laws Log of Drop Time versus Log of Height of Fall

13. Power Laws Log-Log Plots

14. Daily Real-Dollar Rate Power Law in Mean

15. Daily Real-Dollar Rate Power Law for the Means of Increasing Return Time Lags

16. Daily Real-Dollar Rate Power Law in Standard Deviation I

17. Daily Real-Dollar Rate Power Law in Standard Deviation II

18. Hurst Exponent

19. Hurst Exponent and Efficiency Single returns ( ) Hurst exponent Daily data: Intraday data: Such figures are compatible with weak efficiency in the real-dollar market

20. Daily Real-Dollar Rate Power Law in Hurst Exponent I

21. Daily Real-Dollar Rate Power Law in Hurst Exponent II

22. Daily Real-Dollar Rate Power Law in Hurst Exponent III

23. Hurst Exponent Over Time Daily Data

24. Hurst Exponent Over Time Histogram of Daily Data

25. Hurst Exponent Over Time Intraday Data

26. Hurst Exponent Over Time Histogram of Intraday Data

27. Daily Real-Dollar Rate Power Law in Autocorrelation Time

28. LZ Complexity

29. Daily Real-Dollar Rate Power Law in Relative LZ Complexity

30. 15-Minute Real-Dollar Rate Power Law in Mean

31. 15-Minute Real-Dollar Rate Power Law in Standard Deviation

32. 15-Minute Real-Dollar Rate Power Law in Hurst Exponent I

33. 15-Minute Real-Dollar Rate Power Law in Hurst Exponent II

34. 15-Minute Real-Dollar Rate Power Law in Hurst Exponent III

35. 15-Minute Real-Dollar Rate Power Law in Autocorrelation Time

36. 15-Minute Real-Dollar Rate Power Law in Relative LZ Complexity

37. Lévy Distributions Lévy-stable distributions were introduced by Paul Lévy in the early 1920s The Lévy distribution is described by four parameters (1) an index of stability  (2) a skewness parameter (3) a scale parameter (4) a location parameter. Exponent  determines the rate at which the tails of the distribution decay. The Lévy collapses to a Gaussian if  = 2. If  > 1 the mean of the distribution exists and equals the location parameter. But if  < 2 the variance is infinite. The pth moment of a Lévy-stable random variable is finite if p < . The scale parameter determines the width, whereas the location parameter tracks the shift of the peak of the distribution.

38. Lévy Distributions Since returns of financial series are usually larger than those implied by a Gaussian distribution, research interest has revisited the hypothesis of a stable Pareto-Lévy distribution Ordinary Lévy-stable distributions have fat power-law tails that decay more slowly than an exponential decay Such a property can capture extreme events, and that is plausible for financial data But it also generates an infinite variance, which is implausible

39. Lévy Distributions Truncated Lévy flights are an attempt to overturn such a drawback The standard Lévy distribution is thus abruptly cut to zero at a cutoff point The TLF is not stable though, but has finite variance and slowly converges to a Gaussian process as implied by the central limit theorem A canonical example of the use of the truncated Lévy flight for real-world financial data is that of Mantegna and Stanley for the S&P 500

40. Power Laws in Return Tails Stock Markets Index α of the Lévy is the negative inverse of the power law slope of the probability of return to the origin This shows how to reveal self-similarity in a non-Gaussian scaling α = 2: Gaussian scaling α < 2: non-Gaussian scaling For the S&P 500 stock index α = 1.4 For the Bovespa index α = 1.6

41. S&P 500 Probability Density Functions

42. S&P 500 Power Law in the Probability of Return to the Origin

43. S&P 500 Probability Density Functions Collapsed onto the ∆t = 1 Distribution

44. S&P 500 Comparison of the ∆t = 1 Distribution with a Theoretical Lévy and a Gaussian

45. Lévy Flights Owing to the sharp truncation, the characteristic function of the TLF is no longer infinitely divisible as well However, it is still possible to define a TLF with a smooth cutoff that yields an infinitely divisible characteristic function: smoothly truncated Lévy flight In such a case, the cutoff is carried out by asymptotic approximation of a stable distribution valid for large values Yet the STLF breaks down in the presence of positive feedbacks

46. Lévy Flights But the cutoff can still be alternatively combined with a statistical distribution factor to generate a gradually truncated Lévy flight Nevertheless that procedure also brings fatter tails The GTLF itself also breaks down if the positive feedbacks are strong enough This apparently happens because the truncation function decreases exponentially

47. Lévy Flights Generally the sharp cutoff of the TLF makes moment scaling approximate and valid for a finite time interval only; for longer time horizons, scaling must break down And the breakdown depends not only on time but also on moment order Exponentially damped Lévy flight: a distribution might be assumed to deviate from the Lévy in both a smooth and gradual fashion in the presence of positive feedbacks that may increase

48. Probability of Return to the Origin

50. Lévy Flights

54. Exponentially Damped Lévy Flights

56. Exponentially Damped Lévy Flight

58. Multiscaling Whether scaling is single or multiple depends on how a Lévy flight is broken While the abruptly truncated Lévy flight (the TLF itself) exhibits mere single scaling, the smoothly TLF shows multiscaling

59. Multiscaling

67. Log-Periodicity What if extreme events are not in the Lévy tails, and are outliers? Sornette and colleagues put forward the sanguine hypothesis that crashes are deterministic and governed by log-periodic formulas Their one-harmonic log-periodic function is where And the two-harmonic log-periodic function is given by We suggest a three-harmonic log-periodic formula, i.e. Parameter values are estimated by nonlinear least squares

68. Log-Periodicity