The Econophysics of the Brazilian Real-US Dollar Rate

1
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

2

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

3
Data

• Daily and intraday
• Daily series
• 2 January 1995 to 31 December 2003
• 15-minute series
• 930AM of 19 July 2001
• to 430PM of 14 January 2003

4
Raw Daily Series
5
Daily Returns
6
Raw Intraday Series
7
Intraday Returns
8
Discoveries

• Related to regularities found in the study of
  returns
• for increasing

9
Power LawsLog-Log Plots

• Newtonian law of motion governing free fall can
  be thought of as a power law
• Dropping an object from a tower

10
Power LawsDrop Time versus Height of Free Fall
The relation between height and drop time is no
linear
11
Power LawsLog of Drop Time versus Log of Height
of Fall
12
Power LawsLog-Log Plots
13
Daily Real-Dollar RatePower Law in Mean
14
Daily Real-Dollar RatePower Law for the Means of
Increasing Return Time Lags
15
Daily Real-Dollar RatePower Law in Standard
Deviation I
16
Daily Real-Dollar RatePower Law in Standard
Deviation II
17
Hurst Exponent
18
Hurst Exponent and Efficiency

• Single returns ( )
• Hurst exponent
• Daily data
• Intraday data
• Such figures are compatible with weak efficiency
  in the real-dollar market

19
Daily Real-Dollar RatePower Law in Hurst
Exponent I
20
Daily Real-Dollar RatePower Law in Hurst
Exponent II
21
Daily Real-Dollar RatePower Law in Hurst
Exponent III
22
Hurst Exponent Over TimeDaily Data
23
Hurst Exponent Over TimeHistogram of Daily Data
24
Hurst Exponent Over TimeIntraday Data
25
Hurst Exponent Over TimeHistogram of Intraday
Data
26
Daily Real-Dollar RatePower Law in
Autocorrelation Time
27
LZ Complexity
28
Daily Real-Dollar RatePower Law in Relative LZ
Complexity
29
15-Minute Real-Dollar RatePower Law in Mean
30
15-Minute Real-Dollar RatePower Law in Standard
Deviation
31
15-Minute Real-Dollar RatePower Law in Hurst
Exponent I
32
15-Minute Real-Dollar RatePower Law in Hurst
Exponent II
33
15-Minute Real-Dollar RatePower Law in Hurst
Exponent III
34
15-Minute Real-Dollar RatePower Law in
Autocorrelation Time
35
15-Minute Real-Dollar RatePower Law in Relative
LZ Complexity
36
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 ? gt 1 the mean of the distribution exists and
  equals the location parameter.
• But if ? lt 2 the variance is infinite.
• The pth moment of a Lévy-stable random variable
  is finite if p lt ?.
• The scale parameter determines the width, whereas
  the location parameter tracks the shift of the
  peak of the distribution.

37
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

38
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 SP 500

39
Power Laws in Return TailsStock Markets

• Index a 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
• a 2 Gaussian scaling
• a lt 2 non-Gaussian scaling
• For the SP 500 stock index a 1.4
• For the Bovespa index a 1.6

40
SP 500Probability Density Functions
41
SP 500Power Law in the Probability of Return to
the Origin
42
SP 500Probability Density Functions Collapsed
onto the ?t 1 Distribution
43
SP 500Comparison of the ?t 1 Distribution
with a Theoretical Lévy and a Gaussian
44
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

45
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

46
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

47
Probability of Return to the Origin
48
Probability of Return to the Origin
49
Lévy Flights
50
Lévy Flights
51
Lévy Flights
52
Lévy Flights
53
Exponentially Damped Lévy Flights
54
Exponentially Damped Lévy Flights
55
Exponentially Damped Lévy Flight
56
Exponentially Damped Lévy Flight
57
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

58
Multiscaling
59
Multiscaling
60
Multiscaling
61
Multiscaling
62
Multiscaling
63
Multiscaling
64
Multiscaling
65
Multiscaling
66
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

67
Log-Periodicity
68
Log-Periodicity