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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
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Sergio Da Silva
Department of Economics, Federal University of Rio Grande Do Sul
Department of Statistics, University of Brasilia
Department of Physics, Federal University of Alagoas
Department of Physics, University of Brasilia
and the associated paper are available at
Daily and intraday
2 January 1995 to 31 December 2003
9:30AM of 19 July 2001
to 4:30PM of 14 January 2003
Related to regularities found in the study of returns
Newtonian law of motion governing free fall can be thought of as a power law
Dropping an object from a tower
The relation between height and drop time is no linear
Single returns ( )
Such figures are compatible with weak efficiency in the real-dollar market
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.
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
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
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
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
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
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
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
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
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