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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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the econophysics of the brazilian real us dollar rate

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

this presentation and the associated paper are available at sergiodasilva com
This presentation

and the associated paper are available at

SergioDaSilva.com

slide3
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

discoveries
Discoveries

Related to regularities found in the study of returns

for increasing

power laws log log plots
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

power laws drop time versus height of free fall
Power LawsDrop Time versus Height of Free Fall

The relation between height and drop time is no linear

hurst exponent and efficiency
Hurst Exponent and Efficiency

Single returns ( )

Hurst exponent

Daily data:

Intraday data:

Such figures are compatible with weak efficiency in the real-dollar market

l vy distributions
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.

l vy distributions37
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

l vy distributions38
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

power laws in return tails stock markets
Power Laws in Return TailsStock 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

l vy flights
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

l vy flights45
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

l vy flights46
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

multiscaling
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

log periodicity
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