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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 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
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 Related to regularities found in the study of returns for increasing
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 LawsDrop Time versus Height of Free Fall The relation between height and drop time is no linear
Daily Real-Dollar RatePower Law for the Means of Increasing Return Time Lags
Hurst Exponent and Efficiency Single returns ( ) Hurst exponent Daily data: Intraday data: Such figures are compatible with weak efficiency in the real-dollar market
15-Minute Real-Dollar RatePower Law in Relative LZ Complexity
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.
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
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 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
S&P 500Probability Density Functions Collapsed onto the ∆t = 1 Distribution
S&P 500Comparison of the ∆t = 1 Distribution with a Theoretical Lévy and a Gaussian
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
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
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
