تجزیه و تحلیل جدید متناوب، تغییر ناپذیری مقیاس و مقیاس مشخصه اعمال شده برای رفتار شاخص های مالی در نزدیکی یک سقوط

This work is devoted to the study of the relation between intermittence and scale invariance, and applications to the behavior of financial indices near a crash. We developed a numerical analysis that predicts the critical date of a financial index, and we apply the model to the analysis of several financial indices. We were able to obtain optimum values for the critical date, corresponding to the most probable date of the crash. We only used data from before the true crash date in order to obtain the predicted critical date. The good numerical results validate the model.

During the last years the study of log-periodic structures and characteristic scales and the relation with the concept of scale invariance had grown due to the great amount of physical systems presenting log-periodic structures: fluid turbulence [1] and [2], diamond Ising model [3], earthquakes [4], materials rupture [5], black holes [6] and gravitational collapses [7] among others. In a mathematical context, we recall constructions as the Cantor fractal [3] and [8], with a discrete scale changes invariant. The presence of logarithmic periods in physical systems was noted by Novikov in 1966 [9], with the discovery of intermittence effect in turbulent fluids. The relation between both effects has been deeply studied, but it has not been formalized yet. At the same time, the complexity of international finance has grown enormously with the development of new markets and instruments for transferring risks. This growth in complexity has been accompanied by an expanded role for mathematical models to value derivative securities, and to measure their risks. A new discipline Econophysics, has been developed [10]. This discipline was introduced in 1995, see Stanley et al. [11]. It studies the application of mathematical tools that are usually applied to physical models, to the study of financial models. Simultaneously, there has been a growing literature in financial economics analyzing the behavior of major Stock indices [10], [12], [13], [14] and [15]. The Statistical Mechanics theory, like phase transitions and critical phenomena have been applied by many authors to the study of the speculative bubbles preceding a financial crash (see for example [16] and [17]). In these works the main assumption is the existence of log-periodic oscillation in the data. The scale invariance in the behavior of financial indices near a crash has been studied in Refs. [18], [19] and [20]. This work is organized as follows: In Section 2 we give a short introduction to the relation between intermittence and scale invariance, the conditions that a function has to satisfy when both effects are present. We analyze the relation with characteristic scales, and we finally present a method that detects characteristic scales in different systems using the previous results. In Section 3 we present a model that predicts the existence of intermittence and characteristic scales in the behavior of a financial index near a crash [20]. In Section 4 we develop the methods that we will use for our numerical analysis. Finally, we apply the model to the analysis of the behavior of several financial indices: the S&P500 index near the October 1987 crash, and the Argentina MERVAL index as well as the Brazil BOVESPA index and the Mexico MXX index near the October 1997 Asian crash.

The effects of certain local crisis on various and distant markets have largely been cited. The collapse of the crashes of 1987 (S&P500) dragged the collapse of markets worldwide. However, not every crisis has sufficient strength as to drag the fall of leading indices in other countries. In Ref. [16] it has been shown that the crashes of Asian indices had consequences on emergent markets: the Asian crisis had sufficient strength as to drag the fall of leading Latin American indices. Clearly, all these indices crashed in similar dates due to a dragging correlated effect, which most likely started with the instability of the HSI index. This signals the likelihood of the events in different markets and different economic realities which strengthens the hypothesis of imitation and long range correlations among traders. About the stability of the method, we want to remark that, even if the maximal and minimal values for the error are apparently similar, the graphs are plotted in log-domain. Furthermore, we want to remark that we believe the noise in the market can actually change the crashing date, and the change may be significant. We believe the market is a very complex system. The crash might be delayed due to some market force, so it happened sometime after the market top, but that is out the scope of this work. Our goal in this work is to develop a method for finding the market top which leads to a crash. We also want to remark that we only use the data up to a couple of weeks before the crash in all the cases to estimate the critical time. The estimations have errors of around only 10 trading days. The excellent results validate the method.