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This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the information contained in the precursory patterns before large stock market crashes. The chapter first considers how models of cooperative behaviors resulting from imitation between agents organized within a hierarchical structure exhibit the announced critical phenomena decorated with “log-periodicity.” It then examines the underlying hierarchical structure of social networks, critical behavior in hierarchical networks, a hierarchical model of financial bubbles, and discrete scale invariance. It also discusses a technique, called the “renormalization group,” and a simple model exhibiting a finite-time singularity due to a positive feedback induced by trend following investment strategies. Finally, it looks at scenarios leading to discrete scale invariance and log-periodicity.

This chapter examines the universal nature of the critical log-periodic precursory signature of stock market crashes. It considers the crash of October 1987 and of October 1929; the Hong Kong crashes of 1987, 1994, and 1997; the crash of October 1997 and its resonance on the U.S. market; currency crashes; and the crash of August 1998. It also discusses a nonparametric test of log-periodicity, the slow crash of 1962 ending the so-called “tronics boom,” and the Nasdaq crash of April 2000. Finally, it looks at “antibubbles,” taking into account the “bearish” regime on the Nikkei starting from January 1, 1990, the price of gold after the burst of the bubble in 1980. The chapter shows that large stock market crashes are analogous to critical points studied in the statistical physics community in relation to magnetism, melting, and similar phenomena.

This chapter examines how to predict stock market crashes and other large market events as well as the limitations of forecasting, in particular in terms of the horizon of visibility and expected precision. Several case studies are presented in detail, with a careful count of successes and failures. After providing an overview of the nature of predictions, the chapter explains how to develop and interpret statistical tests of log-periodicity. It then considers the concept of an “antibubble,” using as an example the Japanese collapse from the beginning of 1990 to the present. It also describes the first guidelines for prediction, a hierarchy of prediction schemes that includes the simple power law, and the statistical significance of the forward predictions.