Black Swans and fractals: a better way to understand financial markets?

At the beginning of the twentieth century, the French mathematician Louis Bachelier discussed his doctoral thesis entitled Théorie de la Spéculation: the innovative concepts and insights he proposed were afterwards improved and formalised by several researchers and among them in particular by the American economist Eugène Fama, whose greatest contribution is the famous Efficient Market Hypothesis. The works of Bachelier and Fama are strictly connected to each other and they can be imagined as the foundation stones of the entire house of modern finance, which with a more formal and technical expression is defined as capital market theory and has been long since developed and enriched both by economists and by mathematicians: in this sense, a first main augmentation has been brought by Harry Markowitz, whose Modern Portfolio Theory suggests how the problem of diversification can be solved and how efficient portfolios can be identified; a second important improvement has been provided by William Sharpe, whose Capital Asset Pricing Model represents a fundamental technique for valuing stocks and projects of different companies; a third meaningful addition has been supplied by Fischer Black and Myron Scholes, whose option pricing formula allows investors to instantaneously figure out what the fair price of an option has to be.
From a general and global perspective, what actually links and relates to one another all above mentioned ideas, tools and instruments – but also additional contributions provided by several other economists and mathematicians which have not been discussed in my dissertation, although they have a central role in the financial world – are the three main principles on which capital market theory is based, namely rational investors, efficient markets and random walk. Investors are assumed to be rational in the sense that they seek for mean-variance efficiency and search for assets that yield the highest profit for a chosen level of risk, or vice-versa for assets that guarantee the lowest riskiness for a targeted level of return; moreover, they tend to evaluate potential returns based on probabilistic methods, assume that risk can be easily measured according to the Gaussian bell curve and are generally supposed to have both an aversion to risk and the same unique investment horizon. Markets are assumed to be efficient in the sense that they are supposed to automatically and instantaneously discount in prices all relevant information available: this implies that changes in prices are completely unrelated to one another, although the possibility of a form of short-term dependence which dissipates quickly is not completely excluded. Random walks is instead a direct consequence of both the former assumptions, namely rational investors and efficient markets: the idea that prices and returns follow a random walk has a very strong implication, namely that the probability distribution is approximately normal and it is characterised by a mean and a variance which are both finite (Peters 1991:25-26).
However, particularly during the last decades, economists and mathematicians have begun to observe that in the real world the three main insights on which the entire capital market theory is based in several circumstances do not hold. On the one hand, some people have considered unexpected results and unconventional outcomes to be nothing but rare monsters not to be taken in serious consideration and have actually ignored empirical evidence, thus maintaining classical and well-established ideas, tools and instruments as a kind of mantra to be followed regardless of possible complications. On the other hand instead, some other researchers have considered odd, unusual and bizarre empirical findings in a much more serious way and have focused their attentions on such contradictory situations, fearing that the entire house of modern finance could have revealed itself as nothing but a castle built on sand – a partially justified fear, as experiments and tests performed in this direction and exposed in the next paragraphs actually show.