Abstract
Financial markets continuously evolve in a nonlinear, dynamic fashion, affected by destabilizing events and auto-reinforced mechanisms or memory effects. In spite of its complexities, the financial market is not entirely incomprehensible; although market cycles continuously change, their underlying mechanisms remain the same. This chapter provides an explanation of how fractal geometry helps us to understand the mechanisms underlying financial markets. One of the main features of financial markets is the alternation of periods of large price changes with periods of smaller changes. Fluctuations in volatility are unrelated to the predictability of future returns. This statement implies that there is autocorrelation structures dependence in the absolute values of returns. The multifractal model of asset returns combines the properties of L-stable processes (stationary and independent stable increments) and fractional Brownian Motions (tendency of price changes to be followed by changes in the same (or opposite) direction) to allow for long tails, correlated volatilities, and either unpredictability or long memory in returns.
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Notes
- 1.
Classical financial theory assumes that financial markets are efficient. A review of the efficient market hypothesis, however, shows that many of its required conditions are unachievable in reality. First introduced by Louis Bachelier in the 1900s, the concept of an efficient market assumes that competition among a large number of rational investors eventually lead to equilibrium and the resulting equilibrium prices reflect the information content of past or anticipated events. In other words, equilibrium stock prices must be, in principle, equal to their fundamental value and deviations from these equilibrium prices reflect the level of uncertainty at any given moment. As such, the principles of financial markets are identical to those of a roulette game where the players do not have memory or if they do, is too short to be able to take past experiences into consideration. The efficient market approach affirms the unpredictable nature of the financial markets; yields are independent and prices follow a random pattern, that is, a Brownian motion. This randomness obeys a law of probability often attributed to the Laplace Gauss Law of normal distribution (Hayek 2010).
- 2.
The inverse is good, as liquidity shifts to the long side, it is the end of the bear market, the bottom has been reached.
- 3.
John Graham, Campbell Harvey and Shivaram Rajgopal has shown that managers are making real decisions—such as decreasing spending on research and development, maintenance and hiring of critical employees—in order to hit quarterly earnings targets they have provided as part of their own guidance (Graham, November/December 2006).
- 4.
The human being becomes economic during the consolidation phases following a market correction. This behavior is explained by the fact that the variety of financial agents is reduced to professionals who have been re-oriented toward the fundamental financial and economic data.
- 5.
Mandelbrot initially presented a brief description of the multifractal concept, which he then expanded in 1975.
- 6.
It is worth noting here that Mandelbrot's inspiration started with his study of the distribution of cotton prices (Mandelbrot 1963): he noticed that instead of a bell curve, daily price changes yielded a different graph, with a large peak around zero and large deviations to the left and to the right, resulting from rarely occurring large price changes. Mandelbrot’s own work on cotton prices and his other analyses of more recent data on equity markets, suggests that there are, in practice, far more significant price moves than the Neo-classical theory suggests. He established a mathematical framework which allowed him to model the incomprehensible variation of cotton prices that is far from the “polite Gaussian average” (Mandelbrot 2008, p. 169).
- 7.
The Hurst exponent (H) was named in honor of both Harold Edwin Hurst (1880–1978) and Ludwig Otto Holder (1859–1937) by Mandelbrot (2008, pp. 187, 297).
- 8.
Empirical research shows that although the exponent H(t) varies from one system to another, it remains stable for a given system, for whatever the time scale considered. To verify whether the Hurst exponent effectively measures the structure of the correlations of the series studied, choose any series with an exponent H > 0.75 and scramble its data. Then, calculate H again for the scrambled series and H will be equal to 0.5. This means that the system (memory) and its structure were completely destroyed by the scrambling operation.
- 9.
Multifractal measures were introduced in Mandelbrot (1972). A good understanding of multifractal measures can be found in (Mandelbrot, Fisher, and Calvet, A Multifractal Model of Asset Returns, 1997) paper. This paper introduces the concept of Multifractality to economics as it focus on a very concrete aspect of Multifractality.
- 10.
In the case of the solar system, it is the time necessary so that an error in positioning or a disturbance of motion can be multiplied by 10. Thus for a period where: \( {\text{t}}\,<\,{\text{T}}\)we can follow the system by the calculation and \({\text{t}}\,>\,{\text{T}}\) we completely lose the trajectory of the system.
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Hayek Kobeissi, Y. (2013). Turbulence in the Financial Markets. In: Multifractal Financial Markets. SpringerBriefs in Finance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4490-9_1
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DOI: https://doi.org/10.1007/978-1-4614-4490-9_1
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