Multifractal Financial Markets

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.

The concept of “noisy chaos” is introduced in this chapter, based on the processes of bifurcation, entropy, and convergence which occur at the heart of the instability in the financial markets and take into account the sensitivities of the financial systems. The key here is to identify and measure indicators that allow us to construct a model that accounts for extreme consensus factors (undisseminated information, correlated investment horizons, and high leverage) in estimating market reversals. Instability is a relatively subjective notion. If we think in calendar time, the system is unstable as the daily fluctuations seem erratic when compared to periods of months and years. But if we think in intrinsic time, it is as if we are looking at the week as a year, the day as a month, and the minutes as days… in doing so, and as markets are “self-affine” then the L-stable process can be found at the day level and the erratic fluctuations will be at the seconds level.

The sense of time and financial players’ behavior is the central theme of this chapter. The notion of “intrinsic time”, a dimensionless time scale that counts the number of trading opportunities that occur regardless of the calendar time that passes between them, is explained to highlight the difference in investors’ perceptions and how to use this fact as a tool in understanding the processes at play and the biases to identify and avoid. As new information is constantly entering the market financial participants need to revise their expectations according to their own utility perception. As such the study of utility is important to understand the financial marketplace. The key element in any information content is the surprise element. Surprise is experienced only if an unexpected outcome occurs from which a new or different utility per individual is derived. Bearing in mind that information is a decreasing function of probability, we introduce an innovative subjective utility theory as per the findings of Viole and Nawrocki: the “Multiple Heterogeneous Benchmark Utility Functions”. Bayes’ theorem and fuzzy logic that has found application in many contexts are presented as a device to effectively account for “probabilities” in the decision-making process under conditions of uncertainty.

Economic cycles are the strange attractor of the financial process. Economic indicators and their inter-relationships with each other are reviewed in this chapter. The specifics of each cycle make it unique; economic indicators by themselves are not as valuable, it is their causes and the magnitude of their effects that provide value and insight. With this interconnectedness in mind, we are enabled to explore ways to perceive the cycle ahead. Since financial markets strive to foresee economic fluctuations, it is crucial to be able to identify the characteristics of the economic cycles and to analyze those specific characteristics. The identification of macrostate factors (economic cycles, disequilibrium, and changes) of market’s characteristics and the accurate evaluation of the sensitivities of a portfolio to those same risk factors are fundamental tools for all investors wanting to guard against the sudden reversals of trends.

Each market phase has its trading opportunities. A dynamic management approach for trading in multifractal financial markets is introduced in this chapter to allow us to profit from a market’s characteristics. An offensive approach is presented based on the notion of diversification at the strategy level between directional and volatility strategies; and of a macro-design approach. Tools such as cyclical and psychological analysis, fundamental convergent analysis, and the estimation of risks, allow us to evaluate the market biases in order to establish an accurate estimation of the prevailing state of the system and the risk toward which it is heading. Once markets’ characteristics are grasped, risk forecasting models can be enhanced. Models can be built on the basis of multifractal markets but are not limited to using only fractal tools such as, for example, the Hurst exponent. In fact, fractal thinking allows us to discern the most appropriate way of developing models. Be it technical analysis, behavioral finance, cycle analysis, power laws, thermodynamic, and econophysics, etc.…all of these are useful as long as we know how to implement them in our models while remaining aware of their limits. A strategic investment decision must not only be based on the best information available, but also on the possibility of error in the systems of calculation and the development of management strategies. The art of successful tail risk management lies in the ability to hedge against sudden market drifts or any specific micro market risk as well as against long periods of low volatility; and to “time” the volatility to profit from its clustering behavior without having to rely on seismic events to gain profits. That said; total hedging is not possible in absolute terms and if so it does not work at all time. It is important to think in terms of affordable risks before thinking of potential gains. The chapter includes a discussion of recent developments in the various techniques in forecasting risk highlighting their advantages, applications, and limitations.

Today, we are in the midst of an undesirable mix of tight economic and financial conditions. Opaque derivative products which include among others Collateralized-Debt-Obligations and Exchange-Traded Funds continue to shape the markets. Economists need to rethink certain economic concepts and relationships. Actually, fiscal stimulus spending is not stimulating anything. The deadly embrace between over-indebted sovereigns and over-leveraged banks has created a vicious cycle. Faced with these new economic and financial market imbalances, portfolio management has become much more difficult. We need to arm ourselves with how to manage tail risks against the inevitable and recurring loss of confidence which comes from market instability. We must adapt continuously along with the market whatever the pressure. Acknowledging errors and adjusting to them is crucial. Understanding market events and their effects on the market system is an essential element toward building a financial warning system. To this end, we need to identify all pre-switching points; implement them in algorithms that can trigger a warning signal. If we think of Keynes levels, the sixth level would be to have a drummer like mind … to be ahead of the beat. In the end what we achieve by understanding markets’ fractality is more than a tool; it is a way of thinking.