Turbulence is inherently complex. The complexity of financial markets arises due to two factors. First, they are dynamic, and as such are described by more than a single equation. Second, they are nonlinear, which means that whatever equations are involved, they are complicated by the presence of exponents. Furthermore, we don’t even know what the equations are! On the surface, this state of affairs might seem rather bleak. All is not lost, however. Just imagine, were markets purely linear, then Newtonian mechanics would provide tools sufficient for apprehension. This would allow for everyone to “find edge”, and in turn no one would, ironically.
Time series analysis, can provide a ray of hope where market forecasting is concerned. Market forecasting can be viewed as a special case of time series prediction. Specifically, Rescaled Range analysis (R/S analysis) should be applied as an initial filter to differentiate the general type of series under observation. In general there are three time series classifications, which vary according to the particular range of Hurst exponents. For H=0.50, the series is random, where the range increases with the square root of time. For 0<H<0.50, the series is anti-persistent or mean reverting. For 0.50<H<1.00, the series is persistent, or trend reinforcing (correlations exist). Deriving a value for H, also provides insight as to average cycle length. Persistent time series are often chaotic, which makes their projection problematic. However their inherent persistency allows for trends to occur. The chief forecasting challenge that chaotic series present, is their “sensitive dependence on initial conditions.” The real world consequence of this attribute is that whatever the amount of error (however small) in the measurement process of a series, the magnitude of the error expands exponentially, thus adversely impacting the forecast’s reliability. It is this error expansion phenomenon, which is the core hindrance to all turbulent system prediction. Quantitatively, a spectrum of Lyapunov exponents can measure the limits of forecasting accuracy. However, to derive the relevant Lyapunov exponents, the equation, or system of equations for the attractor which generates the time series being studied, must be known. As mentioned above, on this note we come up short.
Minus the precise knowledge of a system’s underlying equations (which is the case for financial markets), it is sometimes possible to reconstruct a phase space for whatever “attractor” is generating the visible time series. Empirical adjustment of the trajectory of the original time series will theoretically enable a mapping to the underlying attractor. Using the formula ” m * t= Q” is implicit to determine probable amounts of lag of the starting time series. “m” in this formula is the embedding dimension, and as a required condition, should be greater than the dimension of the actual system attractor. “Q”, is the mean orbital period, which can be approached through the R/S analysis as indicated above.
Assuming a state or phase space from the time series can be constructed and plotted, minimally, verification of the system’s Lyapunov’s spectrum becomes possible, providing insight concerning the systems susceptibility to sensitive dependence on initial measurements. Also a three dimensional plot (in 2-space) could graphically indicate, say to, an option seller for example, regions of price return, not likely to be visited by the time/price plot.
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