Table 15.1 Comparisons of predictable, unpredictable, and partially predictable phenomena

Predictability

Predictable

Partially Predictable

Unpredictable

Partially Predictable

Predictable

Comments

Mathematics

Correlation

Brownian Motion H

Time series persistence

Equilibrium states

Finance

Behavior

Cigar smoke pattern

Orbits

Exactly the opposite as predicted

Linear

A 3-year-old child

Nonlii

Fractional H<0.5

Antipersistent

New reality

Completely random

Linear 0

H= 0.5 Independent

None

Old assumption

Person with Alzheimers

Curlicues

Multiple moons (3 body problem)

Nonlinear

Fractional H>0.5

Persistent

Partial New reality General public

Twisted spiral

Exactly as predicted

Linear + 1

Perfect

Robot Laminar

Typical orbit Earths moon

and randomness. They are nonlinear. By this definition, standard financial theory assumes linearity because it assumes that asset prices follow a completely random walk, referred to in the scientific literature as Brownian motion (Bm).

The twisted flow, as a simple physical model and as a nonlinear relationship, lies between linearity and chaos and best depicts the reality of everyday markets. They are in a kind of equilibrium that is dynamic and changing in time. Twisted flows are partially predictable although predictability decreases the more it becomes disordered over time.

Any order within a system bears redundancy or patterns that can be exploited.

Imagine a word partially represented by "B ddh ." Whats the full word? If you

guessed Buddha, you are correct. With % of the information you received 100 percent of the message. Patterns in traded assets are no different. Yet, standard financial theory assumes that no patterns exist (i.e., the movements in time are random). This is where nonlinear pricing comes into play.

Nonlinear Pricing

As a recognized leader in the field of nonlinearity, Kriya, Inc, has proffered the following definition: Nonlinear pricing is any technological trading aid that acknowledges the nonlinearities exhibited by markets to more accurately characterize patterns exhibited by traded assets. Nonlinear pricing comprises new technologies such as chaos theory, abductive logic, fuzzy logic, genetic algorithms, and neural nets to model the new paradigm of evolution or adaptation. Soros theory of reflexivity is congruent with nonlinear pricing.

Nonlinear pricing assumes the markets are adaptive and its goal is to quantify the relationship in time between asset price movements, to gain some degree of predictability over probable future prices. Nonlinear pricing has been used successfully since July 1994.

The opportunity to profit is the arbitrage between the pattern depicted by nonlinear pricing and the markets inability to detect the pattern just as accurately. Moreover, this opportunity is unlikely to be arbitraged away because of the number of variables involved, the varying investment horizons, the technology gap, and the fact that the dissemination of Black-Scholes built the derivatives industry on a foundation of linear assumptions.

For a comparison, consider that Wall Street typically makes investment decisions, especially on equities, in one or a combination of three ways: fundamental analysis, technical analysis, and via the "whisper circuit." In a fundamental valuation, consistent earnings are a plus and in classical analysis, its effect on price is heuristically or subjectively interpreted. In contrast, nonlinear pricing actually quantifies this concept, modeling the dynamics of how those fundamentals interact in time and their effect on price.

Measurements

When adopting the nonlinear paradigm, new measurable aspects of time series become available. Some of the more popular concepts are presented in this section.

Persistence and the Hurst Exponent

The basic process is to identify when an equitys price has persistence (the "what"). Then attribute a business reason (the "why") to that behavior. For example, strong price persistence may be reliably detected during strong earnings. Invest when the "what" and "why" agree. Monitor the position and remove when they do not agree. Looking at factors that affect price instead of at price itself makes for more sophisticated relationships.

Cutland et al. found3 that, statistically, random walks occur only part of the time and there is a relationship between variables in time and that relationship changes over time. Peters4 identified the relationship in time using the Hurst exponent H.

The Hurst coefficient, H, is a scaling factor in time. If we were to view a time-series at a faster than normal rate, in hypertime, like using fast-forward on our VCR, H would be an acceleration factor. H tells us something about the functions behavior. Using H is like putting time in fast-forward to discern something about the stocks future tendencies. Mathematically, it looks like the following formula where the equality sign means "equal in distribution," and not "equal point for point along the time series":

X(r) = UaHxX(at)

where

X(t) is a function in time, UaHis the scaling factor,

X(at) is the process X(t) speeded up by a factor of a

When a time series is a , H = 0.5. When H is greater or less than 0.5, the time series is persistent or antipersistent, respectively. If a time series H = 0.7, then it may be said that the time series has a greater probability of continuing its current direction and a lesser probability of reversing itself. The converse is true for H < 0.5 (an antipersistent time series). Rescaled range is one way to measure the Hurst coefficient.5

One of the underlying premises of the portfolio theory of the 1970s is that the movements in time of all traded assets are random 100 percent of the time. In contrast, our research shows that U.S. equities are random only about 10 percent of the time at transient points. In other words, classical theory is inaccurate about 90 percent of the time in its assumption that a financial time series is Bm.

For example, in Figure 15-2, begin at the left-hand side of the graph with the line intersecting H = 0.8. This line measures the Hurst coefficient for the US equity Conseco (NYSE: CNC). If standard financial theory accurately depicted reality, the line intersecting the left-hand side of the graph at 0.8 would have been a constant 0.5. But as we see, from 12 April to 1 October, most of the time the stock price was either persistent or antipersistent-not random.

In simple terms, this graph crystallizes the disagreement between academics and practitioners about the assumption of randomness of stock prices. Practitioners quickly learn that linearity is an inaccurate paradigm. Straight lines are not good forecasting tools, nor are completely random walks. Based on a continuum from randomness (i.e., no relationship in time) to predictability (i.e., perfect relationship in time) many natural phenomena, as well as market behavior, exist most of the time between the two extremes.

Reflexivity

In many important respects, nonlinear pricing is a practical implementation of Soros theory of reflexivity, which he first wrote about in 1987.6 The theory of reflexivity holds that equilibrium-the assumed norm in classical economics-is but a