FIGURE 15.1 A NORMAL DISTRIBUTION CURVE OF PRICE CHANGES AND A STABLE DISTRIBUTION CURVE. THE HORIZONTAL AXIS REPRESENTS HE SPREAD OF POSSIBLE FUTURE PRICES OF AN UNDERLYING SECURITY, WITH THE CENTER OF THE CURVE REPRESENTING TODAYS PRICE.

Gaussian Distribution

Stable Distribution

of our mathematical tools, from simple arithmetic through differential calculus to algebraic topology, rely on the assumption of linearity." In short, we dont have a sufficient set of well-understood tools for nonlinear analysis. Fortunately, this is changing. In 1995, famed investor George Soros wrote:

There has been a recent development in science, variously called the science of complexity, evolutionary systems theory, or chaos theory. To understand historical processes, this approach is much more useful than the traditional approach, which is analytical. Unfortunately, our view of the world has been shaped by analytical sdence to a greater extent than is good for us. Economics seeks to be an analytical science. But all historical processes, including financial markets, are complex and cannot be understood on the basis of analytical science. We need a whole new approach and my theory of reflexivity is just the first step in this direction.1

The very existence of changing interest rates, business and political cycles runs contrary to standard financial theory. These cycles mathematically imply that the dynamics of the market are changing with time, and thus exhibit nonlinear behavior.

Murray Gell-Mann, Nobel laureate in physics, adds:

Some dogmatic neodassical economists had kept claiming that fluctuations around so-called fundamentals in financial markets amounted to a random walk, and they produced some evidence to support their assertion. But in the last few years, it has been shown-I believe quite convincingly-that, in fact, various markets show fluctuations that are not entirely random. They are pseudo-random and that pseudo-randomness can be exploited.2

How can this new understanding be exploited? Heres an example from personal experience.

Years ago, I was a derivative arbitrageur hedging Japanese warrants against their underlying equities at Baring Securities in Tokyo. A warrant is a long dated call option on an equity. The call option and its equity are equivalent, though not identical. Their relationship is intuitive; if the price of an equity rises, then one would correctly expect the call option to increase as well and vice versa. The relationship between the two instruments is measured statistically using the popular Black-Scholes option pricing model.

We encountered a problem though. Warrants typically have a three- to five-year life and the Black-Scholes model becomes very inaccurate after about 90 days. In the simplest terms, if reality and current theory are not congruent, then an arbitrage opportunity exists. Implementing new techniques that more accurately depicted reality led to an arbitrage between the pattern nonlinear pricing can detect and the market, which is less informed. The next question is, can the arbitrage be systematically and profitably exploited? I believe the answer is yes.

Wall Streets View

The language of mathematics may be arbitrarily divided into two camps: the linear or the additive and the nonlinear or the nonadditive. Historically, linear mathematics is the type with which we are most familiar.

For example, if a candy bar costs one dollar, a linear or additive relationship means that three candy bars costs three dollars. Another example of linear addition is 2 + 2 = 4. Linear relationships are not time dependent: 2 + 2 = 4 will remain true tomorrow as well as today and yesterday. Deterministic systems, where the relationships in time are fixed, are linear. Car washes and soda vending machines are good examples: money in and 5 seconds later soda out, ad infinitum. It is easy to understand this simple deterministic or linear relationship, which has been the dominant paradigm for scientific interpretation of phenomena in the West.

The linear view persisted probably because of practical necessity, since technology was not advanced enough to detect modeling errors. Basic laws of physics were simple, widely accepted, and did not allow for the small changes-the nonlinearities- and without the technology to measure them practically, the linear or deterministic viewpoint became dominant.

Like their predecessors, economists borrowed from existing lines of linear thought. Even though the theories may not always have been accurate in the eyes of the practitioner, they fit together neatly in terms of linear math. Probability theory and the partial differential equations of equilibrium physics are the root of classical economics.

When applying classical economic theory and the portfolio theory of the 1970s, we need to make assumptions about the future behavior of asset prices and how the returns of assets are distributed. In keeping with the linear mind-set, theoreticians assumed that the movement in time of any traded asset was random and that its returns were normally (Gaussian) distributed. While this is still the prevailing view, it is wrong.

Why? Trend and momentum are two concepts in technical analysis that are most common to Wall Street. They have their basis in linear math (e.g., 2 + 2 = 4, where the relationship is not time dependent). Unlike momentum, persistence is the tendency of a time series to continue its current direction (up or down) and antipersis-tence is the tendency of a time series to reverse itself rather than to continue its current direction. Momentum cannot characterize time series that are antipersistent. Persistence is also not trend. Trend is a perspective in the present that looks back in time. Persistence is a perspective in the present that looks forward in time to give a likelihood of future price movements.

The problem with the prevailing view is that the time series of financial markets are complex affairs that do not lend themselves to simple interpretations of trend and momentum. Nor are they random most of the time. Most of the time, traded asset prices persist or antipersist. That is, they have the tendency to follow their current path or reverse themselves. Therefore, stock price movements are dependent, in part, on past action and time. The goal, then, is to model this dependency using nonlinear techniques.

N0NL1NEAR1TY

Modeling Phenomena

In terms of modeling phenomena, one may visualize three basic states of the world: predictable, unpredictable, and partially predictable. Table 15-1 summarizes some nomenclature and concepts in terms of those three states. Various terms in the table will be explained throughout this chapter.

The bulk of Western thinking on economics, science, and mathematics is linear and is depicted in the center column and the columns at either extreme. But the readers reward will come from considering the columns in between.

Picmre a lit cigar. For the first three inches or so above the ash, the smoke is smooth or laminar. It then gets a little twisted and about three to five inches up it bends into curlicues. Parallels exist between cigar smoke and dynamic system behavior. Nonlinear mathematics better depicts dynamical systems or systems that change with time. Orbits, behavior, and cigar smoke are dynamical systems. So are markets. Mathematically, nonlinearity means that results are nonadditive. A small difference today can become a bigger difference tomorrow. Results are time dependent.

The laminar flow of smoke approximates a linear relationship-by looking at the smoke particles, you can estimate where they will be a moment or two later.

The curlicues are turbulence. Mathematically, these flows are unpredictable. Knowing the position of any given particle tells you nothing about where it may be in the next instant. Unpredictable systems are out of control because no relationship in time exists. A chaotic system is, in the long term, unpredictable.

Oddly enough, we define linearity by those conditions that are either completely random or perfectly correlated (either positively or negatively) in time. All conditions not at either extreme are a special mixture of persistence or antipersistence