The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 574 Friday, March 12, 2004 12:40 PM

574 The Mathematics of Financial Modeling and Investment Management

The key point is that Takens theorem and all approximation

schemes work only if the dynamic is simple.^13 A number of tests have

been devised to check if economic and financial quantities can be effec-

tively be represented as a simple chaotic laws. Among the tests, in par-

ticular the BDS test (see Chapter 9) is popular amongst economists. The

results of tests are generally negative. There is no compelling evidence

that reasonably simple chaotic dynamics can explain financial processes.

Despite these negative theoretical results, technical rules based on

neural networks or directly on the Takens theorem have been proposed

and continue to be proposed.

These rules have shown some result. This is not necessarily in con-

trast with the negative theoretical finding. One might find some profit-

ability in trading rules even if the dynamics is theoretically not simple.

Technical Analysis and Statistical Nonlinear Pattern Recognition

Technical analysis can also be cast in terms of statistical pattern recogni-

tion. A number of models that fundamentally differ from a random

walk or a martingale model have been proposed. Pair trading and coin-

tegration-based strategies are perhaps the best known examples of sta-

tistical models that exploit statistical patterns.

The empirical literature offers contradicting evidence. There is

agreement that asset price processes offer some level of forecastability.^14

There are also theoretical reasons to believe that price processes in a

finite economy must exhibit cointegration^15 and therefore recognizable

patterns. ARCH and GARCH behavior is another source of nonlinear

statistical patterns. What is not clear, however, is the profitability that

can be associated to these statistical findings once the trading costs are

taken into account.

(^13) Simple dynamics means that there is a low-dimensionality attractor. Chaos theory
is a complex subject. The interested reader should consult Robert C. Hilborn, Chaos
and Nonlinear Dynamics (New York: Oxford University Press, 2000).
(^14) See W. Brock, J. Lakonishok, and B. LeBaron, “Simple Technical Trading Rules
and the Stochastic Properties of Stock Returns,” Working paper 90–22, Wisconsin
Madison Social Systems; and John Campbell, Andrew Lo, and Craig MacKinlay,
The Econometrics of Financial Markets (Princeton, NJ: Princeton University Press,
1997).
(^15) See Marlene Cerchi and Arthur Havenner, “Cointegration and Stock Prices: The
Random Walk on Wall Street Revisited,” Journal of Economic Dynamics and Con-
trol 12 (1988), pp. 333–346; Peter Bossaerts, “Common Nonstationary Compo-
nents of Asset Prices,” Journal of Economic Dynamics and Control 12 (1988), pp.
347–364; and, Barr Rosenberg and J.A. Ohlson, “The Stationary Distribution of Re-
turns and Portfolio Separation in Capital Markets: A Fundamental Contradiction,”
Journal of Financial and Quantitative Analysis 11 (1976), pp. 393–401.