Quantitative Finance for Physicists

Chapter 7

Nonlinear Dynamical Systems

7.1 MOTIVATION

It is well known that many nonlinear dynamical systems, including

seemingly simple cases, can exhibit chaotic behavior. In short, the

presence of chaos implies that very small changes in the initial condi-

tions or parameters of a system can lead to drastic changes in its

behavior. In the chaotic regime, the system solutions stay within the

phase space region named strange attractor. These solutions never

repeat themselves; they are not periodic and they never intersect.

Thus, in the chaotic regime, the system becomes unpredictable. The

chaos theory is an exciting and complex topic. Many excellent books

are devoted to the chaos theory and its applications (see, e.g., refer-

ences in Section 7.7). Here, I only outline the main concepts that may

be relevant to quantitative finance.

The first reason to turn to chaotic dynamics is a better understand-

ing of possible causes of price randomness. Obviously, new infor-

mation coming to the market moves prices. Whether it is a

company’s performance report, a financial analyst’s comments, or a

macroeconomic event, the company’s stock and option prices may

change, thus reflecting the news. Since news usually comes unexpect-

edly, prices change in unpredictable ways.

1

But is new information the

only source reason for price randomness? One may doubt this while

observing the price fluctuations at times when no relevant news is

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released. A tempting proposition is that the price dynamics can be

attributed in part to the complexity of financial markets. The possi-

bility that the deterministic processes modulate the price variations

has a very important practical implication: even though these pro-

cesses can have the chaotic regimes, their deterministic nature means

that prices may be partly forecastable. Therefore, research of chaos in

finance and economics is accompanied with discussion of limited

predictability of the processes under investigation [1].

There have been several attempts to find possible strange attractors

in the financial and economic time series (see, e.g., [1–3] and refer-

ences therein). Discerning the deterministic chaotic dynamics from a

‘‘pure’’ stochastic process is always a non-trivial task. This problem is

even more complicated for financial markets whose parameters may

have non-stationary components [4]. So far, there has been little (if

any) evidence found of low-dimensional chaos in financial and eco-

nomic time series. Still, the search of chaotic regimes remains an

interesting aspect of empirical research.

There is also another reason for paying attention to the chaotic

dynamics. One may introduce chaos inadvertently while modeling

financial or economic processes with some nonlinear system. This

problem is particularly relevant in agent-based modeling of financial

markets where variables generally are not observable (see Chapter

12). Nonlinear continuous systems exhibit possible chaos if their

dimension exceeds two. However, nonlinear discrete systems (maps)

can become chaotic even in the one-dimensional case. Note that the

autoregressive models being widely used in analysis of financial time

series (see Section 5.1) are maps in terms of the dynamical systems

theory. Thus, a simple nonlinear expansion of a univariate autore-

gressive map may lead to chaos, while the continuous analog of this

model is perfectly predictable. Hence, understanding of nonlinear

dynamical effects is important not only for examining empirical

time series but also for analyzing possible artifacts of the theoretical

modeling.

This chapter continues with a widely popular one-dimensional

discrete model, the logistic map, which illustrates the major concepts

in the chaos theory (Section 7.2). Furthermore, the framework for the

continuous systems is introduced in Section 7.3. Then the three-

dimensional Lorenz model, being the classical example of the low-

70 Nonlinear Dynamical Systems