is needed for preserving the conservative character of the cascade.

Secondly, the values of mi can be randomized rather than assigned

fixed values. A cascade with randomized mi is called canonical. In this

case, the condition (6.2.5) is satisfied only on average, that is

" #

X

NÀ1

mi ¼ 1

E (6:2:6)

0

An example of the randomized cascade that has an explicit expres-

sion for the multifractal spectrum is the lognormal cascade [6]. In this

process, the multiplier that distributes the mass of the interval, M, is

determined with the lognormal distribution (i.e., log2 (M) is drawn

from the Gaussian distribution). If the Gaussian mean and variance

are l and s, respectively, then the conservative character of the

cascade E[M] ¼ 0.5 is preserved when

s2 ¼ 2(l À 1)= ln (2) (6:2:7)

The multifractal spectrum of the lognormal cascade that satisfies

(6.2.7) equals

(a À l)2

f(a) ¼ 1 À (6:2:8)

4(l À 1)

Note that in contrast to the binomial cascade, the lognormal

cascade may yield negative values of f(a), which requires interpret-

ation of f(a) other than the fractal dimension.

Innovation of multifractal process, DX ¼ X(t þ Dt) À X(t), is de-

scribed with the scaling rule

E[j(DX)jq ] ¼ c(q)(Dt)t(q)þ1 (6:2:9)

where c(q) and t(q) (so-called scaling function) are deterministic func-

tions of q. It can be shown that the scaling function t(q) is always

concave. Obviously, t(0) ¼ À1. A self-affine process (6.1.4) can be

treated as a multifractal process with t(q) ¼ Hq À 1. In particular, for

the Wiener processes, H ¼ 1„2 and tw (q) ¼ q=2 À 1. The scaling func-

tion of the binomial cascade can be expressed in terms of its multi-

pliers

t(q) ¼ log2 (m0 q þ m1 q ) (6:2:10)

67

Fractals

The scaling function t(q) is related to the multifractal spectrum f(a)

via the Legendre transformation

t(q) ¼ min[qa À f(a)] (6:2:11)

a

which is equivalent to

f(a) ¼ arg min[qa À t(q)] (6:2:12)

q

Note that f(a) ¼ q(a À H) þ 1 for the self-affine processes.

In practice, the scaling function of a multifractal process X(t) can

be calculated using so-called partition function

X

NÀ1

jX(t þ Dt) À X(t)jq

Sq (T, Dt) ¼ (6:2:13)

i¼0

where the sample X(t) has N points within the interval [0, T] with the

mesh size Dt. It follows from (6.2.9) that

log {E[Sq (T, Dt)]} ¼ t(q) log (Dt) þ c(q) log T (6:2:14)

Thus, plotting log {E[Sq (T, Dt)]} against log (Dt) for different values

of q reveals the character of the scaling function t(q). Multifractal

models have become very popular in analysis of the financial time

series. We shall return to this topic in Section 8.2

6.3 REFERENCES FOR FURTHER READING

The Mandelbrot™s work on scaling in the financial time series is

compiled in the collection [1]. Among many excellent books on frac-

tals, we choose [2] for its comprehensive material that includes a

description of relations between chaos and fractals and an important

chapter on multifractals [5].

6.4 EXERCISES

*1. Implement an algorithm that draws the Sierpinski triangle with

r ¼ 0:5 (see Figure 6.2).

Hint: Choose the following fixed points: (0, 0), (0, 100), (100,

0). Use the following method for the randomized choice of the

68 Fractals

fixed point: i ¼ [10 rand()] %3 where rand() is the uniform

distribution within [0, 1] and % is modulus (explain the ration-

ale behind this method). Note that at least 10000 iterations are

required for a good-quality picture.

*2. Reproduce the first five steps of the binomial cascade with

m0 ¼ 0:6 (see Figure 6.3). How will this cascade change if

m0 ¼ 0:8?

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

69

70 Nonlinear Dynamical Systems

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