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85
Nonlinear Dynamical Systems

The generic characteristic of the strange attractor is its fractal
dimension. In fact, the non-integer (i.e., fractal) dimension of an
attractor can be used as the definition of a strange attractor. In
Chapter 6, the box-counting fractal dimension was introduced.
A computationally simpler alternative, so-called correlation dimen-
sion, is often used in nonlinear dynamics [3, 5].
Consider a sample with N trajectory points within an attractor. To
define the correlation dimension, first the relative number of points
located within the distance R from the point i must be calculated
X N
1
pi (R) Â¼ u(R Ã€ jxj Ã€ xi j) (7:6:6)
N Ã€ 1 j Â¼ 1, j 6Â¼ i

In (7.6.6), the Heaviside step function u equals

0, x < 0
uÂ¼ (7:6:7)
1, x ! 0
Then the correlation sum that characterizes the probability of finding
two trajectory points within the distance R is computed

1X N
C(R) Â¼ pi (R) (7:6:8)
N iÂ¼1

It is assumed that C(R) \$ RDc . Hence, the correlation dimension Dc
equals
Dc Â¼ lim [ ln C(R)= ln R] (7:6:9)
R!0

There is an obvious problem of finding the limit (7.6.9) for data
samples on a finite grid. Yet, plotting ln[C(R)] versus ln(R) (which
is expected to yield a linear graph) provides an estimate of the
correlation dimension.
An interesting question is whether a strange attractor is always
chaotic, in other words, if it always has a positive Lyapunov expo-
nent. It turns out there are rare situations when an attractor may be
strange but not chaotic. One such example is the logistic map at the
period-doubling points: Its Lyapunov exponent equals zero while the
fractal dimension is about 0.5. Current opinion, however, holds that
the strange deterministic attractors may appear in discrete maps
rather than in continuous systems [5].
86 Nonlinear Dynamical Systems

7.7 REFERENCES FOR FURTHER READING
Two popular books, the journalistic report by Gleick [8] and the
â€˜â€˜first-handâ€™â€™ account by Ruelle [9], offer insight into the science of
chaos and the people behind it. The textbook by Hilborn [5] provides
a comprehensive description of the subject. The interrelations be-
tween the chaos theory and fractals are discussed in detail in [10].

7.8 EXERCISES
1. Consider the quadratic map Xk Â¼ XkÃ€1 2 Ã¾ C, where C > 0.
(a) Prove that C Â¼ 0:25 is a bifurcation point.
(b) Find fixed points for C Â¼ 0:125. Define what point is an
attractor and what is its attraction basin for X > 0.
2. Verify the equilibrium points of the Lorenz model (7.4.3).
*3. Calculate the Lyapunov exponent of the logistic map as a
function of the parameter A.
*4. Implement the algorithm for simulating the Lorenz model.
(a) Reproduce the â€˜â€˜butterflyâ€™â€™ trajectories depicted in Figure
7.8.
(b) Verify existence of the periodicity window at r Â¼ 150.
(c) Verify existence of the limit cycle at r Â¼ 350.
Hint: Use a simple algorithm: Xk Â¼ XkÃ€1 Ã¾ tF(XkÃ€1 , YkÃ€1 , ZkÃ€1 )
where the time step t can be assigned 0.01.
Chapter 8

Scaling in Financial Time
Series

8.1 INTRODUCTION
Two well-documented findings motivate further analysis of financial
time series. First, the probability distributions of returns often deviate
significantly from the normal distribution by having fat tails and excess
kurtosis. Secondly, returns exhibit volatility clustering. The latter effect
has led to the development of the GARCH models described in Section
5.3.1 In this chapter, we shall focus on scaling in the probability distri-
butions of returns, the concept that has attracted significant attention
from economists and physicists alike.
Alas, as the leading experts in Econophysics, H. E. Stanley and
R. Mantegna acknowledged [2]:
â€˜â€˜No model exists for the stochastic process describing the
time evolution of the logarithm of price that is accepted by
all researchers.â€™â€™
There are several reasons for the status quo.2 First, different financial
time series may have varying non-stationary components. Indeed, the
stock price reflects not only the current value of a companyâ€™s assets
but also the expectations of the companyâ€™s growth. Yet, there is no
general pattern for evolution of a business enterprise.3 Therefore,

87
88 Scaling in Financial Time Series

empirical research often concentrates on the average economic in-
dexes, such as the S&P 500. Averaging over a large number of
companies certainly smoothes noise. Yet, the composition of these
indicators is dynamic: Companies may be added to or dropped from
indexes, and the companyâ€™s contribution to the economic index usu-
ally depends on its ever-changing market capitalization.
Foreign exchange rates are another object frequently used in empir-
ical research.4 Unfortunately, many of the findings accumulated during
the 1990s have become somewhat irrelevant, as several European cur-
rencies ceased to exist after the birth of the Euro in 1999. In any case, the
foreign exchange rates, being a measure of relative currency strength,
may have statistical features that differ among themselves and in com-
parison with the economic indicators of single countries.
Another problem is data granularity. Low granularity may under-
estimate the contributions of market rallies and crashes. On the other
hand, high-frequency data are extremely noisy. Hence, one may
expect that universal properties of financial time series (if any exist)
have both short-range and long-range time limitations.
The current theoretical framework might be too simplistic to ac-
curately describe the real world. Yet, important advances in under-
standing of scaling in finance have been made in recent years. In the
next section, the asymptotic power laws that may be recovered from
the financial time series are discussed. In Section 8.3, the recent
developments including the multifractal approach are outlined.

8.2 POWER LAWS IN FINANCIAL DATA
The importance of long-range dependencies in the financial time series
was shown first by B. Mandelbrot [6]. Using the R/S analysis (see Section
6.1), Mandelbrot and others have found multiple deviations of the
empirical probability distributions from the normal distribution [7].
Early research of universality in the financial time series [6] was
based on the stable distributions (see Section 3.3). This approach,
however, has fallen out of favor because the stable distributions have
infinite volatility, which is unacceptable for many financial applica-
tions [8]. The truncated Levy flights that satisfy the requirement for
finite volatility have been used as a way around this problem [2, 9, 10].
One disadvantage of the truncated Levy flights is that the truncating
89
Scaling in Financial Time Series

distance yields an additional fitting parameter. More importantly,
the recent research by H. Stanley and others indicates that the asymp-
totic probability distributions of several typical financial time series
resemble the power law with the index a close to three [11â€“13]. This
means that the probability distributions examined by Stanleyâ€™s team
are not stable at all (recall that the stable distributions satisfy the
condition 0 < a 2). Let us provide more details about these interest-
ing findings.
In [11], returns of the S&P 500 index were studied for the period
1984â€“1996 with the time scales Dt varying from 1 minute to 1 month.
It was found that the probability distributions at Dt < 4 days were
consistent with the power-law asymptotic behavior with the index
a % 3. At Dt > 4 days, the distributions slowly converge to the
normal distribution. Similar results were obtained for daily returns
of the NIKKEI index and the Hang-Seng index. These results are
complemented by another work [12] where the returns of several
thousand U.S. companies were analyzed for Dt in the range from
five minutes to about four years. It was found that the returns of
individual companies at Dt < 16 days are also described with the
power-law distribution having the index a % 3. At longer Dt, the
probability distributions slowly approach the normal form. It was
also shown that the probability distributions of the S&P 500 index
and of individual companies have the same asymptotic behavior due
to the strong cross-correlations of the companiesâ€™ returns. When these
cross-correlations were destroyed with randomization of the time
series, the probability distributions converged to normal at a much
faster pace.
The theoretical model offered in [13] may provide some explan-
ation to the power-law distribution of returns with the index a % 3.
This model is based on two observations: (a) the distribution of the
trading volumes obeys the power law with an index of about 1.5; and
(b) the distribution of the number of trades is a power law with an
index of about three (in fact, it is close to 3.4). Two assumptions were
made to derive the index a of three. First, it was assumed that the
price movements were caused primarily by the activity of large mutual
funds whose size distribution is the power law with index of one (so-
called Zipfâ€™s law [4]). In addition, it was assumed that the mutual fund
managers trade in an optimal way.
90 Scaling in Financial Time Series

Another model that generates the power law distributions is the
stochastic Lotka-Volterra system (see [14] and references therein).
The generic Lotka-Volterra system is used for describing different
phenomena, particularly the population dynamics with the predator-
prey interactions. For our discussion, it is important that some agent-
based models of financial markets (see Chapter 12) can be reduced to
the Lotka-Volterra system [15]. The discrete Lotka-Volterra system
has the form
1X N
wi (t) (8:2:1)
wi (t Ã¾ 1) Â¼ l(t)wi (t) Ã€ aW(t) Ã€ bwi (t)W(t), W(t) Â¼
N iÂ¼1

where wi is an individual characteristic (e.g., wealth of an investor i;
i Â¼ 1, 2, . . . , N), a and b are the model parameters, and l(t) is a
random variable. The components of this system evolve spontan-
eously into the power law distribution f(w, t) \$ wÃ€(1Ã¾a) . In the
mean time, evolution of W(t) exhibits intermittent fluctuations that
can be parameterized using the truncated Levy distribution with the
same index a [14].
Seeking universal properties of the financial market crashes is
another interesting problem explored by Sornette and others (see
[16] for details). The main idea here is that financial crashes are
caused by collective trader behavior (dumping stocks in panic),
which resembles the critical phenomena in hierarchical systems.
Within this analogy, the asymptotic behavior of the asset price S(t)
has the log-periodic form
S(t) Â¼ A Ã¾ B(tc Ã€ t)a {1 Ã¾ C cos [w ln (tc Ã€ t) Ã€ w]} (8:2:2)
where tc is the crash time; A, B, C, w, a, and w are the fitting
parameters. There has been some success in describing several market
crashes with the log-periodic asymptotes [16]. Criticism of this ap-
proach is given in [17] and references therein.

8.3 NEW DEVELOPMENTS
So, do the findings listed in the preceding section solve the problem
of scaling in finance? This remains to be seen. First, B. LeBaron has
shown how the price distributions that seem to have the power-law
form can be generated by a mix of the normal distributions with
91
Scaling in Financial Time Series

different time scales [18]. In this work, the daily returns are assumed
to have the form
R(t) Â¼ exp [gx(t) Ã¾ m]e(t) (8:3:1)
where e(t) is an independent random normal variable with zero mean
and unit variance. The function x(t) is the sum of three processes with
different characteristic times
x(t) Â¼ a1 y1 (t) Ã¾ a2 y2 (t) Ã¾ a3 y3 (t) (8:3:2)
The first process y1 (t) is an AR(1) process
y1 (t Ã¾ 1) Â¼ r1 y1 (t) Ã¾ Z1 (t Ã¾ 1) (8:3:3)
where r1 Â¼ 0:999 and Z1 (t) is an independent Gaussian adjusted so
that var[y1 (t)] Â¼ 1. While AR(1) yields exponential decay, the chosen
value of r1 gives a long-range half-life of about 2.7 years. Similarly,
y2 (t Ã¾ 1) Â¼ r2 y2 (t) Ã¾ Z2 (t Ã¾ 1) (8:3:4)
where Z2 (t) is an independent Gaussian adjusted so that
var[y2 (t)] Â¼ 1. The chosen value r2 Â¼ 0:95 gives a half-life of about
2.5 weeks. The process y3 (t) is an independent Gaussian with unit
variance and zero mean, which retains volatility shock for one day.
The normalization rule is applied to the coefficients ai
a1 2 Ã¾ a2 2 Ã¾ a3 2 Â¼ 1: (8:3:5)
The parameters a1 , a2 , g, and m are chosen to adjust the empirical data.
This model was used for analysis of the Dow returns for 100 years
(from 1900 to 2000). The surprising outcome of this analysis is retrieval
of the power law with the index in the range of 2.98 to 3.33 for the data
aggregation ranges of 1 to 20 days. Then there are generic comments by
T. Lux on spurious scaling laws that may be extracted from finite
financial data samples [19]. Some reservation has also been expressed
about the graphical inference method widely used in the empirical
research. In this method, the linear regression equations are recovered
from the log - log plots. While such an approach may provide correct
asymptotes, at times it does not stand up to more rigorous statistical
hypothesis testing. A case in point is the distribution in the form
f(x) Â¼ xÃ€a L(x) (8:3:6)
where L(x) is a slowly-varying function that determines behavior of
the distribution in the short-range region. Obviously, the â€˜â€˜universalâ€™â€™
92 Scaling in Financial Time Series

scaling exponent a Â¼ Ã€log [f(x)]= log (x) is as accurate as L(x) is close
to a constant. This problem is relevant also to the multifractal scaling
analysis that has become another â€˜â€˜hotâ€™â€™ direction in the field.
The multifractal patterns have been found in several financial time
series (see, e.g., [20, 21] and references therein). The multifractal
framework has been further advanced by Mandelbrot and others.
They proposed compound stochastic process in which a multifractal
cascade is used for time transformations [22]. Namely, it was assumed
that the price returns R(t) are described as
R(t) Â¼ BH [u(t)] (8:3:7)
where BH [] is the fractional Brownian motion with index H and u(t) is
a distribution function of multifractal measure (see Section 6.2). Both
stochastic components of the compound process are assumed inde-
pendent. The function u(t) has a sense of â€˜â€˜trading timeâ€™â€™ that reflects
intensity of the trading process. Current research in this direction
shows some promising results [23â€“26]. In particular, it was shown
that both the binomial cascade and the lognormal cascade embedded
into the Wiener process (i.e., into BH [] with H Â¼ 0:5) may yield a more
accurate description of several financial time series than the GARCH
model [23]. Nevertheless, this chapter remains â€˜â€˜unfinishedâ€™â€™ as new
findings in empirical research continue to pose new challenges for
theoreticians.

8.4 REFERENCES FOR FURTHER READING
Early research of scaling in finance is described in [2, 6, 7, 9, 17].
For recent findings in this field, readers may consult [10â€“13, 23â€“26].

8.5 EXERCISES
**1. Verify how a sum of Gaussians can reproduce a distribution
with the power-law tails in the spirit of [18].
**2. Discuss the recent polemics on the power-law tails of stock
prices [27â€“29].
**3. Discuss the scaling properties of financial time series reported
in [30].
Chapter 9

Option Pricing

This chapter begins with an introduction of the notion of financial
derivative in Section 9.1. The general properties of the stock options
are described in Section 9.2. Furthermore, the option pricing theory is
presented using two approaches: the method of the binomial trees
(Section 9.3) and the classical Black-Scholes theory (Section 9.4).
A paradox related to the arbitrage free portfolio paradigm on which
the Black-Scholes theory is based is described in the Appendix section.

9.1 FINANCIAL DERIVATIVES
In finance, derivatives1 are the instruments whose price depends
on the value of another (underlying) asset [1]. In particular, the
stock option is a derivative whose price depends on the underlying
stock price. Derivatives have also been used for many other assets,
including but not limited to commodities (e.g., cattle, lumber,
copper), Treasury bonds, and currencies.
An example of a simple derivative is a forward contract that obliges
its owner to buy or sell a certain amount of the underlying asset at a
specified price (so-called forward price or delivery price) on a specified
date (delivery date or maturity). The party involved in a contract as a
buyer is said to have a long position, while a seller is said to have a short
position. A forward contract is settled at maturity when the seller

93
94 Option Pricing

delivers the asset to the buyer and the buyer pays the cash amount at
the delivery price. At maturity, the current (spot) asset price, ST , may
differ from the delivery price, K. Then the payoff from the long
position is ST Ã€ K and the payoff from the short position is K Ã€ ST .
Future contracts are the forward contracts that are traded on
organized exchanges, such as the Chicago Board of Trade (CBOT)
and the Chicago Mercantile Exchange (CME). The exchanges deter-
mine the standardized amounts of traded assets, delivery dates, and
the transaction protocols.
In contrast to the forward and future contracts, options give an
option holder the right to trade an underlying asset rather than the
obligation to do this. In particular, the call option gives its holder the
right to buy the underlying asset at a specific price (so-called exercise
price or strike price) by a certain date (expiration date or maturity).
The put option gives its holder the right to sell the underlying asset at a
strike price by an expiration date. Two basic option types are the
European options and the American options.2 The European options
can be exercised only on the expiration date while the American
options can be exercised any time up to the expiration date. Most of
the current trading options are American. Yet, it is often easier to
analyze the European options and use the results for deriving proper-
ties of the corresponding American options.
The option pricing theory has been an object of intensive research
since the pioneering works of Black, Merton, and Scholes in the
1970s. Still, as we shall see, it poses many challenges.

9.2 GENERAL PROPERTIES OF STOCK OPTIONS
The stock option price is determined with six factors:
Current stock price, S
.
Strike price, K
.
Time to maturity, T
.
Stock price volatility, s
.
Risk-free interest rate,3 r
.
Dividends paid during the life of the option, D.
.
Let us discuss how each of these factors affects the option price
providing all other factors are fixed. Longer maturity time increases
95
Option Pricing

the value of an American option since its holders have more time to
exercise it with profit. Note that this is not true for a European option
that can be exercised only at maturity date. All other factors, how-
ever, affect the American and European options in similar ways.
The effects of the stock price and the strike price are opposite for
call options and put options. Namely, payoff of a call option increases
while payoff of a put option decreases with rising difference between
the stock price and the strike price.
Growing volatility increases the value of both call options and put
options: it yields better chances to exercise them with higher payoff.
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