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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-
71
Nonlinear Dynamical Systems



dimensional continuous chaotic system, is described (Section 7.4).
Finally, the main pathways to chaos and the chaos measures are
outlined in Section 7.5 and Section 7.6, respectively.

7.2 DISCRETE SYSTEMS: THE LOGISTIC MAP
The logistic map is a simple discrete model that was originally used
to describe the dynamics of biological populations (see, e.g., [5] and
references therein). Let us consider a variable number of individuals
in a population, N. Its value at the k-th time interval is described with
the following equation
Nk ¼ ANkÀ1 À BNkÀ1 2 (7:2:1)
Parameter A characterizes the population growth that is determined
by such factors as food supply, climate, etc. Obviously, the popula-
tion grows only if A > 1. If there are no restrictive factors (i.e., when
B ¼ 0), the growth is exponential, which never happens in nature for
long. Finite food supply, predators, and other causes of mortality
restrict the population growth, which is reflected in factor B. The
maximum value of Nk equals Nmax ¼ A=B. It is convenient to intro-
duce the dimensionless variable Xk ¼ Nk =Nmax . Then 0 Xk 1,
and equation (7.2.1) has the form
Xk ¼ AXkÀ1 (1 À XkÀ1 ) (7:2:2)
A generic discrete equation in the form
Xk ¼ f(XkÀ1 ) (7:2:3)
is called an (iterated) map, and the function f(XkÀ1 ) is called the
iteration function. The map (7.2.2) is named the logistic map. The
sequence of values Xk that are generated by the iteration procedure
is called a trajectory. Trajectories depend not only on the iteration
function but also on the initial value X0 . Some initial points turn out
to be the map solution at all iterations. The value XÃ that satisfies the
equation
XÃ ¼ f(XÃ ) (7:2:4)
is named the fixed point of the map. There are two fixed points for the
logistic map (7.2.2):
72 Nonlinear Dynamical Systems



XÃ ¼ 0, and XÃ ¼ (A À 1)=A (7:2:5)
1 2

If A 1, the logistic map trajectory approaches the fixed point XÃ 1
from any initial value 0 X0 1. The set of points that the trajec-
tories tend to approach is called the attractor. Generally, nonlinear
dynamical systems can have several attractors. The set of initial values
from which the trajectories approach a particular attractor are called
the basin of attraction. For the logistic map with A < 1, the attractor
is XÃ ¼ 0, and its basin is the interval 0 X0 1.
1
If 1 < A < 3, the logistic map trajectories have the attractor
Ã
X2 ¼ (A À 1)=A and its basin is also 0 X0 1. In the mean time,
the point XÃ ¼ 0 is the repellent fixed point, which implies that any
1
trajectory that starts near XÃ tends to move away from it.
1
A new type of solutions to the logistic map appears at A > 3.
Consider the case with A ¼ 3:1: the trajectory does not have a single
attractor but rather oscillates between two values, X % 0:558 and
X % 0:764. In the biological context, this implies that the growing
population overexerts its survival capacity at X % 0:764. Then the
population shrinks ˜˜too much™™ (i.e., to X % 0:558), which yields
capacity for further growth, and so on. This regime is called period-
2. The parameter value at which solution changes qualitatively is
named the bifurcation point. Hence, it is said that the period-doubling
bifurcation occurs at A ¼ 3. With a further increase of A, the oscilla-
tion amplitude grows until A approaches the value of about 3.45. At
higher values of A, another period-doubling bifurcation occurs
(period-4). This implies that the population oscillates among four
states with different capacities for further growth. Period doubling
continues with rising A until its value approaches 3.57. Typical tra-
jectories for period-2 and period-8 are given in Figure 7.1. With
further growth of A, the number of periods becomes infinite, and
the system becomes chaotic. Note that the solution to the logistic map
at A > 4 is unbounded.
Specifics of the solutions for the logistic map are often illustrated
with the bifurcation diagram in which all possible values of X are
plotted against A (see Figure 7.2). Interestingly, it seems that there is
some order in this diagram even in the chaotic region at A > 3:6. This
order points to the fractal nature of the chaotic attractor, which will
be discussed later on.
73
Nonlinear Dynamical Systems



0.95
Xk

0.85


0.75


0.65


0.55


0.45


0.35
A = 2.0
A = 3.1 k
A = 3.6
0.25
1 11 21 31 41
Figure 7.1 Solution to the logistic map at different values of the
parameter A.


0 X 1
3




A




4
Figure 7.2 The bifurcation diagram of the logistic map in the parameter
region 3 A < 4.
74 Nonlinear Dynamical Systems



Another manifestation of universality that may be present in cha-
otic processes is the Feigenbaum™s observation of the limiting rate at
which the period-doubling bifurcations occur. Namely, if An is the
value of A at which the period-2n occurs, then the ratio
dn ¼ (An À AnÀ1 )=(Anþ1 À An ) (7:2:6)
has the limit
lim dn ¼ 4:669 . . . : (7:2:7)
n!1

It turns out that the limit (7.2.7) is valid for the entire family of maps
with the parabolic iteration functions [5].
A very important feature of the chaotic regime is extreme sensitiv-
ity of trajectories to the initial conditions. This is illustrated with
Figure 7.3 for A ¼ 3:8. Namely, two trajectories with the initial
conditions X0 ¼ 0:400 and X0 ¼ 0:405 diverge completely after 10


1
Xk



0.8




0.6




0.4




0.2

X0 = 0.4
X0 = 0.405
k
0
1 11 21
Figure 7.3 Solution to the logistic map for A ¼ 3.8 and two initial condi-
tions: X0 ¼ 0:400 and X0 ¼ 0:405.
75
Nonlinear Dynamical Systems



iterations. Thus, the logistic map provides an illuminating example of
complexity and universality generated by interplay of nonlinearity
and discreteness.


7.3 CONTINUOUS SYSTEMS
While the discrete time series are the convenient framework for
financial data analysis, financial processes are often described using
continuous presentation [6]. Hence, we need understanding of the
chaos specifics in continuous systems. First, let us introduce several
important notions with a simple model of a damped oscillator (see,
e.g., [7]). Its equation of motion in terms of the angle of deviation
from equilibrium, u, is
d2 u du
þ g þ v2 u ¼ 0 (7:3:1)
dt2 dt
In (7.3.1), g is the damping coefficient and v is the angular frequency.
Dynamical systems are often described with flows, sets of coupled
differential equations of the first order. These equations in the vector
notations have the following form
dX
¼ F(X(t)), X ¼ (X1 , X2 , . . . XN )0 (7:3:2)
dt
We shall consider so-called autonomous systems for which the func-
tion F in the right-hand side of (7.3.2) does not depend explicitly on
time. A non-autonomous system can be transformed into an autono-
mous one by treating time in the function F(X, t) as an additional
variable, XNþ1 ¼ t, and adding another equation to the flow
dXNþ1
¼1 (7:3:3)
dt
As a result, the dimension of the phase space increases by one. The
notion of the fixed point in continuous systems differs from that of
discrete systems (7.2.4). Namely, the fixed points for the flow (7.3.2)
are the points XÃ at which all derivatives in its left-hand side equal
zero. For the obvious reason, these points are also named the equilib-
rium (or stationary) points: If the system reaches one of these points,
it stays there forever.
76 Nonlinear Dynamical Systems



Equations with derivatives of order greater than one can be also
transformed into flows by introducing additional variables. For
example, equation (7.3.1) can be transformed into the system
du dw
¼ Àgw À v2 u
¼ w, (7:3:4)
dt dt
Hence, the damped oscillator may be described in the two-dimen-
sional phase space (w, u). The energy of the damped oscillator, E,
E ¼ 0:5(w2 þ v2 u2 ) (7:3:5)
evolves with time according to the equation
dE
¼ Àgw2 (7:3:6)
dt
It follows from (7.3.6) that the dumped oscillator dissipates energy
(i.e., is a dissipative system) at g > 0. Typical trajectories of the
dumped oscillator are shown in Figure 7.4. In the case g ¼ 0, the
trajectories are circles centered at the origin of the phase plane. If
g > 0, the trajectories have a form of a spiral approaching the origin
of plane.2 In general, the dissipative systems have a point attractor in
the center of coordinates that corresponds to the zero energy.
Chaos is usually associated with dissipative systems. Systems with-
out energy dissipation are named conservative or Hamiltonian


2.5 2.5
PSI
PSI b)
a)
2
2
1.5
1.5
1
1
0.5
FI
0.5 0
FI ’1.5 ’1 ’0.5 0 0.5 1 1.5
’0.5
0
’1.5 ’0.5 0.5 1.5 ’1
’0.5
’1.5
’1
’2
’1.5
’2.5
’2
’2.5
Figure 7.4 Trajectories of the damped oscillator with v ¼ 2: (a) g ¼ 2; (b)
g ¼ 0.
77
Nonlinear Dynamical Systems



systems. Some conservative systems may have the chaotic regimes,
too (so-called non-integrable systems) [5], but this case will not be
discussed here. One can easily identify the sources of dissipation in
real physical processes, such as friction, heat radiation, and so on. In
general, flow (7.3.2) is dissipative if the condition
X @F
N
div(F)  <0 (7:3:7)
@Xi
i¼1

is valid on average within the phase space.
Besides the point attractor, systems with two or more dimensions
may have an attractor named the limit cycle. An example of such an
attractor is the solution of the Van der Pol equation. This equation
describes an oscillator with a variable damping coefficient
d2 u du
þ g[(u=u0 )2 À 1] þ v2 u ¼ 0 (7:3:8)
dt2 dt
In (7.3.8), u0 is a parameter. The damping coefficient is positive at
sufficiently high amplitudes u > u0 , which leads to energy dissipation.
However, at low amplitudes (u < u0 ), the damping coefficient be-
comes negative. The negative term in (7.3.8) has a sense of an energy
source that prevents oscillations from complete decay. If one intro-
p¬¬¬¬¬¬¬¬
duces u0 v=g as the unit of amplitude and 1=v as the unit of time,
then equation (7.3.8) acquires the form
d2 u du
þ (u2 À e2 ) þ u ¼ 0 (7:3:9)
dt2 dt
where e ¼ g=v is the only dimensionless parameter that defines the
system evolution. The flow describing the Van der Pol equation has
the following form
du dw
¼ (e2 À u2 ) w À u
¼ w, (7:3:10)
dt dt
Figure 7.5 illustrates the solution to equation (7.3.1) for e ¼ 0:4.
Namely, the trajectories approach a closed curve from the initial
conditions located both outside and inside the limit cycle. It should
be noted that the flow trajectories never intersect, even though
their graphs may deceptively indicate otherwise. This property
follows from uniqueness of solutions to equation (7.3.8). Indeed, if the
78 Nonlinear Dynamical Systems



1.5
PSI

1



0.5

FI
0
M2 M1
’1.2 ’0.8 ’0.4 0 0.4 0.8 1.2 1.6 2


’0.5



’1



’1.5
Figure 7.5 Trajectories of the Van der Pol oscillator with e ¼ 0:4. Both
trajectories starting at points M1 and M2, respectively, end up on the same
limit circle.


trajectories do intersect, say at point P in the phase space, this implies
that the initial condition at point P yields two different solutions.
Since the solution to the Van der Pol equation changes qualita-
tively from the point attractor to the limit cycle at e ¼ 0, this point is a
bifurcation. Those bifurcations that lead to the limit cycle are named
the Hopf bifurcations.
In three-dimensional dissipative systems, two new types of attractors
appear. First, there are quasi-periodic attractors. These trajectories are
associated with two different frequencies and are located on the surface
of a torus. The following equations describe the toroidal trajectories
(see Figure 7.6)
x(t) ¼ (R þ r sin (wr t)) cos (wR t)
y(t) ¼ (R þ r sin (wr t)) sin (wR t)
z(t) ¼ r cos (wr t) (7:3:11)
In (7.3.11), R and r are the external and internal torus radii, respect-
ively; wR and wr are the frequencies of rotation around the external
79
Nonlinear Dynamical Systems



12

10

8

6

4

2

0
’12 ’10 ’8 ’6 ’4 ’2 0 2 4 6 8 10 12
’2

’4

’6

’8

’10

’12
Figure 7.6 Toroidal trajectories (7.3.11) in the X-Y plane for R ¼ 10, r ¼ 1,
wR ¼ 100, wr ¼ 3.

and internal radii, respectively. If the ratio wR =wr is irrational, it is
said that the frequencies are incommensurate. Then the trajectories
(7.3.11) never close on themselves and eventually cover the entire
torus surface. Nevertheless, such a motion is predictable, and thus it
is not chaotic. Another type of attractor that appears in three-dimen-
sional systems is the strange attractor. It will be introduced using the
famous Lorenz model in the next section.

7.4 LORENZ MODEL
The Lorenz model describes the convective dynamics of a fluid
layer with three dimensionless variables:
dX
¼ p(Y À X)
dt
dY
¼ ÀXZ þ rX À Y
dt
dZ
¼ XY À bZ (7:4:1)
dt
80 Nonlinear Dynamical Systems



In (7.4.1), the variable X characterizes the fluid velocity distribution,
and the variables Y and Z describe the fluid temperature distribution.
The dimensionless parameters p, r, and b characterize the thermo-
hydrodynamic and geometric properties of the fluid layer. The Lorenz
model, being independent of the space coordinates, is a result of signifi-
cant simplifications of the physical process under consideration [5, 7].
Yet, this model exhibits very complex behavior. As it is often done in
the literature, we shall discuss the solutions to the Lorenz model for
the fixed parameters p ¼ 10 and b ¼ 8=3. The parameter r (which is the
vertical temperature difference) will be treated as the control parameter.
At small r 1, any trajectory with arbitrary initial conditions ends
at the state space origin. In other words, the non-convective state at
X ¼ Y ¼ Z ¼ 0 is a fixed point attractor and its basin is the entire
phase space. At r > 1, the system acquires three fixed points. Hence,
the point r ¼ 1 is a bifurcation. The phase space origin is now repel-
lent. Two other fixed points are attractors that correspond to the
steady convection with clockwise and counterclockwise rotation, re-
spectively (see Figure 7.7). Note that the initial conditions define


10
YZ
8

B
6

4

C
2
D
X
0
’8 ’6 ’4 ’2 0 2 4 6 8
’2
= ’1
A : X-Y, Y(0)
= ’1
B : X-Z, Y(0)
’4 C : X-Y, Y(0) =1
A D : X-Z, Y(0) =1
’6

’8
Figure 7.7 Trajectories of the Lorenz model for p ¼ 10, b ¼ 8/3, r ¼ 6, X(0)
¼ Z(0) ¼ 0, and different Y(0).
81
Nonlinear Dynamical Systems



which of the two attractors is the trajectory™s final destination. The
locations of the fixed points are determined by the stationary solution
dX dY dZ
¼ ¼ ¼0 (7:4:2)
dt dt dt
Namely,
p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
Y ¼ X, Z ¼ 0:5X2 , X ¼ Æ b(r À 1) (7:4:3)
When the parameter r increases to about 13.93, the repelling
regions develop around attractors. With further growth of r, the
trajectories acquire the famous ˜˜butterfly™™ look (see Figure 7.8). In
this region, the system becomes extremely sensitive to initial condi-
tions. An example with r ¼ 28 in Figure 7.9 shows that the change of
Y(0) in 1% leads to completely different trajectories Y(t). The system
is then unpredictable, and it is said that its attractors are ˜˜strange.™™
With further growth of the parameter r, the Lorenz model reveals
new surprises. Namely, it has ˜˜windows of periodicity™™ where the
trajectories may be chaotic at first but then become periodic. One of
the largest among such windows is in the range 144 < r < 165. In this
parameter region, the oscillation period decreases when r grows. Note


60
Y Z
50

40

30

20

10
X
0
’20 ’15 ’10 ’5 0 5 10 15 20 25
’10

’20
X-Y
X-Z
’30
Figure 7.8 Trajectories of the Lorenz model for p ¼ 10, b ¼ 8/3 and r ¼ 28.
82 Nonlinear Dynamical Systems



40
Y(t)




20




0
t
0 2 4 6 8 10 12 14




’20


Y(0) = 1.00
Y(0) = 1.01
’40
Figure 7.9 Sensitivity of the Lorenz model to the initial conditions for p ¼
10, b ¼ 8/3 and r ¼ 28.


that this periodicity is not described with a single frequency, and the
maximums of its peaks vary. Finally, at very high values of
r (r > 313), the system acquires a single stable limit cycle. This fascin-
ating manifold of solutions is not an exclusive feature of the Lorenz
model. Many nonlinear dissipative systems exhibit a wide spectrum of
solutions including chaotic regimes.


7.5 PATHWAYS TO CHAOS
A number of general pathways to chaos in nonlinear dissipative
systems have been described in the literature (see, e.g., [5] and refer-
ences therein). All transitions to chaos can be divided into two major
groups: local bifurcations and global bifurcations. Local bifurcations
occur in some parameter range, but the trajectories become chaotic
when the system control parameter reaches the critical value. Three
types of local bifurcations are discerned: period-doubling, quasi-peri-
odicity, and intermittency. Period-doubling starts with a limit cycle at
some value of the system control parameter. With further change of
83
Nonlinear Dynamical Systems



this parameter, the trajectory period doubles and doubles until it
becomes infinite. This process was proposed by Landau as the main
turbulence mechanism. Namely, laminar flow develops oscillations at
some sufficiently high velocity. As velocity increases, another (incom-
mensurate) frequency appears in the flow, and so on. Finally, the
frequency spectrum has the form of a practically continuous band. An
alternative mechanism of turbulence (quasi-periodicity) was proposed
by Ruelle and Takens. They have shown that the quasi-periodic
trajectories confined on the torus surface can become chaotic due to
high sensitivity to the input parameters. Intermittency is a broad
category itself. Its pathway to chaos consists of a sequence of periodic
and chaotic regions. With changing the control parameter, chaotic
regions become larger and larger and eventually fill the entire
space.
In the global bifurcations, the trajectories approach simple attract-
ors within some control parameter range. With further change of the
control parameter, these trajectories become increasingly complicated
and in the end, exhibit chaotic motion. Global bifurcations are parti-
tioned into crises and chaotic transients. Crises include sudden
changes in the size of chaotic attractors, sudden appearances of the
chaotic attractors, and sudden destructions of chaotic attractors and
their basins. In chaotic transients, typical trajectories initially behave
in an apparently chaotic manner for some time, but then move to
some other region of the phase space. This movement may asymptot-
ically approach a non-chaotic attractor.
Unfortunately, there is no simple rule for determining the condi-
tions at which chaos appears in a given flow. Moreover, the same
system may become chaotic in different ways depending on its par-
ameters. Hence, attentive analysis is needed for every particular
system.


7.6 MEASURING CHAOS
As it was noticed in in Section 7.1, it is important to understand
whether randomness of an empirical time series is caused by noise or
by the chaotic nature of the underlying deterministic process. To
address this problem, let us introduce the Lyapunov exponent. The
major property of a chaotic attractor is exponential divergence of its
84 Nonlinear Dynamical Systems



nearby trajectories. Namely, if two nearby trajectories are separated
by distance d0 at t ¼ 0, the separation evolves as
d(t) ¼ d0 exp (lt) (7:6:1)
The parameter l in (7.6.1) is called the Lyapunov exponent. For the
rigorous definition, consider two points in the phase space, X0 and
X0 þ Dx0 , that generate two trajectories with some flow (7.3.2). If the
function Dx(X0 , t) defines evolution of the distance between these
points, then
1 jDx(X0 , t)j
l ¼ lim ln , t ! 1, Dx0 ! 0 (7:6:2)
jDx0 j
t
When l < 0, the system is asymptotically stable. If l ¼ 0, the system
is conservative. Finally, the case with l > 0 indicates chaos since the
system trajectories diverge exponentially.
The practical receipt for calculating the Lyapunov exponent is as
follows. Consider n observations of a time series x(t): x(tk ) ¼ xk , k ¼ 1,
. . . , n. First, select a point xi and another point xj close to xi . Then
calculate the distances
d0 ¼ jxi À xj j, d1 ¼ jxiþ1 À xjþ1 j, . . . , dn ¼ jxiþn À xjþn j (7:6:3)
If the distance between xiþn and xjþn evolves with n accordingly with
(7.6.1), then
1 dn
l(xi ) ¼ ln (7:6:4)
n d0
The value of the Lyapunov exponent l(xi ) in (7.6.4) is expected to be
sensitive to the choice of the initial point xi . Therefore, the average
value over a large number of trials N of l(xi ) is used in practice

1X N
l¼ l(xi ) (7:6:5)
N i¼1

Due to the finite size of empirical data samples, there are limitations
on the values of n and N, which affects the accuracy of calculating the
Lyapunov exponent. More details about this problem, as well as other
chaos quantifiers, such as the Kolmogorov-Sinai entropy, can be
found in [5] and references therein.
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) ¼ u(R À jxj À xi j) (7:6:6)
N À 1 j ¼ 1, j 6¼ i

In (7.6.6), the Heaviside step function u equals

0, x < 0
u¼ (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) ¼ pi (R) (7:6:8)
N i¼1

It is assumed that C(R) $ RDc . Hence, the correlation dimension Dc
equals
Dc ¼ 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
˜˜first-hand™™ 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 ¼ XkÀ1 2 þ C, where C > 0.
(a) Prove that C ¼ 0:25 is a bifurcation point.
(b) Find fixed points for C ¼ 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 ˜˜butterfly™™ trajectories depicted in Figure
7.8.
(b) Verify existence of the periodicity window at r ¼ 150.
(c) Verify existence of the limit cycle at r ¼ 350.
Hint: Use a simple algorithm: Xk ¼ XkÀ1 þ tF(XkÀ1 , YkÀ1 , ZkÀ1 )
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]:
˜˜No model exists for the stochastic process describing the
time evolution of the logarithm of price that is accepted by
all researchers.™™
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 company™s assets
but also the expectations of the company™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 company™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 [11“13]. This
means that the probability distributions examined by Stanley™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
1984“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 companies™ 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 Zipf™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 þ 1) ¼ l(t)wi (t) À aW(t) À bwi (t)W(t), W(t) ¼
N i¼1

where wi is an individual characteristic (e.g., wealth of an investor i;
i ¼ 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) $ wÀ(1þa) . 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) ¼ A þ B(tc À t)a {1 þ C cos [w ln (tc À t) À 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) ¼ exp [gx(t) þ 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) ¼ a1 y1 (t) þ a2 y2 (t) þ a3 y3 (t) (8:3:2)
The first process y1 (t) is an AR(1) process
y1 (t þ 1) ¼ r1 y1 (t) þ Z1 (t þ 1) (8:3:3)
where r1 ¼ 0:999 and Z1 (t) is an independent Gaussian adjusted so
that var[y1 (t)] ¼ 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 þ 1) ¼ r2 y2 (t) þ Z2 (t þ 1) (8:3:4)
where Z2 (t) is an independent Gaussian adjusted so that
var[y2 (t)] ¼ 1. The chosen value r2 ¼ 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 þ a2 2 þ a3 2 ¼ 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) ¼ xÀa 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 ˜˜universal™™
92 Scaling in Financial Time Series



scaling exponent a ¼ Àlog [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 ˜˜hot™™ 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) ¼ 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 ˜˜trading time™™ that reflects
intensity of the trading process. Current research in this direction
shows some promising results [23“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 ¼ 0:5) may yield a more
accurate description of several financial time series than the GARCH
model [23]. Nevertheless, this chapter remains ˜˜unfinished™™ 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 [10“13, 23“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 [27“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 À K and the payoff from the short position is K À 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.
In the mean time, potential losses cannot exceed the option price.
The effect of the risk-free rate is not straightforward. At a fixed
stock price, the rising risk-free rate increases the value of the call
option. Indeed, the option holder may defer paying for shares and
invest this payment into the risk-free assets until the option matures.
On the contrary, the value of the put option decreases with the risk-
free rate since the option holder defers receiving payment from selling
shares and therefore cannot invest them into the risk-free assets.
However, rising interest rates often lead to falling stock prices,
which may change the resulting effect of the risk-free rate.
Dividends effectively reduce the stock prices. Therefore, dividends
decrease value of call options and increase value of put options.
Now, let us consider the payoffs at maturity for four possible
European option positions. The long call option means that the in-
vestor buys the right to buy an underlying asset. Obviously, it makes
sense to exercise the option only if S > K. Therefore, its payoff is
PLC ¼ max [S À K, 0] (9:2:1)
The short call option means that the investor sells the right to buy an
underlying asset. This option is exercised if S > K, and its payoff is
PSC ¼ min [K À S, 0] (9:2:2)
The long put option means that the investor buys the right to sell an
underlying asset. This option is exercised when K > S, and its payoff
is
PLP ¼ max [K À S, 0] (9:2:3)
The short put option means that the investor sells the right to sell an
underlying asset. This option is exercised when K > S, and its payoff is
96 Option Pricing



PSP ¼ min [S À K, 0] (9:2:4)
Note that the option payoff by definition does not account for the
option price (also named option premium). In fact, option writers sell
options at a premium while option buyers pay this premium. There-
fore, the option seller™s profit is the option payoff plus the option
price, while the option buyer™s profit is the option payoff minus the
option price (see examples in Figure 9.1).
The European call and put options with the same strike price
satisfy the relation called put-call parity. Consider two portfolios.
Portfolio I has one European call option at price c with the strike
price K and amount of cash (or zero-coupon bond) with the present
value Kexp[Àr(T À t)]. Portfolio II has one European put option at
price p and one share at price S. First, let us assume that share does
not pay dividends. Both portfolios at maturity have the same value:
max (ST , K). Hence,
c þ Kexp[Àr(T À t)] ¼ p þ S (9:2:5)
Dividends affect the put-call parity. Namely, the dividends D being
paid during the option lifetime have the same effect as the cash future
value. Thus,
c þ D þ K exp [Àr(T À t)] ¼ p þ S (9:2:6)
Because the American options may be exercised before maturity, the
relations between the American put and call prices can be derived
only in the form of inequalities [1].
Options are widely used for both speculation and risk hedging.
Consider two examples with the IBM stock options. At market
closing on 7-Jul-03, the IBM stock price was $83.95. The (American)
call option price at maturity on 3-Aug-03 was $2.55 for the strike
price of $85. Hence, the buyer of this option at market closing on 7-
Jul-03 assumed that the IBM stock price would exceed $(85 þ 2.55) ¼
$87.55 before or on 3-Aug-03. If the IBM share price would reach say
$90, the option buyer will exercise the call option to buy the share for
$85 and immediately sell it for $90. The resulting profit4 is
$(90À87.55) ¼ $2.45. Thus, the return on exercising this option equals
2:45=2:55Ã 100% ¼ 96%. Note that the return on buying an IBM
share in this case would only be (90 À 83:95)=83:95Ã 100% ¼ 7:2%.
97
Option Pricing



(a) 20
Profit

15


10

Short Call
5
Stock price
0
0 5 10 15 20 25 30 35 40

’5
Long Call

’10


’15


’20



(b) 20
Profit

15
Long Put
10


5
Stock price
0
0 5 10 15 20 25 30 35 40

’5


’10
Short Put
’15


’20
Figure 9.1 The option profits for the strike price of $25 and the option
premium of $5: (a) calls, (b) puts.
98 Option Pricing



If, however, the IBM share price stays put through 3-Aug-03, an
option buyer incurs losses of $2.45 (i.e., 100%). In the mean time, a
share buyer has no losses and may continue to hold shares, hoping
that their price will grow in future.
At market closing on 7-Jul-03, the put option for the IBM share
with the strike price of $80 at maturity on 3-Aug-03 was $1.50. Hence,
buyers of this put option bet on price falling below $(80À1.50) ¼
$78.50. If, say the IBM stock price falls to $75, the buyer of the put
option has a gain of $(78:50 À 75) ¼ $3.50.
Now, consider hedging in which the investor buys simultaneously
one share for $83.95 and a put option with the strike price of $80 for
$1.50. The investor has gains only if the stock price rises above
$(83:95 þ 1:50) ¼$85:45. However, if the stock price falls to say $75,
the investor™s loss is $(80 À 85:45) ¼ À$5:45 rather than the loss of
$(75 À 83:95) ¼ À$8:95 incurred without hedging with the put
option. Hence, in the given example, the hedging expense of $1.50
allows the investor to save $(À5:45 þ 8:95) ¼$3:40.


9.3 BINOMIAL TREES
Let us consider a simple yet instructive method for option pricing
that employs a discrete model called the binomial tree. This model is
based on the assumption that the current stock price S can change at
the next moment only to either the higher value Su or the lower value
Sd (where u > 1 and d < 1). Let us start with the first step of the
binomial tree (see Figure 9.2). Let the current option price be equal to
F and denote it with Fu or Fd at the next moment when the stock price
moves up or down, respectively. Consider now a portfolio that con-
sists of D long shares and one short option. This portfolio is risk-free
if its value does not depend on whether the stock price moves up or
down, that is,
SuD À Fu ¼ SdD À Fd (9:3:1)
Then the number of shares in this portfolio equals
D ¼ (Fu À Fd )=(Su À Sd) (9:3:2)
The risk-free portfolio with the current value (SD À F) has the future
value (SuD À Fu ) ¼ (SdD À Fd ). If the time interval is t and the risk-
99
Option Pricing



Su2

Fuu


Su

Fu


Sud
S
F
Fud


Sd

Fd


Sd2

Fdd
Figure 9.2 Two-step binomial pricing tree.


free interest rate is r, the relation between the portfolio™s present value
and future value is
(SD À F) exp(rt) ¼ SuD À Fu (9:3:3)
Combining (9.3.2) and (9.3.3) yields
F ¼ exp(Àrt)[pFu þ (1 À p)Fd ] (9:3:4)
where
p ¼ [ exp (rt) À d]=(u À d) (9:3:5)
The factors p and (1 À p) in (9.3.4) have the sense of the probabilities
for the stock price to move up and down, respectively. Then, the
expectation of the stock price at time t is
E[S(t)] ¼ E[pSu þ (1 À p)Sd] ¼ S exp (rt) (9:3:6)
This means that the stock price grows on average with the risk-free
rate. The framework within which the assets grow with the risk-free
rate is called risk-neutral valuation. It can be discussed also in terms of
the arbitrage theorem [4]. Indeed, violation of the equality (9.3.3)
100 Option Pricing



implies that the arbitrage opportunity exists for the portfolio. For
example, if the left-hand side of (9.3.3) is greater than its right-hand
side, one can immediately make a profit by selling the portfolio and
buying the risk-free asset.
Let us proceed to the second step of the binomial tree. Using
equation (9.3.4), we receive the following relations between the option
prices on the first and second steps
Fu ¼ exp (Àrt)[pFuu þ (1 À p)Fud ] (9:3:7)
Fd ¼ exp (Àrt)[pFud þ (1 À p)Fdd ] (9:3:8)
The combination of (9.3.4) with (9.3.7) and (9.3.8) yields the current
option price in terms of the option prices at the next step
F ¼ exp (À2rt)[p2 Fuu þ 2p(1 À p)Fud þ (1 À p)2 Fdd ] (9:3:9)
This approach can be generalized for a tree with an arbitrary number
of steps. Namely, first the stock prices at every node are calculated
by going forward from the first node to the final nodes. When the
stock prices at the final nodes are known, we can determine the
option prices at the final nodes by using the relevant payoff relation
(e.g., (9.2.1) for the long call option). Then we calculate the option
prices at all other nodes by going backward from the final nodes to
the first node and using the recurrent relations similar to (9.3.7) and
(9.3.8).
The factors that determine the price change, u and d, can be
estimated from the known stock price volatility [1]. In particular, it
is generally assumed that prices follow the geometric Brownian
motion
dS ¼ mSdt þ sSdW (9:3:10)
where m and s are the drift and diffusion parameters, respectively, and
dW is the standard Wiener process (see Section 4.2). Hence, the price
changes within the time interval [0, t] are described with the lognor-
mal distribution
p¬¬
ln S(t) ¼ N( ln S0 þ (m À s2 =2)t, s t) (9:3:11)
In (9.3.11), S0 ¼ S(0), N(m, s) is the normal distribution with mean
m and standard deviation s. It follows from equation (9.3.11) that the
expectation of the stock price and its variance at time t equal
101
Option Pricing



E[S(t)] ¼ S0 exp (mt) (9:3:12)
Var[S(t)] ¼ S0 2 exp (2mt)[ exp (s2 t) À 1] (9:3:13)
In addition, equation (9.3.6) yields
exp (rt) ¼ pu þ (1 À p)d (9:3:14)
Using (9.3.13) and (9.3.14) in the equality (y) ¼ E[y2 ] À E[y]2 , we
obtain the relation
exp (2rt þ s2 t) ¼ pu2 þ (1 À p)d2 (9:3:15)
The equations (9.3.14) and (9.3.15) do not suffice to define the three
parameters d, p, and u. Usually, the additional condition
u ¼ 1=d (9:3:16)
is employed. When the time interval Dt is small, the linear approxi-
mation to the system of equations (9.3.14) through (9.3.16) yields
p ¼ [ exp (rDt) À d]=(u À d), u ¼ 1=d ¼ exp [s(Dt)1=2 ] (9:3:17)
The binomial tree model can be generalized in several ways [1]. In
particular, dividends and variable interest rates can be included. The
trinomial tree model can also be considered. In the latter model, the
stock price may move upward or downward, or it may stay the same.
The drawback of the discrete tree models is that they allow only for
predetermined innovations of the stock price. Moreover, as it was
described above, the continuous model of the stock price dynamics
(9.3.10) is used to estimate these innovations. It seems natural then to
derive the option pricing theory completely within the continuous
framework.

9.4 BLACK-SCHOLES THEORY
The basic assumptions of the classical option pricing theory are
that the option price F(t) at time t is a continuous function of time
and its underlying asset™s price S(t)
F ¼ F(S(t), t) (9:4:1)
and that price S(t) follows the geometric Brownian motion (9.3.10) [5,
6]. Several other assumptions are made to simplify the derivation of
the final results. In particular,
102 Option Pricing



. There are no market imperfections, such as price discreteness,
transaction costs, taxes, and trading restrictions including those
on short selling.
. Unlimited risk-free borrowing is available at a constant rate, r.
. There are no arbitrage opportunities.
. There are no dividend payments during the life of the option.
Now, let us derive the classical Black-Scholes equation. Since it is
assumed that the option price F(t) is described with equation (9.4.1)
and price of the underlying asset follows equation (9.3.10), we can use
the Ito™s expression (4.3.5)
 
@F @F s2 2 @ 2 F @F
dF(S, t) ¼ mS þ þS dt þ sS dW(t) (9:4:2)
@S2
2
@S @t @S
Furthermore, we build a portfolio P with eliminated random contri-
@F
bution dW. Namely, we choose À1 (short) option and shares of
@S
5
the underlying asset,
@F
P ¼ ÀF þ S (9:4:3)
@S
The change of the value of this portfolio within the time interval dt
equals
@F
dP ¼ ÀdF þ dS (9:4:4)
@S
Since there are no arbitrage opportunities, this change must be equal to
the interest earned by the portfolio value invested in the risk-free asset
dP ¼ rP dt (9:4:5)
The combination of equations (9.4.2)“(9.4.5) yields the Black-Scholes
equation
@F s2 2 @ 2 F
@F
þ rS þS À rF ¼ 0 (9:4:6)
@S2
2
@t @S
Note that this equation does not depend on the stock price drift
parameter m, which is the manifestation of the risk-neutral valuation.
In other words, investors do not expect a portfolio return exceeding
the risk-free interest.
103
Option Pricing



The Black-Scholes equation is the partial differential equation with
the first-order derivative in respect to time and the second-order de-
rivative in respect to price. Hence, three boundary conditions deter-
mine the Black-Scholes solution. The condition for the time variable is
defined with the payoff at maturity. The other two conditions for the
price variable are determined with the asymptotic values for the zero
and infinite stock prices. For example, price of the put option equals
the strike price when the stock price is zero. On the other hand, the put
option price tends to be zero if the stock price approaches infinity.
The Black-Scholes equation has an analytic solution in some
simple cases. In particular, for the European call option, the Black-
Scholes solution is
c(S, t) ¼ N(d1 )S(t) À KN(d2 ) exp[Àr(T À t)] (9:4:7)
In (9.4.7), N(x) is the standard Gaussian cumulative probability
distribution
d1 ¼ [ ln (S=K) þ (r þ s2 =2)(T À t)]=[s(T À t)1=2 ],
(9:4:8)
1=2
d2 ¼ d1 À (T À t)
The Black-Scholes solution for the European put option is
p(S, t) ¼ K exp[Àr(T À t)] N(Àd2 ) À S(t)N(Àd1 ) (9:4:9)
The value of the American call option equals the value of the Euro-
pean call option. However, no analytical expression has been found
for the American put option. Numerical methods are widely used for
solving the Black-Scholes equation when analytic solution is not
available [1“3].
Implied volatility is an important notion related to BST. Usually,
the stock volatility used in the Block-Scholes expressions for the
option prices, such as (9.4.7), is calculated with the historical stock
price data. However, formulation of the inverse problem is also
possible. Namely, the market data for the option prices can be used
in the left-hand side of (9.4.7) to recover the parameter s. This
parameter is named the implied volatility. Note that there is no
analytic expression for implied volatility. Therefore, numerical
methods must be employed for its calculation. Several other functions
related to the option price, such as Delta, Gamma, and Theta (so-
called Greeks), are widely used in the risk management:
104 Option Pricing



@2F
@F @F
D¼ ,G¼ 2,Q¼ (9:4:10)
@S @S @t
The Black-Scholes equation (9.4.6) can be rewritten in terms of
Greeks
s2 2
Q þ rSD þ S G À rF ¼ 0 (9:4:11)
2
Similarly, Greeks can be defined for the entire portfolio. For example,

@P @S
the portfolio™s Delta is . Since the share™s Delta equals unity,
@S @S
Delta of the portfolio (9.4.3) is zero. Portfolios with zero Delta are
called delta-neutral. Since Delta depends on both price and time,
maintenance of delta-neutral portfolios requires periodic rebalancing,
which is also known as dynamic hedging. For the European call and
put options, Delta equals, respectively
Dc ¼ N(d1 ), Dp ¼ N(d1 ) À 1 (9:4:12)
Gamma characterizes the Delta™s sensitivity to price variation. If

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