ñòð. 3 |

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 Â¼ ANkÃ€1 Ã€ BNkÃ€1 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 Â¼ 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 Â¼ A=B. It is convenient to intro-

duce the dimensionless variable Xk Â¼ Nk =Nmax . Then 0 Xk 1,

and equation (7.2.1) has the form

Xk Â¼ AXkÃ€1 (1 Ã€ XkÃ€1 ) (7:2:2)

A generic discrete equation in the form

Xk Â¼ f(XkÃ€1 ) (7:2:3)

is called an (iterated) map, and the function f(XkÃ€1 ) 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 XÃƒ that satisfies the

equation

XÃƒ Â¼ f(XÃƒ ) (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

XÃƒ Â¼ 0, and XÃƒ Â¼ (A Ã€ 1)=A (7:2:5)

1 2

If A 1, the logistic map trajectory approaches the fixed point XÃƒ 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 XÃƒ Â¼ 0, and its basin is the interval 0 X0 1.

1

If 1 < A < 3, the logistic map trajectories have the attractor

Ãƒ

X2 Â¼ (A Ã€ 1)=A and its basin is also 0 X0 1. In the mean time,

the point XÃƒ Â¼ 0 is the repellent fixed point, which implies that any

1

trajectory that starts near XÃƒ 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 Â¼ 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 â€˜â€˜too muchâ€™â€™ (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 Â¼ 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 Feigenbaumâ€™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 Â¼ (An Ã€ AnÃ€1 )=(AnÃ¾1 Ã€ An ) (7:2:6)

has the limit

lim dn Â¼ 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 Â¼ 3:8. Namely, two trajectories with the initial

conditions X0 Â¼ 0:400 and X0 Â¼ 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 Â¼ 3.8 and two initial condi-

tions: X0 Â¼ 0:400 and X0 Â¼ 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

Ã¾ g Ã¾ v2 u Â¼ 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

Â¼ F(X(t)), X Â¼ (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, XNÃ¾1 Â¼ t, and adding another equation to the flow

dXNÃ¾1

Â¼1 (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 XÃƒ 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

Â¼ Ã€gw Ã€ v2 u

Â¼ 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 Â¼ 0:5(w2 Ã¾ v2 u2 ) (7:3:5)

evolves with time according to the equation

dE

Â¼ Ã€gw2 (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 Â¼ 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 âˆ’1.5 âˆ’1 âˆ’0.5 0 0.5 1 1.5

âˆ’0.5

0

âˆ’1.5 âˆ’0.5 0.5 1.5 âˆ’1

âˆ’0.5

âˆ’1.5

âˆ’1

âˆ’2

âˆ’1.5

âˆ’2.5

âˆ’2

âˆ’2.5

Figure 7.4 Trajectories of the damped oscillator with v Â¼ 2: (a) g Â¼ 2; (b)

g Â¼ 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

iÂ¼1

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

Ã¾ g[(u=u0 )2 Ã€ 1] Ã¾ v2 u Â¼ 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-

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ

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

Ã¾ (u2 Ã€ e2 ) Ã¾ u Â¼ 0 (7:3:9)

dt2 dt

where e Â¼ 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

Â¼ (e2 Ã€ u2 ) w Ã€ u

Â¼ w, (7:3:10)

dt dt

Figure 7.5 illustrates the solution to equation (7.3.1) for e Â¼ 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

âˆ’1.2 âˆ’0.8 âˆ’0.4 0 0.4 0.8 1.2 1.6 2

âˆ’0.5

âˆ’1

âˆ’1.5

Figure 7.5 Trajectories of the Van der Pol oscillator with e Â¼ 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 Â¼ 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) Â¼ (R Ã¾ r sin (wr t)) cos (wR t)

y(t) Â¼ (R Ã¾ r sin (wr t)) sin (wR t)

z(t) Â¼ 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

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

âˆ’2

âˆ’4

âˆ’6

âˆ’8

âˆ’10

âˆ’12

Figure 7.6 Toroidal trajectories (7.3.11) in the X-Y plane for R Â¼ 10, r Â¼ 1,

wR Â¼ 100, wr Â¼ 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

Â¼ p(Y Ã€ X)

dt

dY

Â¼ Ã€XZ Ã¾ rX Ã€ Y

dt

dZ

Â¼ XY Ã€ 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 Â¼ 10 and b Â¼ 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 Â¼ Y Â¼ Z Â¼ 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 Â¼ 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

âˆ’8 âˆ’6 âˆ’4 âˆ’2 0 2 4 6 8

âˆ’2

= âˆ’1

A : X-Y, Y(0)

= âˆ’1

B : X-Z, Y(0)

âˆ’4 C : X-Y, Y(0) =1

A D : X-Z, Y(0) =1

âˆ’6

âˆ’8

Figure 7.7 Trajectories of the Lorenz model for p Â¼ 10, b Â¼ 8/3, r Â¼ 6, X(0)

Â¼ Z(0) Â¼ 0, and different Y(0).

81

Nonlinear Dynamical Systems

which of the two attractors is the trajectoryâ€™s final destination. The

locations of the fixed points are determined by the stationary solution

dX dY dZ

Â¼ Â¼ Â¼0 (7:4:2)

dt dt dt

Namely,

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ

Y Â¼ X, Z Â¼ 0:5X2 , X Â¼ Ã† b(r Ã€ 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 â€˜â€˜butterflyâ€™â€™ look (see Figure 7.8). In

this region, the system becomes extremely sensitive to initial condi-

tions. An example with r Â¼ 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 â€˜â€˜strange.â€™â€™

With further growth of the parameter r, the Lorenz model reveals

new surprises. Namely, it has â€˜â€˜windows of periodicityâ€™â€™ 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

âˆ’20 âˆ’15 âˆ’10 âˆ’5 0 5 10 15 20 25

âˆ’10

âˆ’20

X-Y

X-Z

âˆ’30

Figure 7.8 Trajectories of the Lorenz model for p Â¼ 10, b Â¼ 8/3 and r Â¼ 28.

82 Nonlinear Dynamical Systems

40

Y(t)

20

0

t

0 2 4 6 8 10 12 14

âˆ’20

Y(0) = 1.00

Y(0) = 1.01

âˆ’40

Figure 7.9 Sensitivity of the Lorenz model to the initial conditions for p Â¼

10, b Â¼ 8/3 and r Â¼ 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 Â¼ 0, the separation evolves as

d(t) Â¼ 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 Ã¾ 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 Â¼ lim ln , t ! 1, Dx0 ! 0 (7:6:2)

jDx0 j

t

When l < 0, the system is asymptotically stable. If l Â¼ 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 ) Â¼ xk , k Â¼ 1,

. . . , n. First, select a point xi and another point xj close to xi . Then

calculate the distances

d0 Â¼ jxi Ã€ xj j, d1 Â¼ jxiÃ¾1 Ã€ xjÃ¾1 j, . . . , dn Â¼ jxiÃ¾n Ã€ xjÃ¾n j (7:6:3)

If the distance between xiÃ¾n and xjÃ¾n evolves with n accordingly with

(7.6.1), then

1 dn

l(xi ) Â¼ 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

lÂ¼ l(xi ) (7:6:5)

N iÂ¼1

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) 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.

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 Â¼ max [S Ã€ 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 Â¼ min [K Ã€ 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 Â¼ max [K Ã€ 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 Â¼ min [S Ã€ 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 sellerâ€™s profit is the option payoff plus the option

price, while the option buyerâ€™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[Ã€r(T Ã€ 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 Ã¾ Kexp[Ã€r(T Ã€ t)] Â¼ p Ã¾ 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 Ã¾ D Ã¾ K exp [Ã€r(T Ã€ t)] Â¼ p Ã¾ 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 Ã¾ 2.55) Â¼

$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

$(90Ã€87.55) Â¼ $2.45. Thus, the return on exercising this option equals

2:45=2:55Ãƒ 100% Â¼ 96%. Note that the return on buying an IBM

share in this case would only be (90 Ã€ 83:95)=83:95Ãƒ 100% Â¼ 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

âˆ’5

Long Call

âˆ’10

âˆ’15

âˆ’20

(b) 20

Profit

15

Long Put

10

5

Stock price

0

0 5 10 15 20 25 30 35 40

âˆ’5

âˆ’10

Short Put

âˆ’15

âˆ’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 $(80Ã€1.50) Â¼

$78.50. If, say the IBM stock price falls to $75, the buyer of the put

option has a gain of $(78:50 Ã€ 75) Â¼ $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 Ã¾ 1:50) Â¼$85:45. However, if the stock price falls to say $75,

the investorâ€™s loss is $(80 Ã€ 85:45) Â¼ Ã€$5:45 rather than the loss of

$(75 Ã€ 83:95) Â¼ Ã€$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 $(Ã€5:45 Ã¾ 8:95) Â¼$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 Ã€ Fu Â¼ SdD Ã€ Fd (9:3:1)

Then the number of shares in this portfolio equals

D Â¼ (Fu Ã€ Fd )=(Su Ã€ Sd) (9:3:2)

The risk-free portfolio with the current value (SD Ã€ F) has the future

value (SuD Ã€ Fu ) Â¼ (SdD Ã€ 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 portfolioâ€™s present value

and future value is

(SD Ã€ F) exp(rt) Â¼ SuD Ã€ Fu (9:3:3)

Combining (9.3.2) and (9.3.3) yields

F Â¼ exp(Ã€rt)[pFu Ã¾ (1 Ã€ p)Fd ] (9:3:4)

where

p Â¼ [ exp (rt) Ã€ d]=(u Ã€ d) (9:3:5)

The factors p and (1 Ã€ 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)] Â¼ E[pSu Ã¾ (1 Ã€ p)Sd] Â¼ 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 Â¼ exp (Ã€rt)[pFuu Ã¾ (1 Ã€ p)Fud ] (9:3:7)

Fd Â¼ exp (Ã€rt)[pFud Ã¾ (1 Ã€ 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 Â¼ exp (Ã€2rt)[p2 Fuu Ã¾ 2p(1 Ã€ p)Fud Ã¾ (1 Ã€ 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 Â¼ mSdt Ã¾ 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

pï¬ƒï¬ƒ

ln S(t) Â¼ N( ln S0 Ã¾ (m Ã€ s2 =2)t, s t) (9:3:11)

In (9.3.11), S0 Â¼ 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)] Â¼ S0 exp (mt) (9:3:12)

Var[S(t)] Â¼ S0 2 exp (2mt)[ exp (s2 t) Ã€ 1] (9:3:13)

In addition, equation (9.3.6) yields

exp (rt) Â¼ pu Ã¾ (1 Ã€ p)d (9:3:14)

Using (9.3.13) and (9.3.14) in the equality (y) Â¼ E[y2 ] Ã€ E[y]2 , we

obtain the relation

exp (2rt Ã¾ s2 t) Â¼ pu2 Ã¾ (1 Ã€ 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 Â¼ 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 Â¼ [ exp (rDt) Ã€ d]=(u Ã€ d), u Â¼ 1=d Â¼ 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 assetâ€™s price S(t)

F Â¼ 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 Itoâ€™s expression (4.3.5)

@F @F s2 2 @ 2 F @F

dF(S, t) Â¼ mS Ã¾ Ã¾S dt Ã¾ 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 Ã€1 (short) option and shares of

@S

5

the underlying asset,

@F

P Â¼ Ã€F Ã¾ S (9:4:3)

@S

The change of the value of this portfolio within the time interval dt

equals

@F

dP Â¼ Ã€dF Ã¾ 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 Â¼ rP dt (9:4:5)

The combination of equations (9.4.2)â€“(9.4.5) yields the Black-Scholes

equation

@F s2 2 @ 2 F

@F

Ã¾ rS Ã¾S Ã€ rF Â¼ 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) Â¼ N(d1 )S(t) Ã€ KN(d2 ) exp[Ã€r(T Ã€ t)] (9:4:7)

In (9.4.7), N(x) is the standard Gaussian cumulative probability

distribution

d1 Â¼ [ ln (S=K) Ã¾ (r Ã¾ s2 =2)(T Ã€ t)]=[s(T Ã€ t)1=2 ],

(9:4:8)

1=2

d2 Â¼ d1 Ã€ (T Ã€ t)

The Black-Scholes solution for the European put option is

p(S, t) Â¼ K exp[Ã€r(T Ã€ t)] N(Ã€d2 ) Ã€ S(t)N(Ã€d1 ) (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 [1â€“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

DÂ¼ ,GÂ¼ 2,QÂ¼ (9:4:10)

@S @S @t

The Black-Scholes equation (9.4.6) can be rewritten in terms of

Greeks

s2 2

Q Ã¾ rSD Ã¾ S G Ã€ rF Â¼ 0 (9:4:11)

2

Similarly, Greeks can be defined for the entire portfolio. For example,

@P @S

the portfolioâ€™s Delta is . Since the shareâ€™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 Â¼ N(d1 ), Dp Â¼ N(d1 ) Ã€ 1 (9:4:12)

Gamma characterizes the Deltaâ€™s sensitivity to price variation. If

ñòð. 3 |