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Nonlinear Dynamical Systems

The generic characteristic of the strange attractor is its fractal

dimension. In fact, the non-integer (i.e., fractal) dimension of an

attractor can be used as the definition of a strange attractor. In

Chapter 6, the box-counting fractal dimension was introduced.

A computationally simpler alternative, so-called correlation dimen-

sion, is often used in nonlinear dynamics [3, 5].

Consider a sample with N trajectory points within an attractor. To

define the correlation dimension, first the relative number of points

located within the distance R from the point i must be calculated

X N

1

pi (R) Â¼ u(R Ã€ jxj Ã€ xi j) (7:6:6)

N Ã€ 1 j Â¼ 1, j 6Â¼ i

In (7.6.6), the Heaviside step function u equals

0, x < 0

uÂ¼ (7:6:7)

1, x ! 0

Then the correlation sum that characterizes the probability of finding

two trajectory points within the distance R is computed

1X N

C(R) Â¼ pi (R) (7:6:8)

N iÂ¼1

It is assumed that C(R) $ RDc . Hence, the correlation dimension Dc

equals

Dc Â¼ lim [ ln C(R)= ln R] (7:6:9)

R!0

There is an obvious problem of finding the limit (7.6.9) for data

samples on a finite grid. Yet, plotting ln[C(R)] versus ln(R) (which

is expected to yield a linear graph) provides an estimate of the

correlation dimension.

An interesting question is whether a strange attractor is always

chaotic, in other words, if it always has a positive Lyapunov expo-

nent. It turns out there are rare situations when an attractor may be

strange but not chaotic. One such example is the logistic map at the

period-doubling points: Its Lyapunov exponent equals zero while the

fractal dimension is about 0.5. Current opinion, however, holds that

the strange deterministic attractors may appear in discrete maps

rather than in continuous systems [5].

86 Nonlinear Dynamical Systems

7.7 REFERENCES FOR FURTHER READING

Two popular books, the journalistic report by Gleick [8] and the

â€˜â€˜first-handâ€™â€™ account by Ruelle [9], offer insight into the science of

chaos and the people behind it. The textbook by Hilborn [5] provides

a comprehensive description of the subject. The interrelations be-

tween the chaos theory and fractals are discussed in detail in [10].

7.8 EXERCISES

1. Consider the quadratic map Xk Â¼ XkÃ€1 2 Ã¾ C, where C > 0.

(a) Prove that C Â¼ 0:25 is a bifurcation point.

(b) Find fixed points for C Â¼ 0:125. Define what point is an

attractor and what is its attraction basin for X > 0.

2. Verify the equilibrium points of the Lorenz model (7.4.3).

*3. Calculate the Lyapunov exponent of the logistic map as a

function of the parameter A.

*4. Implement the algorithm for simulating the Lorenz model.

(a) Reproduce the â€˜â€˜butterflyâ€™â€™ trajectories depicted in Figure

7.8.

(b) Verify existence of the periodicity window at r Â¼ 150.

(c) Verify existence of the limit cycle at r Â¼ 350.

Hint: Use a simple algorithm: Xk Â¼ XkÃ€1 Ã¾ tF(XkÃ€1 , YkÃ€1 , ZkÃ€1 )

where the time step t can be assigned 0.01.

Chapter 8

Scaling in Financial Time

Series

8.1 INTRODUCTION

Two well-documented findings motivate further analysis of financial

time series. First, the probability distributions of returns often deviate

significantly from the normal distribution by having fat tails and excess

kurtosis. Secondly, returns exhibit volatility clustering. The latter effect

has led to the development of the GARCH models described in Section

5.3.1 In this chapter, we shall focus on scaling in the probability distri-

butions of returns, the concept that has attracted significant attention

from economists and physicists alike.

Alas, as the leading experts in Econophysics, H. E. Stanley and

R. Mantegna acknowledged [2]:

â€˜â€˜No model exists for the stochastic process describing the

time evolution of the logarithm of price that is accepted by

all researchers.â€™â€™

There are several reasons for the status quo.2 First, different financial

time series may have varying non-stationary components. Indeed, the

stock price reflects not only the current value of a companyâ€™s assets

but also the expectations of the companyâ€™s growth. Yet, there is no

general pattern for evolution of a business enterprise.3 Therefore,

87

88 Scaling in Financial Time Series

empirical research often concentrates on the average economic in-

dexes, such as the S&P 500. Averaging over a large number of

companies certainly smoothes noise. Yet, the composition of these

indicators is dynamic: Companies may be added to or dropped from

indexes, and the companyâ€™s contribution to the economic index usu-

ally depends on its ever-changing market capitalization.

Foreign exchange rates are another object frequently used in empir-

ical research.4 Unfortunately, many of the findings accumulated during

the 1990s have become somewhat irrelevant, as several European cur-

rencies ceased to exist after the birth of the Euro in 1999. In any case, the

foreign exchange rates, being a measure of relative currency strength,

may have statistical features that differ among themselves and in com-

parison with the economic indicators of single countries.

Another problem is data granularity. Low granularity may under-

estimate the contributions of market rallies and crashes. On the other

hand, high-frequency data are extremely noisy. Hence, one may

expect that universal properties of financial time series (if any exist)

have both short-range and long-range time limitations.

The current theoretical framework might be too simplistic to ac-

curately describe the real world. Yet, important advances in under-

standing of scaling in finance have been made in recent years. In the

next section, the asymptotic power laws that may be recovered from

the financial time series are discussed. In Section 8.3, the recent

developments including the multifractal approach are outlined.

8.2 POWER LAWS IN FINANCIAL DATA

The importance of long-range dependencies in the financial time series

was shown first by B. Mandelbrot [6]. Using the R/S analysis (see Section

6.1), Mandelbrot and others have found multiple deviations of the

empirical probability distributions from the normal distribution [7].

Early research of universality in the financial time series [6] was

based on the stable distributions (see Section 3.3). This approach,

however, has fallen out of favor because the stable distributions have

infinite volatility, which is unacceptable for many financial applica-

tions [8]. The truncated Levy flights that satisfy the requirement for

finite volatility have been used as a way around this problem [2, 9, 10].

One disadvantage of the truncated Levy flights is that the truncating

89

Scaling in Financial Time Series

distance yields an additional fitting parameter. More importantly,

the recent research by H. Stanley and others indicates that the asymp-

totic probability distributions of several typical financial time series

resemble the power law with the index a close to three [11â€“13]. This

means that the probability distributions examined by Stanleyâ€™s team

are not stable at all (recall that the stable distributions satisfy the

condition 0 < a 2). Let us provide more details about these interest-

ing findings.

In [11], returns of the S&P 500 index were studied for the period

1984â€“1996 with the time scales Dt varying from 1 minute to 1 month.

It was found that the probability distributions at Dt < 4 days were

consistent with the power-law asymptotic behavior with the index

a % 3. At Dt > 4 days, the distributions slowly converge to the

normal distribution. Similar results were obtained for daily returns

of the NIKKEI index and the Hang-Seng index. These results are

complemented by another work [12] where the returns of several

thousand U.S. companies were analyzed for Dt in the range from

five minutes to about four years. It was found that the returns of

individual companies at Dt < 16 days are also described with the

power-law distribution having the index a % 3. At longer Dt, the

probability distributions slowly approach the normal form. It was

also shown that the probability distributions of the S&P 500 index

and of individual companies have the same asymptotic behavior due

to the strong cross-correlations of the companiesâ€™ returns. When these

cross-correlations were destroyed with randomization of the time

series, the probability distributions converged to normal at a much

faster pace.

The theoretical model offered in [13] may provide some explan-

ation to the power-law distribution of returns with the index a % 3.

This model is based on two observations: (a) the distribution of the

trading volumes obeys the power law with an index of about 1.5; and

(b) the distribution of the number of trades is a power law with an

index of about three (in fact, it is close to 3.4). Two assumptions were

made to derive the index a of three. First, it was assumed that the

price movements were caused primarily by the activity of large mutual

funds whose size distribution is the power law with index of one (so-

called Zipfâ€™s law [4]). In addition, it was assumed that the mutual fund

managers trade in an optimal way.

90 Scaling in Financial Time Series

Another model that generates the power law distributions is the

stochastic Lotka-Volterra system (see [14] and references therein).

The generic Lotka-Volterra system is used for describing different

phenomena, particularly the population dynamics with the predator-

prey interactions. For our discussion, it is important that some agent-

based models of financial markets (see Chapter 12) can be reduced to

the Lotka-Volterra system [15]. The discrete Lotka-Volterra system

has the form

1X N

wi (t) (8:2:1)

wi (t Ã¾ 1) Â¼ l(t)wi (t) Ã€ aW(t) Ã€ bwi (t)W(t), W(t) Â¼

N iÂ¼1

where wi is an individual characteristic (e.g., wealth of an investor i;

i Â¼ 1, 2, . . . , N), a and b are the model parameters, and l(t) is a

random variable. The components of this system evolve spontan-

eously into the power law distribution f(w, t) $ wÃ€(1Ã¾a) . In the

mean time, evolution of W(t) exhibits intermittent fluctuations that

can be parameterized using the truncated Levy distribution with the

same index a [14].

Seeking universal properties of the financial market crashes is

another interesting problem explored by Sornette and others (see

[16] for details). The main idea here is that financial crashes are

caused by collective trader behavior (dumping stocks in panic),

which resembles the critical phenomena in hierarchical systems.

Within this analogy, the asymptotic behavior of the asset price S(t)

has the log-periodic form

S(t) Â¼ A Ã¾ B(tc Ã€ t)a {1 Ã¾ C cos [w ln (tc Ã€ t) Ã€ w]} (8:2:2)

where tc is the crash time; A, B, C, w, a, and w are the fitting

parameters. There has been some success in describing several market

crashes with the log-periodic asymptotes [16]. Criticism of this ap-

proach is given in [17] and references therein.

8.3 NEW DEVELOPMENTS

So, do the findings listed in the preceding section solve the problem

of scaling in finance? This remains to be seen. First, B. LeBaron has

shown how the price distributions that seem to have the power-law

form can be generated by a mix of the normal distributions with

91

Scaling in Financial Time Series

different time scales [18]. In this work, the daily returns are assumed

to have the form

R(t) Â¼ exp [gx(t) Ã¾ m]e(t) (8:3:1)

where e(t) is an independent random normal variable with zero mean

and unit variance. The function x(t) is the sum of three processes with

different characteristic times

x(t) Â¼ a1 y1 (t) Ã¾ a2 y2 (t) Ã¾ a3 y3 (t) (8:3:2)

The first process y1 (t) is an AR(1) process

y1 (t Ã¾ 1) Â¼ r1 y1 (t) Ã¾ Z1 (t Ã¾ 1) (8:3:3)

where r1 Â¼ 0:999 and Z1 (t) is an independent Gaussian adjusted so

that var[y1 (t)] Â¼ 1. While AR(1) yields exponential decay, the chosen

value of r1 gives a long-range half-life of about 2.7 years. Similarly,

y2 (t Ã¾ 1) Â¼ r2 y2 (t) Ã¾ Z2 (t Ã¾ 1) (8:3:4)

where Z2 (t) is an independent Gaussian adjusted so that

var[y2 (t)] Â¼ 1. The chosen value r2 Â¼ 0:95 gives a half-life of about

2.5 weeks. The process y3 (t) is an independent Gaussian with unit

variance and zero mean, which retains volatility shock for one day.

The normalization rule is applied to the coefficients ai

a1 2 Ã¾ a2 2 Ã¾ a3 2 Â¼ 1: (8:3:5)

The parameters a1 , a2 , g, and m are chosen to adjust the empirical data.

This model was used for analysis of the Dow returns for 100 years

(from 1900 to 2000). The surprising outcome of this analysis is retrieval

of the power law with the index in the range of 2.98 to 3.33 for the data

aggregation ranges of 1 to 20 days. Then there are generic comments by

T. Lux on spurious scaling laws that may be extracted from finite

financial data samples [19]. Some reservation has also been expressed

about the graphical inference method widely used in the empirical

research. In this method, the linear regression equations are recovered

from the log - log plots. While such an approach may provide correct

asymptotes, at times it does not stand up to more rigorous statistical

hypothesis testing. A case in point is the distribution in the form

f(x) Â¼ xÃ€a L(x) (8:3:6)

where L(x) is a slowly-varying function that determines behavior of

the distribution in the short-range region. Obviously, the â€˜â€˜universalâ€™â€™

92 Scaling in Financial Time Series

scaling exponent a Â¼ Ã€log [f(x)]= log (x) is as accurate as L(x) is close

to a constant. This problem is relevant also to the multifractal scaling

analysis that has become another â€˜â€˜hotâ€™â€™ direction in the field.

The multifractal patterns have been found in several financial time

series (see, e.g., [20, 21] and references therein). The multifractal

framework has been further advanced by Mandelbrot and others.

They proposed compound stochastic process in which a multifractal

cascade is used for time transformations [22]. Namely, it was assumed

that the price returns R(t) are described as

R(t) Â¼ BH [u(t)] (8:3:7)

where BH [] is the fractional Brownian motion with index H and u(t) is

a distribution function of multifractal measure (see Section 6.2). Both

stochastic components of the compound process are assumed inde-

pendent. The function u(t) has a sense of â€˜â€˜trading timeâ€™â€™ that reflects

intensity of the trading process. Current research in this direction

shows some promising results [23â€“26]. In particular, it was shown

that both the binomial cascade and the lognormal cascade embedded

into the Wiener process (i.e., into BH [] with H Â¼ 0:5) may yield a more

accurate description of several financial time series than the GARCH

model [23]. Nevertheless, this chapter remains â€˜â€˜unfinishedâ€™â€™ as new

findings in empirical research continue to pose new challenges for

theoreticians.

8.4 REFERENCES FOR FURTHER READING

Early research of scaling in finance is described in [2, 6, 7, 9, 17].

For recent findings in this field, readers may consult [10â€“13, 23â€“26].

8.5 EXERCISES

**1. Verify how a sum of Gaussians can reproduce a distribution

with the power-law tails in the spirit of [18].

**2. Discuss the recent polemics on the power-law tails of stock

prices [27â€“29].

**3. Discuss the scaling properties of financial time series reported

in [30].

Chapter 9

Option Pricing

This chapter begins with an introduction of the notion of financial

derivative in Section 9.1. The general properties of the stock options

are described in Section 9.2. Furthermore, the option pricing theory is

presented using two approaches: the method of the binomial trees

(Section 9.3) and the classical Black-Scholes theory (Section 9.4).

A paradox related to the arbitrage free portfolio paradigm on which

the Black-Scholes theory is based is described in the Appendix section.

9.1 FINANCIAL DERIVATIVES

In finance, derivatives1 are the instruments whose price depends

on the value of another (underlying) asset [1]. In particular, the

stock option is a derivative whose price depends on the underlying

stock price. Derivatives have also been used for many other assets,

including but not limited to commodities (e.g., cattle, lumber,

copper), Treasury bonds, and currencies.

An example of a simple derivative is a forward contract that obliges

its owner to buy or sell a certain amount of the underlying asset at a

specified price (so-called forward price or delivery price) on a specified

date (delivery date or maturity). The party involved in a contract as a

buyer is said to have a long position, while a seller is said to have a short

position. A forward contract is settled at maturity when the seller

93

94 Option Pricing

delivers the asset to the buyer and the buyer pays the cash amount at

the delivery price. At maturity, the current (spot) asset price, ST , may

differ from the delivery price, K. Then the payoff from the long

position is ST Ã€ K and the payoff from the short position is K Ã€ ST .

Future contracts are the forward contracts that are traded on

organized exchanges, such as the Chicago Board of Trade (CBOT)

and the Chicago Mercantile Exchange (CME). The exchanges deter-

mine the standardized amounts of traded assets, delivery dates, and

the transaction protocols.

In contrast to the forward and future contracts, options give an

option holder the right to trade an underlying asset rather than the

obligation to do this. In particular, the call option gives its holder the

right to buy the underlying asset at a specific price (so-called exercise

price or strike price) by a certain date (expiration date or maturity).

The put option gives its holder the right to sell the underlying asset at a

strike price by an expiration date. Two basic option types are the

European options and the American options.2 The European options

can be exercised only on the expiration date while the American

options can be exercised any time up to the expiration date. Most of

the current trading options are American. Yet, it is often easier to

analyze the European options and use the results for deriving proper-

ties of the corresponding American options.

The option pricing theory has been an object of intensive research

since the pioneering works of Black, Merton, and Scholes in the

1970s. Still, as we shall see, it poses many challenges.

9.2 GENERAL PROPERTIES OF STOCK OPTIONS

The stock option price is determined with six factors:

Current stock price, S

.

Strike price, K

.

Time to maturity, T

.

Stock price volatility, s

.

Risk-free interest rate,3 r

.

Dividends paid during the life of the option, D.

.

Let us discuss how each of these factors affects the option price

providing all other factors are fixed. Longer maturity time increases

95

Option Pricing

the value of an American option since its holders have more time to

exercise it with profit. Note that this is not true for a European option

that can be exercised only at maturity date. All other factors, how-

ever, affect the American and European options in similar ways.

The effects of the stock price and the strike price are opposite for

call options and put options. Namely, payoff of a call option increases

while payoff of a put option decreases with rising difference between

the stock price and the strike price.

Growing volatility increases the value of both call options and put

options: it yields better chances to exercise them with higher payoff.

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