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U.U.D.M. Project Report 2010:18 Examensarbete i matematik, 30 hp Handledare: Stamatios C. Nicolis Examinator: David J.T. Sumpter Maj 2010 Department of Mathematics Uppsala University Stochastic Models of Stock Market Dynamics Na Li

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Page 1: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios

U.U.D.M. Project Report 2010:18

Examensarbete i matematik, 30 hpHandledare: Stamatios C. Nicolis Examinator: David J.T. Sumpter

Maj 2010

Department of MathematicsUppsala University

Stochastic Models of Stock Market Dynamics

Na Li

Page 2: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios
Page 3: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios

Stochastic Models of Stock

Market Dynamics

Master in Computational Science

Na Li

Supervised by

David J. T. Sumpter

Stamatios C. Nicolis

June 17, 2010

Mathematics Department, Uppsala UniversitySE-751 05 Uppsala, Sweden

Page 4: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios

Abstract

Stock market crashes in the global econmoy have repeatedly made the headlines recently,

and are something that cannot be ignored by investors and traders. In this report, the

classical economic models and ’econo-physics’ models are first discussed before using a

novel equation free approach to capture the salient features of fluctuations of the Dow

Jones Industrial Average over the past 80 years. A stochastic model is then proposed

and analyzed in details which reproduce the dynamics of the stock market over different

time scales, reconciliating therefore the classical and ”eono-physics” views. The concept

of drawdowns is also introduced in order to relate crashes to extreme events and conse-

quently to outliers. Major stock indices are then analyzed and confronted to our model.

A large part of the physics literature views stock market as complex adaptive system, we

also studied the non-linear based on the ’econo-physics’ properties to compare with the

dynamical model. Finally, we discussed general stochastic models with different types

of noise to show that noise play a crucial role in the dynamics.

i

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Contents

Abstract i

Contents ii

1 Introduction 1

1.1 Standard Model-Diffusion for Stock Price . . . . . . . . . . . . . . . . . 2

1.2 Why the Standard Model might be Wrong . . . . . . . . . . . . . . . . . 4

1.3 Challenge to the Traditional Finance Theory . . . . . . . . . . . . . . . 7

2 Finding a Empirical Model for Returns 8

2.1 Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Data 12

4 Model Description 15

4.1 Model State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Fokker-Planck Analysis in Ito Interpretation . . . . . . . . . . . . . . . . 15

4.3 Trying the Model to Empirical Data . . . . . . . . . . . . . . . . . . . . 17

4.4 Other Major Stock Market Indices . . . . . . . . . . . . . . . . . . . . . 19

4.5 Individual Shares of Dow Jones Industrial Average Index . . . . . . . . 21

5 Study Drawdown and Drawup Distributions of Stock Market Indices 26

5.1 Drawdown and Drawup . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2 Major Financial Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Study Non-Linear Model of Stock Market Return 29

6.1 Non-Linear Model of Stock Market Return . . . . . . . . . . . . . . . . 29

6.2 Fokker-Planck Analysis in Ito Interpretation . . . . . . . . . . . . . . . . 30

7 General Stochastic Models 32

ii

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Contents iii

7.1 General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.2 Linear Model with Additive Noise . . . . . . . . . . . . . . . . . . . . . 34

7.3 Linear Model with Multiplicative Noise . . . . . . . . . . . . . . . . . . 35

7.4 Nonlinear Model with Additive Noise . . . . . . . . . . . . . . . . . . . . 36

7.5 Nonlinear Model with Multiplicative Noise . . . . . . . . . . . . . . . . . 37

7.6 Linear Model with Additive and Multiplicative Noise . . . . . . . . . . . 38

8 Conclusion 40

Acknowledgements 42

Bibliography 43

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Chapter 1

Introduction

Synopsis

On October 29, 1929, the stock market dropped 11.5%, bringing the

Dow Jones Industrial Average 39.6% off its high and leading to a total of

14 billion dollars of loss. On October 19, 1987, the Dow Jones Industrial

Average dropped 22.6% in a single trading day. This was a drop of 36.7%

from its high on August 25, 1987. During this crash, 1/2 trillion dollars of

wealth were erased. From September 2000 to January 2, 2001, the Nasdaq

dropped 45.9%. A total of 8 trillion dollars of wealth was lost in the crash of

2000. These disasters attracted a great deal of attention and currently many

studies are devoted to stock market crashes.

The classical views of a Brownian motion model under the efficient market

hypothesis holds that market returns are independent of each other and mar-

ket crashes operate at the shortest time scales. In contrast ’econo-physicists’

emphasize that market returns follow long range correlations and scaling laws

over time scales spanning several orders of magnitude with a Levy stable dis-

tribution and provide on this basis an explanation on how crash may arise.

Over the years these two competing Schools have remained contentious. Re-

cently a new perspective of financial market the behavioral finance raised

that concerned the market uncertain comes from the human judgment and

decision-marking.

Starting from the definition of stock market return we discuss in this

chapter two different theoretic frameworks: standard model under hypoth-

esis of efficient market and ’econophysics’ views proposed by fractal market

hypothesis. A brief introduction of the behavioral finance will be given at

the end of this chapter.

1

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Chapter 1. Introduction 2

1.1 Standard Model-Diffusion for Stock Price

”Efficient Market Hypothesis” (EMH) is a controversial topic in the field of Finance.

It can be traced to the work of French mathematician Louis Bachelier on the stock

price fluctuation in the early 1900. On his PhD thesis The Theory of Speculation [1],

he assumed that the stock price dynamics follows a Brownian motion without drift

(resulting in a Gaussian stock prices distribution). The major flaw of his hypothesis is

that one allows for negative value of stock prices. Still, five years before Einstein [2], he

derived most of the theory of diffusion process opening the way to the works of Wiener

[3], Kolmogorov [4] or Ito [5]. For many, Bachelier is considered as the father of financial

mathematics and a pioneer of the ”Efficient Market Hypothesis”.

The EMH is closely related to the random walk hypothesis and is therefore an important

assumption in financial mathematics. The random walk hypothesis stating that the

prices of the stock market cannot be predicted because economists believe that stock

prices are completely random because of the efficiency of the market. There are other

economists who believe that the market is predictable. These people believe that prices

may move in trends and that the study of past prices can be used to forecast future

price direction. More than half century later, Bachelier’s idea of the market was taken

out of the grave by Samuelson [6] who provided evidence that the capital price is a

martingale – in other words, that future earning is unpredictable. Samuelson modified

the Bachelier model (also known as the arithmetic Brownian motion model) assuming

that the return rates, instead of the stock prices, follow a Brownian motion (also known

as the geometric Brownian motion model or the economic Brownian motion model). As

a result of the geometric random walk the stock prices follow a log-normal distribution,

instead of a normal distribution as assumed by Bachelier (1900). Application of the

random walk model to the logged series implies that the forecast for the next value

of the original series will equal the previous value plus a constant percentage increase.

Geometric random walk is useful in modeling stock prices over times. This view also

contributed to Fischer Black and Myron Scholes (1973) formulation of the Option Price

model known as the famous Black-Scholes mode [7]. This model has a widespread use

in finance today.

At the same period as Samuelson, Fama [8] elaborated on the EMH, arguing that if

the market can fully and quickly reflect all the available information, then it is efficient.

In this highly competitive and rational market, no one can forecast the price changes

from the past information, the price changes will depend on future information, in other

words, the market has no memory. At first, some evidences seemed to support this view.

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Chapter 1. Introduction 3

Today it is realized that the fully and effective market is unrealistic : price fluctuations

sometimes cannot immediately reflect all the information, traders and investors are

probably not relying only on the current price when they are doing transactions.

And the price and return series independence to each other also produced important

assumption about statistic properties in financial market. The return series is normal

distribution or log-normal distribution. But nowadays the random walk hypothesis has

been refute by extensive empirical evidence [9]. The random walk remains the most

important and basic assumption in the finance community.

Based on efficient market hypothesis theory, the change of an asset price can be written

as:

y(t+1) = y(t)(1 + λ+ σε(t)) (1.1)

where y(t) and y(t+1) are the stock market index at time t and t + 1, λ is a drift and

the average growth rate of the market, σ is the volatility of the stock price and ε(t) is a

white noise process.

On another hand, the relative stock market return x(t) is defined as:

x(t) =y(t+1) − y(t)

y(t)=y(t+1)

y(t)− 1 (1.2)

Substituting (1.1) into (1.2) we can get the standard model for market return:

x(t) = λ+ σε(t)

ory(t+1)

y(t)− 1 = λ+ σε(t) (1.3)

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Chapter 1. Introduction 4

1.2 Why the Standard Model might be Wrong

Unfortunately, the EMH has lots of limitations to explain the financial market. It not

be supported by events such as the US stock market crash in October 1987. The rea-

sons are the axiom of expected price formation based on rational, all-knowing investors

and traders that they can response the information with linear pattern, they can im-

mediately respond to a given information and their action will not be misunderstood,

missed or delayed. Moreover, the EMH assumes that trader can not use information get

benefit from the market. As a corollary stock prices are independent to each other, the

price fluctuation follow the Markov process is random walks, the distribution is normal

distribution(or log-normal distribution). However, the real stock markets is not simple,

ordered which is chaos and complex.

Although the works cited in Sect.1.1 turned out to be in agreement with many data,

they couldn’t capture market dynamics at some historical periods. More specifically, the

heavy tails in the stock prices distribution reflecting the crashes couldn’t be described

with a Gaussian Distribution. In 1962 Mandelbrot [10] first proposed that these tails

follow a power law, thereby casting doubts on the validity of the classical model. In stock

market, volatility is not constant and behaves stochastically. The shortcoming of the

classical model is dramatically illustrated by i.e. a normal distribution for log-returns,

completely missing the observed tail behavior.

Most approaches to explain crashes investigate for possible mechanisms or effects that

operate at very short time scales (hours, days or weeks at most) [6][8]. Another different

view is that the underlying cause of a crash must be investigated months or years before

it happens, in the progressive increasing build-up of market cooperatively or effective

interactions between investors, often translated into accelerating ascent of the market

price (the bubble) [11].

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Chapter 1. Introduction 5

Figure 1.1: Distribution of daily returns for the DJIA and the Nasdaq index for the

period Jan. 2nd, 1990 till Sept. 29, 2000.

Crashes are different from normal market days. A traditional way to characterize them

is to look at the frequency distribution of the daily returns. Fig.1.1 [11] shows the

distribution of daily returns of the DJIA and of the Nasdaq index for the period January

2nd, 1990 till September 29, 2000. It shows the distribution of daily returns has fatter

tails. This connected with ’econo-physicists’ emphasize the market returns follow scaling

laws consistent over several orders of magnitude with a Levy stable distribution. One

important characteristic of the Levy-stable distribution is that has a power law tail with

exponent less than 3, thus predicting the very large fluctuations, crashes and rallies in

the stock market.

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Chapter 1. Introduction 6

Figure 1.2: Normalized natural logarithm of the cumulative distribution of drawdowns

and of the complementary cumulative distribution of drawups for the Dow Jones Indus-

trial Average index (US stock market). The two continuous lines show the fits of these

two distributions with the stretched exponential distribution.

Another way is to analyze frequency distribution of drawdown Fig.1.2 [11] and shows

that the large financial crashes are outliers [12]. A drawdown is defined as a persistent

decrease in the price over consecutive days. A drawdown is thus the cumulative loss from

the last maximum to the next minimum of the price. Their distribution thus captures

the way successive drops can influence each other and construct in this way a persistent

process, which is not measured by the distribution of returns. In fact, distribution

fatter tail phenomenon shows high degree autocorrelation of financial time series reflect

the stock market’s complexity. The fluctuations in returns have similarity and long

memory characteristics base on the Dow Jones Index Return for short, medium and

long time. The higher order correlations and dependence that may caused the largest

drawdown. The main mechanisms leading to positive feedbacks, i.e. self-reinforcement,

such as investors and traders the imitative and herding behavior (when the individual

can not understand the operational status of the market, they tend to follow the others.)

Positive feedbacks provide the fuel for the development of speculative bubbles, preparing

the instability for a major crash.

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Chapter 1. Introduction 7

1.3 Challenge to the Traditional Finance Theory

With development of the financial market, the traditional theory of finance is facing chal-

lenges. EMH assumption for the rational investors are raised more and more doubts.

In 20th century, a new theory Behavioral Finance came out which attempts to bet-

ter describe the decision-making behavior of investors. There are some doubts to the

assumption of the rational investors: [13–19]

• Traders are not immediately responding when they receive the information. They

usually respond to the information after information has been confirmed.

• Only a few traders are rational. There is large evidence shows that most of traders

follow each other to make decision in the stock market.

• Traders are not avoiding risk all times.

• Sometimes the traders are over confident, expecting more to their own information.

This belong to the subjective biased.

Behavioral finance theory and observation of a large number of empirical research re-

sults show that: traders are not always rational for decision making, they usually have

cognitive bias in reality, this bias usually affect the behavior of the traders. Behavioral

finance study behavior of the traders and the link to the trader’s psychological process.

Different from the traditional theory, behavioral finance shows: traders are not rational

but are human beings. Large number of empirical evidence shows, first, that traders can

not rational and unbiased response the information. Second, traders are heterogeneous.

Because different traders hold different information, have different investment cost, use

different investment portfolio and also investment period is different. Traders in the

market are not homogenous. Third, traders do not want to loss, sometimes they want

to take risk. They choose take risk to protect loss. Fourth, trader’s psychological char-

acteristics are different when they face risk. Someone would like to take adventurous,

someone would like to keep conservative. Fifth, the market is not efficient.

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Chapter 2

Finding a Empirical Model for

Returns

2.1 Model I

In this chapter I discuss two basic discrete time models of stock market dynamics which

incorporate positive feedback.

Positive feedback occur in the stock market. If a stock is more welcome than other

stock means more people would like to buy, interaction between the traders, more and

more people will buy this stock. This loop make large fluctuation in stock market. The

probability distribution is leptokurtosis and with heavy tailed.

The idea that the stock market can not be predicted has been refuted by extensive

empirical evidence. Indeed, in oder to decide to buy or to sell, it is useful study the

origin of the price changes- price returns. From the time series of the price returns we

found the return large fluctuation usually follow the larger fluctuation than the last time.

The price change has correlation relationship. From the behavior view, crashes based

on the ides of ”rational imitation” or ”herding”: traders copy the buy and sell decisions

of their peers leading to a build up of correlations with the market that may sometimes

accumulate as a crash [21–25] . We will then be considered by another variable of interest

namely returns rates instead of only predicted the stock price for the classical model.

And investors un-immediately actions will as memory reflects on the returns series. The

correlation occured between returns on one day and returns on the previous day. In

order to capture the behavior stochastic the drift term will not be constant. We assume

a linear relationship between returns on consecutive days and add Markov noise process

generated by the environment to arrive the Model I.

8

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Chapter 2. Finding a Empirical Model for Returns 9

The relative return x(t+1) at time t+ 1 is defined as:

x(t+1) = (1− a)x(t) + ε(t) (2.1)

Here (1-a) is a drift and ε(t) is a white noise process.

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.110

−4

10−3

10−2

10−1

returns

log(F

requ

en

cy)

Figure 2.1: Probability Distribution for Model I with parameter a=0.1

The return probability distribution obtained from Model I show in Fig.2.1. Although

Fig.2.1 shows the return series still is log-normal distribution it can reflects the auto-

correlation between daily returns.

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Chapter 2. Finding a Empirical Model for Returns 10

2.2 Model II

The critical assumption for efficient market is traders are rational. They can accurate,

immediately and unbiased reflect to the received the information. They response to the

information is linear pattern. The real stock market not all the traders are rational,

there are also include some noise traders. A noise trader also known informally as idiot

trader is described in the literature of financial research as a stock trader whose decisions

to buy, sell, or hold are irrational and erratic. The presence of noise traders in financial

markets can then cause prices and risk levels to diverge from expected levels even if

all other traders are rational. Those noise traders one of the reason caused the stock

market internal instability. For model II we assume the a linear relationship between

returns on consecutive days and multiplicative noise that generated by internal stock

market noise traders. The multiplicative noise is valid on the long time scales compare

with the additive only relevant for short time scales.

The relative return x(t+1) at time t+ 1 is defined as:

x(t+1) = (1− a)x(t) + bx(t)ε(t) (2.2)

Here (1-a) is a drift, b is the volatility of the return rates and ε(t) is a white noise

process.

−0.6 −0.4 −0.2 0 0.2 0.4 0.610

−4

10−3

10−2

10−1

returns

log(F

requency)

Figure 2.2: Probability Distribution for Model II with parameter a=0.1, b=-0.9

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Chapter 2. Finding a Empirical Model for Returns 11

As can be seen, the return probability distribution obtained from Model II show in

Fig.2.2 have sharply peaked. From large evidence, it is clear that the return series of the

observed data deviate from the normal distribution, the relatively large value of kurtosis

statistics suggests that the underlying data are fat tailed and sharply peaked about the

mean when compared with the normal distribution.

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Chapter 3

Data

We investigate the Dow Jones Industrial Average index open price from October 1st

1928 to September 11th 2009. There are total 20327 trade days. The Index series and

Return series as show in Fig.3.1. From Fig.3.1b, we can see a line close to year 1990 have

larger motions than other. In fact it represents the huge disaster-stock market crash

occur in 1987. If we see the index series in Fig.3.1 a, there is only a small drop around

year 1990. Studying the return series it is easier to detect the stock market fluctuation.

1928 1950 1970 1990 2010 20300

5000

10000

15000

Year

Ind

ex v

alu

e

a

(a)

1928 1950 1970 1990 2010 2030−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Year

Re

turn

b

(b)

Figure 3.1: Dow Jones Industrial Average from January 1st 1929 to September 11th

2009 index open price series in Figure (a) and return series in Figure (b).

In recent years the ”equation-free” approach has been proposed as a general framework

for developing multiscale methods for efficiently capturing the macroscale behavior of

a system using only the microscale models different from the traditional modeling ap-

proaches [26–28]. For complex system, a persistent feature is the emergence of macro-

scopic coherent behavior from the interactions of microscopic agents such as molecules,

cells, or individuals in a population. It is needed to develop an approach that allows as to

12

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Chapter 3. Data 13

perform macroscopic tasks acting on the microscopic models directly at fine scale.

We are interested on two main variables, namely the mean and standard deviation of the

returns at time t and on how they depend upon the returns on the previous time.

f(x(t)) = E[x(t+ 1)|x(t)] (3.1)

s(x(t), x(t+ 1)) =

√(x(t+ 1)− x(t)− f(x(t))

)2

(3.2)

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

a

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

b

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

c

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

d

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

e

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

f

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

g

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

h

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

i

Figure 3.2: f(x)versus x(t), standard deviation of the drift(s(x)) versus x(t) and proba-

bility distribution of daily, weekly and monthly returns. Owing to highly inhomogeneous

sampling of the data, the convention adopted for the binning is: for negative values−0.1d

where d varies from 1 to 2.8 by steps of 0.1 and symmetrically for positive x′s.

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Chapter 3. Data 14

Fig.3.2 show the mean, standard deviation and probability distribution of daily, weekly

and monthly returns for consecutive time steps. From the Fig.3.2, we can see the stock

market for different time scale has below features:

• The x(t+1) versus x(t) is linear depend between -0.05 and 0.05 in x-axle for daily

return. It reflected the autocorrelation between daily returns and this correlations

become weaken over weeks and months.

• The standard deviations versus x(t) displays a V-like shape for daily return, each

branch of which increases in roughly linear fashion with the size of the previous

return. Notice that the two branches are asymmetries that reflecting the well-

known leverage effect or volatility asymmetry [29, 30]. Volatility seems to rise

when a stock’s price drops and fall when the stock goes up.

• The distribution of daily return is leptokurtosis and with heavy tailed in the daily

distribution. Weekly and monthly return distribution gradually converge to the

normal distribution comparing to the daily one.

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Chapter 4

Model Description

4.1 Model State

From last chapter taken Dow Jones data over 80 years of trading, it shows for small daily

returns the f(x(t)) depends in the mean linearly on x(t) that reflect the autocorrelation

between daily returns. And s(x(t), x(t+1)) displays a V-like dependent on x(t). Consider

above features of the data, for the dynamic model we assume a linear relationship

between returns on consecutive days. We incorporate variability first by modeling the

fluctuations in the slope of this linear law as a Markov noise process F1(t) and second

by allowing for an extra background variability generated by environment in which

the actors are embedded and modeled as an additive Markov noise F2(t). Expect this

model display the probability distribution of daily returns is sharply peaked and have

fat tails.

We write the model on continuous time stochastic differential equation is:

dx

dt= −α(1 + F1(t))x+ F2(t) (4.1)

where F1(t) has the variance q21 and F2(t) has the variance q22.

4.2 Fokker-Planck Analysis in Ito Interpretation

Eq.(4.1) defines a diffusion process in the x-space, whose probability distribution obeys

to a Fokker-Planck type equation [29]. Fokker-Planck type equation is commonly used

in the physics to find the probability distribution function. The Fokker-Planck type

equation is a deterministic partial differential equation which tells how the probabil-

ity distribution function evolves in time. We adopt the Ito interpretation to analyse

Fokker-Planck equation as widely used in the finance literature. We recall that this

interpretation implies that the continuous time description of Eq.(4.1) can be regarded

15

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Chapter 4. Model Description 16

as the limit of a discrete time one whereas in the Stratonovich interpreation used widely

in the physics limit of a colored noise whose correlation time goes to zero [31, 32].

One can write the Fokker-Planck equation as:

∂P

∂t=

∂xαxP +

12∂

∂x2(q21α

2x2 + q22)P (4.2)

where the first term is the dirft and the second one is the diffusion.

The Ito drift-diffusion process can be written as:

dxt = a(x, t)dt+ b(x, t)dw(t) (4.3)

We can use get the process for X(t)n:

dxnt = [nXn−1a(x, t) +12n(n− 1)Xn−2b2(X, t)]dt+ nXn−1b(x, t)dW (t) (4.4)

The ordinary differential equation for the expectation of X(t)n:

dE[xn(t)]dt

= nE[Xn−1a(X, t)] +12n(n− 1)E[Xn−2b2(X, t)] (4.5)

We can use Fokker-Planck equation to calculate some import moments in statistic to

exact parameter value of the dynamic model. In statistician, the mean is the first

moment and is also referred to as the expected value. The variance is the second central

moment, and kurtosis is rescaled fourth central moment.

4.2.1 First moment

when n=1dx

dt= −αx (4.6)

Stable if α=0, xs=0

4.2.2 Second moment

when n=2dx2

dt= −2α(1− αq21

2)x2 + q22 (4.7)

Stable if αq21<0

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Chapter 4. Model Description 17

4.2.3 Fourth moment

when n=4dx4

dt= −4α(1− 3αq21

2)x4 + 6q22x2 (4.8)

Stable if αq21<2/3

4.2.4 Steady-state solution

The steady state solution:

P ≈ exp

[− 2αq22

∫dx

(1 + αq21)x

1 +(q21α2

q22

)x2

]

≈ exp

[− α

q22

1 + q21αq21α

2

q22

ln

(1 +

q21α2

q22x2

)]

≈[1 +

q21α2

q22x2

]− 1+q21α

αq12

(4.9)

This equation is power law form interpolation. If (q21α2/q22)>>1 this distribution has

a power law tail with exponent (−(1 + q21α)/αq12), otherwise it shows the bell shaped.

And we can use this equation to capture the tail feature for the financial data. Because

the power law tail always can used to predicting the very large the fluctuations, crashes

and rallies in the stock market.

4.3 Trying the Model to Empirical Data

Now we summarize the main predictions of the model and compare them with the data.

Because the parameter α is related the slope of the return, q1 is related to the stock

market internal fluctuation and q2 is related to the variability generated by the external

environment. So the first step is fit the these three model parameters α, q1, q2. When

fitting with the data, from the moment 1 equation we can get the slope and set the

equation for moment 2 and 4 to zero and extract the parameters value from the real

data.

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Chapter 4. Model Description 18

slope = exp(−α) (4.10)

−2(

1− α

2q21

)x2 +

q22α

= 0 (4.11a)

−4(

1− 3α2q21

)x4 +

6q22αx2 = 0 (4.11b)

Fig.4.1 shows that the dynamical model worked well from daily to monthly based on

the parameters value have been fitted with the daily data. In the figures, data in red,

model in black, the dashed lines is the predicted distribution of returns.

The model gives a good match over 80 years of the data and capture a number of features

from the data:

• Correlations in returns seen over days disappear over weeks and months;

• Correlations in the variance of the returns weaken but remain visible as the time

scale is lengthened.

• The return distribution transforms from having a steep point and long tail for days

to a wide distribution for moths.

Page 25: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios

Chapter 4. Model Description 19

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

a

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

b

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log

(Fre

que

ncy)

c

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

d

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

e

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log

(Fre

que

ncy)

f

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

g

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

h

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

i

Figure 4.1: f(x)versus x(t), standard deviation of the drift(s(x)) versus x(t) and prob-

ability distribution of daily returns. Parameter values used on the basis of the daily

data and equal to α=3.0095, q21=0.2009, and q22=0.0005. The predicted distribution of

returns (dashed lines) are fitted with the actual values of the parameters: α=3.0095,

q21=0.2009, and q22=0.0005 for daily returns; α=5.6787, q21=0.0914, and q22=0.0054 for

weekly returns; α=2.3547, q21=0.2248, and q22=0.0102 for monthly returns. Binning same

as Fig.3.2.

4.4 Other Major Stock Market Indices

Fig.4.2 shows the model fits well with the other major stock market indices for daily

returns. Data in red, model in black, the dashed lines is the predicted distribution of

returns.

Page 26: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios

Chapter 4. Model Description 20

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

a

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

b

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log

(Fre

que

ncy)

c

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

d

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

e

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log

(Fre

que

ncy)

f

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

g

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

h

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

i

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

j

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

k

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

l

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

m

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

n

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

o

Figure 4.2: (a)-(c)S&P 500(USA) from Jan 3,1950 to Jan 31,2010, (d)-(f)Nasdaq(USA)

from Feb 5, 1971 to Jan 31, 2010, (g)-(i)FTSE 100(UK) from Apr 2, 1984 to Jan 31, 2010,

(j)-(l)Nikkei 225(Japan) from Jan 4, 1984 to Jan 31, 2010. The parameter values are fit-

ted with the actual data: α=2.9278, q21=0.2093, and q22=0.0004 for S&P 500, α=3.6468,

q21=0.1552, and q22=0.0009 for Nasdaq, α=2.6207, q21=0.0916, and q22=0.0005 for FTSE

100, α=2.8902, q21=0.1729, and q22=0.0008 for Nikkei 225. α=3.4229, q21=0.1816, and

q22=0.0015 for Hang Seng. Binning same as Fig.3.2.

Page 27: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios

Chapter 4. Model Description 21

4.5 Individual Shares of Dow Jones Industrial Average In-

dex

We selected ten individual stocks from Dow Jones Industrial Average index (Here GE

is the oldest stock and IBM is the highest price stock now in Dow Jones) to fit with

dynamic model. From Table 4.1, we can see half of them got negative slope. Normally,

the the individual shares are different from the major indices, it usually have several

bonus events during the year. And we compared each individual stock return fluctuation

with their stock splits and dividend events, most of time are corresponding to each

other.

The Dow Jones Industrial Average is a price-weighted index that is calculated by dividing

the sum of the prices of the 30 component stocks. When one individual stock have stock

splits, dividend and so on, the Dow Jones will change the divisor to keep the index value

consistent. The dynamical model fails to predict the negative slope, although it does

reproduce the overall return distribution. We could correct this deficiency by simply

changing the sign of x in the model at the end of each day, but this is a somewhat

clumsy solution. Basically, a negative slope implies non-monotomic behavior in time

which requires in turn an enlarged description involving at least one additional dynamical

variable.

Table 4.1: Basic StatisticsStock Slope α q21 q22

BAC -0.0234 3.7534 0.1009 0.0048

CSCO -0.0455 3.0899 0.1023 0.0056

DIS 0.0766 2.5696 0.2573 0.0020

GE -0.0175 0.0467 0.1019 0.0023

IBM -0.0257 3.6619 0.1041 0.0019

KO 0.0406 3.2034 0.2068 0.0016

MCD 0.0034 5.6787 0.1156 0.0037

MMM 0.0119 4.4300 0.1493 0.0018

MSFT -0.0338 3.3876 0.1036 0.0040

WMT 0.0161 4.1281 0.1592 0.0037

Fig.4.3 to Fig.4.12 show the behavior of the ten individual stocks from Dow Jones

Industrial Average index. Notice that the data slope was negative was taken in absolute

values. The theoretical probability distribution is plotted in dashed black line with

Page 28: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios

Chapter 4. Model Description 22

parameter values taken from the data:

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

1

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)s

(x)

2

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log

(Fre

que

ncy)

3

Figure 4.3: Bank of America Corporation(BAC) from May 29, 1986 to Feb1, 2010. The

theoretical probability distribution is plotted in dashed black line with parameter values

taken from the data: α= 3.7534, q21=0.1009, and q22=0.0048. Binning same as Fig.3.2.

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

4

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

5

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

6

Figure 4.4: Cisco System Inc(CSCO) from April 1, 1990 to Feb 1, 2010. The theoretical

probability distribution is plotted in dashed black line with parameter values taken from

the data: α=3.0899, q21=0.1023, and q22=0.0056. Binning same as Fig.3.2.

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

7

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

8

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

9

Figure 4.5: Walt Disney Co.(DIS) from Jan 2, 1962 to Feb 1, 2010. The theoretical

probability distribution is plotted in dashed black line with parameter values taken

from the data: α=2.5696, q21=0.2573, and q22=0.0020. Binning same as Fig.3.2.

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Chapter 4. Model Description 23

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

10

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

11

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log

(Fre

que

ncy)

12

Figure 4.6: General Electric Co.(GE) from Jan 2, 1962 to Feb 1, 2010. The theoretical

probability distribution is plotted in dashed black line with parameter values taken from

the data: α=4.0467, q21=0.1634, and q22=0.0023. Binning same as Fig.3.2.

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

13

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

14

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

15

Figure 4.7: International Business Machines Corp.(IBM) from Jan 2, 1970 to Jan 2, 2010.

The theoretical probability distribution is plotted in dashed black line with parameter

values taken from the data: α=3.6619, q21=0.1041, and q22=0.0019. Binning same as

Fig.3.2.

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

16

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

17

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

18

Figure 4.8: Coca-Cola Company(KO) from Jan 2, 1962 to Jan 2, 2010. The theoretical

probability distribution is plotted in dashed black line with parameter values taken from

the data: α=3.2034, q21=0.2068 , and q22=0.0016. Binning same as Fig.3.2.

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Chapter 4. Model Description 24

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

19

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

20

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log

(Fre

que

ncy)

21

Figure 4.9: McDonald’s Corp(MCD) from Jan 2, 1970 to Feb 2, 2010. The theoretical

probability distribution is plotted in dashed black line with parameter values taken from

the data: α=5.6787, q21=0.1156, and q22=0.0037. Binning same as Fig.3.2.

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

22

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

23

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

requency)

24

Figure 4.10: 3M Co.(MMM) from Jan 2, 1962 to Jan 2, 2010. The theoretical probability

distribution is plotted in dashed black line with parameter values taken from the data:

α=4.4300, q21=0.1493, and q22=0.0018. Binning same as Figure 3.2.

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

25

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

26

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log(F

reque

ncy)

27

Figure 4.11: Microsoft Corporation(MSFT) from Mar 13, 1986 to Jan 2, 2010. The

theoretical probability distribution is plotted in dashed black line with parameter values

taken from the data: α=3.3876, q21=0.1036, and q22=0.0040. Binning same as Fig.3.2.

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Chapter 4. Model Description 25

−0.1 −0.05 0 0.05 0.1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

x (t)

f(x

)

28

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x (t)

s(x

)

29

−0.3 −0.2 −0.1 0 0.1 0.2 0.310

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

returns

log

(Fre

que

ncy)

30

Figure 4.12: Wal-Mart Stores Inc.(WMT) from Aug 25, 1972 to Jan 2, 2010. The

theoretical probability distribution is plotted in dashed black line with parameter values

taken from the data: α=4.1281, q21=0.1592, and q22=0.0037. Binning same as Fig.3.2.

Page 32: Stochastic Models of Stock Market Dynamics393013/FULLTEXT01.pdfStochastic Models of Stock Market Dynamics Master in Computational Science Na Li Supervised by David J. T. Sumpter Stamatios

Chapter 5

Study Drawdown and Drawup

Distributions of Stock Market

Indices

5.1 Drawdown and Drawup

In order to better characterize and understand crashes as extreme events we will use

Drawdown and Drawup distribution. Their definition are respectively a loss from a

last local maximum to a next local minimum and gain from a last local minimum to

a next local maximum [11]. A crucial problem of daily returns is that it wipes out

the correlation between successive days by assuming independent returns. For instance,

in Fig.5.1 present the characteristics for the 14 largest drawdowns that have occurred

in the Dow Jones Industrial Average in last century. Only three lasted one or two

days, whereas nine lasted four days or more. It suggests the existence of a transient

correlation when large successive drops are observed. If the common approach is followed

to examine probability of the daily distribution of returns in order to estimate a crash

of 30% occurring over three days with three successive losses of exactly 10%, assuming

independence, it is the probability of one daily loss of 10% times the probability of

one daily loss of 10% times the probability of one daily loss of 10%, giving 109. This

corresponds to a 1 event in 1 billion trading days. This utterly impossible result relies

on the incorrect hypothesis that these three events are independent. Simply looking

at daily returns and at their distributions has destroyed the information that the daily

returns may be correlated at special times.

26

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Chapter 5. Study Drawdown and Drawup Distributions of Stock Market Indices 27

Figure 5.1: Characteristics of the 14 largest drawdowns of the DJIA in the twentieth

century.

5.2 Major Financial Indices

Fig.5.2 shows major world financial indices probability distribution of drawdown and

drawup distribution. Data in red, model in black. By analyzing the major financial

indices, it has been found that most of distributions of drawdowns and drawups are

very well fitted with the data.

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Chapter 5. Study Drawdown and Drawup Distributions of Stock Market Indices 28

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.210

−6

10−5

10−4

10−3

10−2

10−1

100

a

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.210

−6

10−5

10−4

10−3

10−2

10−1

100

b

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.210

−6

10−5

10−4

10−3

10−2

10−1

100

c

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.210

−6

10−5

10−4

10−3

10−2

10−1

100

d

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.210

−6

10−5

10−4

10−3

10−2

10−1

100

e

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.210

−6

10−5

10−4

10−3

10−2

10−1

100

f

Figure 5.2: (a) Dow Jones Industrial Average(USA) from Oct 1,1928 to Sep 11,2009,

(b) S&P 500(USA) from Jan 3,1950 to Jan 31,2010, (c) Nasdaq(USA) from Feb 5, 1971

to Jan 31, 2010, (d) FTSE 100(UK) from Apr 2, 1984 to Jan 31, 2010, (e) Nikkei

225(Japan) from Jan 4, 1984 to Jan 31, 2010, (f) Hang Seng(HK) from Dec 31, 1986 to

Jan 31, 2010

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Chapter 6

Study Non-Linear Model of Stock

Market Return

6.1 Non-Linear Model of Stock Market Return

In this chapter, we extend the model described in Sect.4.1 by taking into accounts non-

linearities that can be present. These nonlinearities can be associated to such behaviors

as herding ones. We will take in this chapter the same approach as we did in chapter

4. That is to say a ”Mean field” approach of models where all the qualities usage are

average. We add nonlinear term −βx3 allowing a symmetry-breaking bifurcation as the

parameter α becomes negative. Now, the empirical data analyzed here count against

such an explanation. We find instead that α>1, supporting our idea that crashes are

induced by the variance in returns exceeding the rate of convergence to zero, i.e. αq21>2.

The dynamic model with no additive noise term have already appears in early litera-

ture, i.e. Fredrich et al.[33] fit dynamic model with no noise term with currency exhange

rates. Later Sornette wants to show that the parameter values measures by Fredrich

et al. produces a power law exponents are typically smaller. Sornette’s argument in

two respects: firstly, Eq.(4.11) gives rise to power law behavior only in a range of large

deviations and second, the signature of financial crisis resides not only in the value of

the exponent but also on the parameter αq21 for he existence of a return distribution, the

dynamic model provide a clear relationship between Eq.(4.3) and market crashes.

dx

dt= αx(1 + F1(t))− βx3 (6.1)

where F1(t) has the variance q21.

29

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Chapter 6. Study Non-Linear Model of Stock Market Return 30

6.2 Fokker-Planck Analysis in Ito Interpretation

∂P

∂t= − ∂

∂x(αx− βx3)P +

12∂2

∂x2(α2x2q21)P (6.2)

where the first term is the dirft and the second one is the diffusion.

As in Sect.4.2 we are able to calculate the first, second and fourth moment.

dx

dt= αx− βx3

dx2

dt= 2α(1 +

α

2q21)x2 − 2βx4

dx4

dt= 4α(1 +

32αq21)x4 − 4βx6 (6.3)

And the steady state solution of Eq.(6.2) can be written as

P = C · exp[

2(1− q21α)ln(x)q21α

− βx2

q21α2

]= C · exp

[− βx2

q21α2

]· x

2(1−q21α)

αq12 (6.4)

Finally, setting Eq.(6.3) to zero, we are now in measure to evaluate the parameters of

Eq.(6.4)

f1 = q21α =2(x4

2 − x2x6)

x2x6 − 3x42

f2 =β

α=

−2x2x4

x2x6 − 3x42 (6.5)

Fig.6.1 shows the probability distribution of returns of six major financial indexes. Data

is red, theoretical distribution is black line. Data is measured by the parameters provided

by Eq.(6.5) and the theoretical distribution is obtained from Eq.(6.4). One can see that

the theoretical distribution fits well expect a little bit over-estimate the data. Noticed

that there is a asymmetry between negative and positive return for the theoretical

distribution reflecting that a very strong outlier exists in the negative return.

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Chapter 6. Study Non-Linear Model of Stock Market Return 31

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

a

returns

log

(Fre

que

ncy)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

b

returns

log

(Fre

que

ncy)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

c

returns

log

(Fre

que

ncy)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

d

returns

log

(Fre

que

ncy)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

e

returns

log

(Fre

que

ncy)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

f

returns

log

(Fre

que

ncy)

Figure 6.1: (a) Probability distribution of daily returns for Dow Jones Industrial Aver-

age(USA) from Oct 1, 1929 to Sep 11, 2009. Fitting with the actual data the parameters

equals to f1=-3.4679, f2=-260.0098; (b) Probability distribution of S&P 500(USA) from

Jan 3,1950 to Jan 31,2010, parameters f1=-2.5150, f2=-110.7004. (c) Probability dis-

tribution of Nasdaq(USA) from Feb 5, 1971 to Jan 31, 2010, parameters f1=-13.5864,

f2=-2.4736. (d)Probability distribution of FTSE 100(UK) from Apr 2, 1984 to Jan

31, 2010, parameters f1=-3.1221, f2=-228.1465. (e) Probability distribution of Nikkei

225(Japan) from Jan 4, 1984 to Jan 31, 2010, parameters f1=-5.4251, f2=-1.0462. (f)

Probability distribution of Hang Seng(HK) from Dec 31, 1986 to Jan 31, 2010, param-

eters f1=-3.0281, f2=-54.0530.

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Chapter 7

General Stochastic Models

The study of dynamical system subject to a noise perturbation is very important in

physics, chemistry and biology, as well as in several other areas. In this chapter we have

investigated different type of noise driven dynamical system.

7.1 General Setting

Consider the general form of evolution equation

dx

dt= u(x) + v(x)F1(t) (7.1)

where F1(t) is a Gaussian white noise of variance equal to q21 and u(x) and v(x) are any

linear or nonlinear functions of x. Notice that the specific case u(x)=0, v(x)=1 will not

be considered here as it is well known that it is non stationary.

7.1.1 Fokker-Planck equation in Ito interpretation

Eq.(7.1) defines a diffusion process in the x-space, whose probability distribution obeys

to a Fokker-Planck type equation. We adopt the Ito interpretation as widely used in

the finance literature. One can write thus the Fokker-Planck equation as

∂P

∂t= − ∂

∂xu(x)P +

12∂2

∂x2

(q21v

2(x))P (7.2)

32

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Chapter 7. General Stochastic Models 33

The stationary state is

−2u(x)P +d

dx

(q21v

2(x))P = 0 (7.3)

which we can rewrite

2(q21v(x)v′(x)− u(x)

)P + q21v

2(x)P ′ = 0 (7.4)

and Finally,

P ′ =1q21

1v2(x)

(2u(x)− 2q21v(x)v′(x)

)P

P = C · exp[ ∫

dx1q21

1v2(x)

(2u(x)− 2q21v(x)v′(x)

)](7.5)

We next consider specific forms of interest for u(x) and v(x).

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Chapter 7. General Stochastic Models 34

7.2 Linear Model with Additive Noise

In this case, v(x)=1 and we choose the simplest function u(x)=-αx.

Eq.(7.1) becomesdx

dt= −αx+ F1(t) (7.6)

From Eq.(7.5) we have

P = C · exp[ ∫

dx1q21

(−2αx)]

= C · exp[−αx2

q21

](7.7)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

10−80

10−60

10−40

10−20

100

x

log(F

requency)

Figure 7.1: Probability distribution as obtained from Eq.(7.7). Parameter values are

q21=0.1, α=0.8.

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Chapter 7. General Stochastic Models 35

7.3 Linear Model with Multiplicative Noise

In this case, we choose u(x)=v(x)=-αx.

Eq.(7.1) becomesdx

dt= −αx(1 + F1(t)) (7.8)

and from Eq.(7.5) we have

P = C · x−2(1+q21α)

αq12 (7.9)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

10−80

10−60

10−40

10−20

100

x

log(F

requency)

Figure 7.2: Probability distribution as obtained from Eq.(7.9). Parameter values are

q21=0.1, α=0.8.

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Chapter 7. General Stochastic Models 36

7.4 Nonlinear Model with Additive Noise

Let us take a nonlinear function u(x)=αx-βx3 and v(x)=1.

Eq.(7.1) becomesdx

dt= αx− βx3 + F1(t) (7.10)

And, from Eq.(7.5) we have

P = C · exp[

2x2(α− βx2)q21

](7.11)

x

log(F

requency)

−1 −0.5 0 0.5 1

10−20

100

a

−1 −0.5 0 0.5 1

10−20

100

b

−1 −0.5 0 0.5 1

10−20

100

c

−1 −0.5 0 0.5 1

10−20

100

d

Figure 7.3: Probability distribution as obtained from Eq.(7.11). Parameter values are

q21=0.1, α=0.8, β=100 (a), β=10 (b), β=1 (c), β=0.15 (d).

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Chapter 7. General Stochastic Models 37

7.5 Nonlinear Model with Multiplicative Noise

We have now u(x)=αx-βx3 and v(x)=αx.

Eq.(7.1) becomesdx

dt= αx(1 + F1(t))− βx3 (7.12)

Again, from Eq.(7.5) we have

P = C · exp[

2(1− q21α)ln(x)q21α

− βx2

q21α2

]= C · exp

[− βx2

q21α2

]· x

2(1−q21α)

αq12 (7.13)

x

log(F

requency)

−1 −0.5 0 0.5 1

10−20

100

a

−1 −0.5 0 0.5 1

10−20

100

b

−1 −0.5 0 0.5 1

10−20

100

c

−1 −0.5 0 0.5 1

10−20

100

d

Figure 7.4: Probability distribution as obtained from Eq.(7.13). Parameter values are

q21=0.1, α=0.8, β=100 (a), β=10 (b), β=1 (c), β=0.15 (d).

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Chapter 7. General Stochastic Models 38

7.6 Linear Model with Additive and Multiplicative Noise

In a more general case of the presence of an additive and a multiplicative noise(of

different variances q21 and q22) one can write the model as

dx

dt= u(x) + v(x)F1(t) + F2(t) (7.14)

As in Sect.7.1, one can find the probability distribution of Eq.(7.14) by writing the

Fokker Planck equation

∂P

∂t= − ∂

∂xu(x)P +

12∂

∂x

[q21v(x)

∂xv(x)P + q22

∂xP

](7.15)

we choose here u(x)=v(x)=-αx. Eq.(7.14) can now be written as

dx

dt= −α(1 + F1(t))x+ F2(t) (7.16)

Eq.(7.15) becomes then

P = C · exp

[− 2αq22

∫dx

(1 + αq21)x

1 +(q21α2

q22

)x2

]

= C · exp

[− α

q22

1 + q21αq21α

2

q22

ln

(1 +

q21α2

q22x2

)]

= C ·[1 +

q21α2

q22x2

]− 1+q21α

αq12

(7.17)

Notice that when q1=0 in Eq.(7.15) (F1(t)=0 in Eq.(7.14)) one recovers the same dis-

tribution than Eq.(7.7). Similarly, when q2=0 in Eq.(7.15)) (F2(t)=0 in Eq.(7.14)), one

recovers the distribution of Eq.(7.9).

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Chapter 7. General Stochastic Models 39

x

log(F

requency)

−1 −0.5 0 0.5 1

10−20

100

a

−1 −0.5 0 0.5 1

10−20

100

b

−1 −0.5 0 0.5 1

10−20

100

c

−1 −0.5 0 0.5 1

10−20

100

d

Figure 7.5: Probability distribution as obtained from Eq.(7.17). Parameter values are

q21=0.1, α=0.8, q22=0.005 (a), q22=0.05 (b), q22=0.5 (c), q22=1 (d).

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Chapter 8

Conclusion

Frequent financial crises has become one of the most important factor influence the

world economy in 20th century. In order to management the risk, study the financial

market mechanism, the price fluctuation and trader behavior are very important.

The classical view of stock market returns are independent of each other over anything

but the very shortest of time scales. The ”econo-physicists” emphasize the returns

follow scaling laws consistent over several orders of magnitude with a Levy stable dis-

tribution.

We analyzed several major financial indices in order to capture features of the stock

market fluctuations reflecting investor behavior. We use a novel equation free approach

to identify dynamical features of fluctuations of stock market and then develop a dy-

namical model which captures these features and reflects a weak herding behavior. This

theory links together individual decision and collective behavior. It appears that in some

cases that market crashes are caused by the traders copying the buy and sell decisions

of their peers, leading to the build up of correlations within the market that accumulate

to a crash. The dynamical model gives good math data over 80 years and scales well

between the time scales of days up to the months.

’Econophysics’ classified the stock market crashes as ”extreme events” phenomena, a

subject belongs to the general area of complex system. We studied the drawdown and

analyzed several major stock indices to confronted our model. A characteristic feature of

a complex system is the possible occurrence of coherent large scale collective behaviors,

resulting from the repeated nonlinear interactions among constituents. To account for

this, we also analyzed the non-linear model extension from the our dynamical model

and it gave good matching with sever major stock indices. Finally, we studied several

types noise to show how it play the critical role in the dynamical system.

When we used the dynamical model fitting with the individual shares from the Dow

Jones Industrial Average, the model did not get the good match as several major stock

indices. About half of the individual shares get the negative slope. Our model fails to

40

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Chapter 8. Conclusion 41

predict with the negative slope. Normally, the individual shares different from stock

indices, it usually have several bonus events during the year. With this background, the

individual share for relative stock return x(t) is defined as:

x(t) =y(t+1) − y(t) + z(t)

y(t)(8.1)

Here z(t) is portfolio include bonus, dividend, and other income for individual share at

time t.

Determining the portfolio z(t) will be one of the direction would like to work in the future.

It is interesting to show our dynamical model work on the individual share.

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Acknowledgements

I would like to thank my supervisors David J. T. Sumpter and Stamatios C. Nicolis for

their great help and support which made this thesis possible. Special thanks Stamatios

C. Nicolis for his patience, in-depth comment and invaluable guidance throughout the

work. Finally, I would also like to thank my lovely friends and family, they gave me

constant love, encouragement and support through these years.

42

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