USF Libraries
USF Digital Collections

Stochastic modeling and statistical analysis

MISSING IMAGE

Material Information

Title:
Stochastic modeling and statistical analysis
Physical Description:
Book
Language:
English
Creator:
Wu, Ling
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
Publication Date:

Subjects

Subjects / Keywords:
Hybrid System
Nonlinear Models
Option Pricing
Forecasting
ARIMA
Dissertations, Academic -- Mathematics and Statistics -- Doctoral -- USF   ( lcsh )
Genre:
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: The objective of the present study is to investigate option pricing and forecasting problems in finance. This is achieved by developing stochastic models in the framework of classical modeling approach. In this study, by utilizing the stock price data, we examine the correctness of the existing Geometric Brownian Motion (GBM) model under standard statistical tests. By recognizing the problems, we attempted to demonstrate the development of modified linear models under different data partitioning processes with or without jumps. Empirical comparisons between the constructed and GBM models are outlined. By analyzing the residual errors, we observed the nonlinearity in the data set. In order to incorporate this nonlinearity, we further employed the classical model building approach to develop nonlinear stochastic models. Based on the nature of the problems and the knowledge of existing nonlinear models, three different nonlinear stochastic models are proposed. Furthermore, under different data partitioning processes with equal and unequal intervals, a few modified nonlinear models are developed. Again, empirical comparisons between the constructed nonlinear stochastic and GBM models in the context of three data sets are outlined. Stochastic dynamic models are also used to predict the future dynamic state of processes. This is achieved by modifying the nonlinear stochastic models from constant to time varying coefficients, and then time series models are constructed. Using these constructed time series models, the prediction and comparison problems with the existing time series models are analyzed in the context of three data sets. The study shows that the nonlinear stochastic model 2 with time varying coefficients is robust with respect different data sets. We derive the option pricing formula in the context of three nonlinear stochastic models with time varying coefficients. The option pricing formula in the frame work of hybrid systems, namely, Hybrid GBM (HGBM) and hybrid nonlinear stochastic models are also initiated. Finally, based on our initial investigation about the significance of presented nonlinear stochastic models in forecasting and option pricing problems, we propose to continue and further explore our study in the context of nonlinear stochastic hybrid modeling approach.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2010.
Bibliography:
Includes bibliographical references.
System Details:
Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
Statement of Responsibility:
by Ling Wu.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains X pages.
General Note:
Includes vita.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
usfldc doi - E14-SFE0003387
usfldc handle - e14.3387
System ID:
SFS0027702:00001


This item is only available as the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam 22 Ka 4500
controlfield tag 007 cr-bnu---uuuuu
008 s2010 flu s 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0003387
035
(OCoLC)
040
FHM
c FHM
049
FHMM
090
XX9999 (Online)
1 100
Wu, Ling.
0 245
Stochastic modeling and statistical analysis
h [electronic resource] /
by Ling Wu.
260
[Tampa, Fla] :
b University of South Florida,
2010.
500
Title from PDF of title page.
Document formatted into pages; contains X pages.
Includes vita.
502
Dissertation (Ph.D.)--University of South Florida, 2010.
504
Includes bibliographical references.
516
Text (Electronic dissertation) in PDF format.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
3 520
ABSTRACT: The objective of the present study is to investigate option pricing and forecasting problems in finance. This is achieved by developing stochastic models in the framework of classical modeling approach. In this study, by utilizing the stock price data, we examine the correctness of the existing Geometric Brownian Motion (GBM) model under standard statistical tests. By recognizing the problems, we attempted to demonstrate the development of modified linear models under different data partitioning processes with or without jumps. Empirical comparisons between the constructed and GBM models are outlined. By analyzing the residual errors, we observed the nonlinearity in the data set. In order to incorporate this nonlinearity, we further employed the classical model building approach to develop nonlinear stochastic models. Based on the nature of the problems and the knowledge of existing nonlinear models, three different nonlinear stochastic models are proposed. Furthermore, under different data partitioning processes with equal and unequal intervals, a few modified nonlinear models are developed. Again, empirical comparisons between the constructed nonlinear stochastic and GBM models in the context of three data sets are outlined. Stochastic dynamic models are also used to predict the future dynamic state of processes. This is achieved by modifying the nonlinear stochastic models from constant to time varying coefficients, and then time series models are constructed. Using these constructed time series models, the prediction and comparison problems with the existing time series models are analyzed in the context of three data sets. The study shows that the nonlinear stochastic model 2 with time varying coefficients is robust with respect different data sets. We derive the option pricing formula in the context of three nonlinear stochastic models with time varying coefficients. The option pricing formula in the frame work of hybrid systems, namely, Hybrid GBM (HGBM) and hybrid nonlinear stochastic models are also initiated. Finally, based on our initial investigation about the significance of presented nonlinear stochastic models in forecasting and option pricing problems, we propose to continue and further explore our study in the context of nonlinear stochastic hybrid modeling approach.
590
Advisor: Gangaram S. Ladde, Ph.D.
653
Hybrid System
Nonlinear Models
Option Pricing
Forecasting
ARIMA
690
Dissertations, Academic
z USF
x Mathematics and Statistics
Doctoral.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.3387



PAGE 1

Stochastic Modeling and Statistical Analysis by Ling Wu A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics and Statistics College of Arts and Sciences University of South Florida Major Professor: Gangaram S. Ladde, Ph.D. Kandethody M. Ramachandran, Ph.D. Wonkuk Kim, Ph.D. Marcus Mcwaters, Ph.D. Date of Approval: April 1, 2010 Keywords: Hybrid System, Nonlinear Models, Option Pricing, Forecasting, ARIMA Copyright 2010 Ling Wu

PAGE 2

Dedication I dedicate this dissertation to my husba nd Hu, Xuequn. Without his patience, understanding, support, and most of all love, the completion of this work would not have been possible.

PAGE 3

Acknowledgments I would like to express the deepest appreciation to my major professor Dr. Gangaram S. Ladde, who has supported me throughout my dissertati on with his patience, knowledge, encouragement and effort. Without him this dissertation would not have been written. One simply could not wish for a better or friendlier as an individual with demanding and challenging supervisor. I would like to thank my committee member, Dr. Kandethody M. Ramachandran, Dr. Wonkuk Kim, and Dr. Marcus McWaters, for their suppor t and guidance to complete my research work. In addition, I would like to express my appreciation to Dr. Christos P. Tsokos for his comments, suggestions to this work.

PAGE 4

Table of Contents List of Tables iv List of Figures vii Abstract ix Chapter 1 Review and Basic Concepts 1 1.0 Introduction 1 1.1 Financial Mathematics 1 1.1.1 Fundamental Concepts 1 1.1.2 Option Pricing 3 1.2 Development of Stochastic Modeling 8 1.2.1 Conditions of Stochastic Process Random Walk 8 1.2.2 Mean and Variance of Aggregate Change of Price 10 1.2.3 Wiener Process 12 1.3 General Stochastic Differential Equations 15 1.4 Least Square Method 17 1.5 Maximum Likelihood Estimati on Method and ARIMA Model 17 Chapter 2 Linear Stochastic Models 21 2.0 Introduction 21 2.1 GBM Models 22 2.2 GBM Model on Overall Data 24 2.3 GBM Models under Data Partitioning Schemes without Jumps 27 2.4 GBM Models under Data Par titioning Schemes with Jumps 42 2.5 Illustration of GBM Models to Data Set of Stock Y 49 2.6 Illustration of GBM Models to Data Set of S&P500 Index 52 2.7 Conclusions and Comments 54 Chapter 3 Nonlinear Stochastic Models 57 3.0 Introduction 57 3.1 Stochastic Nonlinear Dynamic Model 1 (Black-Karasinski Model) 57 3.2 Stochastic Nonlinear Dynamic Model 2 67 i

PAGE 5

3.3 Stochastic Nonlinear Dynamic Model 3 76 3.4 Illustration of Nonlinear Stochastic Models to Data Set of Stock Y 87 3.5 Illustration of Nonlinear Stochastic Models to Data Set of S&P500 Index 88 3.6 Conclusions and Comments 89 Chapter 4 Nonlinear Stochastic Mode ls with Time Varying Coefficients 92 4.0 Introduction 92 4.1 Nonlinear Stochastic Dynamic Model 1 with Time Varying Coefficients 92 4.2 Nonlinear Stochastic Dynamic Model 2 with Time Varying Coefficients 100 4.3 Prediction and Comparison on Overall Data Sets 106 4.4 Conclusions 114 Chapter 5 European Option Pricing 115 5.0 Introduction 115 5.1 European Option Pricing for Nonlinear Model 1 116 5.2 European Option Pricing for Nonlinear Model 2 120 5.3 European Option Pricing for Nonlinear Model 3 125 Chapter 6 Option Pricing for Hybrid Models 128 6.0 Introduction 128 6.1 Option Pricing for Hybrid GBM Models 128 6.2 Option Pricing for Hybrid Nonlinear Stochastic Models 133 6.2.1 Hybrid Nonlinear Stochastic Model 1 134 6.2.2 Hybrid Nonlinear Stochastic Model 2 137 6.2.3 Hybrid Nonlinear Stochastic Model 3 141 Chapter 7 Future Research Plan 144 7.1 Data Smoothing Transformation 144 7.1.1 Nonlinear Stochastic Model 1 144 7.1.2 Nonlinear Stochastic Model 2 145 7.2 Forecasting Problem 146 7.3 Option Pricing Problem 146 References 147 Appendices 152 Appendix A1: The Estimated Para meters of Stock X Applying Monthly GBM Model 153 Appendix A2: The Estimated Para meters of Stock X Applying ii

PAGE 6

Unequal Interval GBM Model 155 Appendix A3: The Estimated Jump Coefficient of Stock X Applying Monthly GBM Model 156 Appendix A4: The Estimated Jump Coefficient f Stock X Applying Unequal Interval GBM Model 157 About the Author End Page iii

PAGE 7

List of Tables Table 2.2.1 Basic Statistics for Model in Equation (2.2.1) 26 Table 2.3.1 Mean and Standard Deviation of Da ily Differences of 4 Quarters of Stock X 30 Table 2.3.2 Basic Statistics for Model in Equation (2.3.2) 33 Table 2.3.3 Basic Statistics for Model in Equation (2.2.1), (2.3.3), and (2.3.7) 35 Table 2.3.4 Basic Statistics for Model in Equation (2.2.1), (2.3.3), (2.3.7) and (2.3.10) 39 Table 2.3.5 Basic Statistics for Model in Equation (2.2.1), (2.3.3), (2.3.7), (2.3.10) and (2.3.13) 41 Table 2.4.1 Basic Statistics for Linear Models (2 .2.1), (2.3.3), (2.3.7), (2.3.10), (2.3.13) and with Jumps (2.3.3), (2.3.7) of Stock X 44 Table 2.4.2 Basic Statistics for Linear Models (2.2.1), (2.3.3), (2.3.7), and with and without Jumps Models (2.3.10), (2.3.13) of Stock X 46 Table 2.4.3 Basic Statistics for Linear Models (2.2.1) and Models with and without Jumps (2.3.3), (2.3.7), (2.3.10), (2.3.13) of Stock X 48 Table 2.5.1 Basic Statistics for Linear Models without Jumps (2.2.1), with and Models without Jumps (2.3.3), (2.3.7), (2.3.10), (2.3.13) of Stock Y 51 Table 2.6.1 Basic Statistics for Linear Models without Jumps (2.2.1), and Models with and without Jumps (2.3.3), (2.3.7), (2.3.10), (2.3.13) of S&P 500 Index 53 Table 3.1.1 Estimated Parameters in Model 3.1.1 of Stock X 59 Table 3.1.2 Basic Statistics of Model 3.1.1 62 Table 3.1.3 Estimated Parameters in Model 3.1.2 of Stock X 63 Table 3.1.4 Basic Statistics of Models 2.4.3, 3.1.1, 2.4.4 and 3.1.2 of Stock X 66 Table 3.2.1 Estimated Parameters of Model 3.2.1 of Stock X 69 Table 3.2.2 Basic Statistics of Models 2.4.3, 3.1.1 and 3.2.1 of Stock X 72 Table 3.2.3 Estimated Parameters of Model 3.2.2 of Stock X 73 Table 3.2.4 Basic Statistics of Models 2.4.3, 3.1.1, 3.1.2, 2.4.4, 3.2.1 and 3.2.2 of Stock X 76 Table 3.3.1 Estimated Parameters of Model 3.3.1 of Stock X 78 Table 3.3.2 Basic Statistics of Models 2.4.3, 3.1.1, 3.2.1 and 3.3.1 of Stock X 81 Table 3.3.3 Estimated Parameters of Model 3.3.2 of Stock X 82 Table 3.3.4 Basic Statistics for Models 2.4.3, 3.1.1, 3.2.1, 3.3.1, 3.1.2, 3.2.2 and 3.3.2 of Stock X 86 Table 3.4.1 Basic Statistics for Models of Stock Y 87 Table 3.5.1 Basic Statistics for Models of S&P 500 Index 88 Table 3.6.1 Summary of Models in Chapter 3 89 iv

PAGE 8

Table 4.1.1 AIC of Time Varying Coefficients Nonlinear Model 1 of Different Models of Three Data Sets: Stock X, Stoc k Y and S&P 500 Index 94 Table 4.1.2 Basic Statistics of Time Varying Coefficients Nonlinear Model 1 of Three Data Sets: Stock X, Stock Y a nd S&P 500 Index 96 Table 4.1.3 Basic Statistics of Stochastic M odels 4.1.1 with Different Set of (p, q) under Log-Transformation with Unequal Data Partition, threshold=5% of Stock X 97 Table 4.1.4 Basic Statistics of Stochastic Mode ls 4.1.1 with Different Set of (p, q) Under Log-Transformation with Unequal Data Pa rtition, threshold=4.5% of Stock Y 98 Table 4.1.5 Basic Statistics of Stochastic Mode ls 4.1.1 with Different Set of (p, q) Under Log-Transformation with Unequal Data Partition, threshold=2% of S&P 500 Index 99 Table 4.2.1 Basic Statistics of Time Varying Coefficients Nonlinear Model 2 of Three Data Sets: Stock X, Stock Y and S&P 500 Index 101 Table 4.2.2 Basic Statistics of Stochastic M odels 4.2.1 with Different Set of (p, q) under Transformation N SN t11 with Unequal Data Par tition, threshold=5% of Stock X 104 Table 4.2.3 Basic Statistics of Stochastic M odels 4.2.1 with Different Set of (p, q) under Transformation N SN t11 with Unequal Data Par tition, threshold=4.5% of Stock Y 105 Table 4.2.4 Basic Statistics of Stochastic Models 4.2.1 with Different Set of (p, q) under Transformation N SN t11 with Unequal Data Partition, threshold=2% of S&P 500 Index 106 Table 4.3.1 Comparison Cited Models in Section 4.3 for Stock X 107 Table 4.3.2 Comparison Cited Models in Section 4.3 for Stock Y 108 Table 4.3.3 Comparison Cited Models in Sec tion 4.3 for S&P500 Index 108 Table 4.3.4 Actual and Predicted Price for Stock X 110 Table 4.3.5 Actual and Predicted Price for Stock Y 111 Table 4.3.6 Actual and Predicted Price for S&P 500 Index 112 Table 4.3.7 Basic Statistics by Using Different Predicted Models for Stock X 113 Table 4.3.8 Basic Statistics by Using Different Predicted Models for Stock Y 113 Table 4.3.9 Basic Statistics by Using Different Predicted Models for S&P500 Index 113 Table 4.3.10 Summary of Predictions for Three Data Sets 113 Table 5.1.1 Call and Put Option Price of Nonlinear Model 1 119 v

PAGE 9

Table 5.1.2 Call and Put Option Price of GBM Model 120 Table 5.2.1 Call and Put Option Price of Nonlinear Model 2 124 Table 5.3.1 Call and Put Option Price of Nonlinear Model 3 127 Table 6.1.1 Call and Put Option Price of Hybrid GBM Model 2.4.3 132 Table 6.1.2 Call and Put Option Price of Hybrid GBM Model 2.4.4 133 Table 6.2.1 Call and Put Option Price of Hybrid Nonlinear Model 3.1.1 137 Table 6.2.2 Call and Put Option Price of Hybrid Nonlinear Model 3.1.2 137 Table 6.2.3 Call and Put Option Price of Hybrid Nonlinear Model 3.2.1 141 Table 6.2.4 Call and Put Option Price of Hybrid Nonlinear Model 3.2.2 141 Table 6.2.5 Call and Put Option Price of Hybrid Nonlinear Model 3.3.1 143 Table 6.2.6 Call and Put Option Price of Hybrid Nonlinear Model 3.3.2 143 Table 7.1.1 AIC of Time Varying Coefficients Nonlinear Model 1 (n=3) of Different Models of Three Datasets: Stock X, Stock Y and S&P 500 Index 144 Table 7.1.2 Basic Statistics of Time Varying Coefficients Nonlinear Model 1 (n=3) of Three Data Sets: Stock X, St ock Y and S&P500 Index 145 Table 7.1.3 Basic Statistics of Time Varying Coefficients Nonlinear Model 2 (n=3) of Three Data Sets: Stock X, St ock Y and S&P500 Index 146 vi

PAGE 10

List of Figures Figure 1.1.1 Binomial Tree for a Stochastic Stock Price Process 4 Figure 2.2.1 Daily Adjusted Closing Price Process for Stock X 24 Figure 2.2.2 Prediction and One Possible Path of Stock Xs Price Process Using Model (2.2.1) 26 Figure 2.2.3 Q-Q Plot for Model in Equation (2.2.1) 27 Figure 2.3.1 Daily Difference of Stock X in Q1 28 Figure 2.3.2 Daily Difference of Stock X in Q2 29 Figure 2.3.2 Daily Difference of Stock X in Q3 29 Figure 2.3.2 Daily Difference of Stock X in Q4 30 Figure 2.3.5 Comparison on Model (2.2.1) with Model (2.3.3) of Stock X 32 Figure 2.3.6 Q-Q Plot for Model in Equation (2.3.3) 33 Figure 2.3.7 Comparison on Model (2.2.1), (2.3.3) with Model (2.3.7) of Stock X 35 Figure 2.3.8 Q-Q Plot of Model (2.3.7) 36 Figure 2.3.9 Comparison on Model (2.2.1), (2.3.3), (2.3.7) with Model (2.3.10) of Stock X 38 Figure 2.3.10 Comparison on Model (2.3.1), (2.3.3), (2.3.7), (2.3.10) with Model (2.3.13) of Stock X 41 Figure 2.4.1 Comparison of Models (2.3.3), (2 .3.7) with and without Jumps of Stock X 44 Figure 2.4.2 Figure 2.4.2 Comparison of Models (2.3.10) and (2.3.13) with and without Jumps of Stock X 46 Figure 2.4.3 Comparisons of Models with and without jumps (2.3.10), (2.3.13) and (2.4.3) of Stock X 48 Figure 2.5.1 Daily Adjusted Closing Price for Stock Y 50 Figure 2.5.2 The Best Two Estimated Models of Stock Y 50 Figure 2.6.1 Daily Adjusted Closing Price for S&P 500 Index 52 Figure 2.6.2 The Best Two Estimated Models of S&P 500 Index 53 Figure 2.7.1 Some Residual Plots of Stock X 55 Figure 3.1.1 Comparison of Model 2.4.3 with M odel 3.1.1 of Stock X (Observations 1-300) 61 Figure 3.1.2 Comparison of Model 2.4.3 with Model 3.1.1 of Stock X (Observations 300-600) 61 Figure 3.1.3 Comparison of Model 2.4.3 with Model 3.1.1 of Stock X (Observations 600-848) 62 Figure 3.1.4 Comparison of Model 2.4.3, 2.4.4, 3.1.1 with Model 3.1.2 of Stock X (Observations 1-300) 65 vii

PAGE 11

Figure 3.1.5 Comparison of Model 2.4.3, 2.4.4, 3.1.1 with Model 3.1.2 of Stock X (Observations 300-600) 65 Figure 3.1.6 Comparison of Model 2.4.3, 2.4.4, 3.1.1 with Model 3.1.2 of Stock X (Observations 600-848) 66 Figure 3.2.1 Comparison of Model 2.4.3, 3.1.1 with Model 3.2.1 of Stock X (Observations 1-300) 70 Figure 3.2.2 Comparison of Model 2.4.3, 3.1.1 with Model 3.2.1 of Stock X (Observations 300-600) 71 Figure 3.2.3 Comparison of Model 2.4.3, 3.1.1 with Model 3.2.1 of Stock X (Observations 600-848) 71 Figure 3.2.4 Comparison of Model 2.4.3, 2.4.4, 3.2.1 with Model 3.2.2 of Stock X (Observations 1-300) 74 Figure 3.2.5 Comparison of Model 2.4.3, 2.4.4, 3.2.1 with Model 3.2.2 of Stock X (Observations 300-600) 75 Figure 3.2.6 Comparison of Model 2.4.3, 2.4.4, 3.2.1 with Model 3.2.2 of Stock X (Observations 600-848) 75 Figure 3.3.1 Comparison of Model 2.4.3, 3.1.1, 3.2.1 with Model 3.3.1 of Stock X (Observations 1-300) 80 Figure 3.3.2 Comparison of Model 2.4.3, 3.1.1, 3.2.1 with Model 3.3.1 of Stock X (Observations 300-600) 80 Figure 3.3.3 Comparison of Model 2.4.3, 3.1.1, 3.2.1 with Model 3.3.1 of Stock X (Observations 600-848) 81 Figure 3.3.4 Comparison on Model 2.4.3, 2.4.4, 3.3.1 with Model 3.3.2 of Stock X (Observations 1-300) 84 Figure 3.3.5 Comparison on Model 2.4.3, 2.4.4, 3.3.1 with Model 3.3.2 of Stock X (Observations 300-600) 85 Figure 3.3.6 Comparison on Model 2.4.3, 2.4.4, 3.3.1 with Model 3.3.2 of Stock X (Observations 600-848) 85 Figure 3.6.1 The Predicted Value of Stock X 90 Figure 6.1.1 State Switch Illustration of Hybrid GBM 129 Figure 6.2.1 State Switch Illustration of Hybrid Nonlinear Stochastic Model 134 viii

PAGE 12

Stochastic Modeling and Statistical Analysis Ling Wu ABSTRACT The objective of the present study is to inves tigate option pricing and forecasting problems in finance. This is achieved by deve loping stochastic models in th e framework of classical modeling approach. In this study, by utilizing the stock price da ta, we examine the correctness of the existing Geometric Brownian Motion (GBM) model under standard statistical tests. By recognizing the problems, we attempted to demonstrate the development of modified linear models under different data partitioning processes with or w ithout jumps. Empirical comparisons between the constructed and GBM models are outlined. By analyzing the residual errors, we observed the nonlinearity in the data set. In order to incorporate this nonlinearity, we further employed the classical model building approach to develop nonlinear stochastic m odels. Based on the nature of the problems and the knowledge of existing nonlinear models, three different nonlinear stochastic models are proposed. Furthermore, under different data partitioning processes with equal and unequal intervals, a few modified nonlinear models are developed. Again, empirical comparisons between the constructed nonlinear ix

PAGE 13

stochastic and GBM models in the c ontext of three data sets are outlined. Stochastic dynamic models are also used to pr edict the future dynamic state of processes. This is achieved by modifying the nonlinear stochastic models from constant to time varying coefficients, and then time series models are c onstructed. Using these constructed time series models, the prediction and comparison problems with the existing time series models are analyzed in the context of three data sets. The study shows that the nonlinear stochastic model 2 with time varying coefficients is robus t with respect different data sets. We derive the option pricing formula in the cont ext of three nonlinear stochastic models with time varying coefficients. The option pricing formula in the frame work of hybrid systems, namely, Hybrid GBM (HGBM) and hybrid nonlinear stochastic models are also initiated. Finally, based on our initial investigation about the significance of presented nonlinear stochastic models in forecasting and option pricing problems we propose to continue and further explore our study in the context of nonlinear stochastic hybrid modeling approach. x

PAGE 14

Chapter 1 Review and Basic Concepts 1.0 Introduction Financial mathematics derives and extends the mathematical or numerical models that are suggested by financial economists. Stochastic process is widely used here to obtain the fair price of derivatives of an asset. In this chapter, we first review some financial terminologies and methodologies, in Sections1.1. In Section 1.2, we present the development of stochastic models. General stochastic diffe rential equations and I toDoob formula are reviewed in Section 1.3. Furthermore, the least square estimation method is reviewed to estimate the parameters in Section 1.4. Finally, the maximum likelihood estimation method of time series model (ARMA model) is outlined in Section 1.5. 1.1 Financial Mathematics 1.1.1 Fundamental Concepts During 1600s, Tulip dealing was big business in Holland. Flower growers and dealers were trading in options to guarantee prices. Until 1700s options were declared illegal in London. The Investment Securities Act of 1934 created the Securities and Exchange Commission (SEC), and gave the SEC the power to regulate options. In April 26, 1973, the Chicago Board Option Exchange (CBOE) started trading and listed 16 call options on 16 stocks. A few years later, CBOE began trading put option, and ten years late r, CBOE began trading Index option. On the first day of trading in 1973, 911 contracts traded. Today, the CBOEs average daily volume consistently exceeds one million c ontracts per day [4]. The concept of financial derivatives plays an important role in an interconnected financial world. 1

PAGE 15

Definition 1.1.1 Derivatives : Derivatives are financial instruments whose value is derived from the value of something else. They generally take the form of contracts under which the parties agree to payments between them based upon the value of an underlying asset or other data at a particular point in time [2, 4, 19]. The main types of derivatives are futures, fo rwards, options and swaps. The main use of derivatives is to minimize risk for one party wh ile offering the potential for a high return (at increased risk) to another. In a short term, the main use of derivatives is in risk management. The diverse range of potential underlying assets and payoff alternatives lead to a huge range of derivatives contracts available to be traded in th e market. One of the most important derivatives is option. In the following, we define op tion, and outline different types options. Definition 1.1.2 Options : Options are financial instruments that convey the right, but not the obligation to engage in a future transaction on some underlying security. Financial instruments are cash, evidence of an ownership interest in an entity, or a contractual right to receive, or deliver, cash or another financial instrument [2, 4, 19]. For example, buying a call option provides the right to buy a specified amount of a security at a set strike price at some time on or before expiration, while buying a put option provides the right to sell. There are 4 kinds of options: (i) European option: An option that may only be exercised on expiration. (ii) American option: An option that may be exer cised on any trading day on or before expiration. (iii) Bermuda option: An option that may be ex ercised only on specified dates on or before expiration. (iv) Barrier option: Any option with the general ch aracteristic that the underlying securitys price must reach some trigger level before the exercise can occur. Definition 1.1.3 Strike price (K) : For an option, the strike price (K) or exercise price, is the key variable in a derivatives contract between two par ties. Where the contract requires delivery of the underlying instrument, the trade will be at the stri ke price, regardless of the spot price (market price S) of the underlying instrument at that time. Strike price is the fixed price at which the 2

PAGE 16

owner of an option can purchase, in the case of a ca ll, or the fixed price at which the owner of an option can sell, in the case of a put, the underlying security or commodity [2, 4, 19]. The concepts of payoff for options are defined as below. Definition 1.1.4 Payoff : The payoff for a call option at time T is Max{(ST K); 0}, or formally (ST K)+.The payoff for a put option at time T is Max{(K ST); 0}, or formally (K ST)+. T is the maturity time at which the derivative contract expires [2, 4, 19]. In the following, we define the concept of hedge in finance. Definition 1.1.5 Hedge : A hedge is an investment that is taken out specifically to reduce or cancel out the risk in another investment [2, 4, 19]. Hedging is a strategy designed to minimize e xposure to an unwanted business risk, while still allowing the business to profit from an investment activity. Typically, a hedger might invest in a security that he/she believes to be under-priced rela tive to its "fair value", and combines this with a short sale of a related security or securities. Thus the hedger is indifferent to the movements of the market as a whole, and is interested in only the performance of the 'under-priced' security relative to the hedge. 1.1.2 Option Pricing Modern option pricing techniques, usually using st ochastic calculus, are often considered among the most mathematically complex of all applied areas of finance. In 1959, M. F. M. Osborne wrote a paper "Brownian Motion in the Stock Mark et" [36]. In 1964, another paper, by A. James Boness, focused on options. In his work, entitled "E lements of a Theory of Stock Option Value", Boness developed a pricing model that made a significant theoretical jump from that of his predecessors [8]. More significantly, his work serve d as a precursor to that of Fischer Black and Myron Scholes who in 1973 introduced their landmark option pricing model Black Scholes Model [33] 3

PAGE 17

There are two types of option pricing approaches namely discrete and continuous processes. In the following, we briefly describe the discrete time option pricing process. Discrete Time Option Pricing Process (Binomial Tree): We suppose that the market is observable at times 0 = t0 < t1 < t2 < < tN = T. On each time period [ti, ti+1], the stock price follows the binary model. After i time periods, the stock has 2i possible values. We also suppose that the length of any time period has the same length t We define to be the discounted stock process, such that, where r is the interest rate. Figure 1.1.1 is the binomial tree for a stochastic stock price process. 0{}kkSkrt kSeS k Figure 1.1.1 Binomial Tree for a Stochastic Stock Price Process In the following, before we state very importa nt result, we first give some definitions: In probability theory, when we talk about a ra ndom variable, we specify a probability triple (,,) FP where is the sample apace, F is a collection of subsets of ,also called -field, and P is the probability of each event A F 4

PAGE 18

To specify a stochastic process, we required not only a single -field, F but also an increasing sequence of sub algebras, The collection is called a filtration and the quadruple is called a filtered probability space. 1...nnFFF0},)nP 0{}nnF(,,{ FF In probability theory, suppose that 0(,,{},)nFFP is a filtered probability space. The sequence of random variables is a martingale with respect to P and if 0{}nnX 0{}nnF[||],nEX and 1[|]nnEXFX n for all n. Theorem 1.1.1 (The binomial representation theorem) [19]: Suppose that the measure Q is such that the discounted binomial price process NnnS 0} ~ {0{}nnF is a Q-martingale. If is any other (Q,{ Fn}n 0,)-martingale, then there exists an predictable process 0{}nnNV1{}nn (portfolio process) such that 1 011 0()nn. j jj jVVSS Remark 1.1.1 [19]: From Theorem 1.1.1, we know that if 0} ~ { iiV is the discounted price of a claim (European call or put option), then such a predictable process 1}{ ii (portfolio process) arises as the stock holding when we c onstruct out replicating portfolio. There are three steps to pricing and hedging a claim CT at time T: (i) Find a probability measure Q under which the disc ounted stick price (with its natural filtration) is a martingale. (ii) Form the discounted value process, ]|[ ~ iT rtQ i tri iFCeEVeV (iii) Find a predictable process Nii 1}{ such that iiiSV ~ ~ In the following, we present a very fundamental result in the theory of continuous time option pricing process. Theorem 1.1.2 [19]: The fundamental theorem of continuous option pricing is: (i) There is a probability measure Q equivalent to P under which the discounted stock price 0} ~ { ttSis a martingale. 5

PAGE 19

(ii) Under the probability measure Q, suppose that a claim at time T is given by the non-negative random variable If TTFC ][2T QCEThen, the claim is replicable and the value at ti me t of any replicating portfolio is given by, in particular, the fair price at time 0 for the option is ]|[),()( tT tTrQFCeEtSV] ~ [][0 T Q T rTQCECeEV tS is a stock price process, is the discounted stock price process. tS Theorem 1.1.3 Black-Scholes Model [7,19]: Under the following assumptions: (i) There is no credit risk, we can buy and sell cash bond without credit risk. And there is only market risk, which means the stock price can go up and down arbitrarily. (ii) The market is maximally efficient, that is it is infinitely liquid and does not exhibit any friction. This means all relevant information is fully reflected and priced in the stock price, and there are no any other additional costs. (iii) Continuous trading is possible. (iv) The time evolution of the asset price is st ochastic process and called geometric Brownian motion, the mathematical expression is tttdSSdtSdWt is a stock process, and are constant. tS (v) There is no dividend. (vi) The underlying asset is arbitrarily divisible. And the market is arbitrage free, which means the market prices do not allow for profitable arbitrage. The value at time t of a European option whose payoff at maturity ()TTCfS is (,)tVFtS t, where 2 21 2 ()() () 2(,) ( ) 2y rTtyTt rTte Ftxefxe dy 6

PAGE 20

For European call option, suppose that ()()TT f SSK Then, let Tt 12(,)()()rFtxxdKed, where, is the standard normal cumulative distribution function, (.) 2 1log() 2x r K d and 21dd For European put option, suppose that ()()TT f SKS then let Tt 21(,)()()rFtxKedxd, where, is the standard normal cumulative distribution function, (.) 2 1log() 2 x r K d and 21dd The Black-Scholes formula is based on assumption of log-normal stock diffusion with constant volatility, that is, the stock price process is a stochastic process desc ribed by the following stochastic differential equation of the form: tttdSSdtSdWt This has become the universal benchmark for option pricing. But, we are all aware of that it is flawed. The drift and volatility are not a constant In 1973, Merton first allows the drift and volatility to be a deterministic function of time. Later on, other models allow not only time, but also state dependence of the volatility. This method is called as a local volatility approach. There are some very famous local volatility mode ls. For example, Mertons model (1973) takes the form tdrdtdWt [33]. Vasicek (1977) deriving an e quilibrum model of discount bond 7

PAGE 21

price process by using the Ornstein-Uhle nbeck process [41]. It takes the form ()ttdrrdtdWt Dothan (1978) used model ttdrrdWt in valuing discount bonds [18]. And Brennan and Schwartz (1980) used model ()ttrdtdr trdWt in deriving a numerical model for convertible bond prices [11]. These are called linear models. Other nonlinear models such as Cox-Ingersoll-Ross model [15] a nd Black-Karasinski model [6], take nonlinear functions of the asset price at time t as the drift and/or volatility. In the next section, we will introduce how to develop the stochastic process. 1.2 Development of Stochastic Modeling In this section, by following a real stochastic modeling approach [26], we outline the derivation of stochastic model of stock price. This is based on the basic descriptive statistical approach. It utilizes the Random Walk process to initiate a scope and a development of stochastic models of dynamic processes. Here, a state is a conceptua lly common term and description of processes in the sciences and engineering is used, for example, a state can be distance traveled by an object in the physical process, concentration of a chemical substance in a chemical process, number of species in a biological process, and in social science or this thesis, state is the price of an asset in a sociological process. 1.2.1 Conditions of Stochastic Process Random Walk Let St be a price of a stock at time t. The price of the asset is observed over an interval of [t, t+ t], where t is a small increment in t. Without loss in generality, we assume that t is positive. The price process is under the influence of random pe rturbations. We experimentally observe price process 0ttSS 12,,...,ntttttSSSS of a stock at 0tt 1, tt 22, tt , kktt over the time interval [t, t+ t], where n belongs to {1, 2, 3, } and = t/n. These observations are made under the following conditions: ,nttt C1. The stock price is under the influence of independent and identical random impulses that are taken place at 12,,...,,...,kntttt 8

PAGE 22

C2. The influence of a random impact on the stock price is observed on every time subinterval of length C3. For each k I(1,n)={1,2,,k,n}, it is assumed that the stock price is either increased by or decreased by We refer ktS ktS ktS as a microscopic/local experimental or knowledgebase observed increment to the price of the stock per impact on the subinterval of length C4. It is assumed that is constant for k I(1,n) and is denoted by ktS ktS Zk=Z with |Zk|= S >0. Thus, for each k I(1,n), there is a constant random increment Z of magnitude S to the price of the stock per impact on the subinterval of length In short, from the first three conditions, under n independent and identical random impacts, the initial price and n knowledge-base observed random increments Zk of constant magnitude S in the state at over the given interval [t, t+ t] of length t are: 12,,...,,...,kttttnk n0 10 21 1 11 2...... ......kk nntt tt tt tt ttSS SSZ SSZ SSZ SSZ Zks are defined by ,kSforpositiveincrement Z Sfornegativeincrement The 4th condition implies that they are mutually independent random variables. From the above discussion, the prices and are random impacts at the k-th instance and the final time on the price process respectively. Moreover, they are expressed by: ktSntS9

PAGE 23

1kk tktti iSSS Z and 1 n ttti iSS Z where 1 n i i Z is referred as an aggregate increment to the given price tSS of the stock at the given time t over the interval [t, t+ t] of length t. In this case, the aggregate change of the price of the stock tttSS under n observations of the system over the given interval [t, t+ t] of length t is described by 1 n i i ttt nZ t SSn n S, (1.2.1) where 11n n iS niZ. Sn is the sample average of the aggregate price incremental data. 1.2.2 Mean and Variance of Aggregate Change of Price For each random impact and any real number p, 0
PAGE 24

(0,) (0,)|| 1 [()ii iIn iIn n] Z Z Smnm nmnm 1 [() mSnmS n ] 111 [(2)||]n i imnZ nn 1 [(2)]nmnS n (1.2.2) where (0,) I nand (0,) I n are denoted by (0,){(0,):||}ii I niInZZ and (0,(0 }i){ ,):||i I ni IZ nZ respectively, and 11 ||n ni iSZ n Thus from (1.2.1) and (1.2.2), we get 1 (2)n tttS SSmn n t ). (1.2.3) Furthermore, in this case, the aggregate change of price of the stock ()(SttSt over the time interval of length t under n identical random impacts on the stock over the given interval [t, t+ t] of time is also described by: 1n ttti iSS Z = total amount of positive increment total amount of negative increment 1 ()(2)(2)n nS mSnmSmnSmnt n 11

PAGE 25

This is identical with the expression in (1.2.3). So the mean of the aggregate change of the price of the stock over the interval [t, t+ t] is given by: tttSS[]tttESS 01! (2)(1) ()!n mnm n mS n mnppt nnm = ()nS p qt and the variance is: []tttVarSS 2[()()()]nS ESttStpqt 22 0 21 [(2)]P(,)[()] () 4.n nn m nSS mntmnppqt n S pqt (1.2.4) /nS (or S ) and 2()/nS (or 2() S ) are microscopic or local stock average increment and sample microscopic or local average square increment per unit time over the uniform length of sample subinterval [tk-1, tk], k=1,2, n of interval [t, t+ t]. 1.2.3 Wiener Process In reality we note that there are restrictions on S and Similarly, the parameter p cannot be taken arbitrary. Moreover, the price of the stoc k cannot go to infinity on an interval whose length is small. In view of these considerations, for sufficiently large n, the following conditions seem to be natural: = tttSSnS tn 224()()1(2) p qpqpqpq, 12

PAGE 26

2 0() lim[]2nS D 00limlim[()]n SS p qC and 00limlim41SpqHere C and D are certain constants. C is calle d drift, and D is called diffusion coefficient. Moreover, C can be interpreted as the average/mean/e xpected rate of change of price of the stock per unit time, and D can be interpreted as the mean square rate of price change of the stock per unit time over an interval of length t. From the above discussion, we obtain 00limlim[]ttt SESSCt (1.2.5) and 00limlim[]2ttt SVarSSDt (1.2.6) Now, we define 2() (,,) 4()ttt n nSSnpqS ytnt npqS By Central Limit Theorem [37], we conclude that the process (,,) y tnt is approximated by standard normal random variable for each t. Moreover, from t n we have 2() (,,) () 4n ttt nS SSpq ytnt S pqt t Hence, 00limlim(,,)xytnt 2 00() limlim[ ] () 4n ttt S nS SSpqt S pqt t 2tttSSC Dt t. 13

PAGE 27

For fixed t, the random variable 00limlim(,,)S y tnt has standard normal distribution. Now, by rearranging the above expression, we get tttSS 002[limlim(,,SCtDtytnt)] and denoting 00[limlim(,,)]ttt StytntWW tW it can be rewritten as: 2ttt tSSCtD WS, (1.2.7) where Wt is Wiener process. Thus the aggregate price change of the stock under independent and identical random impacts over the given interval [t, t+ t] is interpreted as the sum of the average/mean/expected price change of the stock C t, and the mean square price change of the stock tttS 2tDWdue the random environm ental perturbations. If t is very small, then its differential dt=t and the It-Doob dS is defined by: 2tdSCdtDdWt, (1.2.8) where C and D are the drift and the coefficients, respectively. Equation in (1.2.8) is called the ItDoob type stochastic differential equation. This Random Walk modeling process can be applied to formulate mathematical model in finance. Let St be either a rate of price/value of an asset per unit item/size and per unit time. The specific rate of price/value or the rate of price/value is observed over an interval of [t, t+ t], where t is a small increment in t. Without loss in generality, we assume that t>0. The process is under the influence of exogenous or endoge nous random perturbations of national/international/commerce/trade/monetary/socia l welfare polices. As the result of this, the St is affected by the random environmental pert urbations. By following the development of the above Random Walk Model, its mathematical description is as described in (1.2.8). 14

PAGE 28

Furthermore for very small t, the Random Walk modeling approach leads to the formulation of mathematical model in finance. Usually it takes the following form: tdSdtdWt (1.2.9) We note that if St is the specific rate of the price/value at time t, then ( =C) is called a measure of the average specific rate (per capital growth/decay rate) of the price/value of the asset at the time t, and ( 2=2D) is called the volatility which measur es the standard deviation of the specific rate (per capital growth/decay rate) of the price/ value at a time t over an interval of small length t=dt. Remark 1.2.1 : Here, for the sake of simplicity, we only assume that ntS is constant. Actually, it need not be constant. Moreover, it can be any smooth function of t or St. The expected value of the increment E[ St+tSt] can be replaced by the conditional expected value E[ St+tSt| St]. C and D may be any smooth function of time t and the state S, satisfying certain conditions. We will discuss this issue in the next section. 1.3 General Stochastic Differential Equations In this section, we outline the fundamental result that assures to undertake the study of dynamic modeling. In financial engineering, it is common to model a continuous time price process described by the I toDoob type stochastic differential equation. A ge neral stochastic differential equation takes the form: 00(,)(,),.tttttdSStdtStdWSS (1.3.1) Here, Wt is a Brownian motion, and St > 0, which is the price process. 0tt Under the following smoothness conditions on functions and one can establish the existence and uniqueness of the so lution of process of (1.3.1). 15

PAGE 29

Theorem 1.3.1 (Existence and Uniqueness Theorem) [23]: Suppose that there exist some constants K, L > 0 such that the functions and in (1.3.1) satisfy the following conditions 22(,)(,)(1) StStKS2 2| (Growth Condition) (1.3.2) and 12121|(,)(,)||(,)(,)|| StStStStLSS (Lipschitz Condition) (1.3.3) Then, it can be shown that the stochastic differe ntial equation in (1.3.1) has a unique solution. This is very important and well known in financial engineering, because the unique solution of the stochastic equation in (1.3.1) is a stochastic process adapted to Brownian filtration [23]. 0}{ ttF These two conditions, growth condition (1.3.2) a nd Lipschitz condition (1.3.3) (named after Rudolf Lipschitz), are sufficient conditions, not necessary conditions for the existence and a uniqueness. In this dissertation, all models represen ted by stochastic differential equations must satisfy these two conditions. In equation (1.3.1), it is known that Wt is a Brownian motion, which is continuous everywhere, but it is not differentiable anywhere. To find the information about the solution of the equation (1.3.1), we need a way to take integr al of a stochastic process. In 1951, I to Kiyoshithe published his very famous I tos formula. Theorem 1.3.2 ( I toDoob Formula) [21]: Let f be a function such that its partial derivatives ],),[[ RRbaCf f t f x and 2 2 f x exist and are continuous. Then we have 2 0 2 0001 (,)(0,)(,)(,)(,) 2ttt ts s sfffs f tWfWsWdWsWdssWds xss (1.3.4) Moreover, I toDoobs formula in differential form is represented by 2 21 (,)(,)(,)(,) 2tt ttdftWftWdWftWdtftWdt StS t (1.3.5) This is fundamental result in I toDoob stochastic calculus [10]. We use this formula frequently. 16

PAGE 30

1.4 Least Square Method The credit for discovery of the method of Least squa res is given to Carl Fridrich Gauss who used the procedure in the early part of the nineteenth cen tury [33]. It is the most widely used technique in data analysis. The least square technique can be interpreted as a method of fitting data. The best fit in the least-squares sense is an instance of the model for which the sum of squared residuals has its least value. A residual is the difference between an observed value and the value predicted by the model. Unlike maximum likeli hood [37], the least square estimation does not require the distribution assumption. When the parame ters appear linearly in an expression, then the estimation problem can be solved in closed form. We recall the formula of the linear model that y is related linearly to the regressor variable xs as: 01122...iiik kyxxxi i (1,2,...,;1) innk (1.4.1) The ideal conditions of the least square model are a) i is model error, with mean zero, b) thei are uncorrelated, and have comm on variance (homogeneous variance). 1.5 Maximum Likelihood Estimation Method of ARIMA Model ARIMA(p,d,q) (autoregressive integrated moving average) process provides a very general class of models for modeling and forecasting dynami c phenomena in science and engineering which can be stationarized by applying transformati ons, namely, difference, logarithm, or other transformations. Here, p stands for the number of autoregressive terms, called autoregressive order; d is the order (or degree) of difference of the time series; and q is the number of lagged forecast errors in the prediction equation, called moving average order. ARIMA(p,d,q) models are ARMA(p,q) models with d th -order difference transformation. First, we introduce the difference filter as follows: dB )1( (1.5.1) where B is called backward shift operator, and ,1 ttzBzmtt mzzB, and is a ntzt,...,1, 17

PAGE 31

time series data set. In ARIMA(p,d,q) models, after taking d th -order difference transformation, we suppose that the time series is stationary. For stationary time series, ARMA(p,q) models have following form: ,... ...11 11 qtq ttptp ttz zz (1.5.2) that is, ,)... 1()... 1(2 21 2 21 t q q t p pB BBzBBB or, t tBzB )()( (1.5.3) where )(B and )(B are polynomials of degree p and q in B [10]. Therefore, ARIMA(p,d,q) model can be represented as t t dBzBB )()1)(( (1.5.4) where d, B, )( B and )( B are as defined above. Even though, the values of p and q can be determined by the number of significant spikes in PACF (partial auto correlation function) and ACF (auto correlation function) plots respectively. There are several models that are adequately re presented by a give time series. Hence, criterions such as AIC (Akaikes information criterion) and BIC (Bayesian information criterion) are used to selecte the best model. In our study, we c hoose AIC, because BIC penalizes more with larger data sets. AIC was defined by Akaike in 1973 and takes the following form [3]: kL AIC 2)ln(2 (1.5.5) where, L is maximized value of the likelihood f unction for the estimated model, k is the number of parameters in the model. If the model errors are assumed to be norm ally and independently distributed, RSS is the residual sum of square and is defined as where n is the number of observations. Maximizing the lik elihood, the AIC can be written as n i iRSS1 2 ]1) 2 [ln(2 n RSS nkAIC After simplification and remove the unaffected constant term, AIC is simplifies to: )][ln(2 n RSS nkAIC (1.5.6) 18

PAGE 32

The unconditional log-likelihood function of a ARMA (p,q) model is defined by Box, Jenkins, and Reinsel in 1994 as follows [10]: 2 ),,( )2ln( 2 )ln(2 2 S n L (1.5.7) where, ),,( S is the unconditional sum of residual square, exampled by n M t n t tz Ez E S2 2)],,,|([)],,,|([),,( (1.5.8) where, ),,,|( z Et is the conditional expectedt given z ,,, M is a large integer such that the backforecast increment |),,, z |),,,|(|1z Et t( E is less than any arbitrary predetermined small value for ).1( M t Then problem of parameters estimation in ARIM A model reduces to the problem of finding out how to estimate of and so that2 t),,( Shas minimum value. For example, the backforecasting for ARMA(1,1): Given 1 1 ttttzz we rewrite as ttt tzz 1 1. If we let0 t by giving and we can recursively solve1 t Then parameters and can be estimated as those value which minimize) ,,( S. After obtaining and which maximize the log-likelihood function (1.5.7), the estimator of is computed by 2 n S ) , ( 2 (1.5.9) Applying (1.5.9) and (1.5.7) in (1.5.5) and reducin g the constant in AIC, (1.5.5) is expressed as k nAIC 2) ln(2 (1.5.10) Therefore, we can choose the ARIMA model with smallest AIC. The estimated parameters ,,...,1 p and q ,...,1 by least square and maximum likelihood are not identical, but the difference is always trivial. 19

PAGE 33

Statistical Model Identification Procedure 1.5.1 [38]: Now, we summarize the development of the ARIMA(p,d,q) model as follows: i. Transform the original observations n tSt,...,2,1, into n tSfVt t,...,2,1),( if necessary. ii. Seasonal differences chosen if needed using a variation on the Canova-Hansen test [14]. iii. Check for stationarity of n tVt,...,2,1, by determining the order of differencing d, according to KPSS test [22]. iv. Set 5qp, 3p and 3 q. List all possible set of ).,( qpv. For each possible set of),( qp, applying maximum likelihood method, to estimate the parameters ,,...,1 p and q ,...,1of each model. vi. Computer AIC for each model. vii. Choose the model which has the smallest AIC. 20

PAGE 34

Chapter 2 Linear Stochastic Models 2.0 Introduction Certain stochastic processes are functions of Brownian motion process and have many applications in financial engineering and sciences. Some special processes are solutions of I toDoob type stochastic differential equations. Moreover, such processes also describe the stochastic behavior of an asset price in finance [23]. In this chapter, we introduce the well-known linear stochastic models, which are also called GBM (Geometric Brownian Motion) models. By followi ng the historical model building process, we attempt to develop a stochastic model for stock market price system. As the part of the model building process, we employ two stock prices selected from Fortune 500 companies and one stock Index S&P500. The first step in the classical model building approach is to draw a sketch of the data set. The second step is to use a proper knowledge of the dynamic process and the given data set to estimate the parameters in functions. In section 2.1, we briefly review a basic concep tual model GBM model. We utilize statistical procedure to sketch a stock price data set and to estimate the parameters in the historical GBM model in Section 2.2. The Q-Q plot of residual error of model in Section 2.2 motivates to seek a modified version of GBM. By using the same modeling procedure, we discuss several different results with different data partitioning processes, in Section 2.3. Again, after studying the Q-Q plots of residual errors of models in Section 2.3, we introduce the different data partitioning schemes combined with jumps in Section 2.4. We give other examples in Section 2.5 and Section 2.6 using the same procedure. A few of conclusions are drawn in Section 2.7. The data sets we applied here are the two stocks selected from Fortune 500 companies and the S&P 500 index. The daily adjusted closing prices can be free downloaded from the website http://finance.yahoo.com 21

PAGE 35

2.1 GBM Models In this dissertation, as we noted before, we w ill be following the classica l model building process. For this purpose we use a data about the dynamic process of interest and some prior information about the dynamic process. In our case, we do have a data set about the stock prices selected from Fortune 500 companies and a prior well-known theoretical model GBM model. Our initial attempt is to use the stock price data, the GBM model and the statistical techniques. To use these three basic components of modeling, first we need to perform the reduction process of converting the GBM model into linear regression equation. This reduction technique is systematically outlined in this section. We initiate the usage of a classical modeling appr oach to develop suitable modified stochastic models for the price movement of individual stoc ks. For this purpose, we first utilize the recent trend in the literature that starts with a conceptu al model, and attempt to fit a dataset into it. We begin with utilizing the existing Geometric Brownian Motion (GBM) model and try to fit a dataset into it, and then use the basic statistics to validate the m odel in the statistical framework. The commonly used benchmark for comparison is the well-known Black-Scholes model which is based on Geometric Brownian motion [32]. St is called GBM (Geometric Brownian Motion) process, that is, the solution of the following linear I toDoob type stochastic differential equation tttdSSdtSdWt (2.1.1) where and are some constants, is called drift, is called volatility, and Wt is a normalized Brownian motion process. Let K be any number greater than (22 ), and L be any number greater than ||| | From this we conclude that equa tion (2.1.1) satisfies conditions (1.3.2) and (1.3.3). Applying I to Doob s formula applied to () lntt f SS we have 21 () 2 0ttW tSSe, (2.1.2) 22

PAGE 36

where, Wt is a Brownian motion process as usual. is also called exponential Brownian motion process, since St takes the exponential form of Wt. As we already introduced in Chapter 1, one of the most important assumptions in Black-Scholes model [7] is that the stock price process is GBM process. tS We want to use the historic stock data set to ex amine the GBM model (2.1.1), that is, we want to estimate the parameters and Here, we try to use the least square method to estimate parameters in the GBM model. In equation (2.1.1), the error term does not have common variance. It is related to St. This means that as the stock price increases, the variance also increases. With a transformation using lntV tS I toDoob s formula, we obtain 2 2 21 (ln)((ln))() 2tt tt ttdVSdS SdS SS t 2 2 222 2 2111 ()() 2 11 () 2 1 (). 2tt tt tt t tdS dS SS dtdWSdW S dtdW t (2.1.3) Thus, 21 (ln)() 2ttdS dtdW (2.1.4) From the Euler type discretization [24] of the st ochastic differential equa tion (2.1.4), we have 2 111 lnln()() 2tt ttSStWW (2.1.5) Let 1lnlntt t y SS, 1tttWW and we know 1 t equation (2.1.5) can be rewritten as 21 2tyt (2.1.6) and are parameters that we want to estimate. 23

PAGE 37

According to the properties of standard Brownian motion process, for each n 1, and any sequence of time the random variables 010......itttt n1iittWW are independent, 1tttWW has the standard normal distribution wi th mean zero and variance 1. Thus the conditions of least square estimations are satisfied. 21 2 can be estimated as the average value of t y which is 1lnlnttSS can be estimated as the standard deviation of [35]. We will use this least square estimation in our work for both linear and nonlinear models. 1lnlnttSS Remark 2.1.1 : An alternative way to estimate the para meters is as described below. Since 1tttWW is standard normal distribution, is log-normally distributed with mean tS 21 2 and variance 2. Then we can estimate the drift and volatility parameters by using the historical price data. Taking the logarithm of we can estimate tS 21 2 as the average values of and can estimate the volatility by taking the standard deviation of [35]. This is exactly the same as what we have estimated by using least square method. 1lnlnttSS1lnlnttSS 2.2 GBM Model on Overall Data In this section, by using fortune 500 compani es price dataset, we estimate the parameters 21 2 and 2 in (2.1.6). This is achieved in the fram ework of the overall price data of stock X. Suppose we let St be the daily adjusted closing values of stock X that we collect form the fortune 500 companies that we mentioned early. A plot of the actual data set is drawn in Figure 2.2.1. 24

PAGE 38

0 200 400 600 800 100200300400500600700 TimePrice Figure 2.2.1 Daily Adjusted Closing Price Process for Stock X We pick one stock X over long period (3 years) of time to build its GBW model. The Figure 2.2.1 shows its daily adjusted closing price process from 8/19/2004 to 12/31/2007. Using the least square method described in Section 1.4, the estimates of drift and volatility are as follows 0.002501028 and 0.02107507 Hence the GBM process for the stock X pri ce is the solution of the following linear I toDoob stochastic differential equation: 0.0025010280.02107507ttdS Sdt SdW t t. (2.2.1) The stock price process is 20.02107507 (0.002501028 )0.02107507 2 0ttW tSSe (2.2.2) In equation (2.2.2), is the initial stock price of the price of the stock process. Wt is Brownian motion, that is, it is a random process. Under direct simulation of the stock price process as we generate the Brownian motion, we get the different values. We first use the Monte Carlo method [34] to predict the stock price process and then calcu late the average of the process. This is a very 0S 25

PAGE 39

general approach that is used in many areas, su ch as physics, chemistry, finance etc. Here, we simulate the stock price process 2000 times. Using Monte Carlo method a plot of the stock price process of (2.2.1) is given in Figure 2.2.2. Th e red curve in Figure 2.2.2 represents the result using Monte Carlo simulation method. The process in blue curve is one resulted from simulation which varies from time to time. 0 200 400 600 800 200400600800 TimePrice TimePrice 0 200 400 600 800 200400600800 0 200 400 600 800 200400600800 TimePrice Raw Data One Possible Path MC Prediction Figure 2.2.2 Prediction and One Possible Path of Stock Xs Price Process Using Model (2.2.1) After we estimate the parameters, is estimated by tS 2 1 2 1 ln ln t tSS. The basic statistics reflecting the accuracy of model in Equation (2.2.1) are mean of the residuals r, variance of the residuals, and standard deviation of the residuals, where residual errors are defined as Table 2.2.1 shows these basic statistics. 2rStrrSttSS Table 2.2.1 Basic Statistics for Model in Equation (2.2.1) r 2 rS rS No. of Parameters 28.29653 8752.84 93.55661 2 26

PAGE 40

To test the homogeneous errors in equation (2.1 .6), actually, we assume that the error term is normally distributed. We use Q-Q plot to test it. In statistics, a Q-Q plot ("Q" stands for quantile ) is a graphical method for diagnosing differences between the probability distribution of a population from which a random sample has been drawn and a comparison distribution. An example of this kind of difference that can be tested is non-normality of the population distribution. The normal distribution is represented by a straight line. The Q-Q plot is in Figure 2.2.3 -3-2-10123 -4-20246 Normal Q-Q plotTheoretical QuantilesSample Quantiles Figure 2.2.3 Q-Q Plot for Model in Equation (2.2.1) Remark 2.2.1 : From the table 2.2.1, the average residua l is 28.29653. This means that zero mean condition obviously is not satisfied. Also, the varian ce is too large. From the Figure 2.2.2, we can see that the prediction line (in red) cannot describ e the stock process. Furthermore, we can see a reverse S shape in the Q-Q plot. All these observati ons suggest that we need a more work to get the better model. 2.3 GBM Models under Data Partitioning Schemes without Jumps The usage of the overall data in estimating the para meters in (2.1.6) suggests to modify the usage of data. The estimated parameters in secti on 2.2 are not realistic. This has been 27

PAGE 41

evidenced by the Q-Q plot test for the homogene ous of error in (2.1.6) over the entire period of the data. As a result of this, it is natural to partition the data, and repeat the procedure outlined in Section 2.2. In this section, we will use the same stock price process under different data partitioning to develop the GBM models. The data is reorganized half-yearly, quarterly, and monthly to build GBM models on different segments of pe riods of the overall period of dataset. If we revise the dataset more closely, we will fi nd some pattern. Figures 2.3. 1-2.3.4 show that the daily difference of stock X in 4 quarters fro m August 2004 to end of year 2007. The daily differences in quarter 2 (Q2) and quarter 3 (Q3) are in the range [-20, 20]. The daily differences in quarter 1 (Q1) and quarter 4 (Q4) are much bi gger than those in quarter 2 (Q2) and quarter 3 (Q3). Also in a particular quarter, most of the daily differences follow the similar pattern. 01 02 03 04 05 06 0 -40-2002040 Dialy Difference (Q1)Difference 01 02 03 04 05 06 0 -40-2002040 Dialy Difference (Q1)Difference 01 02 03 04 05 06 0 -40-2002040 Dialy Difference (Q1)Difference 2005 2006 2007 Figure 2.3.1 Daily Difference of Stock X in Q1 28

PAGE 42

0102030405060 -20-1001020 Dialy Difference (Q2)Difference 0102030405060 -20-1001020 Dialy Difference (Q2)Difference 0102030405060 -20-1001020 Dialy Difference (Q2)Difference 2005 2006 2007 Figure 2.3.2 Daily Difference of Stock X in Q2 0102030405060 -30-20-1001020 Dialy Difference (Q3)Difference 0102030405060 -30-20-1001020 Dialy Difference (Q3)Difference 0102030405060 -30-20-1001020 Dialy Difference (Q3)Difference 2005 2006 2007 Figure 2.3.3 Daily Difference of Stock X in Q3 29

PAGE 43

0102030405060 -40-2002040 Dialy Difference (Q4)Difference 0102030405060 -40-2002040 Dialy Difference (Q4)Difference 0102030405060 -40-2002040 Dialy Difference (Q4)Difference 0102030405060 -40-2002040 Dialy Difference (Q4)Difference 2004 2005 2006 2007 Figure 2.3.4 Daily Difference of Stock X in Q4 Table 2.3.1 also shows the standard deviations in Q1 and Q4 are larger than the standard deviations in Q2 and Q3. This suggests us that we might reorganize the sample dataset into two sub data sets Q2 and Q3 as subset 1 and Q1 and Q4 as subset 2. Furthermore, we also divide the sample dataset into 4 sub datasets -4 quarters. For each subset, we use the same method, which is described in Section 2.2 to develop its GBM model, separately. Table 2.3.1 Mean and Standard Deviation of Daily Differences of 4 Quarters of Stock X 2007 2006 2005 2004 mean s.d. mean s.d. mean s.d. mean s.d. Q1 -0.038 7.468 -0.401 12.496 -0.201 4.489 NA NA Q2 1.024 5.263 0.466 7.511 1.776 4.963 NA NA Q3 0.595 6.573 -0.152 5.569 0.349 4.483 NA NA Q4 1.941 13.637 0.930 7.947 1.562 7.659 0.987 5.843 30

PAGE 44

Data Partition Process 2.3.1 : From the description of constr uction of Figures 2.3.1-2.3.4 and Table 2.3.1, we partition the overall data set into two sub datasets. represent the quarter year time intervals starting Q1, Q2, Q3 and Q4 etc. The sub dataset 1 contains obser vations in the Q1 or the Q4. The sub dataset 2 contains observations in the Q2 or the Q3. 11223[0,),[,),[,), ttttt [3445,),[,)... tttt GBM Model without Jumps 2.3.1(Half Yearly GBM Model without Jumps): The GBM processes without jumps using Data Partition Process 2.3.1 are the solutions of the following linear I toDoob type stochastic differential equation: if t is in Q1 or Q4, 14 1414 1414QQQQQ tttdSSdtSdWt t 00SS 23 2323 2323QQQQQ tttdSSdtSdW, if t is in Q2 or Q3. (2.3.1) 14Q and 23Q are drifts, and 14Q and 23Q are volatility rates for two sub datasets, respectively. By following definition [16, 26, 27], the price pro cess is the solution of (2.3.1), it take the form 2 141414 14 2 232323 23 14 1 2 141414 23 14 3 2 232323 231 (()) 2 00 0 1 (()) 2 11 1 (()) 2 22 1 (()) 2 33,0 lim, lim, limQQQ t QQQ t QQQ t QQQ ttW Q t tW Q Q tt tt tW Q Q t tt tt tW Q t tSSe SSt SSe SStt S SSe SStt SSe S 23 557, ... ......Q t tStt 1 1 3 3 5t t t t (2.3.2) 0S is the initial value of the price process. There are 4 parameter14Q 14Q and 23Q 23Q n to be estimated. eed For stock X, the estimated results are as following 31

PAGE 45

, if t is in Q1 or Q4. 14 14 14 0.0022841410.02447219QQ ttdS Sdt SdW Q t tQ t t 23 23 23 0.0027337290.01671308QQ ttdS Sdt SdW if t is in Q2 or Q3. (2.3.3) And the estimated stock Xs price process is: 2 140.02447219 (0.002284141 )0.02447219 2 0ttW Q tSSe, if ; 1[0,) tt 2 23 14 10.01671308 (0.002733729 )0.01671308 2 (lim)ttW Q Q tt ttSSe if tt; 13[,) t 2 23 14 30.02447219 (0.002284141 )0.02447219 2 (lim)ttW Q Q tt ttSSe if tt; 35[,) t (2.3.4) The prediction result of stock Xs price process of in (2.3.3) is provided in Figure 2.3.5. We see that the blue curve and red curve are very close. Because the two drifts in Equation (2.3.3) are very close. The most different part in Equation (2.3.3) is the volatilities. 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice Raw Data Overall GBM Q14 and Q23 GBM Figure 2.3.5 Comparison on Model (2.2.1) with Model (2.3.3) of Stock X 32

PAGE 46

Table 2.3.2 Basic Statistics for Model in Equation (2.3.2) Model r 2 rS rS No. of Intervals No. of Parameters GBM with Overall Data 28.29653 8752.84 93.55661 1 2 Q14 and Q23 GBM without jumps 29.67727 8836.837 94.00445 8 4 -3 -1 123 -4-20246 Q-Q Plot for Q1 & Q4Theoretical QuantilesSample Quantiles -3-10123 -4-20246 Q-Q Plot for Q2 & Q3Theoretical QuantilesSample Quantiles Figure 2.3.6 Q-Q Plot for Model in Equation (2.3.3) Remark 2.3.1 : From Figure 2.3.6, we still can see there are reverse S shapes in the two Q-Q plots for both Q1 and Q4, and Q2 and Q3. Table 2.3.2 provides the basic statistics. And from Figure 2.3.5, we dont see an improvement from GBM model on overall data. All these suggest that we need more work to get the better model. In the following, we try to reorganize the dataset into 4 sub datasets quarter 1 (Q1), quarter 2 (Q2), quarter 3 (Q3), and quarter 4 (Q4). Data Partition Process 2.3.2 : The time intervals as defined in data partition process 2.3.1. The sub datasets 1, 2, 3, and 4 contain observations in Q1, Q2 Q3, and Q4, respectively. 11223[0,),[,),[,), ttttt [3445,),[,)... tttt 33

PAGE 47

The GBM Model without Jumps 2.3.2 (Quarterly GBM Model without Jumps): The GBM processes without jumps using Data Partition Process 2.3.2 are the solutions of the following linear I toDoob type stochastic differential equation: if t is in Q1. 11111QQQQQ tttdSSdtSdWtttt 22222QQQQQ tttdSSdtSdW, if t is in Q2. 33333QQQQQ tttdSSdtSdW, if t is in Q3. 44444QQQQQ tttdSSdtSdW, if t is in Q4. (2.3.5) 1Q 2Q ,3Q and4Q are drifts, and 1Q ,2Q ,3Q 4Q are volatilities for four quarters respectively. By following definition [16, 26, 27], the price process is the solution of Equation (2.3.5), and takes the form 2 333 3 2 444 3 4 1 2 111 1 4 2 2 222 2 1 31 (()) 2 00 0 1 (()) 2 11 1 (()) 2 22 1 (()) 2 33,0 lim, lim, lim,QQQ t QQQ t QQQ t QQQ ttW Q t tW Q Q tt tt tW QQ tt tt t tW QQ tt tt Q tSSe SSt SSe SStt SSe SStt S SSe SStt S 1 1 2 2 3 3 4t t t t 2 333 2 3 41 (()) 4 2 45 4lim ... ... ...QQQ tQ tW t ttSS ttt Se (2.3.6) There are 8 parameters 1Q 2Q ,3Q ,4Q 1Q ,2Q ,3Q and 4Q These parameters need to be estimated. For stock X, the estimated results are as following: 34

PAGE 48

11 0.00041859440.02459217QQ ttdS Sdt SdW 1Q t t2Q t t3Q t t4Q t t, if t is in Q1. 22 0.0037920020.01712985QQ ttdS Sdt SdW if t is in Q2. 33 0.0018153370.01632730QQ ttdS Sdt SdW if t is in Q3. 44 0.0042386250.02424493QQ ttdS Sdt SdW if t is in Q4. (2.3.7) Following the earlier arguments, Figure 2.3.7 exhibists the result of prediction of stock Xs price process of (2.3.7). We note that the red curve ( quarterly GBM model) is not similar to the blue curve (Overall GBM model) as well as orange curve (Q14 and Q23 GBM model). This is due to obvious reasons 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice Raw Data Overall GBM Q14 and Q23 GBM Q1 Q2 Q3 and Q4 GBM Figure 2.3.7 Comparison on Model (2.2.1), (2.3.3) with Model (2.3.7) of Stock X Table 2.3.3 provides the statistics for 3 models namely, overall GBM Model, Q14 and Q23 GBM model and Quarterly GBM model. 35

PAGE 49

Table 2.3.3 Basic Statistics for Model in Equation (2.2.1) (2.3.3) and (2.3.7) Model r 2 rS rS No. of Intervals No. of Parameters GBM with Overall Data 28.29653 8752.84 93.55661 1 2 Q14 and Q23 GBM without jumps 29.67727 8836.837 94.00445 8 4 Quarterly GBM without jumps 53.49948 5570.643 74.63674 14 8 -3-2-10123 -4-20246 Q-Q Plot for Q1Theoretical QuantilesSample Quantiles -3-2-10123 -4-20246 Q-Q Plot for Q2Theoretical QuantilesSample Quantiles -3-2-10123 -4-20246 Q-Q Plot for Q3Theoretical QuantilesSample Quantiles -3-2-10123 -4-20246 Q-Q Plot for Q4Theoretical QuantilesSample Quantiles Figure 2.3.8 Q-Q Plot of Model (2.3.7) 36

PAGE 50

Remark 2.3.2 : (a) From Figure 2.3.7, we notice the large deviation between the predicted result and the observed data set. (b) From Table 2.3.3, we note that the quarterly partition data set approach gives the least variance with the largest mean of the residuals. (c) Figure 2.3.8 is the QQ plots for model (2.3.7). We observe that there is still a reverse S shape in the Q-Q plot for Q2 and Q4. In Q1 and Q3, most points fall in the normal distributions and there are a few outliers. (d) Again, after careful review of the Figures 2. 3.1-2.3.4, we found some patterns. The daily differences in quarter 1 (Q1) and quarter 4 (Q4) are much larger than those differences with regards to in quarter 2 (Q2) and quarter 3 (Q 3). The daily differences do not follow the same pattern in the same quarter in different year, that is, the dynamic of stock price in the same quarter with different year follows different pattern. As a result of this, we develop two kinds of data partitioning schemes, we dont put the observations in different years together. Data Partition Process 2.3.3 : Let be a monthly sub intervals for m month data set. The sub dataset 1 contains observations in the 1st month, the sub dataset 2 contains observations in the 2nd month, the sub dataset 3 contains observations in the 3rd month, the sub dataset m contains observations in the m-th month. 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,)mmtt GBM Model without Jumps 2.3.3 (Monthly GBM Model without Jumps): Let be the m monthly sub intervals. The GBM process without jumps is the solution of the following linear 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,mmtt) I toDoob type stochastic differential equation: iiiiiMMMMM tttdSSdtSdWt00SS, if 1 iittt 1,..., im (2.3.8) Here,i M and i M , are monthly drift and volatility co efficients, respectively. Again, by following definition [16, 26, 27], the price pr ocess is the solution of (2.4.8), and takes the following form: 1,...,im 37

PAGE 51

2 111 1 2 222 2 1 1 2 1 11 (()) 2 00 0 1 (()) 2 11 1 (()) 2 11, lim, ... ... ... lim,MMM t MMM t MMM mmm t m m mtW M t tW MM tt tt t tW MM tm mtm ttSSe SSttt SSe SSttt S SSe SSttt 0 1 1 2 1 m (2.3.9) There are parameters 2 m i M and i M ,1,..., im need to be estimated. m is the number of month of stock price process. The methods of estimation parameters are the same. For stock X, the estimated results are as following. ).,[, 0.01499199 25900000.0 ...... ... );,[ 0.02181682 0.01146937 );,0[ 0.03775822 0.00321650 4140 21 141 41 41 2 2 2 1 1 1tttifdWS dtS Sd tttifdWS dtS Sd ttifdWS dtS Sdt M t M t M t t M t M t M t t M t M t M t (2.3.10) All estimated parameters i M and i M ,1,..., im are given in Appendix A1. Figure 2.3.9 is the prediction of stock Xs prices process. Table 2.3.4 provides the basic statistics for estimated model corresponding to the original Equation (2.3.8). 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice Raw Data Overall GBM Q14 and Q23 GBM Q1 Q2 Q3 and Q4 GBM Monthly GBM Figure 2.3.9 Comparison on Model (2.2.1), (2.3.3), (2.3.7) with Model (2.3.10) of Stock X 38

PAGE 52

Table 2.3.4 Basic Statistics for Model in Equation (2.2.1) (2.3.3) (2.3.7) and (2.3.10) Model r 2rS rS No. of Intervals No. of Parameters GBM with Overall Data 28.29653 8752.84 93.55661 1 2 Q14 and Q23 GBM without jumps 29.67727 8836.837 94.00445 8 4 Quarterly GBM without jumps 53.49948 5570.643 74.63674 14 8 Monthly GBM without jumps -80.10483 11754.25 108.4170 41 82 Remark 2.3.3 : (a) From Figure 2.3.9 we can see that the monthly GBM model in red really catches the dynamic of the stock price process. (b) The stock price process shows it is always over predicted. The basic statistics in Table 2.3.4 shows that the variance and standard deviation of the residual are very large in this monthly GBM model. Data Partition Processes 2.3.1-2.3.3 have a common character, that is, the length of time interval in each model is exactly the same. For examples, the length of time interval in Data Partition Process 2.3.1, 2.3.2, and 2.3.3 ar e two quarters, one quarter, and one month respectively. If there is a big shock in the stock price in one of the intervals, this kind of equal length model cannot incorporate the effects of the big shock. To a void this problem, we provide a modified data partition process, this allows us to have unequal length of intervals. Data Partition Process 2.3.4 : Let be the data set time decomposition into n time intervals. We suppose all the big shocks come at times The sub dataset 1 contains observations in the 1st time interval, that is in the sub dataset 2 contains observations in the 2nd interval, that is the sub dataset 3 contains observations in the 3rd interval, that is , the sub dataset n contains observations in the nth interval, that is 11223[0,),[,),[,), ttttt [23[,) tt3445,),[,)... tttt12[,) tt1[,)nntt 121,,...nttt 1[0,) t1[,)nntt 39

PAGE 53

GBM Models without Jumps 2.3.4 (Unequal Interval GBM Model without Jumps): By utilizing the above described sub interval decomposition, GBM processe without jumps is the solution of the following linear I toDoob type stochastic differential equation: iiiiiIIIII tttdSSdtSdWt00SS, if 1iittt 1,..., in (2.3.11) iI and iI , are the i-th drift and ith volatility coefficients, respectively. By following the definition [16, 26, 27], The solution to Equation (2.3.11) is, and takes the following form: 1,..., i n 2 111 1 2 222 2 1 1 2 1 11 (()) 2 00 0 1 (()) 2 11 1 (()) 2 11, lim, ... ... ... lim,III t III t III nnn t n n ntW I t tW II tt tt t tW II tn ntn ttSSe SSttt SSe SSttt S SSe SSttt 0 1 1 2 1 n (2.3.12) There are parameters 2 n iI and iI ,1,..., in and these parameters need to be estimated. Now the key issue is how to define unequal length of time interval. The basic idea about defining the time intervals is that we want to identify the dates, having the large daily relative difference. So we need to define the threshold first, that is we need to define the threshold of daily relative difference of stock price. There are two issues that we want to consider. The first issue is that the threshold cannot be either too large or too small. This is because of the fact that if the threshold is too large, then we may have too few intervals, and it cannot incorporate the dynamic of stock price process. Therefore, we cannot have a good mode l. If the threshold is too small, then we may have too many intervals, that is, for some time intervals, there are few observations so that we cannot reasonably develop a model. The second issue is, after defining the threshold, the lengths of some time intervals are still too long. In th is case, we break these time intervals into months, since monthly GBM model shows very good dyna mic character. Once we define unequal length of time intervals, we apply the same procedure to estimate parameters. 40

PAGE 54

The method of estimation parameters is as described in chapter 1. For stock X, the estimated results are as ).,[, 0.014992 25900000.0 ...... ... );,[ 0.028731 0.013391 );,0[ 0.024052 0.006501 3837 21 138 38 38 2 2 2 1 1 1tttifdWS dtS Sd tttifdWS dtS Sd ttifdWS dtS Sdt I t I t I t t I t I t I t t I t I t I t (2.3.13) The estimated parameters iI and iI ,1,..., in are given in Appendix A2. Figure 2.3.10 is the predicted stock Xs price process. Table 2.3.5 provides the basic statistics for estimated model corresponding to (2.3.13). 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice Raw Data Overall GBM Q14 and Q23 GBM Q1 Q2 Q3 and Q4 GBM Monthly GBM Unequal Interval GBM Figure 2.3.10 Comparison on Model (2.3.1), (2.3.3), (2.3.7), (2.3.10) with. Model (2.3.13) of Stock X Table 2.3.5 Basic Statistics for Model in Equation (2.2.1), (2.3.3), (2.3.7), (2.3.10) and (2.3.13) Model r 2 rS rS No. of Intervals No. of Parameters GBM with Overall Data 28.29653 8752.84 93.55661 1 2 41

PAGE 55

Q14 and Q23 GBM without jumps 29.67727 8836.837 94.00445 8 4 Quarterly GBM without jumps 53.49948 5570.643 74.63674 14 8 Monthly GBM without jumps -80.10483 11754.25 108.4170 41 82 Unequal Interval GBM without jumps 24.91557 3992.349 63.18504 39 78 Remark 2.3.4 : Figure 2.3.10 shows that the Monthly GBM model (dashed red curve) and Unequal interval GBM model (solid red curve) are approximations of the true stock price movements in comparison to other linear models However, Table 2.3.5 shows all these 5 models have very large residuals. This is largely due to the accumulated e rrors in models without jumps. When we make a prediction, we only use the stock price at time 0 as the initial value to predict a long time behavior of the stock price. In section 2.4, we will add jumps to this model to reduce the cumulative error. 2.4 GBM Models under Data Partitioning Schemes with Jumps All models in Section 2.3 are without jumps, that is, we take the left limit of the right endpoint of previous time interval as the initial value of the next time interval. This simplistic approach carries the previous time interval error to next time interval. The cumulated error might be very big. Here, we modify the models of Section 2. 3 by adding jumps into the models. The data partition processes and other parameters such as dr ifts and volatilities remain the same. We will not repeat in this section. GBM Model with Jumps 2.4.1 (Half Yearly GBM Model with Jumps): Let be the time intervals as defined in Data Partition Process 2.3.1. By following the argument, the GBM so lution process with jumps of (2.3.1) has the following form: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt 42

PAGE 56

2 1414 14 14 2 2323 23 23 14 1 2 1414 14 23 14 3 2 2323 23 231 (()) 2 00 0 1 (()) 2 11 1 13 1 (()) 2 22 2 35 1 (()) 2 33,0 lim, lim,QQQ t QQQ t QQQ t QQQ ttW Q t tW Q Q tt tt tW Q Q t tt tt tW Q tSSe SSt SSe SStt S SSe SStt SSe S 23 535lim, ... ... ...Q t ttStt 1t t t 7t (2.4.1) Here, 1 2 ,3 are jump coefficients co rresponding to jump times and can be estimated as ,...,,531ttt 1 14 11 limt Q t ttS S, 3 23 32 limt Q t ttS S, 5 14 53 limt Q t ttS S, GBM Model with Jumps 2.4.2 (Quarterly GBM Model with Jumps): Let be the time intervals as defined in Data Partition Process 2.3.2. Again, the GBM solution process with jumps of (2.3.2) has the following form: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt 2 333 3 2 444 3 4 1 1 2 111 1 4 2 2 222 2 1 31 (()) 2 00 0 1 (()) 2 11 1 12 1 (()) 2 22 2 23 1 (()) 2 33 3 3,0 lim, lim, lim,QQQ t QQQ t QQQ t QQQ ttW Q t tW Q Q tt tt tW QQ tt tt t tW QQ t t ttSSe SSt SSe SStt SSe SStt S SSe SSt 1t t t 2 333 2 3 44 1 (()) 4 2 45 44lim ... ... ...QQQ tQ tW Q t tt ttt SS ttt SSe (2.4.2) Here, 1 2 ,3 are jump coefficients co rresponding to the jump time and can be estimated as ,...,,,4321tttt 1 1 11 limt Q t ttS S, 2 2 22 limt Q t ttS S, 3 3 33 limt Q t ttS S, 4 4 44 limt Q t ttS S, 43

PAGE 57

Figure 2.4.1 is the result of prediction of stock Xs price process of (2.3.3) with jumps and of (2.3.7) with jumps. We see that the red and blue curves are not as smooth as green and orange curves, this is because of the fact that there are jumps in green and orange curves of (2.3.3) and (2.3.7), respectively with respect to jumps. It is obvious that models with jumps provide better predicted results. 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice Raw Data Q14 and Q23 GBM with No Jump Q14 and Q23 GBM with Jumps Q1 Q2 Q3 and Q4 GBM with No Jump Q1 Q2 Q3 and Q4 GBM with Jumps Figure 2.4.1 Comparison of Models (2.3.3), (2.3.7) with and without Jumps of Stock X Table 2.4.1 provides the basic statistics that refl ects the accuracy of model (2.3.1), (2.4.1) and (2.3.5) with jumps. Table 2.4.1 Basic Statistics for Linear Models (2.2.1), (2.3.3), (2.3.7), (2.3.10), (2.3.13) and with Jumps (2.3.3), (2.3.7) of Stock X Model r 2 rS rS No. of Intervals No. of Parameters GBM with Overall Data 28.29653 8752.84 93.55661 1 2 Q14 and Q23 GBM without jumps 29.67727 8836.837 94.00445 8 4 44

PAGE 58

Quarterly GBM without jumps 53.49948 5570.643 74.63674 14 8 Monthly GBM without jumps -80.10483 11754.25 108.4170 41 82 Unequal-Interval GBM without jumps 24.91557 3992.349 63.18504 39 78 Q14 and Q23 GBM with jumps 1.759521 3181.759 56.40708 8 11 Quarterly GBM with jumps -10.26338 1450.633 38.08717 14 21 Remark 2.4.1 : From the Figure 2.4.1 and Table 2.4.1, we notice that models with jumps are much better than models without jumps. Moreover, the quarterly data partition has better result than half yearly data partition. GBM Models with Jumps 2.4.3 (Monthly GBM Model with Jumps): Let be the m monthly time intervals as defined in data partition process (2.3.3). The GBM solution pro cess with jumps of (2.3.10) takes the form: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,mmtt) 2 111 1 2 222 2 1 1 2 1 11 (()) 2 00 0 1 (()) 2 11 1 12 1 (()) 2 11 1 1, lim, ... ... ... lim,MMM t MMM t MMM mmm t m m mtW M t tW MM tt tt t tW M M tmm mtm ttSSe SSttt SSe SSttt S SSeSStt 0 1 mt (2.4.3) Here, 1 2 ,3 are jump coefficients and can be estimated as 1 1 11 limt M t ttS S, 2 2 22 limt M t ttS S, 3 3 33 limt M t ttS S, 4 4 44 limt M t ttS S, The methods of estimation parameters are the same as we mentioned in Section 2.3. For stock X, 45

PAGE 59

the estimated parameters i M and i M ,1,..., im are given in Appendix A1, the estimated jump coefficients 1 2 ,3 1m are provided in Appendix A3. Figure 2.4.2 is the prediction of stock Xs prices process. Table 2.4.2 provides the basic statistics of model (2.3.13) with jumps. 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice Raw Data Half Yearly GBM with Jumps Quarterly GBM with Jumps Monthly GBM no Jumps Monthly GBM with Jumps Figure 2.4.2 Comparison of Models (2.3.10) and (2.3.13) with and without Jumps of Stock X Table 2.4.2 Basic Statistics for Linear Models (2 .2.1), (2.3.3), (2.3.7), and with and without Jumps Models (2.3.10), (2.3.13) of Stock X Model r 2 rS rS No. of Intervals No. of Parameters GBM with Overall Data 28.29653 8752.84 93.55661 1 2 Q14 and Q23 GBM without jumps 29.67727 8836.837 94.00445 8 4 Quarterly GBM without jumps 53.49948 5570.643 74.63674 14 8 46

PAGE 60

Monthly GBM without jumps -80.10483 11754.25 108.4170 41 82 Unequal Interval GBM without jumps 24.91557 3992.349 63.18504 39 78 Q14 and Q23 GBM with jumps 1.759521 3181.759 56.40708 8 11 Quarterly GBM with jumps -10.26338 1450.633 38.08717 14 21 Monthly GBM with jumps -1.22683 207.3278 14.3989 41 122 Remark 2.4.2 : The solid red curve in Figure 2.4.2 follows the same dynamic pattern as the dashed red curve. The only difference between these two curves is that Monthly GBM with Jumps model doesnt accumulate large error, while the models without jumps do accumulate large errors. This can also be further confirmed from basic statistics in Table 2.4.2. The monthly GBM model with jumps has the least mean, variance, and standard error of residual error. GBM Model with Jumps 2.4.4 (Unequal Interval GBM Model with Jumps): Let be the n time intervals as defined in Data Partition Process (2.3.4). Similarly, by following defin ition [16, 26, 27], the GBM solution processes with jumps of (2.3.11) has the following form: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,nntt) 2 111 1 2 222 2 1 1 2 1 11 (()) 2 00 0 1 (()) 2 11 1 12 1 (()) 2 11 1 1, lim, ... ...... lim,III t III t III nnn t n n ntW I t tW II tt tt t tW II tnn ntn ttSSe SSttt SSe SSttt S SSeSStt 0 1 nt (2.4.4) There are parameters 2 n iI and iI ,1,...,in and these parameters need to be estimated. n is the number of intervals of stock price process. We adapt the earlier procedure to create unequal 47

PAGE 61

intervals, and estimate the drifts and volatilities as in Section 2.3. Here, 1 2 ,3 are jump coefficients corresponding to jump times at and can be estimated as ,...,,321ttt 1 1 11 limt I t ttS S, 2 2 22 limt I t ttS S, 3 3 33 limt I t ttS S, 4 4 44 limt I t ttS S, For stock X, we use the estimated parameters iI and iI ,1,...,in (Appendix A2), and the estimated jump coefficients 1 2 ,3 1m (Appendix A4). Figure 2.4.3 is the predicted process of stock X. Table 2.4.3 provides the basic statistics for model in Model (2.3.13) with jumps. 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice 0 200 400 600 800 200400600800 TimePrice Raw Data Monthly GBM no Jumps Monthly GBM with Jumps Unequal Interval GBM no Jumps Unequal Interval with Jumps Figure 2.4.3 Comparisons of Models with a nd without jumps (2.3.10), (2.3.13), (2.4.3) of Stock X Table 2.4.3 Basic Statistics for Linear Models (2.2.1) and Models with and without Jumps (2.3.3), (2.3.7), (2.3.10), (2.3.13) of Stock X Model r 2rS rS No. of Intervals No. of Parameters GBM with Overall Data 28.29653 8752.84 93.55661 1 2 48

PAGE 62

Q14 and Q23 GBM without jumps 29.67727 8836.837 94.00445 8 4 Quarterly GBM without jumps 53.49948 5570.643 74.63674 14 8 Monthly GBM without jumps -80.10483 11754.25 108.4170 41 82 Unequal Interval GBM without jumps 24.91557 3992.349 63.18504 39 78 Q14 and Q23 GBM with jumps 1.759521 3181.759 56.40708 8 11 Quarterly GBM with jumps -10.26338 1450.633 38.08717 14 21 Monthly GBM with jumps -1.22683 207.3278 14.3989 41 122 Unequal Interval GBM with jumps -1.962899 258.1040 16.06562 39 116 Remark 2.4.3 : In Table 2.4.3 we remark that overall the Monthly GBM Model with jumps and Unequal Interval GBM model with jumps, relativ ely provides the least mean and the variance of residual error. Generally speaking, for stock X, the GBM models with jumps perform better than those GBM models without jumps. 2.5 Illustration of GBM Models to Data Set of Stock Y Before we make conclusions about this chapter, we apply the de veloped linear stochastic models to the other companys (Y) stock price pro cess. It is more than 22 years and has 5630 observations. Figure 2.5.1 shows its daily adjusted closing price from 9/10/1984 to 12/31/2006. 49

PAGE 63

0 10002000300040005000 020406080 TimePrice Figure 2.5.1 Daily Adjusted Closing Price for Stock Y We apply those linear models, under different data portioning process with or without jumps to the price data set of stock Y. The procedures are exactly the same as those applied to stock X in Sections 2.2, 2.3 and 2.4. To minimize the repetiti on, here we only give Figure 2.5.2 with regard to the best two estimated models and the summary of basic statistics of different linear models of stock Y in Table 2.5.1. 010002000300040005000 02 04 06 08 01 00 TimeP rice 010002000300040005000 02 04 06 08 01 00 TimeP rice 010002000300040005000 02 04 06 08 01 00 TimeP rice Raw Data Monthly GBM with Jumps Unequal Interval GBM with Jumps Figure 2.5.2 The Best Two Estimated Models of Stock Y 50

PAGE 64

Table 2.5.1 Basic Statistics for Lin ear Models without Jumps (2.2.1), with and without Jumps (2.3.3), (2.3.7), (2.3.10), (2.3.13) of Stock Y Model r 2 rS rS No. of Intervals No. of Parameters GBM with Overall Data -10.22182 211.7418 14.55135 1 2 Q14 and Q23 GBM without jumps -10.48387 214.6396 14.65058 45 4 Quarterly GBM without jumps -10.54319 216.0761 14.69953 89 8 Monthly GBM without jumps -0.5712012 137.0789 11.70807 268 536 Unequal Interval GBM without jumps -1.461658 77.70724 8.815171 256 512 Q14 and Q23 GBM with jumps 0.993067 26.28088 5.126488 45 48 Quarterly GBM with jumps 0.4321374 12.24818 3.49974 89 96 Monthly GBM with jumps -0.0098261 1.206479 1.098399 268 803 Unequal Interval GBM with jumps -0.0124816 1.199703 1.095310 256 767 Remark 2.5.1 : From, in Table 2.5.1 we note that for stock Y, two models: the Monthly GBM Model with jumps and Unequal Interval GBM mode l with jumps, both relatively provide the least mean, variance of residual error. Generally speaking, for stock Y, the GBM models with jumps perform better than those GBM models without jumps. 51

PAGE 65

2.6 Illustration of GBM Models to Data Set of S&P 500 Index In our previous estimation, we applied the a bove developed linear stoc hastic models to two individual stock price data sets of X and Y. In this section, we apply the GBM models to S&P500 Index. It is more than 59 years, and has 14844 observations. Figure 2.6.1 shows its daily adjusted closing price from 1/1/1950 to 12/31/2008. 0 5000 10000 15000 0 500 10001500 TimePrice Figure 2.6.1 Daily Adjusted Closing Price for S&P500 Index We apply same linear models, under different data portioning process with or without jumps to the data set of S&P500 Index. The procedures are exactly the same as those applied to stock X in Sections 2.2, 2.3, 2.4, and section 2.5. To mini mize the repetition, here we only give Figure 2.6.2 with regard to the best two estimated models and the summary of basic statistics of different linear models of S&P500 Index in Table 2.6.1. 52

PAGE 66

0 5000 10000 15000 050010001500 TimeP rice 0 5000 10000 15000 050010001500 TimeP rice 0 5000 10000 15000 050010001500 TimeP rice Raw Data Monthly GBM with Jumps Unequal Interval GBM with Jumps Figure 2.6.2 The Best Two Estimated Models of S&P500 Index Table 2.6.1 Basic Statistics for Lin ear Models without Jumps (2.2.1), with and without Jumps (2.3.3), (2.3.7), (2.3.10), (2.3.13) of S&P500 Index Model r 2rS rS No. of Intervals No. of Parameters GBM with Overall Data 141.3899 55048.98 234.6252 1 2 Q14 and Q23 GBM without jumps 141.6477 55031.77 234.5885 119 4 Quarterly GBM without jumps 142.6970 55408.23 235.3895 236 8 Monthly GBM without jumps -0.0171440 148.0943 12.1694 708 1416 Unequal Interval GBM without jumps 2.541547 210.3291 14.50273 570 1140 53

PAGE 67

Q14 and Q23 GBM with jumps -0.2859031 4717.424 68.6835 119 122 Quarterly GBM with jumps -0.8944856 1954.800 44.21312 236 243 Monthly GBM with jumps 58.30902 486.4579 22.05579 708 2123 Unequal Interval GBM with jumps 2.471564 210.2159 14.49882 570 1709 Remark 2.6.1 : Again, from Table 2.6.1 we remark that for S&P500 Index, there are two models: the Monthly GBM Model without jumps and Unequal Interval GBM model with jumps, both relatively, provide the least mean and variance of residual error with the least number of time intervals. Generally speaking, for S&P500 Index, the GBM models with jumps perform better than those GBM models without jumps. 2.7 Conclusions and Comments In this chapter, by employing classical model bu ilding process, we develop the modified version of GBM models under different data partitioning processes and coupled with or without jumps. The main focus was how to modify the existing GBM model in order to have a best fit with least mean and variance of residual error. Based on the study of three data sets in Chapter 2, one can immediately draw a couple of conclusions. (i) The first one is the usage of GBM model of overall dataset might not give us a good fit. Data partitioni ng improves the result. (ii) Also we show that models with jumps perform much better than the ones without jumps. This improvement is largely due to the accumulated errors in the m odel without jumps. Moreover, the environmental random perturbations cause to modify parameters in GBM model. In the next chapters, we will focus on models with jumps using monthly da ta partitioning and unequal interval data partitioning process, since models with these two da ta partitioning with jumps have less mean and variance of residual error. 54

PAGE 68

The GBM process is the solution of a linear stoch astic differential equation. Because the drift and volatility rate functions are linear From the equation (2.1.6), we know that 1lnlntt t y SS is expected to have a random pattern around the 21 2 Moreover, we would like to see the values in the neighborhood of the line y= 21 2 Figure 2.7.1 is a residual plot of monthly GBM model of Stock X. (a) (b) (c) (d) Figure 2.7.1 Some Residual Plots of Stock X 55

PAGE 69

In Figure 2.7.1(a), we see that the residual values st art out close to the line, then deviate from it. In Figure 2.7.1(b), there are a lot of runs of many negative residuals in a row. In Figure 2.7.1(c), we see there is a trend of the residuals. The magnitude of the residuals gets bigger as time goes on. Moreover, in Figure 2.7.1(b) and (d), we see the number of positive points are much larger than the number of negative points. From these obser vations and the Q-Q plots for model (2.2.1) (Figure 2.2.3), (2.3.3) (Figure 2.3.6) and (2.3.7) (Figure 2.3.8) suggest that the linear GBM model and its generalized models are inadequate to represent the stock price models. All these indicate that the linear model might not be good enough to fit the dataset. To build more precise models for competitive business processes, even a small diffe rence is important. In Chapter 3, we find a remedy to partially solve the cited limitations by developing the nonlinear stochastic models. 56

PAGE 70

Chapter 3 Nonlinear Stochastic Models 3.0 Introduction In Chapter 2, we initiated the development of stochastic modeling by using the classical modeling procedure in a systematic way. We made an attempt to modify the GBM model. The developed modified GBM models raised the issue about the st ochastic linear models of stock price processes. This was eluded in Section 2.7. There are many nonlinear stochastic models that describe the stochastic behavior of asset price in finance. In this chapter, we will focus on the nonlinear stochastic models. In Chapter 2, we have alr eady seen that modified GBM models with monthly and unequal interval data partiti oning process with jumps have better results in terms of minimum mean and variance of residual error, even though, we needed to estimate more parameters. Here, we will just focus on monthly and unequal interv al data partitioning processes with jumps. In Sections 3.1, 3.2 and 3.3, we develop three di fferent nonlinear stochastic models to our three datasets. In each section, we will first introduce the nonlinear stochastic model. We then develop the monthly and unequal interval nonlinear models with jumps based on each data set. Furthermore, we analyze and compare the nonlinear models with corresponding modified GBM models. In Sections 3.4 and 3.5, we illustrate non linear stochastic models in the context of data sets stock Y and S&P 500 Index respectively. Finally, conclusions are drawn in Section 3.6. 3.1 Stochastic Nonlinear Dynamic Model 1 (Black-Karasinski Model) Black-Karasinski (BK) model [6] describes a short-term interest rate process. It takes the following form 2(ln) 2ttttdSSSdtSdW t (3.1.1) where, and are parameters and is Brownian motion. tW 57

PAGE 71

To test the existence of a unique solution, let K be any number greater than 22 2 1) 2 ( M, and L be any number greater than || 2 ||||2 2 M, where and are sufficiently large constants such that and 1M2MtSM ln1 |lnln|1 2 1 22 t t t tS S S SM It is obvious that equation (3.1.1) satisfy the conditions (1.3.2) and (1.3.3). is the unique solution of (3.1.1). Even though, The BK mode l usually describes a short-term interest rate process, it may also be applied to the short-term stock price process. tS We note that the volatility function is linear and dr ift function is nonlinear. In order to derive the regression equation, we use the following transformation t tSV ln and apply differential formula (1.3.5) to obtain, DooboIt 2 2 21 (ln)((ln))() 2tt tt ttdVSdS SdS SS t 2 2 2 222 2111 ()() 2 11 ((ln) ) () 22 (ln)tt tt ttt tt tt ttdS dS SS SSdtSdWSdW SS SdtdW 1t Then, t t tdWdtVdV )( (3.1.2) By using the Euler type discretization process [24] stochastic differential equation (3.1.2) can be reduced to 11()(ttt ttVVVtWW1) (3.1.3) From 1 tttWW and equation (3.1.3) can be rewritten as 1 t t t tV V 1)1( (3.1.4) where, and are as defined in (3.1.1). By applying the least square regression method [35] and using above cited data sets, we can estimate these parameters. 58

PAGE 72

Nonlinear Stochastic Model 3.1.1 (Monthly Nonlinear Model 1 with Jumps): Let be the m monthly time intervals as defined in Data Partition Process (2.3.3). The nonlinear stochastic model is described by following stochastic differential equation: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,mmtt) 2() (ln ) 2i iiii iiiM MMMMMMM tttt t 00SS dSS SdtSdW if 1 iitt t 1,..., im (3.1.5) i M i M and i M are parameters. These parameters need to be estimated. By following definition [16, 26, 27], the solution of (3.1.5) takes the form 1,..., i m2 1 110001 1211 12 1100 1111112(,,), (,,), ,lim(,,) () ... ... ... (,,), ,lim(,,)mtt mmmmmmmmmn ttSttSttt SttStttSSttS St SttStttSSttS (3.1.6) Here, is the initial value of the stock price process. 0S12,1,...,m are jumps. These jumps are estimated by 11 1 m tSmlim ,..., lim lim 1 2 2 1 12 2 1 1 tt m tt t tt tS S S S Sm The estimated parameters in Monthly Nonlinear Stochastic Model (3.1.1) of stock X are presented in Table 3.1.1. The AIC (Akaike's inform ation criterion) criterion [3] defined in (1.5.6). Here, we use AIC as the criterion whenever we n eed to compare different models. The preferred model is the model with the lowest AIC value. Table 3.1.1 Estimated Parameters in Model 3.1.1 of Stock X Interval Index Monthly GBM Model with Jumps Monthly Nonlinear Model 1 with Jumps AIC AIC 1 0.003217 0.037758 24.99217 -0.93259 4.347902 0.025463 20.74442 2 0.011469 0.021817 42.50585 0.010445 -0.03801 0.021799 44.52993 3 0.01924 0.041526 83.16599 -0.00148 0.025789 0.041526 85.15694 4 -0.00152 0.037226 82.10229 -0.27051 1.398991 0.034175 80.71667 5 0.002806 0.019117 57.44675 -0.05069 0.26617 0.018992 59.16818 6 0.001162 0.029425 72.08337 -0.3736 1.966097 0.026578 69.95869 59

PAGE 73

7 -0.0017 0.027966 67.90378 -0.32282 1.700644 0.02585 66.93144 8 -0.00177 0.012578 39.10143 -0.23453 1.217925 0.011507 37.27279 9 0.009641 0.020969 63.75357 -0.03145 0.175542 0.020858 65.44605 10 0.01115 0.016296 61.93046 0.089916 -0.48049 0.015445 61.48279 11 0.002921 0.021646 83.43298 -0.33056 1.873092 0.019655 81.20437 12 -0.00096 0.016438 66.92268 -0.2209 1.257716 0.015708 67.08642 13 -0.00017 0.013734 66.01481 -0.19786 1.119452 0.013038 65.65693 14 0.004932 0.015047 67.11831 -0.14534 0.835089 0.014276 66.79189 15 0.008161 0.02977 98.04718 0.027813 -0.15265 0.029718 99.9177 16 0.004194 0.01872 87.811 -0.18623 1.118515 0.017263 86.41869 17 0.001243 0.013153 74.5625 -0.22105 1.33551 0.012174 73.30687 18 0.002672 0.033797 109.8641 -0.32346 1.974105 0.030375 107.9387 19 -0.00865 0.035981 101.9126 -0.39163 2.309854 0.028169 93.75399 20 0.003501 0.025925 105.6795 -0.07197 0.426261 0.025685 107.2806 21 0.003837 0.019752 83.25841 -0.26766 1.615277 0.017746 81.18764 22 -0.00514 0.018633 90.55914 -0.24664 1.46339 0.016451 86.83802 23 0.005609 0.01691 85.99189 -0.12762 0.767304 0.016503 87.09986 24 -0.00399 0.011988 65.98114 -0.04066 0.240058 0.011892 67.73263 25 -0.00081 0.014567 81.26636 -0.55654 3.301192 0.01207 74.65778 26 0.003112 0.015259 75.33679 -0.1616 0.969446 0.01435 74.72001 27 0.007971 0.022015 103.9302 -0.05786 0.35943 0.021685 105.1114 28 0.000937 0.014354 85.01112 -0.15455 0.956602 0.01374 85.10525 29 -0.0025 0.012368 73.32039 -0.11528 0.707825 0.012047 74.33965 30 0.004407 0.016734 87.08846 -0.41656 2.58325 0.013689 81.1518 31 -0.00564 0.016148 80.05819 -0.38312 2.351359 0.013406 74.47795 32 0.000967 0.013749 83.07253 -0.2835 1.7344 0.012763 81.86601 33 0.001484 0.011091 69.15906 -0.39337 2.423194 0.009461 64.82723 34 0.00256 0.011958 79.16138 -0.01537 0.097133 0.011955 81.16398 35 0.002363 0.009975 71.65205 -0.21045 1.315889 0.00921 70.31184 36 -0.00107 0.014438 88.62695 -0.07007 0.438718 0.014276 90.13775 60

PAGE 74

37 0.000524 0.012519 88.16149 -0.34231 2.134238 0.011438 85.94986 38 0.00511 0.00978 65.80358 -0.02092 0.136596 0.009749 67.64959 39 0.009694 0.015521 108.0988 -0.02538 0.173118 0.015449 109.955 40 -0.00057 0.027629 125.7364 -0.15358 0.99984 0.026446 125.9705 41 2.59E-06 0.014992 96.70551 -0.28643 1.874382 0.013882 95.61875 Figures 3.1.13.1.3 are the plots of predicted value of Monthly Nonlinear Model 1 of stock X with Jumps with observations ranging from 1 to 300, 300 to 600 and 600 to 848 respectively. 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Figure 3.1.1 Comparison of Model 2.4.3 with Model 3.1.1 of Stock X (Observations 1-300) 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Figure 3.1.2 Comparison of Model 2.4.3 with Model 3.1.1 of Stock X (Observations 300-600) 61

PAGE 75

600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Figure 3.1.3 Comparison of Model 2.4.3 with Model 3.1.1 of Stock X (Observations 600-848) Table 3.1.2 shows the overall basic statistics of Monthly GBM Model and Monthly Nonlinear Model 3.1.1. Table 3.1.2 Basic Statistics of Model 3.1.1 of Stock X Model r 2 rS rS No. of Intervals No. of Parameters Monthly GBM with Jumps -1.242020 207.264 14.39667 41 122 Monthly Nonlinear Model 1 with Jumps -1.928296 141.1754 11.88173 41 163 Remark 3.1.1 : From Table 3.1.2 we can see that overa ll, the Monthly Nonlinear Model 3.1.1 with Jumps has less variance of the residual error. From the Table 3.1.1 and Figures 3.1.1 3.1.3, we remark that for some months, GBM Model is better than Nonlinear Model 3.1.1 in terms of AIC. For example, in the 2nd, 3rd, 5th, 9th, 12th, 15th month etc, GBM model has less AIC than Nonlinear Model 3.1.1. There are 17 out of 41 months (41%), that GBM model has less AIC than Nonlinear Model 1. We further note that the N onlinear Model 3.1.1 has 3 parameters and the GBM model has 2 parameters. 62

PAGE 76

Nonlinear Stochastic Model 3.1.2 (Unequal Interval Nonlinear Model 1 with Jumps): Let be the n time intervals as defined in Data Partition Process 2.3.4. The nonlinear stochastic differential equation is described by: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,nntt) 2() (ln ) 2i iiii iiiI IIIIIII ttttdSS SdtSdW t, 00SS if 1 iittt 1,..., in (3.1.7) iI iI and iI , are parameters which can be estimated by the method as described above. 1,...,in2 By following definition [16, 26, 27], the solution of (3.1.7) is given by: (3.1.8) 1 110001 121112 1100 1111112(,,), (,,), ,lim(,,) () ... ... ... (,,),,lim(,,)ntt nmnnnnnnnn ttSttSttt SttStttSSttS St SttStttSSttS 0S is the initial value of the stock price process. 12,1,...,n are jumps. These jumps are estimated by: lim11 1 n tt tS Sn n ,..., lim lim 1 2 2 1 12 2 1 1 n tt t tt tS S S S The parameters of stochastic model (3.1.7) are presented in Table 3.1.3. Furthermore, the AIC for both GBM and nonlinear model are also included in Table 3.1.3. Table 3.1.3 Estimated Parameters in Model 3.1.2 of Stock X Interval Index Unequal Interval GBM Model with Jumps Unequal Interval Nonlinear Model 1 with Jumps AIC AIC 1 0.006621 0.024521 53.16678 -0.06936 0.331231 0.024124 54.29912 2 0.013391 0.028731 52.68493 -0.28443 1.411569 0.023982 48.97729 3 0.009891 0.055846 66.41971 -0.48125 2.509755 0.042146 61.5785 4 0.001149 0.030116 76.19733 -0.41257 2.131512 0.026637 72.7967 5 0.009663 0.019472 36.22862 -0.24579 1.292223 0.016518 34.07066 63

PAGE 77

6 -0.00134 0.026352 49.0021 -0.70337 3.70968 0.021988 45.88822 7 0.005189 0.038095 50.92597 -0.27218 1.436821 0.035407 51.11064 8 -0.00261 0.015574 80.28809 -0.13185 0.68589 0.014794 78.59092 9 0.008553 0.017281 92.29718 -0.03304 0.185145 0.017021 93.15485 10 0.00884 0.022777 99.00801 -0.20055 1.137652 0.020094 95.10469 11 -0.00053 0.014418 40.64565 -0.69145 3.933416 0.011311 36.36812 12 7.88E-05 0.015385 117.7922 -0.12929 0.733401 0.014874 117.2144 13 0.000596 0.015815 94.64219 -0.36491 2.091368 0.014159 90.2432 14 0.016518 0.033196 71.0436 -0.35994 2.137839 0.02067 61.21107 15 0.009311 0.013909 44.40171 -0.14129 0.856515 0.012957 44.72932 16 -0.00076 0.016247 90.84893 -0.25676 1.548757 0.015142 89.6624 17 0.004541 0.025039 60.94867 -0.41581 2.549001 0.019093 56.78929 18 -0.01211 0.036582 105.6407 -0.0507 0.290682 0.036318 107.3503 19 0.000358 0.028831 120.3625 -0.20871 1.226962 0.027295 119.5099 20 0.010446 0.022202 84.47223 -0.31116 1.870249 0.015605 74.16229 21 -0.00389 0.02337 57.79999 -0.10737 0.642615 0.022928 59.33579 22 -0.0026 0.019065 78.89382 -0.26636 1.579654 0.017477 77.70388 23 0.002168 0.015421 115.5262 -0.15501 0.931566 0.014359 113.3126 24 -0.00417 0.014351 64.01523 -0.2814 1.66841 0.012898 62.11432 25 0.003034 0.012816 66.33031 -0.52939 3.146627 0.01112 62.77867 26 0.003202 0.014617 99.57111 -0.20284 1.224103 0.013662 98.0609 27 0.007015 0.020763 116.315 -0.37375 2.313722 0.015966 106.9751 28 -0.00327 0.012388 57.10168 -0.88324 5.459753 0.005951 36.7315 29 1.37E-05 0.015938 96.23449 -0.116 0.715536 0.015467 96.88494 30 -0.00115 0.017478 101.1193 -0.18984 1.169378 0.016698 100.9772 31 -0.00045 0.014767 97.74304 -0.37187 2.273761 0.013167 94.06035 32 2.02E-05 0.01084 100.6316 -0.36305 2.234232 0.009826 96.81189 33 0.007892 0.011226 46.63383 -0.07094 0.445688 0.011061 48.32431 34 0.0025 0.008967 101.7473 -0.04038 0.255452 0.008882 103.1137 35 -0.00407 0.016643 80.90792 -0.77061 4.807935 0.008814 59.38289 64

PAGE 78

36 0.003471 0.010888 113.8111 0.010153 -0.06017 0.01088 115.7619 37 0.008705 0.015797 75.73326 -0.11488 0.745197 0.015096 76.28284 38 0.010025 0.015063 59.36543 -0.13148 0.86973 0.014223 60.01012 39 -0.00136 0.022851 199.8702 -0.21082 1.373688 0.021244 196.409 Figures 3.1.4 3.1.6 are the plots of predicted value of Unequal Interval Nonlinear Model 3.1.2 of stock X with observations ranging from 1 to 300, 300 to 600 and 600 to 848 respectively. 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 1 with Jumps Figure 3.1.4 Comparison of Model 2.4.3, 2.4.4, 3.1.1 with Model 3.1.2 of Stock X (Observations 1-300) 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 1 with Jumps Figure 3.1.5 Comparison of Model 2.4.3, 2.4.4, 3.1.1 with Model 3.1.2 of Stock X (Observations 300-600) 65

PAGE 79

600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 1 with Jumps Figure 3.1.6 Comparison of Model 2.4.3, 2.4.4, 3.1.1 with Model 3.1.2 of Stock X (Observations 600-848) Table 3.1.4 shows the overall basic statistics of monthly GBM model 2.4.3 with jumps, Monthly Nonlinear Model 3.1.1 with Jumps, Unequal Inte rval GBM model 2.4.4 with Jumps and Unequal Interval Nonlinear Model 3.1.2 with jumps. Table 3.1.4 Basic Statistics of Models 2. 4.3, 3.1.1, 2.4.4 and 3.1.2 of Stock X Model r 2 rS rS No. of Intervals No. of Parameters Monthly GBM with Jumps -1.242020 207.264 14.39667 41 122 Monthly Nonlinear Model 1 with Jumps -1.928296 141.1754 11.88173 41 163 Unequal Interval GBM with Jumps 1.962899 258.1040 16.06562 39 116 Unequal Interval Nonlinear Model 1 with Jumps 0.5315015 131.2354 11.4558 39 155 66

PAGE 80

Remark 3.1.2 : From Table 3.1.4, we note that, the Un equal Interval Nonlinear Model 3.1.2 with Jumps has least mean and variance of the residual error. From the Table 3.1.3 and Figures 3.1.4 3.1.6 we conclude that on some intervals, the GBM model is better than Nonlinear Model 3.1.2. In addition the GBM model is better than Nonlinear Model 3.1.2 in terms of AIC. For example, on the 7th, 9th, 15th, 21st, 29th, 33rd, intervals, the GBM model has less AIC than Nonlinear Model 3.1.2. There are 12 out of 39 intervals (31%), on which the GBM model has less AIC than Nonlinear Model 3.1.2. 3.2 Stochastic Nonlinear Dynamic Model 2 This nonlinear stochastic model 2 [ 26] is described by the following DooboIt differential equation t N t N t N tt tdWSdtS N SSdS ) 2 (122 (3.2.1) where, N ,, and are parameters; moreover 1,2.10 NN, and is Brownian motion. It is easy to verify that rate functions in (3.2.1) satisfies the conditions for existence and uniqueness of solution [23,28]. We note that the volatility and drift functions are nonlinear functions of In order to derive the regression equation, we use the following transformation tWtS N SN t1Vt 1 and apply differential formula to obtain DooboIt 2 1 2 2 1)))( 1 (( 2 1 ) 1 (t N t t t N t t tdS N S S dS N S S dV t N t t N t N t t N t N t N tt N t t N t t N tdWdtS dWSS N dWSdtS N SSS dSSdSS )( )( 2 ) ) 2 ( )()( 2 11 2 221 122 2 Then, t t tdWdtVN dV ))1(( (3.2.2) Again by using Euler type discretization process [2 4], stochastic differential equation (3.2.2) can be reduced to 67

PAGE 81

)())1((1 1 1 tt t ttWWtVN VV (3.2.3) From 1 tttWW and equation (3.2.3) can be rewritten as 1 t t t tVN V )))1(1((1 (3.2.4) where, and are as defined in (3.2.1). For given N, by applying the least square regression method [35] and using above cited data set, these parameters can be estimated, analogously. N is estimated by the value, under which the model has least variance of residual error. Nonlinear Stochastic Model 3.2.1 (Monthly Nonlinear Model 2 with Jumps): Let be the m monthly time intervals as defined in stochastic model 3.1.1. The nonlinear stochastic model 3.2.1 with jumps takes the following form of nonlinear 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,mmtt) I toDo ob type stochastic differential equation: t N M t M N M t M M N M t M M t M M tdWSdt S N S S dSi M i i i M i i i i M i i ii i)())()( 2 )( (12 2 00SS if 1 iittt 1,...,im (3.2.5) i M i M and i M are parameters. These parameters are estimated as described above. As before, following definition [16, 26, 27], the solution of (3.2.5) takes the form: 1,..., i m2 1 110001 1211 12 1100 1111112(,,), (,,), ,lim(,,) () ... ... ... (,,), ,lim(,,)mtt mmmmmmm mmn ttSttSttt SttStttSSttS St SttStttSSttS (3.2.6) Again, is the initial value of the stock price process. 0S12,1,...,m are jumps, and can. lim ,..., lim lim 1 1 2 2 1 11 1 2 2 1 1 m tt t m tt t tt tS S S S S Sm m Table 3.2.1 gives estimated parameters by applyi ng Monthly Nonlinear Stochastic Model 3.2.1 of stock X. 68

PAGE 82

Table 3.2.1 Estimated Parameters of Model 3.2.1 of Stock X Interval Index Monthly Nonlinear Model 2 with Jumps N 1 -0.93097 98.58582 2.685764 0 2 0.015167 -0.40094 2.562583 0 3 0.006386 0.702974 2.43304 0.2 4 -0.26424 46.59828 6.056676 0 5 -0.04732 9.06878 3.426131 0 6 3.111272 -1.65455 0.014153 1.12 7 -0.32461 63.03162 5.104841 0 8 -0.23319 41.98315 2.08146 0 9 0.162624 -0.05328 0.007219 1.2 10 -0.43324 0.148857 0.005128 1.2 11 -0.32678 94.48073 5.651904 0 12 -3.1569 4.702766 0.02342 0.93 13 -0.19667 56.36633 3.738925 0 14 0.726705 -0.2303 0.004546 1.2 15 0.227674 -0.4653 0.06296 0.87 16 0.934944 -0.28125 0.005202 1.2 17 1.106195 -0.33042 0.003641 1.2 18 -0.30599 136.999 13.06886 0 19 1.943516 -0.59744 0.008646 1.2 20 -0.06734 25.27551 9.241028 0 21 1.339686 -0.4007 0.005306 1.2 22 1.223408 -0.37344 0.004986 1.2 23 -0.11434 46.91534 6.455683 0 24 -0.04108 15.00859 4.799298 0 25 -0.55537 209.269 4.546397 0 26 0.805277 -0.24258 0.004332 1.2 27 0.291471 -0.08414 0.006396 1.2 69

PAGE 83

28 0.770865 -0.22354 0.00398 1.2 29 -0.11439 53.08718 5.661491 0 30 2.085035 -0.60321 0.003964 1.2 31 1.903402 -0.55777 0.003924 1.2 32 -0.27964 126.9381 5.734823 0 33 -0.39253 185.8715 4.464326 0 34 -0.00631 4.182385 5.648348 0 35 -0.20829 108.206 4.736808 0 36 0.348545 -0.09964 0.004068 1.2 37 1.709708 -0.49133 0.003289 1.2 38 0.107688 -0.02919 0.002777 1.2 39 -0.01282 14.14102 9.792642 0 40 -0.15131 101.7773 17.81769 0 41 1.431732 -0.38677 0.003751 1.2 Figures 3.2.13.2.3 are the plots of predicted value of Monthly Nonlinear Model 3.2.1 of stock X with observation ranging from 1 to 300, 300 to 600 and 600 to 848 respectively. 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Monthly Nonlinear Model 2 with Jumps Figure 3.2.1 Comparison of Model 2.4.3, 3.1.1 with Model 3.2.1 of Stock X (Observations 1-300) 70

PAGE 84

300350400450500550600 300350400450500550 TimePrice 300350400450500550600 300350400450500550 TimePrice 300350400450500550600 300350400450500550 TimePrice 300350400450500550600 300350400450500550 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Monthly Nonlinear Model 2 with Jumps Figure 3.2.2 Comparison of Model 2.4.3, 3.1.1 with Model 3.2.1 of Stock X (Observations 300-600) 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Monthly Nonlinear Model 2 with Jumps Figure 3.2.3 Comparison of Model 2.4.3, 3.1.1 with Model 3.2.1 of Stock X (Observations 600-848) Table 3.2.2 shows the overall basic statistics of Monthly GBM Model 2.4.3, Monthly Nonlinear Model 3.1.1 and Monthly Nonlinear Model 3.2.1 with Jumps of Stock X. 71

PAGE 85

Table 3.2.2 Basic Statistics of Models 2.4.3, 3.1.1 and 3.2.1 of Stock X Model r 2 rS rS No. of Intervals No. of Parameters Monthly GBM with Jumps -1.242020 207.264 14.39667 41 122 Monthly Nonlinear Model 1 with Jumps -1.928296 141.1754 11.88173 41 163 Monthly Nonlinear Model 2 with Jumps -2.090806 143.2248 11.96765 41 204 Remark 3.2.1 : From Table 3.2.2, we observe that under the same data partition process, Monthly Nonlinear Model 3.2.1 with Jumps has less variance than Monthly GBM with Jumps. Overall, the Monthly Nonlinear Model 3.1.1 with Jumps has less variance of the residual error than Monthly GBM Model and Monthly Nonlinear Model 3.2.1 with Jumps. Nonlinear Stochastic Model 3.2.2 (Unequal Interval Nonlinear Model 2 with Jumps): Let be the n time intervals defined in stochastic model 3.1.2. The nonlinear stochastic model 3.2.2 with jumps takes the following form: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,nntt) t N I t I N I t I I N I t II t I I tdWSdt S N SS dSi I ii i I i i i i I ii ii i)())()( 2 )( (12 2 , if 00SS 1 iittt 1,..., in (3.2.7) iI iI and iI , are parameters and can be estimated as described before. 1,...,in2 By following definition [16, 26, 27], the solution of (3.2.7) takes the form 1 110001 121112 1100 1111112(,,), (,,), ,lim(,,) () ... ... ... (,,),,lim(,,)ntt nmnnnnnnnn ttSttSttt SttStttSSttS St SttStttSSttS (3.2.8) Here, is the initial value of the stock price process. 0S12,1,...,n are jumps and can be estimated as 1 12 12 11. S 121 12 ,,... limlimlimn nt tt n n tt tt ttS SS SS 72

PAGE 86

Table 3.2.3 gives estimated parameters with rega rd to Unequal Interval Nonlinear Stochastic Model 3.2.2 of stock X. Table 3.2.3 Estimated Parameters of Model 3.2.2 of Stock X Interval Index Unequal Interval Nonlinear Model 2 with Jumps N 1 -0.06298 7.518172 2.580372 0 2 -0.2657 38.08205 3.391367 0 3 -0.44948 82.83006 7.48971 0 4 -0.41314 72.45667 4.66966 0 5 -0.23482 45.135 3.059264 0 6 -0.75107 106.8789 3.141901 0.06 7 -0.27195 53.40573 6.840905 0 8 0.656408 -0.23191 0.005193 1.2 9 0.172972 -0.05645 0.005866 1.2 10 -0.18794 54.75371 5.654817 0 11 -0.68875 203.5307 3.345713 0 12 0.646214 -0.20781 0.004773 1.2 13 -0.36426 112.3249 4.348781 0 14 -0.32689 124.4562 7.29626 0 15 -0.13146 56.60131 5.240336 0 16 -0.25507 106.2544 6.30183 0 17 -0.40544 186.3563 8.661997 0 18 0.239471 -0.07623 0.010924 1.2 19 1.039288 -0.32073 0.008402 1.2 20 -0.29476 120.2958 6.163061 0 21 0.531576 -0.16061 0.00685 1.2 22 -0.26539 99.89832 6.775433 0 23 -0.14965 60.98254 5.730283 0 24 1.402875 -0.42858 0.003933 1.2 25 -0.52427 199.9783 4.240122 0 73

PAGE 87

26 -0.1963 82.03641 5.598122 0 27 -0.34739 169.7543 7.612169 0 28 4.412648 -1.28169 0.001728 1.2 29 -0.1149 54.87902 7.372088 0 30 0.945964 -0.27597 0.004852 1.2 31 -0.37018 167.4691 5.936408 0 32 -0.36119 170.0127 4.617445 0 33 -0.05807 31.62257 5.295368 0 34 -0.03837 21.49953 4.633074 0 35 3.847553 -1.10474 0.002529 1.2 36 0.01336 -5.18139 5.645615 0 37 0.58191 -0.15902 0.00418 1.2 38 -0.12128 90.81585 9.907542 0 39 1.050197 -0.2853 0.005779 1.2 Figures 3.2.4, 3.2.5 and 3.2.6 are the plots of predicted value of Unequal Interval Nonlinear Model 3.2.2 of stock X with Jumps with observations ranging from 1 to 300, 300 to 600 and 600 to 848 respectively. 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 2 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 2 with Jumps Figure 3.2.4 Comparison of Model 2.4.3, 2.4.4, 3.2.1 with Model 3.2.2 of Stock X (Observations 1-300) 74

PAGE 88

300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 2 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 2 with Jumps Figure 3.2.5 Comparison of Model 2.4.3, 2.4.4, 3.2.1 with Model 3.2.2 of Stock X (Observations 300-600) 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 2 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 2 with Jumps Figure 3.2.6 Comparison of Model 2.4.3, 2.4.4, 3.2.1 with Model 3.2.2 of Stock X (Observations 600-848) 75

PAGE 89

Table 3.2.4 shows the overall basic statistics of Monthly GBM model 2.4.3, Nonlinear Model 3.1.1 and 3.1.2 with Jumps, Unequal GBM model 2.4.4, and Unequal Nonlinear Model 3.2.1 and 3.2.2 with Jumps. Table 3.2.4 Basic Statistics of Models 2.4.3, 3. 1.1, 3.1.2, 2.4.4, 3.2.1 and 3.2.2 of Stock X Model r 2 rS rS No. of Intervals No. of Parameters Monthly GBM with Jumps -1.242020 207.264 14.39667 41 122 Monthly Nonlinear Model 1 with Jumps -1.928296 141.1754 11.88173 41 163 Monthly Nonlinear Model 2 with Jumps -2.090806 143.2248 11.96765 41 204 Unequal GBM with Jumps -1.962899 258.1040 16.06562 39 116 Unequal Nonlinear Model 1 with Jumps -0.5315015 131.2354 11.4558 39 155 Unequal Nonlinear Model 2 with Jumps -0.6097021 131.3068 11.45892 39 194 Remark 3.2.2 : From Table 3.2.4 we observe that under the same data partition processes, Nonlinear Models 3.1.1 and 3.2. 1 have less variance of the r esidual error than Monthly GBM model, and Nonlinear Models 3.1.2 and 3.2.2 also have less variance of the residual than Unequal GBM model. Overall, the nonlinear models 3.1.2 and 3.2.2 under the Unequal Interval data partition process have less variance of the residua l error than the nonlinear models 3.1.1 and 3.2.1under the monthly data partition process. 3.3 Stochastic Nonlinear Dynamic Model 3 This nonlinear stochastic model 3 [ 26] is described by the following DooboIt differential equation 22()tttttdSSSSdtSdWt (3.3.1) 76

PAGE 90

where, and are parameters and is Brownian motion. It is easy to check that rate functions in (3.3.1) satisfies the conditions fo r existence and uniqueness of solution [23,28]. In order to derive a regression equation, we use the following transformation tW 1t tV S and applying differential formula to obtain Doob oIt 2 2 2111 ()(())() 2tt tt ttdVdS dS SSSS t 222 222 3 2 2 111 ()() 2 1 (( )) () 2 ()tttt ttttttttt tt tSdSSdS SSSSdtSdWSSdW SdtSdW 2 Then, tt t tdWVdtVdV )( (3.3.2) Again, the Euler type discretized version of (3.3.2) is as follows 111()(ttt tttVVVtVWW1) (3.3.3) From the definition of V, we note that 11 11ttt t ttVVS y VS 1 tttWW and With this notation, equation (3.3.3) can be rewritten as 1 t 11 ()t ty Vt (3.3.4) Then, parameters, and can be estimated using least square method [35]. Nonlinear Stochastic Model 3.3.1 (Monthly Nonlinear Model 3 with Jumps): Let be the m monthly time intervals as defined in stochastic model 3.1.1. The nonlinear stochas tic model 3.3.1 takes the form of following nonlinear 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,mmtt) I toDoob type stochastic differential equation: 22(()())iiiiiiiiiMMMMMMMMM tttttdSSS SdtSdWt, 00SS if 1 iittt 1,..., im (3.3.5) 77

PAGE 91

i M i M and i M are parameters and are estimated as described above. Thus, the solution of equation (3.3.5) is given by 1,..., i m2 1 110001 1211 12 1100 1111112(,,), (,,), ,lim(,,) () ... ... ... (,,), ,lim(,,)mtt mmmmmmm mmn ttSttSttt SttStttSSttS St SttStttSSttS (3.3.6) where is the initial value of the stock price process. 0S12,1,...,m are jumps. These jumps are estimated as 11 1m tSmlim ,..., lim lim 1 2 2 1 12 2 1 1 tt m tt t tt tS S S S Sm The estimated parameters of Monthly Nonlinear St ochastic Model 3.3.1 of stock X are recorded in Table 3.3.1. Table 3.3.1 Estimated Parameters of Model 3.3.1 of Stock X Interval Index Monthly Nonlinear Model 3 with Jumps 1 0.922514 -0.0087161 0.025344 2 0.004041 6.17E-05 0.021409 3 0.023693 -4.18E-05 0.039823 4 0.261409 -0.0014855 0.034493 5 0.050772 -0.0002666 0.018895 6 0.37251 -0.0019314 0.026447 7 0.318094 -0.0016404 0.025629 8 0.229969 -0.0012777 0.011528 9 0.041487 -0.0001638 0.020631 10 -0.07527 0.00036332 0.015148 11 0.330384 -0.0011435 0.019702 78

PAGE 92

12 0.219936 -0.0007408 0.015711 13 0.196132 -0.0006846 0.013041 14 0.153618 -0.0004918 0.014218 15 -0.0255 0.00010249 0.028538 16 0.190213 -0.0004688 0.017376 17 0.222578 -0.0005293 0.012198 18 0.312192 -0.0006983 0.031076 19 0.381663 -0.001048 0.028307 20 0.071652 -0.0001924 0.025384 21 0.269717 -0.0006458 0.017525 22 0.246151 -0.0006521 0.016515 23 0.12528 -0.0003065 0.016356 24 0.03558 -9.80E-05 0.011957 25 0.552865 -0.0014675 0.012074 26 0.167816 -0.0004166 0.014242 27 0.06565 -0.0001331 0.021147 28 0.158273 -0.0003247 0.013793 29 0.111985 -0.0002415 0.01214 30 0.422469 -0.0008563 0.013585 31 0.381775 -0.0008249 0.013505 32 0.278473 -0.0006137 0.012737 33 0.392775 -0.0008295 0.009392 34 0.01369 -2.39E-05 0.011873 35 0.210991 -0.0004062 0.009164 36 0.071916 -0.0001373 0.014521 37 0.345019 -0.0006763 0.011447 38 0.025629 -3.84E-05 0.009695 39 0.033435 -3.82E-05 0.01535 40 0.150713 -0.0002246 0.026663 41 0.287854 -0.0004142 0.013927 79

PAGE 93

Figures 3.3.1, 3.3.2 and 3.3.3 are the plots of predicted value of Monthly Nonlinear Model 3.3.1 of stock X with observations ranging from 1 to 300, 300 to 600 and 600 to 848 respectively. 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Monthly Nonlinear Model 2 with Jumps Monthly Nonlinear Model 3 with Jumps Figure 3.3.1 Comparison of Model 2.4.3, 3.1.1, 3.2.1 with Model 3.3.1 of Stock X (Observations 1-300) 300350400450500550600 300350400450500550 TimePrice 300350400450500550600 300350400450500550 TimePrice 300350400450500550600 300350400450500550 TimePrice 300350400450500550600 300350400450500550 TimePrice 300350400450500550600 300350400450500550 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Monthly Nonlinear Model 2 with Jumps Monthly Nonlinear Model 3 with Jumps Figure 3.3.2 Comparison of Model 2.4.3, 3.1.1, 3.2.1 with Model 3.3.1 of Stock X (Observations 300-600) 80

PAGE 94

600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 1 with Jumps Monthly Nonlinear Model 2 with Jumps Monthly Nonlinear Model 3 with Jumps Figure 3.3.3 Comparison of Model 2.4.3, 3.1.1, 3.2.1 with Model 3.3.1 of Stock X (Observations 600-848) Table 3.3.2 shows the overall basic statistics of Monthly GBM Model 2.4.3, Monthly Nonlinear Model 3.1.1, 3.2.1, and Monthly Nonlinear Model 3.3.1 with Jumps. Table 3.3.2 Basic Statistics of Models 2. 4.3, 3.1.1, 3.2.1 and 3.3.1 of Stock X Model r 2 rS rS No. of Intervals No. of Parameters Monthly GBM with Jumps -1.242020 207.264 14.39667 41 122 Monthly Nonlinear Model 1 with Jumps -1.928296 141.1754 11.88173 41 163 Monthly Nonlinear Model 2 with Jumps -2.090806 143.2248 11.96765 41 204 Monthly Nonlinear Model 3 with Jumps -1.731151 139.2792 11.80166 41 163 81

PAGE 95

Remark 3.3.1 : From Table 3.3.2 we remark that ove rall, the Monthly Nonlinear Model 3.3.1 with Jumps has least variance of the residual error than Monthly GBM Model 2.4.3, Nonlinear Model 3.1.1 and 3.2.1. Nonlinear Stochastic Model 3.3.2 (Unequal Interval Nonlinear Model 3 with Jumps): Let be the n time intervals as defined stochastic model 3.1.2. The nonlinear stochastic model 3.3.2 is described by: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,nntt)t 22(()())iiiiiiiiiMMMMIIIII tttttdSSSSdtSdW, if 00SS 1 iittt 1,...,in (3.3.7) where,iI iI and iI , are parameters as defined and estimated. The solution of equation (3.3.7) is represented by: 1,..., i n2 1 110001 121112 1100 1111112(,,), (,,), ,lim(,,) () ... ... ... (,,),,lim(,,)ntt nmnnnnnnnn ttSttSttt SttStttSSttS St SttStttSSttS (3.3.8) 0S is the initial value of the stock price process. 12,1,...,n are jumps and can be estimated as 1 12 121121 12 1 ,,... limlimlimn nt tt n n tt tt ttS SS SSS The estimated parameters of Unequal Interval N onlinear Stochastic Model 3.3.2 of stock X are recorded in Table 3.3.3. Table 3.3.3 Estimated Parameters of Model 3.3.2 of Stock X Interval Index Monthly Nonlinear Model 3 with Jumps 1 0.071226 -0.0006014 0.023908 2 0.29716 -0.0020805 0.023786 3 0.471856 -0.0025641 0.042281 82

PAGE 96

4 0.409774 -0.0023388 0.026435 5 0.251018 -0.0013076 0.016441 6 0.700165 -0.0035874 0.021907 7 0.270348 -0.0013796 0.034988 8 0.130079 -0.0007162 0.014833 9 0.040871 -0.0001544 0.01686 10 0.210273 -0.000724 0.020032 11 0.689476 -0.0023335 0.011359 12 0.129436 -0.0004453 0.014872 13 0.364516 -0.0011824 0.014153 14 0.362752 -0.0009541 0.020535 15 0.147044 -0.0003428 0.01289 16 0.257365 -0.0006181 0.015308 17 0.422811 -0.0009202 0.019309 18 0.043869 -0.0001438 0.036822 19 0.213002 -0.0005966 0.027236 20 0.322719 -0.0007918 0.015437 21 0.106569 -0.0002685 0.022847 22 0.262288 -0.0006972 0.017497 23 0.153312 -0.0003764 0.014273 24 0.278518 -0.0007413 0.012968 25 0.527504 -0.0013832 0.011062 26 0.200628 -0.0004803 0.013615 27 0.36656 -0.0007506 0.015936 28 0.88476 -0.0018291 0.005951 29 0.114746 -0.0002405 0.01549 30 0.19202 -0.0004058 0.016691 31 0.368976 -0.0008158 0.013194 32 0.358247 -0.0007612 0.009788 83

PAGE 97

33 0.074069 -0.0001383 0.010944 34 0.041831 -7.49E-05 0.008841 35 0.766581 -0.001496 0.008805 36 -0.00713 2.00E-05 0.010855 37 0.123638 -0.000189 0.01503 38 0.137906 -0.0001851 0.014094 39 0.21229 -0.0003142 0.021393 Figures 3.3.4, 3.3.5 and 3.3.6 are plots of predic tions of Unequal Interval Nonlinear Model 3.3.2 of stock X with observations ranging from 1 to 300, 300 to 600 and 600 to 848 respectively. 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice 0 50 100 150 200 250 300 100150200250300350400450 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 3 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 3 with Jumps Figure 3.3.4 Comparison on Model 2.4.3, 2.4.4, 3.3.1 with Model 3.3.2 of Stock X (Observations 1-300) 84

PAGE 98

300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice 300 350 400 450 500 550 600 300350400450500550 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 3 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 3 with Jumps Figure 3.3.5 Comparison on Model 2.4.3, 2.4.4, 3.3.1 with Model 3.3.2 of Stock X (Observations 300-600) 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice 600 650 700 750 800 850 400500600700800 TimePrice Raw Data Monthly GBM with Jumps Monthly Nonlinear Model 3 with Jumps Unequal Interval GBM with Jumps Unequal Interval Nonlinear Model 3 with Jumps Figure 3.3.6 Comparison on Model 2.4.3, 2.4.4, 3.3.1 with Model 3.3.2 of Stock X (Observations 600-848) 85

PAGE 99

Table 3.3.4 shows the overall basic statistics of Monthly GBM Model 2.4.3, Monthly Nonlinear Model 3.1.1, 3.2.1 and 3.3.1 with jumps, and Unequal Nonlinear Model 3.1.2, 3.2.2 and 3.3.2 with Jumps. Table 3.3.4 Basic Statistics for Models 2.4.3, 3.1.1, 3.2.1, 3.3.1, 3.1.2, 3.2.2 and 3.3.2 of Stock X Model r 2 rS rS No. of Intervals No. of Parameters Monthly GBM with Jumps -1.22683 207.3278 14.3989 41 122 Monthly Nonlinear Model 1 with Jumps -1.928296 141.1754 11.88173 41 163 Monthly Nonlinear Model 2 with Jumps -2.090806 143.2248 11.96765 41 204 Monthly Nonlinear Model 3 with Jumps -1.731151 139.2792 11.80166 41 163 Unequal Interval GBM with Jumps -1.962899 258.1040 16.06562 39 116 Unequal Interval Nonlinear Model 1 with Jumps -0.531502 131.2354 11.4558 39 155 Unequal Interval Nonlinear Model 2 with Jumps -0.609702 131.3068 11.45892 39 194 Unequal Interval Nonlinear Model 3 with Jumps -0.402368 132.16 11.49609 39 155 Remark 3.3.2 : From Table 3.3.4 we conclude that overall, the Nonlinear Model 3.1.2 with Unequal Interval has less variance among all Mode ls (Monthly and Unequal Intervals) With Monthly data partitioning, Nonlinear Model 3.3.1 with Jumps has the least mean and variance of residual error. With Unequal Interval data par titioning, all Nonlinear M odels 3.1.2, 3.2.2 and 3.3.2 have less mean and variance of residual error than GBM (linear) model. 86

PAGE 100

3.4 Illustration of Nonlinear Stochastic Models to Data Set of Stock Y In this section, we apply the Monthly Nonlinear M odels 1, 2 and 3 with jumps, that is, Nonlinear Model 3.1.1, 3.2.1 and 3.3.1 to stock Y. We also apply the Unequal Interval Nonlinear Models 1, 2 and 3 with jumps, that is, Nonlinear Model 3.1.2, 3.2.2 and 3.3.2 to stock Y. To minimize the repetition, here we only give the summary of these 6 models in Table 3.4.1. The price data set of stock Y is relative larger than the price data set of stock X. There are 5630 observations over the past 22 years from September 1984 to December 2006. The Monthly Nonlinear Models have 268 monthly intervals, and the Unequal Interval Models have 256 intervals with the daily relative difference = 3.5% as the threshold. Table 3.4.1 Basic Statistics for Models of Stock Y Model r 2 rS rS No. of Intervals No. of Parameters Monthly GBM with Jumps -0.009826 1.206479 1.098399 268 803 Monthly Nonlinear Model 1 with Jumps 0.020068 1.469688 1.212307 268 1071 Monthly Nonlinear Model 2 with Jumps -0.002057 1.155363 1.074878 268 1339 Monthly Nonlinear Model 3 with Jumps 0.026632 1.295051 1.138003 268 1071 Unequal Interval GBM with Jumps -0.012482 1.199703 1.095310 256 767 Unequal Interval Nonlinear Model 1 with Jumps -0.004258 0.606470 0.778762 256 1023 Unequal Interval Nonlinear Model 2 with Jumps -0.011478 0.603385 0.776779 256 1279 Unequal Interval Nonlinear Model 3 with Jumps 0.006577 0.612513 0.782632 256 1023 87

PAGE 101

Remark 3.4.1 : Table 3.4.1 shows the overall basic statistics of Stock Y with respect to all stated Monthly and Unequal Interval Nonlinear Models. Under the Monthly data partitioning, Nonlinear Model 2 has the least mean and variance of resi dual error. Under the Unequal Interval data partitioning, all stated Nonlinear Models 1, 2 an d 3 have less mean and variance of residual error than the GBM model. Moreover, Nonlinear Model 2 with Unequal Interval has the least variance and standard deviation of resi dual error among all models. Furt hermore, this unequal data partitioning process has less number of subinterva ls than the monthly data partitioning process. 3.5 Illustration of Nonlinear Stochastic Models to Data Set of S&P 500 Index In this section, we apply the Monthly Nonlinear Models 1, 2 and 3, that is, Nonlinear Model 3.1.1, 3.2.1 and 3.3.1 of S&P 500 Index. We also apply the Unequal Interval Nonlinear Models 1, 2 and 3, that is, Nonlinear Model 3.1.2, 3.2.2 and 3.3.2 on S&P 500 Index. Again, to minimize the repetition, here we only give the summary of these 6 models in Table 3.5.1. Since the dataset is too large, here we only provide the summary of the models. The dataset of SP500 Index is larger than the previous datasets of stocks X and Y. There are 14844 observations over the past 59 years starting from January 1950 to December 2008. The Monthly Nonlinear Models have 708 monthly intervals, and the Unequal Interval Models have 570 intervals with the daily relative difference = 0.8% as the threshold. Table 3.5.1 Basic Statistics for Models of S&P 500 Index Model r 2 rS rS No. of Intervals No. of Parameters Monthly GBM with Jumps 58.30902 486.4579 22.05579 708 2123 Monthly Nonlinear Model 1 with Jumps 4.027517 281.7153 16.78438 708 2831 Monthly Nonlinear Model 2 with Jumps 4.084275 330.1271 16.78393 708 3539 Monthly Nonlinear Model 3 with Jumps 4.274907 282.7780 16.81600 708 2831 88

PAGE 102

Unequal Interval GBM with Jumps 2.471564 210.2159 14.49882 570 1709 Unequal Interval Nonlinear Model 1 with Jumps 0.6186245 79.46592 8.914366 570 2279 Unequal Interval Nonlinear Model 2 with Jumps 0.5835638 78.5180 8.861039 570 2849 Unequal Interval Nonlinear Model 3 with Jumps 0.6590607 79.5725 8.920342 570 2279 Remark 3.5.1 : Table 3.5.1 shows the overall basic statistics of S&P 500 Index for all stated Monthly and Unequal Interval Nonlinear Models. Under the Monthly data partitioning, Nonlinear Model 1 has the least mean and variance of resi dual error. Under the Unequal Interval data partitioning, all stated Nonlinear Models 1, 2 an d 3 have less mean and variance of residual error than the GBM model. Nonlinear Model 2 with Unequal Interval has the least mean and the variance of residual error among all stated models Furthermore, this unequal interval nonlinear model 2 has the least variance of residual error and the number of intervals. 3.6 Conclusions and Comments In this chapter, we presented three nonlinear stochastic models. By using classical model building process, we developed the modified version of nonlinear stochastic models under equal and unequal data partitioning processes with jumps. Ba sed on our study, in the following, we draw a few important conclusions. (a) The Following Table 3.6.1 provides the summary of results for all three datasets. It shows that Nonlinear Model 2 ranks No.1 in both monthly and unequal interval data pa rtitioning models of 2 out 3 data sets (stock X, stock Y and S&P 500 Index). Table 3.6.1 Summary of Models in Chapter 3 Stock Monthly Interval Unequal Interval Rank1 Rank2 Rank3 Rank4 Rank1 Rank2 Rank3 Rank4 X Non.3 Non.2 Non.1 GBM Non.1 Non.2 Non.3 GBM Y Non.2 GBM Non.3 Non.1 Non.2 Non.1 Non.3 GBM S&P500 Non.2 Non.1 Non.3 GBM Non.2 Non.1 Non.3 GBM 89

PAGE 103

The three data sets in our study, both stocks X and Y are from world Fortune 500 companies; S&P 500 Index is a stock Index. From Table 3.6.1, we notice that for two data sets (stock Y and S&P 500 Index) with both Monthly Interval and Unequal Interval data partitioning processes, Nonlinear Model 2 is the best model. These two data sets (stock Y and S&P 500 Index) share a common characteristic. Comparing to data set stock X (848 observations), both of these two dataset are very large, having 5630 and 14844 observations, respectively. (b) Tables 3.3.4, 3.4.1 and 3.5.1 show the overall basic statistics of different models of stocks X, Y and S&P 500 Index. For the monthly data partitioning, Nonlinear Model 2 is better than GBM Model for Stock Y and S&P 500 Index, and Nonl inear Model 3 is better than GBM model for stock X. For the unequal interval data partitioni ng process, Nonlinear Model 2 is better than GBM Model for Stock Y and S&P 500 Index, and Nonlinear Model 3 is better than GBM model for stock X. (i) For monthly data partitioning, all three non linear models are better than GBM model for all three price data sets. The Nonlinear Mode l 2 is better than GBM model and Nonlinear Models 1 and 3. (ii) Under unequal interval data partitioning process and for all three stock data sets, all nonlinear models are better than GBM model. (iii) The unequal data partitioning approach is s uperior than the monthly data partitioning approach. (iv) Under both equal and unequal data partitioning approach, the Nonlinear Model 2 is the best for stock Y and S&P 500 Index, and Nonlinear Model 1 is best for stock X. 05101520253035 260280300320340 The 13th IntervalIndexPrice 05101520253035 260280300320340 IndexPrice 05101520253035 260280300320340 IndexPrice 05101520253035 260280300320340 IndexPrice 05101520253035 260280300320340 IndexPrice Raw Data GBM Nonlinear Model 1 Nonlinear Model 2 Nonlinear Model 3 51 01 5 160180200220240 The 6th IntervalIndexPrice 51 01 5 160180200220240 IndexPrice 51 01 5 160180200220240 IndexPrice 51 01 5 160180200220240 IndexPrice 51 01 5 160180200220240 IndexPrice Raw Data GBM Nonlinear Model 1 Nonlinear Model 2 Nonlinear Model 3 (a) (b) Figure 3.6.1 The Predicted Value of Stock X 90

PAGE 104

(c) Again, from three tables 3.3.4, 3.4.1 and 3.5. 1, we observe that the performance of Nonlinear Models 1, 2, and 3 are very similar. The predic ted values for a particular interval are in Figure 3.6.1(a) and (b). In Figure 3.6.1 (a) and (b), we notice that the 3 red curve, orange curve and green curve are overlapped on each other. The bl ue curve represents the predicted value using GBM model. Furthermore, from the plot (the 13th interval and the 6th interval), we conclude that in this particular intervals Nonlinear Models estimates the stock price better than the GBM model. The reason is in that particular time interval every possible environmental information often leads to wild movements in stock price. The drift and volatility are not constant any more in that particular time interval. Hence the nonlinear model can describe the stock price process much better than the GBM model. (d) So far, we focused our attention to build stochastic models for stock price data sets. This modeling approach can also be used for any othe r type of data sets. Furthermore, our preceding stochastic modeling analysis of stock price confirms that a stock price process is nonlinear and non stationary stochastic models. However, the next important problem in modeling is to predict the future dynamic state of processes, in particular, stock market price. The study of this problem is focused in the next chapter. 91

PAGE 105

Chapter 4 Nonlinear Stochastic Models with Time Varying Coefficients 4.0 Introduction Stochastic dynamic models described in Chapters 2 and 3 were applicable to piece-wise timeinvariant dynamic processes. In this chapter, based on our study of three stochastic nonlinear models, we generalize our stochastic modeling d ynamic process by using the nonlinear stochastic differential equations with time varying coefficien ts. We focus our attention to only nonlinear models 1 and 2 that have been exhib ited better than mode l 3 in Chapter 3. Corresponding to nonlinear time invariant models 1 and 2, we present nonlinear stochastic models with time varying coefficient in Secti on 4.1 and 4.2, respectively. Using these nonlinear time varying models, we derive corresponding time series models. These time series models are tested by the three data sets, stock X, Y and S& P 500 Index. Furthermore, they are compared to the existing time series models [10, 12, 13, 38, 39 ] in Section 4.3. Finally, conclusions are drawn in Section 4.4. 4.1 Nonlinear Stochastic Dynamic Mode l 1 with Time Varying Coefficients In Chapters 3, nonlinear stochastic dynamic models with constant coefficients were investigated. In this section, we assume that the rate parame ters in the nonlinear stochastic dynamic model 1 (in Section 3.1, Chapter 3) are functions of time. Nonlinear Stochastic Model 1 on Overall Data: Let us consider a stochastic nonlinear model corresponding to equation (3.1.1) as tttt t ttttdWSdtS S dS ) 2 ln(2 0)0( SS (4.1.1) 92

PAGE 106

where, parameters and are time varying smooth functions [6, 26]. We note that the existence and uniqueness of solution process of (4 .1.1) follows by following similar arguments used in Section 3.1. By following the arguments and using the transformation t tSV ln we obtain tt ttttdWdtVdV )(. (4.1.2) To estimate the time varying parameters and we first use a numerical integration applied (4.1.2) as: t t ss kt kt ss t t sss kt kt sss ktt t kt ss t kt sss t kt sdW dW dsV dsV VV dW dsV dV1 1 1 1... )(...)( )( t ktk kt ktkktkV VV 1 1 1 11 11... ... ... where, k is any positive integer and ),1,0(~1NWWititit for .1,...,1,0 kiBy denoting11... kk, and rearranging terms in the equation, we have the following equation t ktk t ktkktktV VV V 1 1 11 11... ... )1( (4.1.3) This is exactly a time series ARIMA model with or der (p,q), where p=k and q=k-1. The constant term can be eliminated by taking the first order difference filter (d=1). Obviously, we notice that when k=1, we have the constant coefficients case (3.1.4) in Chapter 3. If k=2, equation (4.1.3) is equivalent to ARIMA(2,1). If we assume that k=2 and02 then equation (4.1.3) is equivalent to ARIMA(2,0). Under the transformation and following the Statistical Model Identification Procedure 1.5.1 described in Section 1.5, the AICs of AR IMA models of three data sets (stock X, Y and S&P 500 Index) are presented in Table 4.1.1. t tSV ln 93

PAGE 107

Table 4.1.1 AIC of Time Varying Coefficients Nonlinear Model 1 of Different Models of Three Dataset s: Stock X, Stock Y and S&P 500 Index Model Stock X Stock Y S&P 500 Index AIC AIC AIC (3,1,2) -4122.57 -22693.15 -96126.87 (3,1,1) -4124.70 -22694.84 -96126.3 (3,1,0) -4124.20 -22687.87 -96128.11 (2,1,3) -4127.76 -22692.57 -96126.82 (2,1,2) -4125.36 -22685.91 -96128.74 (2,1,1) -4126.42 -22682.35 -96130.18 (2,1,0) -4126.20 -22683.14 -96126.53 (1,1,3) -4124.86 -22694.39 -96127.68 (1,1,2) -4124.19 -22682.03 -96130.20 (1,1,1) -4126.21 -22681.05 -96120.59 (1,1,0) -4128.10 -22683.05 -96082.23 (0,1,3) -4124.30 -22687.57 -96129.77 (0,1,2) -4126.20 -22682.90 -96130.02 (0,1,1) -4127.92 -22683.06 -96086.05 From Table 4.1.1, we notice that for stock X, ARIMA model (1,1,0) giv es us the minimum AIC, that is, a mix model of a first order autoregressive with a first difference filter. The model is written as t0.02110687 )1)(0872.01( tVBB. After expanding the autoregressive operato r and the difference filter, we have t tVBB0.02110687 )0872.00872.11(2 which implies t t t tV V V 0.02110687 0872.00872.12 1 By letting 0 t we have the one day ahead forecasting formula of of stock X as tV2 10872.00872.1 t t tV V V. (4.1.4) 94

PAGE 108

Then, by applying the inverse transformation of ln, we get The residual error ) exp( t tVS iiiSSr is computed, and its basic statistics is recorded in Table 4.1.2. Similarly, for data set Stock Y, the fitted ARIM A model (3,1,1) gives us the minimum AIC. The model is t tB VBB BB)6638.0 03220248.0()1)(0523.00156.06575.01(3 2 By following above argument, we have 1 4 3 2 16638.0 03220248.00523.00367.06419.03425.0 t t t t t t tV V V V V By letting 0 t we obtain the one day ahead forecasting formula of of stock Y as tV1 4 3 2 16638.00523.00367.06419.03425.0 t t t t t tV V V V V. (4.1.5) Again, by applying the inverse transformation of ln, we get The residual error is computed, and its basic statistics are recorded in Table 4.1.2. ) exp( t tVS iiiSSr For data set S&P 500 Index, the fitted ARIMA models (1,1,2) gives us the minimum AIC, and the model is t tBB VBB)0438.02787.02 0.00949052 ()1)(2297.01(2 By following above argument, we have 2 1 2 10438.02787.0 009490522.02297.07703.0 t t t t t tV V V By letting 0 t we obtain the one day ahead forecasting formula of of S&P 500 Index as tV2 1 2 10438.02787.0 2297.07703.0 t t t t tV V V (4.1.6) Again, by applying the inverse transformation of ln, we get The residual error is computed, and its basic statistics are recorded in Table 4.1.2. ) exp( t tVS iiiSSr 95

PAGE 109

Table 4.1.2 Basic Statistics of Time Varying Coefficients Nonlinear Model 1 of Three Data Sets: Stock X, Stock Y and S&P 500 Index Data Set Model Mean of residual Vari ance of residual Standard deviation of residual Stock X (1,1,0) 0.628727 57.38475 7.575272 Stock Y (3,1,1) 0.015286 0.344827 0.587220 S&P 500 Index (1,1,2) 0.058922 46.90737 6.848895 Remark 4.1.1 : For data set stock X, from Tables 3.3.4 a nd 4.1.2, the nonlinear stochastic model 1 with time varying coefficients has the minimum variance of residual error. This is the same as for stock Y (Tables 3.4.1 and 4.1.2) and S&P 500 Index (Tables 3.5.1 and 4.1.2). We note that the nonlinear stochastic model 1 with time varying coe fficients is applied to overall data set. The study in Chapter 3 is with regard to the unequal interval data partitioning process. In the following we apply the unequal interval Data Partitioning Process 2.3.4 for nonlinear stochastic model 1 with time varying coefficients. Nonlinear Stochastic Model 4.1.1 (Unequal Interval Nonlinear Model 1 with Time Varying Coefficients): Let be the n time intervals as defined in Data Partition Process 2.3.4. The nonlinear stochastic differential equation is described by: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,)nntt t I t I t I t I t I t I t I t I tdWSdtS S dSii i i i i i i ) 2 )( ln(2, 00SS if 1iittt 1,..., in (4.1.7) iiII, and , are time varying parameters. iI1,..., i n As before, by imitating the time series definition process, we arrive at (4.1.8) ii ii ii ii ii i i iI t I I kt I k I kt I k I t I I kt I k I kt I k I tV VV V 1 11 1 11 11... ... )1( Furthermore, and are defined analogous to (3.1.8). tS1,...,2,1, nii 96

PAGE 110

Stochastic Model Identification Procedure 4.1.1: In the following, we present a modified version of Statistical Model Identification Procedure 1.5.1 [10,12,38]. It is as follows: i. By following the Data Partition Process 2.3.4, the entire data set is decomposed into n sub data sets. ii. For every sub data set, use the transformationi, 1,...,iI t I tSV ln in iii. For every sub data sets, repeat steps ii v in Stochastic Model Identification Procedure 1.5.1. iv. For every sub data set, and for each possibl e set of (p, q), compute the predicted value),(qpI tiV, and then compute the predicted value),(qpI tiS, by using the inverse of ln transformation, that is, ) exp( ),( ),(qpI t qpI ti iV S v. For every sub data set and for each possible models, compute the residual error .,...,2,1. ,, 1 ),( ),(nitttSSri i qpI tt qp ti vi. For all possible set of (p, q), compute mean, variance and standard deviation of overall residual error The model provides the smallest variance of residual is the fitted model. 1,),(Ttrqp t Table 4.1.3, 4.1.4 and Table 4.1.5 exhibit the basic statistics of the residuals using different value of k with unequal interval data partitioning of three datasets: Stock X, Y and S&P 500 Index respectively. Here the thresholds of daily rela tive difference for three data sets are set to 5%, 4.5% and 2%, respectively, and the corresponding number of intervals are 10, 66 and 87. Table 4.1.3 Basic Statistics of Stochastic Models 4.1.1 with Different Set of (p, q) Under Log-Transformation with Unequal Data Partition, threshold=5% of Stock X Model Mean of Residual Variance of Residual Standard Deviation of Residual Number of Intervals (3,1,2) 0.535531 44.01918 6.634695 10 (3,1,1) 0.477112 46.09286 6.789173 10 (3,1,0) 0.584083 47.90231 6.92115 10 97

PAGE 111

(2,1,3) 0.571359 43.62668 6.60505 10 (2,1,2) 0.588372 44.5358 6.673515 10 (2,1,1) 0.648873 47.5926 6.898739 10 (2,1,0) 0.652255 48.23812 6.945367 10 (1,1,3) 0.479333 45.94483 6.778261 10 (1,1,2) 0.558626 46.4712 6.816979 10 (1,1,1) 0.562882 46.79851 6.840944 10 (1,1,0) 0.636871 48.66251 6.975852 10 (0,1,3) 0.640242 48.47799 6.962613 10 (0,1,2) 0.64442 48.24343 6.945749 10 (0,1,1) 0.640242 48.47799 6.962613 10 Table 4.1.4 Basic Statistics of Stochastic Models 4.1.1 with Different Set of (p, q) Under Log-Transformation with Unequal Data Partition, threshold=4.5% of Stock Y Model Mean of Residual Variance of Residual Standard Deviation of Residual Number of intervals (3,1,2) 0.013137 0.284444 0.533333 66 (3,1,1) 0.010776 0.303817 0.551196 66 (3,1,0) 0.011263 0.305855 0.553041 66 (2,1,3) 0.012679 0.287968 0.536626 66 (2,1,2) 0.00985 0.296015 0.544072 66 (2,1,1) 0.010492 0.306466 0.553593 66 (2,1,0) 0.012846 0.308965 0.555846 66 (1,1,3) 0.010388 0.302722 0.550202 66 (1,1,2) 0.010206 0.306028 0.553198 66 (1,1,1) 0.013911 0.308552 0.555475 66 (1,1,0) 0.013396 0.311613 0.558223 66 (0,1,3) 0.013494 0.31157 0.558185 66 (0,1,2) 0.013209 0.30873 0.555635 66 (0,1,1) 0.013494 0.31157 0.558185 66 98

PAGE 112

Table 4.1.5 Basic Statistics of Stochastic Models 4.1.1 with Different Set of (p, q) Under Log-Transformation with Unequal Data Partition, threshold=2% of S&P 500 Index Model Mean of Residual Variance of Residual Standard Deviation of Residual Number of intervals (3,1,2) 0.116613 39.73163 6.303303 87 (3,1,1) 0.107649 41.31667 6.427805 87 (3,1,0) 0.108677 42.04793 6.484438 87 (2,1,3) 0.130816 37.94544 6.159987 87 (2,1,2) 0.118426 40.47064 6.361654 87 (2,1,1) 0.105609 41.83156 6.467732 87 (2,1,0) 0.107766 42.51392 6.52027 87 (1,1,3) 0.112032 41.55797 6.446547 87 (1,1,2) 0.115318 41.9422 6.47628 87 (1,1,1) 0.113373 42.86889 6.547434 87 (1,1,0) 0.110322 43.49516 6.595086 87 (0,1,3) 0.105709 43.30008 6.580279 87 (0,1,2) 0.105485 42.55877 6.523708 87 (0,1,1) 0.105709 43.30008 6.580279 87 From Table 4.1.3 and Table 4.1.5, we can see th at the model (2,1,3) has minimum variance and standard deviation of residuals, for stock X a nd S&P 500 Index. From Table 4.1.4 we see that the model (3,1,2) is the best model which provides th e minimum variance of the residual. We further note that ARIMA model (2,1,3) is th e best for three all data sets. Remark 4.1.2 : For stock X, we compare Table 3.3.4, 4.1.2, with Table 4.1.3, we notice that nonlinear model 1 with time varying coefficients under unequal interval data partitioning process provides least variance and standard deviation of residual error. Similarly, for stock Y, comparing Table 3.4.1, 4.1.2 with Table 4.1.4; for S&P 500 Index, comparing Table 3.5.1, 4.1.2 with Table 4.1.5, we have the same conclusion. 99

PAGE 113

4.2 Nonlinear Stochastic Dynamic Mode l 2 with Time Varying Coefficients In this section, we assume that the rate parame ters in the nonlinear stochastic dynamic model 2 (in Section 3.2, Chapter 3) are not constants, that is, the rates, and are functions of time, and N is still a constant. Nonlinear Stochastic Model 2 on Overall Data: Let us consider a stochastic nonlinear model corresponding to equation (3.2.1) as t N tt N tt N tttttdWSdtS N SSdS ) 2 (122 (4.2.1) 0)0( SS where, coefficients and are time varying smooth functions [26]. We note that the existence and uniqueness of solution process of (4 .2.1) follows by following similar arguments used in Section 3.2. By following the arguments and using the transformation N S VN t t 11, we obtain tt tt ttdWdtVN dV ))1((. (4.2.2) To estimate the time varying parameters and we first use a numerical integration applied to (4.2.2) as follows: t t ss kt kt ss t t ss s kt kt ss s ktt t kt ss t kt ss s t kt sdW dW dsVN dsVN VV dW dsVN dV1 1 1 1... ))1((...))1(( ))1(( t ktk kkt ktkVN VN 1 1 11 11... ... )1(...)1( where, k is any positive integer and ),1,0(~1NWWititit for .1,...,1,0 kiBy denoting11... kk and rearranging terms in the equation, we have the following equation t ktk t ktk ktk tVN VNVNV 1 11 11... )1(... )1()1)1(( (4.2.3) This is also a time series ARIMA model with order (p,q), where p equals k and q equals k-1. The constant term can be eliminated by taking the first order difference filter (d=1). 100

PAGE 114

Similarly, we notice that when k=1, we have the constant coefficients case (3.2.4) in Chapter 3. If k=2, equation (4.2.3) is equivalent to ARIMA(2,1). If we assume that k=2 and02 then equation (4.2.3) is equivalent to ARIMA(2,0). Stochastic Model Identification Procedure 4.2.1: The difference between the nonlinear model 1 and 2 is that nonlinear model 2 has a parameter N that can not be estimated directly. In the following, we present a modified version of Statistical Model Identification Procedure 1.5.1. It is as follows: i. Let 2.1 and N 1. 0 Nii. For each value of N, say NN, use the transformation, Tt N S VN t t,...,2,1, 1 1 iii. Follow the Stochastic Model Identification Procedures 1.5.1 (ii-vii). iv. By knowing the best model ) (),(Nqpfor each value of N, compute the predicted value of price process by applying the inverse transformation of .) ) 1(( 1 1 ) ( N t N tVN S v. Computer the residual error T. tSSrN ttt,...,2,1, ) (vi. Repeat the steps (ii-v) for each given ]2.1,1(. )1,0[ NNvii. The value Nand the corresponding model provides the smallest variance of residual error (T ,1) is the estimated N and fitted model. trt, ,...,2 We apply the Stochastic Model Identification Pro cedure 4.2.1 to three data sets and the result is exhibited in Table 4.2.1. Table 4.2.1 Basic Statistics of Time Varying Coefficients Nonlinear Model 2 of Three Data Sets: Stock X, Stock Y and S&P 500 Data Set Model N Mean of residual Variance of residual Standard deviation of residual Stock X (3,1,2) 0.07 0.623089 56.59861 7.523205 Stock Y (2,1,2) 0.03 0.013675 0.340876 0.583846 S&P 500 Index (3,1,2) 0.03 0.071906 46.27549 6.802609 101

PAGE 115

Table 4.2.1 shows basic statistics of time varyin g coefficients of nonlinear stochastic model 2 for three Data Sets: Stock X, Stock Y and S&P 500. From the table, we can see that for stock X, ARIMA model (3,1,2) gives us the minimum variance of residual. The model is t tBB VBB BB)999.06804.0 4.942671()1)(0841.09055.05837.01(2 3 2 After expanding the autoregressive operato r and the difference filter, we have t tBB VB B BB)999.06804.0 4.942671()0841.08214.03218.05837.11(2 4 3 2 which implies .999.06804.0 4.942671 0841.08214.03218.05837.12 1 4 3 2 1 t t t t t t t tV V V V V By letting 0 t we have the one day ahead forecasting formula of of stock X as tV.999.06804.00841.08214.03218.05837.1 2 1 4 3 2 1 t t t t t t tV V V V V (4.2.4) Then, by applying the inverse transformation, N t tVNS 1 1) ) 1(( the residual error is computed, and its basic statistics are recorded in Table 4.2.1. 07.0 NiiiSSr Similarly, for data set Stock Y, the fitted AR IMA model (2,1,2) gives us the minimum variance of residual. The model is t tBB VBBB)7420.03207.1 0.5260228()1)(6948.02999.11(2 2 By following above argument, we have 2 1 3 2 17420.03207.1 0.5260228 6448.09947.12999.0 t t t t t t tV V V V By letting 0 t we obtain the one day ahead forecasting formula of of stock Y as tV2 1 3 2 17420.03207.16448.0 9947.12999.0 t t t t t tV V V V (4.2.5) Again, by applying the inverse transformation, N t tVNS 1 1) ) 1(( the residual error is computed, and its basic statistics are recorded in Table 4.2.1. 03.0 NiiiSSr For data set S&P 500 Index, the fitted ARIMA mode ls (3,1,2) gives us the minimum variance of residual, and the model is 102

PAGE 116

t tBB VBB BB)2477.03932.0 5.517246()1)(0385.01991.03286.01(2 3 2 By following above argument, we have 2 1 4 3 2 12477.03932.0 517246.50385.01606.01295.06714.0 t t t t t t t tV V V V V By letting 0 t we obtain the one day ahead forecasting formula of of S&P 500 Index as tV2 1 4 3 2 12477.03932.00385.01606.01295.06714.0 t t t t t t tV V V V V (4.2.6) Again, by applying the inverse transformation, N t tVNS 1 1) ) 1(( the residual error is computed, and its basic statistics are recorded in Table 4.2.1. 03.0 NiiiSSr Remark 4.2.1 : For data set stock X, comparing Table 3.3.4, 4.1.1 and Table 4.2.1, nonlinear stochastic model 2 with time varying coefficien ts has the minimum variance of residual error. This is the same as for stock Y (Table 3.4.1, 4.1.1 and Table 4.2.1) and S&P 500 Index (Table 3.5.1, 4.1.1and Table 4.2.1). In the following we apply the unequal interval Data Partitioning Process 2.3.4 for nonlinear stochastic model 2 with time varying coefficients. Nonlinear Stochastic Model 4.2.1 (Unequal Interval Nonlinear Model 2 with Time Varying Coefficients): Let be the n time intervals as defined in Data Partition Process 2.3.4. The nonlinear stochastic differential equation is described by: 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,)nntt t N I t I t N I t I t N I t I t I t I t I tdWSdtS N SSdSii i i ii ii i)())()( 2 )( (122 00SS if 1 iittt 1,...,in (4.2.7) iiII, and , are time varying parameters, N is constant.[26,27,29]. iI1,..., i nAs before, by imitating the time series definition process, we arrive at (4.2.8) ii ii i i iI t I I kt I k I kt I k I tVN VNVN V11 11)1(... )1()1)1(( ii iiI t I I kt I k 1 1... Furthermore, and are defined analogous to (3.2.8). tS1,...,2,1, nii 103

PAGE 117

Stochastic Model Identification Procedure 4.2.2: In the following, we present a modified version of Statistical Model Identification Procedure 1.5.1. It is as follows: i. By following the Data Partition Process 2.3.4, th e entire data set is decomposed into n sub data sets. ii. For each sub data set, follow steps i-ii of th e Stochastic Model Identification Procedures 4.2.1. iii. For each sub data set, follow the Stochastic Mode l Identification Procedures 1.5.1 steps ii-v. iv. Using estimated parameters in step iii and compute the residual error T for all possible (p,q). tSSrqpN tt qpN t,...,2,1, ),)( ( ),)( ( v. Repeat steps ii-iv for ]2.1,1(. )1,0[ NNvi. For, a given set of (p, q), we compute overall sum of squared error for every value of N by T t qpN t qpNr RSS1 2 ),)( ( ),)( (vii. For the given (p, q) in step vi, we find th e best N, corresponding to the minimum RSS. viii. Repeat steps vi vii for all possible model (p,q ), we find the best Ns with respect to the minimum RSS. ix. From viii we choose the model with corresponding N, which provides the smallest RSS. Table 4.2.2, 4.2.3 and Table 4.2.4 show the basic statistics of the residual error using different set of (p, q) with unequal interval data partitioning of three dataset s: Stock X, Y and S&P 500 Index, respectively. Here the thresholds of daily rela tive difference for three data sets are set to 5%, 4.5% and 2%, respectively, such that the sub intervals have enough observations to estimate the parameters. The residual error is defined as well as for all observations. iiiSSr Table 4.2.2 Basic Statistics of Stochastic Models 4.2.1 with Different Set of (p, q) under Transformation N SN t11 with Unequal Data Partition, threshold=5% of Stock X Model Mean of Residual Variance of Residual Standard Deviation of Residual Number of intervals (3,1,2) 0.48359 42.8776 6.548099 10 (3,1,1) 0.531298 45.89411 6.774519 10 104

PAGE 118

(3,1,0) 0.571061 47.81292 6.914689 10 (2,1,3) 0.527294 41.97917 6.479133 10 (2,1,1) 0.631136 47.1654 6.867707 10 (2,1,0) 0.636309 48.16542 6.940131 10 (1,1,3) 0.520179 45.68867 6.75934 10 (1,1,2) 0.532497 46.36265 6.809013 10 (1,1,0) 0.619884 48.62252 6.972985 10 (0,1,3) 0.623101 48.43584 6.959586 10 (0,1,2) 0.627815 48.17912 6.941118 10 (0,1,1) 0.623101 48.43584 6.959586 10 Table 4.2.3 Basic Statistics of Stochastic Models 4.2.1 with Different Set of (p, q) under Transformation N SN t11 with Unequal Data Partition, threshold=4.5% of Stock Y Model Mean of Residual Variance of Residual Standard Deviation of Residual Number of intervals (3,1,2) 0.013494 0.280789 0.529895 66 (3,1,1) 0.012033 0.301285 0.548894 66 (3,1,0) 0.011385 0.305035 0.552300 66 (2,1,3) 0.012897 0.281243 0.530324 66 (2,1,0) 0.012724 0.308477 0.555407 66 (1,1,3) 0.009512 0.301548 0.549134 66 (1,1,2) 0.011650 0.305205 0.552454 66 (1,1,0) 0.012968 0.311111 0.557773 66 (0,1,3) 0.012975 0.311157 0.557814 66 (0,1,2) 0.013045 0.308202 0.555160 66 (0,1,1) 0.012975 0.311157 0.557814 66 105

PAGE 119

Table 4.2.4 Basic Statistics of Stochastic M odels 4.2.1 with Different Set of (p, q) under Transformation N SN t11 with Unequal Data Partition, threshold=2% of S&P 500 Index Model Mean of Residual Variance of Residual Standard Deviation of Residual Number of intervals (3,1,1) 0.084646 41.14148 6.414162 87 (3,1,0) 0.093201 41.90288 6.473243 87 (2,1,3) 0.114031 37.68062 6.138454 87 (2,1,0) 0.091409 42.38462 6.510347 87 (1,1,3) 0.099771 41.4015 6.43440 87 (1,1,2) 0.093127 41.76005 6.462202 87 (1,1,0) 0.095328 43.36592 6.585281 87 (0,1,3) 0.08993 43.17434 6.570719 87 (0,1,2) 0.090155 42.4362 6.514308 87 (0,1,1) 0.08993 43.17434 6.570719 87 Remark 4.2.2 : For stock X, we compare Table 3.3.4, 4.2.1, with Table 4.2.2, we notice that nonlinear model 2 with time varying coefficients under unequal interval data partitioning process provides least variance and standard deviation of residual error. Similarly, for stock Y, comparing Table 3.4.1, 4.2.1 with Table 4.2.3; for S&P 500 Index, comparing Table 3.5.1, 4.2.1 with Table 4.2.4, we have the same conclusion. 4.3 Prediction and Comparison on Overall Data Sets In Sections 4.1 and 4.2, using nonlinear continuo us time varying stochas tic models, we derived time series models. In this section, we compare our study of Sections 4.1 and 4.2 with the existing time series models, namely, k-th moving averag e model, k-th weighted and k-th exponential weighted moving average models [13,38,39]. A comparative study is made in the context of three overall data sets. In fact, the following models are compared with each other. Time series model (ARIMA) [10,12] k-th moving average model (Shis model 1) [13,38,39] k-th weighted moving average model (Shis model 2) [13,38,39] 106

PAGE 120

k-th exponential weighted moving average model (Shis model 3) [13,38,39] Nonlinear Models with constant coefficients, Chapter 3 Nonlinear Stochastic Model 1 on Overall Data Set, Section 4.1 Nonlinear Stochastic Model 2 on Overall Data Set, Section 4.2 We summary the results for stock X, stock Y a nd S&P 500 Index in Table 4.3.1 Table 4.3.2 and Table 4.3.3 respectively. Table 4.3.1 Comparison Cited Mode ls in Section 4.3 for Stock X Model Mean of Residual Variance of Residual Standard Deviation of Residual ARIMA 0.6385010 57.39102 7.575686 k-th Moving Average Model 0.6342918 57.03750 7.552318 k-th Weighted Moving Average Model 0.6359891 57.14087 7.559158 k-th Exponential Weighted Moving Average Model 0.8944923 64.64898 8.040459 Nonlinear Models with Constant Coefficients -0.6097021 131.2354 11.45580 Nonlinear Model 1 with Time Varying Coefficients 0.628727 57.38475 7.575272 Nonlinear Model 2 with Time Varying Coefficients 0.623089 56.59861 7.523205 Remark 4.3.1 : For stock X, five models perform pretty much close to each other. They are ARIMA model, k-th moving average model, k-th weighted moving average model, nonlinear stochastic models 1 and 2 with time vary ing coefficients. Among these models, nonlinear stochastic model 2 with time varying coefficients has the least variance and standard deviation. 107

PAGE 121

Table 4.3.2 Comparison Cited Mode ls in Section 4.3 for Stock Y Model Mean of Residual Variance of Residual Standard Deviation of Residual ARIMA 0.00725343 0.3419923 0.5848011 k-th Moving Average Model 0.00748872 0.3418268 0.5846595 k-th Weighted Moving Average Model 0.00741209 0.3411141 0.5840498 k-th Exponential Weighted Moving Average Model 0.01503370 0.3773675 0.6143024 Nonlinear Models with Constant Coefficients -0.01147757 0.6033852 0.776779 Nonlinear Model 1 with Time Varying Coefficients 0.015286 0.344827 0.587220 Nonlinear Model 2 with Time Varying Coefficients 0.013675 0.340876 0.583846 Remark 4.3.2 : Like stock X, for stock Y, five models perform pretty much close to each other. They are also ARIMA model, k-th moving average model, k-th weighted moving average model, nonlinear stochastic models 1 and 2 with time varying coefficients. Among these models, nonlinear stochastic model 2 with time varying coe fficients has the least variance and standard deviation. Table 4.3.3 Comparison Cited Models in Section 4.3 for S&P 500 Index Model Mean of Residual Variance of Residual Standard Deviation of Residual ARIMA 0.07225937 46.2575 6.801286 k-th Moving Average Model 0.08731528 59.17083 7.692258 108

PAGE 122

k-th Weighted Moving Average Model 0.07014848 46.25595 6.801173 k-th Exponential Weighted Moving Average Model 0.09544027 56.68555 7.52898 Nonlinear Models with Constant Coefficients 0.5835638 78.5180 8.861039 Nonlinear Model 1 with Time Varying Coefficients 0.058922 46.90737 6.848895 Nonlinear Model 2 with Time Varying Coefficients 0.071906 46.27549 6.802609 Remark 4.3.3 : In Table 4.3.3, for S&P 500 Index, four models perform pretty much close to each other. There are ARIMA model, k-th weighted moving average model, nonlinear stochastic models 1 and 2 with time varying coefficien ts. Among these models, k-th weighted moving average model has least variance and standard de viation of residual error. We note that our nonlinear model 2 is reasonably close to linear weighted model. From above discussion, we draw a few conclusions: For all three datasets, nonlinear stochastic models with time vary coefficient have less variance and standard deviation of residual than the nonlinear models with constant coefficients. Nonlinear stochastic model 2 with time varyin g coefficients has the least variance and standard deviation of residual among all models for two data sets, namely, stocks X and Y. Dr. Shis k-th weighted moving average mode l [38] has the least variance and standard deviation of residual among all models for one dataset S&P 500 Index. We remark the standard deviation of nonlinear stochastic m odel 2 is larger 0.01954, that is, about 0.04% larger than Dr. Shis k-th weighted moving average model. Knowing the performance of nonlinear stochastic m odels with constant coefficients, we present Tables 4.3.4, 4.3.5 and 4.3.6 for remaining si x models, namely, ARIMA model, k-th moving average model, k-th weighted moving average model, k-th exponential weighted moving average model, nonlinear stochastic models 1 and 2 with time varying coefficients. These tables contain the actual and forecasted values for three data sets. 109

PAGE 123

Table 4.3.4 Actual and Predicted Price for Stock X t Actual Value Predicted Value ARIMA k-th k-th Weighted k-th Exp. Weighted Nonlinear Model 1 Nonlinear Model 2 848 685.19 690.5668 683.002 687.8663 688.95 690.5245 688.9735 849 685.33 684.6626 669.0717 678.8684 680.7026 684.642 684.7898 850 657 685.3417 675.2173 683.2993 684.0164 685.3423 685.6791 851 649.25 654.6294 634.8802 645.5674 648.7488 654.5775 655.224 852 631.68 648.5779 613.5021 636.6892 640.2909 648.5692 647.6274 853 653.2 630.1105 601.3531 622.4698 625.3764 630.1352 630.6804 854 646.73 654.9494 649.3253 655.5306 655.3039 655.0986 654.9312 855 638.25 646.2265 659.2641 650.889 649.8597 646.1798 647.0794 856 653.82 637.5815 630.9644 632.4086 633.6497 637.5274 635.8812 857 637.65 655.0054 655.3506 657.6908 656.6988 655.16 654.9123 858 615.95 636.504 639.1148 636.6878 637.1356 636.3224 637.4456 859 600.79 614.2723 587.4148 601.4316 605.362 614.1321 612.8671 860 600.25 599.5327 562.7062 588.4312 591.9794 599.4835 599.1939 861 584.35 600.2053 576.8988 595.1241 597.1468 600.2029 601.133 862 548.62 583.0321 566.9171 576.6775 579.0589 582.9814 583.1959 863 574.49 545.2863 505.6626 528.6499 533.91 545.4584 544.1784 864 566.4 576.4021 550.7285 573.7575 574.4719 576.695 574.846 865 555.98 565.8405 575.6357 571.1122 570.1095 565.7497 566.9464 866 550.52 555.2441 549.3242 548.4349 550.4871 555.1378 553.4377 867 548.27 550.1295 532.6924 545.7815 547.0418 550.0745 549.7962 868 564.3 548.1087 536.8028 545.8932 546.8686 548.0856 546.3439 869 515.9 565.4388 572.9851 569.0178 567.9075 565.6321 565.2503 870 495.43 513.1611 496.6788 502.3305 506.0323 512.4411 512.5852 871 506.8 493.965 434.953 471.6746 478.8219 493.7756 491.6517 872 501.71 507.562 478.7335 505.019 506.1324 507.7206 505.5978 110

PAGE 124

Table 4.3.5 Actual and Predicted Price for Stock Y t Actual Value Predicted Value ARIMA k-th k-th Weighted k-th Exp. Weighted Nonlinear Model 1 Nonlinear Model 2 5631 83.8 84.82902 86.77076 85.95206 85.54406 84.87698 84.7722 5632 85.66 84.00733 87.0601 84.66872 84.41756 83.74178 83.90168 5633 85.05 85.66035 87.22635 86.23666 86.03169 85.51144 85.77717 5634 85.47 85.15029 86.28798 85.54065 85.43127 85.16966 85.14353 5635 92.57 85.52184 85.80557 85.64303 85.62554 85.30488 85.46407 5636 97 92.53652 97.42856 94.78931 93.93159 92.758 92.42399 5637 95.8 97.25259 107.4367 100.8205 99.56975 96.75397 97.02509 5638 94.62 96.03867 102.1023 97.29214 96.84044 95.48694 96.09939 5639 97.1 94.72032 94.26917 94.38129 94.52896 94.60442 95.00356 5640 94.95 97.146 97.17758 97.85285 97.66013 97.20943 97.29875 5641 89.07 95.14956 95.76419 94.89111 94.93671 94.90279 95.07715 5642 88.5 89.20548 84.0502 86.72816 87.58486 88.98043 89.02443 5643 86.79 88.50399 81.45928 86.72795 87.39757 88.74691 88.18962 5644 85.7 86.87246 83.05968 85.62383 85.99651 86.91702 86.35764 5645 86.7 85.75254 83.14882 84.53992 84.89636 85.66368 85.31255 5646 86.25 86.74345 85.69431 86.68557 86.72493 86.83622 86.48165 5647 85.38 86.34851 86.61938 86.425 86.38849 86.1994 86.26917 5648 85.94 85.45339 84.8209 84.94762 85.08256 85.36444 85.54056 5649 85.55 85.9905 85.31589 85.98491 86.02736 85.99153 86.15072 5650 85.73 85.63656 85.58075 85.61227 85.6082 85.54971 85.74118 5651 84.74 85.80065 85.76531 85.66177 85.67719 85.70833 85.81162 5652 84.75 84.82062 84.19112 84.5681 84.68931 84.76484 84.6882 5653 83.94 84.80058 83.82978 84.54765 84.62527 84.73972 84.61102 5654 84.15 84.01544 83.17927 83.60801 83.7308 83.99208 83.83675 5655 86.15 84.19971 83.51475 84.04115 84.10393 84.12795 84.05082 111

PAGE 125

Table 4.3.6 Actual and Predicted Price for S&P 500 Index t Actual Value Predicted Value ARIMA k-th k-th Weighted k-th Exp. Weighted Nonlinear Model 1 Nonlinear Model 2 14845 931.8 900.8314 922.4829 909.6238 905.9727 902.5525 900.9227 14846 927.45 929.2406 970.9482 944.2084 939.3075 932.9931 929.0251 14847 934.7 925.1714 953.0097 932.0198 929.1136 925.6639 925.1775 14848 906.65 933.7874 944.5406 936.7494 935.8587 935.8555 933.9431 14849 909.73 906.9402 895.3576 899.2609 901.1965 904.6217 907.0545 14850 890.35 910.8315 889.0191 904.3632 906.3103 911.6803 911.0914 14851 870.26 891.0103 870.3993 884.6703 886.3622 888.6707 891.0485 14852 871.79 873.1138 843.0015 860.8976 865.2716 870.6269 873.2177 14853 842.62 873.4106 852.11 868.7196 870.5928 872.5018 873.1921 14854 843.74 844.7466 810.8334 832.4831 837.2048 840.9775 845.0018 14855 850.12 846.7019 835.1808 841.1104 843.4486 845.4405 846.3571 14856 805.22 850.2321 836.0299 848.9394 849.8067 849.7196 850.1915 14857 840.24 808.6526 790.8013 796.2332 801.0583 803.0576 808.5679 14858 827.5 841.9541 837.8833 841.4893 841.914 844.4779 841.5228 14859 831.95 825.9884 813.0403 827.1792 827.3513 824.1323 826.4057 14860 836.57 834.0853 855.8398 834.8218 834.2957 833.7654 833.6063 14861 845.71 836.0695 841.6322 838.2363 838.3141 835.7427 835.7924 14862 874.09 844.725 838.8276 847.1184 846.0498 846.1652 845.2083 14863 845.14 872.1378 915.9827 886.4861 881.3798 874.839 871.5874 14864 825.88 844.9611 849.1547 841.3719 842.5598 842.5651 845.1329 14865 825.44 828.8902 812.1273 818.3743 822.268 826.9521 828.9131 14866 838.51 825.9724 788.507 816.5074 819.2051 825.8489 826.2782 14867 832.23 837.8741 832.4872 841.8058 840.0885 839.0741 838.1304 14868 845.85 832.6112 852.194 834.7648 833.7577 831.2401 832.112 14869 868.6 845.546 846.6754 849.1995 848.8302 847.0508 845.5924 112

PAGE 126

Then we compute the basic statistics for residual errors using different predicted models for the three data sets. Table 4.3.7, 4.3.8 and 4.3.9 contain the results. Table 4.3.7 Basic Statistics by Using Different Predicted Models for Stock X Stat. ARIMA k-th k-th Weighted k-th Exp. Weighted Nonlinear Model 1 Nonlinear Model 2 Mean 6.979054 -9.14724 1.737706 3.528108 6.95156 6.495499 Var. 312.6866 868.8794 461.1024 406.0678 312.5214 327.0268 S.D. 17.68295 29.47676 21.4733 20.15112 17.67828 18.08388 Table 4.3.8 Basic Statistics by Using Different Predicted Models for Stock Y Stat. ARIMA k-th k-th Weighted k-th Exp. Weighted Nonlinear Model 1 Nonlinear Model 2 Mean 0.031855 0.247946 0.096411 0.067641 -0.0583 -0.0523 Var. 5.767871 16.25066 6.582993 5.9614 5.6971 5.741183 S.D. 2.401639 4.03121 2.565734 2.441598 2.38686 2.396077 Table 4.3.9 Basic Statistics by Using Different Predicted Modelsfor S&P 500 Index Stat. ARIMA k-th k-th Weighted k-th Exp. Weighted Nonlinear Model 1 Nonlinear Model 2 Mean 1.73382 -0.96296 0.579749 1.01513 1.362997 1.717323 Var. 406.8795 850.3763 463.5857 430.691 423.539 404.9443 S.D. 20.17126 29.16121 21.53104 20.7531 20.58006 20.12323 Table 4.3.10 Summary of Predictions for Three Data Sets Data Set ARIMA k-th k-th Weighted k-th Exp. Weighted Nonlinear Model 1 Nonlinear Model 2 Stock X 5 5 5 3 3 4 Stock Y 4 4 5 1 6 5 S&P 500 5 8 5 1 2 4 113

PAGE 127

Remark 4.3.4 : From Table 4.3.7, 4.3.8 and 4.3.9, we conclude that the statistics of using different model to predict a future value, the nonlinear stochastic model 1 with time varying coefficients model has less sta ndard deviation of residual for two data sets stock X and stock Y. For S&P 500 Index, our nonlinear stochastic model 2 with time varying coefficients has the least standard deviation of residual. Table 4.3.10 summarizes the frequency of best model when predicting three data sets using different models. We further note that Table 4.3.10 summarizes the frequency of the best performance of models under three data sets predicted values. This summary in the context of Table 4.3.7, 4.3.8 and 4.3.9 suggests that nonlinear stochastic model 2 with time varying coefficients is robus t with respect different data sets. 4.4 Conclusions In Section 4.3, we studied prediction and co mparison about the performance of presented and existing models. This was based on three overall data sets. So far the formulations of stochastic nonlinear Models 4.1.1 and 4.2.1 with time varying coefficient were utilized for the data fitting. We note that the performance of these models in the framework of data fitting is superior than the existing time series models nonlinear stochastic mode ls 1 and 2 for overall data. Due to the nature of these models, the forecasting problem is open. This problem will be part of our future research plan. 114

PAGE 128

Chapter 5 European Option Pricing 5.0 Introduction In this chapter, we investigat e the option pricing problem in the frame of nonlinear stochastic models described in Chapters 3 and 4. By employing the nonlinear stochastic models of stock price process, the formulas of option pricing are de rived. In particular, we derive the European call and put option pricing formulas of nonlinear st ochastic models 1, 2 and 3. These results are presented in Sections 5.1, 5.2 and 5.3. 5.1 European Option Pricing for Nonlinear Stochastic Model 1 The probabilistic approach to pricing options will result in a price expressed as the discounted expected value of a claim with respect to a probability measure. The solution process of stochastic differential equation in (1.3.1) is a stochastic process adapted to Brownian filtration Under conditions (1.3.2) and (1.3.3) it has a unique solution process [23, 28]. We recall that (4.1.1) has a unique solution of equation (4.1.1). 0}{ ttF The nonlinear stochastic model 1 (Section 4.1) with time varying coefficients, takes the following form tttt t ttttdWSdtS S dS ) 2 ln(2, 0)0( SS where, coefficients and are time varying smooth functions, and is Brownian motion. tWIn Section 4.1, by using the transformation t tSY ln equation (4.1.1) is transformed into linear form tt ttttdWdtYdY )(. 115

PAGE 129

The solution to this stochastic differential equation is t t ssst t t ssttttdW ds YY0 0 0, ,, where and )exp(0t t s tds )exp(,t s u stdu Then can be written as tY t t ss t s u t t s t s u t t s ttdWdu dsdu ds YY0 0 0 0)exp( )exp()exp( (5.1.1) Then by using the inverse transformation of ln, we obtain t t ss t s u t t s t s u t t s t tdWdu dsdu ds YS0 0 0 0)exp( )exp()exp(exp (5.1.2) Remark 5.1.1 : For nonlinear stochastic model 1 with constant coefficients (3.1.1) and00 t, (5.1.1) and (5.1.2) reduce to t s st t st t t tdWedseYeSY0 )( 0 )( 0ln t s st t tdWeeeYe0 0)1( (5.1.3) and t s s t t tdWee e e teSS0)1( 0 (5.1.4) respectively. Now, let V be the European option for a stock with respect to nonlinear stochastic model 1 with time varying coefficients (4.1.1). is the value of the option at time t, where is the stock price defined in (5.1.2). The strike price K and maturity time T are as defined in Section 1.1, and r is fixed interest rate. Appl ying to Theorem 1.1.2, we have (,)VSttS 116

PAGE 130

]|[),()( tT tTrQFCeEtSV, where (5.1.5) put SK callKS CT T T},0), max{( },0), max{( There is no a simple formula to compute the value of To compute the numerical value of we use equations (5.1.2) to simulate the value of and then compute the expected value in (5.1.5). (,) VStTS(,) VSt From (5.1.1), knowing T and tST TSY ln let ttSStt 0,0 andtT for nonlinear stochastic model 1 with constant coefficients, we note that is normally distributed with TY T t s sT tT t tT TdWe eSeEYE)( )( )()1(ln ][ T t s sT tdWEe eSe)()1(ln )1(ln eSet, and T t s sT tT t tT TdWe eSeVarYVar)( )( )()1(ln )( T t s sTdWVare)(22 T t sTdse)(22 1 22 2 e. Thus, for European call op tion, (5.1.5) reduces to ]|[),()( tT tTrQFCeEtSV K eeEZe eSe tTrQt1 2 )1(ln )(2 2 (5.1.6) where Z is standard normal random variable. 117

PAGE 131

First, we establish for the range of values of Z the integrand is non-zero. 01 2 )1(ln2 2 K eZe eSet is equivalent to 1 2 )1(lnln2 2 e eSeK Zt. By letting 1 2 )1(lnln2 2 e eSeK dt, (5.1.6) reduces to d z Ze eSe rdz e K eetSVt 2 ),(2 1 2 )1(ln2 2 2 d z r d z Ze eSe rdz e Kedz e eet 2 22 2 1 2 )1(ln2 2 2 2 )( 21 22 )1(ln2 22dKedz e er d Ze z reSet )( 22 )1 2 ( 1 4 )1(ln2 2 2 2 2dKedz e er d ez ereSet )( 1 22 2 1 4 )1(ln2 2dKede er ereSet (5.1.7) Similarly, the formula corresponding to a European put option is ]|[),()( tT tTrQFCeEtSV Ze eSe tTrQteKeE1 2 )1(ln )(2 2 (5.1.8) where Z is standard normal random variable. Again, first we establish for the range of values of Z the integrand is non-zero. 118

PAGE 132

01 2 )1(ln2 2 Ze eSeteK is equivalent to d e eSeK Zt 1 2 )1(lnln2 2 From the above discuss, (5.1.8) reduces to d z Ze eSe rdz e eKetSVt 2 ),(2 1 2 )1(ln2 2 2 1 2 )(),(2 2 1 4 )1(ln2 2 ed edKetSVereSe rt. (5.1.9) Illustration 5.1.1 : In the following, we outline an illustr ation to exhibit the usefulness of the resented result. Suppose the yearly interest rate %5.6 r, by applying (5.1.7) and (5.1.9), the call and put option price are computed and recorded in Table 5.1.1 for three data sets. Similarly, the call and put option price of GBM model are computed and recorded in Table 5.1.2 for three data sets. Table 5.1.1 Call and Put Option Price of Nonlinear Model 1 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 18.17 32.71 2.96 7.76 13.56 17.27 60 27.53 52.99 6.63 10.76 27.60 25.29 100 30.55 65.45 9.21 12.71 38.22 29.96 200 30.75 83.54 13.83 15.93 59.61 36.77 119

PAGE 133

Table 5.1.2 Call and Put Option Price of GBM Model T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 23.18 29.25 2.96 7.81 13.64 17.21 60 44.42 45.63 6.69 10.92 27.91 25.16 100 59.71 56.09 9.39 12.99 38.80 29.77 200 89.00 73.46 14.48 16.54 61.05 36.52 5.2 European Option Pricing for Nonlinear Stochastic Model 2 The nonlinear stochastic model 2 (Section 4.2) with time varying coefficients, takes the following form t N tt N tt N tttttdWSdtS N SSdS ) 2 (122, 0)0( SS where, coefficients and are time varying smooth functions, N is a constant and is Brownian motion. An argument about the existence and uniqueness of solutions of this equation can be reformulated. 1,2.10 NNtWIn Section 4.2, using the transformation N S YN t t 11, equation (4.2.1) was transformed into linear form tt ttt tdWdtYN dY ))1((. The solution to this stochastic differential equation is as follows t t ssst t t ssttttdW ds YYo 0 0, ,, where, and t t s tdsN0)1(exp t s u stduN )1(exp,120

PAGE 134

Then can be written as tY t t ss t s u t s t s u t t s ttdWduN dsduN dsN YY0 0 0)1(exp )1(exp )1(exp0 (5.2.1) Then by using the inverse transformation of N S YN t t 11 N t t s t s u t t s t s u t t s N t tdWduN dsduN dsN N S NS 1 1 10 0 0 0)1(exp )1(exp )1(exp 1 )1( (5.2.2) Remark 5.2.1 : For nonlinear stochastic model 2 with constant coefficients (3.2.1) and 00 t, (5.2.1) and (5.2.2) reduce to t s stN t stN tN N t tdW edseVe N S Y0 )()1( 0 )()1( 0 )1( 11 t s stN tN tN NdW e e N e N S0 )()1( )1( )1( 1 0)1( )1( 1 (5.2.3) and N t s stN tN tNN tdW eN e eSS 1 1 0 )()1( )1( )1(1 0)1()1 ( (5.2.4) respectively. Now, let V be the European option on a stock with respect to nonlinear stochastic model 2 with time varying coefficients. is the value of the option at time t, where is the stock price process defined in (5.2.2). The strike price K and maturity time T are as defined in Section 1.1. r is fixed interest rate. Applyi ng to Theorem 1.1.2, we have (,) VSttS ]|[),()( tT tTrQFCeEtSV, where (5.2.5) put SK callKS CT T T},0), max{( },0), max{( 121

PAGE 135

There is no a simple formula to compute the value of To compute the numerical value of we use equations (5.2.2) to simulate the value of and then compute the expected value in (5.2.5). (,) VStTS(,) VSt For nonlinear stochastic model 2 with constant coefficients, from (5.2.1), knowing and T, letting andtSttSStt 0,0tT we note that is normally distributed with TY T t s sTN tTN N t tTN TdW e e NN S eEYE)()1( )()1( 1 )()1()1 ( )1(1 ][ T t s sTN N N t NdW eE e NN S e)()1( )1( 1 )1()1( )1(1 )1( )1(1)1( 1 )1( N N t Ne NN S e, and T t s sTN tTN N t tTN TdW e e NN S eVarYVar)()1( )()1( 1 )()1()1 ( )1(1 )( T t sTNds e)()1(22 )1( )1(2)1(2 2 Ne N. Hence, for European call option, (5.2.5) reduces to ]|[),()( tT tTrQFCeEtSV KZ e N N e eSeEN N N NN t rQ 1 1 )1(2 2 )1( )1(11 )1(2 )1()1( (5.2.6) where Z is standard normal random variable. 122

PAGE 136

First, we establish for the range of values of Z the integrand is non-zero. 0 1 )1(2 )1()1(1 1 )1(2 2 )1( )1(1 KZ e N N e eSN N N NN t is equivalent to 1 )1(2 )1( )1()1(2 2 )1( )1(1 1 N N NN t Ne N N e eSK Z. Setting 1 )1(2 )1( )1()1(2 2 )1( )1(1 1 N N NN t Ne N N e eSK d, (5.2.6) reduces to d z N N N NN t rdz e KZ e N N e eSe tSV 2 1 )1(2 )1()1( ),(2 1 1 )1(2 2 )1( )1(12 d z N N N NN t rdz e Z e N N e eSe 2 1 )1(2 )1()1(2 1 1 )1(2 2 )1( )1(12 (5.2.7) )( dKer Similarly, the formula corresponding to a European put option is ]|[),()( tT tTrQFCeEtSV NN t rQeSKeE)1(1 N N NZ e N N e1 1 )1(2 2 )1(1 )1(2 )1()1( (5.2.8) where Z is standard normal random variable. We establish for the range of values of Z the integrand is non-zero. 01 )1(2 )1()1(1 1 )1(2 2 )1( )1(1 N N N NN tZ e N N e eSK 123

PAGE 137

is equivalent to d e N N e eSK ZN N NN t N 1 )1(2 )1( )1()1(2 2 )1( )1(1 1 d z N N N NN t rdz e Z e N N e eSKe tSV 2 1 )1(2 )1()1( ),(2 1 1 )1(2 2 )1( )1(12 )( dKer d NN t reSe1(1 z N N Ndz e Z e N N e 2 1 )1(2 )1()1(2 1 1 )1(2 2 )1( )2 (5.2.9) Illustration 5.2.1 : In the following, we illustration the usef ulness of the above presented result. In (5.2.7) and (5.2.9), ,,r and N are estimated from observations. Other parameters such as the yearly interest rate is set to by applying (5.2.7) and (5.2.9), the call and put option price are computed and recorded in Table 5.2.1 for three data sets. %5.6 Table 5.2.1 Call and Put Option Price of Nonlinear Model 2 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 13.41 13.37 0.13 3.46 9.17 15.67 60 31.89 15.32 2.06 1.65 17.98 23.99 100 47.57 15.26 5.13 0.86 24.00 29.53 200 81.73 13.25 14.61 0.20 34.77 39.12 124

PAGE 138

5.3 European Option Pricing for Nonlinear Stochastic Model 3 Similarly, the nonlinear stochastic model 3, with time varying coefficients, takes the following form ttt tttttttdWSdtSSSdS ) (22, 0)0( SS (5.3.1) where, coefficients and are time varying smooth functions, and is Brownian motion. We note that the existence and uniqueness of solutio n process of (5.3.1) is justied in Section 3.3. tW In Section 3.3 (Chapter 3), by using the transformation equation (5.3.1) was transformed into linear form 1 ttSYttt ttt tdWYdtYdY )(. The solution to this stochastic differential equation is t t ssttttds YY0 0,, where, t t ss t t s s tdW ds0 0) 2 1 (exp2 and t s uu t s u u stdW du ) 2 1 (exp2 ,. Then can be written as tY t t t s uu t s u u s t t ss t t s s t tds dWdu dWds Y Y0 0 0 0) 2 1 (exp ) 2 1 (exp2 2 (5.3.2) Then by using the inverse transformation of 1 ttSY 1 2 20 0 0 0) 2 1 (exp ) 2 1 (exp t t t s uu t s u u s t t ss t t s s t tds dW du dW ds Y S (5.3.3) 125

PAGE 139

Remark 5.3.1 : For nonlinear stochastic model 3 with constant coefficients (3.3.1) and 00 t, (5.3.2) and (5.3.3) reduce to t dWst dWt ttds e eYSYt s t s0 ))( 2 ( ) 2 ( 0 12 0 2 t sWs Wtds eY et0 )() 2 ( 0 ) 2 (2 2 (5.3.4) and t sWs Wt tds e S e St0 )() 2 ( 0 ) 2 (2 21 (5.3.5) respectively. Similarly, let V be the European option on a stock with respect to nonlinear stochastic model 3 with time varying coefficients. is the value of the option at time t, where is the stock price process defined in (5.3.3). The strike price K and maturity time T are as defined in Section 1.1. r is fixed interest rate. Appl ying to Theorem 1.1.2, we have (,) VSttS ]|[),()( tT tTrQFCeEtSV, where (5.3.6) put SK callKS CT T T},0), max{( },0), max{( There is no a simple formula to compute the value of (,) VSt Illustration 5.3. 1: In the following, we illustrate the usef ulness of the above presented result. To compute the numerical value of we use equations (5.3.3) or (5.3.5) to simulate the value of and then compute the expected value in (5.3.6). Suppose the yearly interest rate (,) VStTS %5.6 r, the call and put option price are computed and r ecorded in Table 5.3.1 for three data sets. 126

PAGE 140

Table 5.3.1 Call and Put Option Price of Nonlinear Model 3 T Stock X 0S=691.48 K=600 Stock Y 0S=84.84 K=70 S&P 500 Index 0S=903.25 K=800 call put call put call put 5 58.92 5.18 16.16 0.56 101.56 0.02 10 2.73 65.75 15.39 0.58 92.16 0.06 20 0 286.2 11.57 1.03 56.24 0.91 60 0 525.13 0.11 17.25 0 210.66 100 0 556.44 0 38.04 0 431.09 127

PAGE 141

Chapter 6 Option Pricing for Hybrid Models 6.0 Introduction We studied GBM models and nonlinear stochastic models under the different data partitioning processes in Chapter 2, 3 and 4. In Chapter 5, we derived the European option pricing formulas for three nonlinear stochastic models, and apply to th ree data sets. In this chapter, we first derive the European call and put option pricing formulas in Section 6.1 for Hybrid GBM Models. In Section 6.2, we present option pricing formul as for hybrid nonlinear stochastic models. 6.1 Option Pricing for Hybrid GBM Models In 2003, G.Yin, et proposed a hybrid GBM m odel (HGBM). In HGBM model, drift and volatility are not deterministic functions anymore. They are perturbed by stochastic process such as a Markov Chain. By following development of a class of stochastic hybrid GBM system [16,44]: ,,)(,))(,())(,(kkktttStSSdWttSdtttdS 1111((,,,),),kkkkkkkSGSttS 00(), StS 10 0(,),(),(1,),kkMStkI (6.1.1) where, is a continuous price of the stock, S))(,(tt and))(,(tt are drift and volatility governed by the underlying discrete events th at can be modeled by a stochastic process )( t with a finite state. Figure 6.1.1 illustrate system switching from state k to state k+1 when at time event occurs, a jump kt)( t also occurs here. 128

PAGE 142

Figure 6.1.1 State Switch Illustration of Hybrid GBM In Chapter 2, we develop several modified GBM models which are HGBM models. The solution process of these HGBM models takes the general form: m m W t W WW tt m WWtt Wt Wtttt eS ttt eS tt eStm mmt m ttmm m m tt t t1 )0()0)( 2 1 (...)())( 2 1 ( 011 2 1 )())( 2 1 () 2 1 ( 01 1 ) 2 1 ( 01 1 2 1 1 11 2 11 1 21 2 2211 2 11 1 2 11... ... ... 0 Let m m mmmttttttttttt ,..., ,0211 122 11, 1 2 1 1 2 1, ,..., ,1 20 1 m m mttm tt m tt tWWWWWWWWWWWW, the price process is represented as following m m Wt WWtt m i i WWtt Wt Wt tttt eS ttt e eS tt eS Sm i m i iiiii m ttmmmm tt t1 ) 2 1 ()())( 2 1 ( 1 1 0 2 1 )())( 2 1 ( ) 2 1 ( 01 1 ) 2 1 ( 01 11 2 1 1 2 1 21 2 22111 2 11 1 2 11... ... 0 (6.1.2) where, be any one of data partition processes which are defined in Chapter 2. 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,mmtt) 129

PAGE 143

Now, let V be the European option on a stock with respect to hybrid GBM model (6.1.1). is the value of the option at time t, where is the stock price process defined in (6.1.2). The strike price K and maturity time T are as defined in Section 1.1. r is fixed interest rate. Applying to Theorem 1.1.2, we have (,) VSttS ]|[),()( tT tTrQFCeEtSV, where (6.1.3) put SK callKS CT T T},0), max{( },0), max{( For hybrid GBM model, from (6.1.3), by knowing and tStT we note that 1 1 0lnm i i T TS S Y is normal distributed with 1 11 2 1 2) 2 1 ()())( 2 1 (][1m i m i iiiii ttm m m m TWt WWtT EYEm 1 1 2 1 2) 2 1 ())( 2 1 (m i iii m m mt tT = m i iiit1 2) 2 1 (, and m i ii m i iii TW t VarYVar1 1 2) 2 1 (()( m i ii m i ii m mtt tT1 2 1 1 2 1 2)( Hence, for European call option, (6.1.3) reduces to ]|[),()( tT tTrQFCeEtSV K eSeEZt t m i i tTrQm i ii m i iii 1 2 1 2) 2 1 ( 1 1 0 )( (6.1.4) where Z is standard normal random variable. First, we establish for the range of values of Z the integrand is non-zero. 130

PAGE 144

01 2 1 2) 2 1 ( 1 1 0 K eSZt t m i im i ii m i iii is equivalent to m i ii m i iii m i it t S K Z1 2 1 2 1 1 0) 2 1 ( ln Let m i ii m i iii m i it t S K d1 2 1 2 1 1 0) 2 1 ( ln (6.1.4) reduces to d z Zt t m i i rdz e K eSetSVm i ii m i iii 2 ),(2 ) 2 1 ( 1 1 02 1 2 1 2 d z r d z Zt t m i i rdz e Kedz e eSem i ii m i iii 2 22 2 ) 2 1 ( 1 1 02 2 1 2 1 2 )( 21 2 2 1 22 1 1 0 ) 2 1 (dKedz e S er d Zt z m i i rtm i ii m i iii )( 22 ) ( 1 1 0 ) 2 1 (2 1 2 1 2 1 2dKedz e S er d tz m i i rttm i ii m i ii m i iii )() (1 2 1 1 0 ) 2 1 (1 2dKedt S er m i ii m i i rtm i iii (6.1.5) Similarly, the corresponding formula for a European put option is ]|[),()( tT tTrQFCeEtSV Zt t m i i tTrQm i ii m i iiieSKeE1 2 1 2) 2 1 ( 1 1 0 )( (6.1.6) where Z is standard normal random variable. 131

PAGE 145

First, we establish for the range of values of Z the integrand is non-zero. 01 2 1 2) 2 1 ( 1 1 0 Zt t m i im i ii m i iiieSK is equivalent to d t t S K Zm i ii m i iii m i i 1 2 1 2 1 1 0) 2 1 ( ln (6.1.6) reduces to d z Zt t m i i rdz e eSKetSVm i ii m i iii 2 ),(2 ) 2 1 ( 1 1 02 1 2 1 2 ) ( )(),(1 2 1 1 0 ) 2 1 (1 2 m i ii m i i rt rt d S edKetSVm i iii (6.1.7) Illustration 6.1.1 : In the following, we illustrate the usefulness of the above presented result. Suppose the yearly interest rate %5.6 r, by applying (6.1.5) and (6.1.7), the call and put option price of three data sets are computed and recorded in Tables 6.1.1 and 6.1.2 for Hybrid GBM models 2.4.3 and 2.4.4, respectively. Table 6.1.1 Call and Put Option Price of Hybrid GBM Model 2.4.3 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 22.19 12.99 2.06 5.30 11.57 106.96 60 95.38 12.97 5.99 6.04 34.26 130.94 100 165.04 8.68 9.56 6.32 50.87 148.54 200 373.42 3.76 18.20 6.37 80.60 180.52 132

PAGE 146

Table 6.1.2 Call and Put Option Price of Hybrid GBM Model 2.4.4 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 28.75 15.66 4.71 4.07 44.70 48.50 60 107.67 8.25 8.25 5.35 51.33 87.26 100 185.16 6.60 11.96 5.86 64.30 106.90 200 486.84 1.34 20.79 6.37 81.72 136.09 6.2 Option Pricing for Hybrid Nonlinear Stochastic Models By following development of a class of stochastic hybrid dynamic system [16]: ,,)(,))(,,())(,,(0 kkkt ettStSdWtStFdttStFdS 1111((,,,),)kkkkkkSGSttS ,k 00(), StS 10 0(,),(),(1,),kkMStkI (6.2.1) where, S is a continuous price of the stock, for nonlinear model 1, t t tttS S F ) 2 ln(2 )( )( )( 0 and tteSF)( for nonlinear model 2, 122 )( )( )(02N tt N ttttS N SSF and N tteSF)(and for nonlinear model 3, and ttttttSSSF2 )( 2 )( )(0 tteSF)( ))(,,(0tStF and ))(,,( tStFe are governed by the underlying discrete events that can be modeled by a stochastic process )( t with a finite state. 133

PAGE 147

Figure 6.2.1 illustrate system switching from state k to state k+1 when at time event occurs, a jump kt)( t also occurs here. Figure 6.2.1 State Switch Illustration of Hybrid Nonlinear Stochastic Model In Chapter 3, we develop several nonlinear st ochastic models which are hybrid nonlinear stochastic models. The solution process of these hybrid stochastic models takes the following form: m mt tt m mmmm t tt ttttS SSttS tttSS SttS tt SS SttS Sm1 1 11 1 2 1 1 1121 1 00 001,lim ),,( ... ... ... ,lim ),,( 0, ),,(1 1 (6.2.2) where, be any one of data partition processes which are defined in Chapter 2. 11223[0,),[,),[,), ttttt [3445,),[,)... tttt1[,mmtt) 6.2.1 Hybrid Nonlinear Stochastic Model 1 The solution process of the hybrid nonlinear stochastic models 3.1.1 and 3.1.2 is given by, m mt tt m dWe e e mm t tt dWe e e dWe e e ttttS S eS tttSS eS tt SS eS Sm t m t s st m m m tt m m m m tt m t t s s t tt tt t s st t t1 1 )1 ( 11 2 1 1 )1( 11 1 00 )1( 0,lim ... ... ... ,lim 0,1 1 )( ) 1 ( ) 1 ( 1 1 ) (2 2 ) 1 ( 2 2 2 )1 ( 2 0 )( 1 1 1 1 1 1 (6.2.3) Recursively, we have 134

PAGE 148

0 ... )(ln ln11111S eSt t tt tmm mm ( 2 )( 1ln ln1 tt m tt mmm mme e 1 ... )( 3 )ln ...ln33 111 111 t t tt m tmm mm mme )1( ...)1( )1(11 22111 11 1 1... )( 1 1 )( 1 1 )( t t t tt t tt m m tt m me e ee emmmm mm mm mm 1 0 1 22 111 1 2 1 1 1)( ... )( 1 )( )( 1 )(...t t s st t t tt t t s st tt m t t s st mdWe e dWeedWemm mm m m m mm m m Let m mmmmttttttttttt ,..., ,0211 122 11, 1 2 1 0ln lnln ln2 2 2 1m i im t m t ti j jmjm m i iie SeS 1 1 )(1 11 1)1( )1 (m i t t im im tt m mi j jmjm imim mme e e 1 1 )( )(1 1 11 1m i t t s st t im t t s st mim im imim i j jmjm m mdW e e dWe (6.2.4) Now, let V be the European option on a stock with respect to hybrid nonlinear model 1. is the value of the option at time t, where is the stock price process defined in (6.2.4). The strike price K and maturity time T are as defined in Section 1.1. r is fixed interest rate. Applying to Theorem 1.1.2, we have (,) VStTS]|[),()( tT tTrQFCeEtSV, where put SK callKS CT T T},0), max{( },0), max{( For hybrid nonlinear model 1, by knowing and tStT we note that is normal distributed with TSln 1 2 1 0ln lnln][ln2 2 2 1m i im t m t Ti j jmjm m i iie SeSE )( )1( )1 (1 1 )(1 11 1T e e em i t t im im tT m mi j jmjm imim mm (6.2.5) 135

PAGE 149

and 1 1 )( )(1 1 11 1)(lnm i t t s st t im T t s st m Tim im imim i j jmjm m mdW e e dWeVarSVar )()1 ( 2 )1 ( 22 1 1 2 2 2 )(2 21 11 1T e e em i t t im im tT m mimim i j jmjm mm (6.2.6) Hence, for European call option, ]|[),()( tT tTrQFCeEtSV K eeEZTTtTrQ )()()(, where Z is standard normal random variable. First, we establish for the range of values of Z the integrand is non-zero. 0)()(K eZTT is equivalent to d T TK Z )( )(ln where, )( T and )( T are defined in (6.2.5) and (6.2.6) respectively. Now we compute a European call option as d z ZTTrdz e K eetSV2 ),(2 )()(2)())(()(dKedT er rT (6.2.7) Similarly, the corresponding formula for a European put option is d z ZTT rdz e eKetSV 2 ),(2 )()(2))(( )()(TdedKerT r (6.2.8) Illustration 6.2.1 : In the following, we illustrate the usefulness of the above presented result. Suppose the yearly interest rate %5.6 r, by applying (6.2.7) and (6.2.8), the call and put option price of three data sets are computed and recorded in Tables 6.2.1 and 6.2.2 for Hybrid nonlinear models 3.1.1 and 3.1.2, respectively. 136

PAGE 150

Table 6.2.1 Call and Put Option Price of Hybrid Nonlinear Model 3.1.1 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 3.54 16.17 0 3.27 1.98 35.93 60 8.03 10.40 0 9.75 61.27 0.64 100 0 92.73 0 12.58 170.64 0 200 0 169.69 0 21.40 262.25 0 Table 6.2.2 Call and Put Option Price of Hybrid Nonlinear Model 3.1.2 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 9.27 7.66 0 12.76 0.40 31.67 60 9.10 8.86 0 15.74 3.08 31.02 100 0 123.62 0 19.09 61.14 0.36 200 0 171.43 0 21.37 192.99 0 6.2.2 Hybrid Nonlinear Stochastic Model 2 Similarly, the solution process of the hybrid nonlin ear stochastic models 3.2.1 and 3.2.2 is given by, 137

PAGE 151

m mt tt m N t t s stN m ttN m m ttN N mm t tt N t t s stN ttN ttN N N t s stN tN tN N ttttSSdWeN e eS tttSS dWeN e eS tt SS dWeN eeS Sm m m mm mm1 1 1 1 )()1( )()1( )()1( 1 11 21 1 1 1 )()1( 2 )()1( 2 2 )()1( 1 11 1 00 1 1 0 )()1( 1 )1( 1 1 )1( 1 0,lim )1()1 ( ... ... ... ,lim )1()1( 0, )1()1(1 1 1 1 1 1 2 12 12 1 1 1 (6.2.9) Let m mmmmttttttttttt ,..., ,0211 122 11, recursively, we have 111 111 1 1 2 1)1(... )1()()1( 1 1 0 1 1 1 1 1 11 ... 1tN t N ttN N N N m m N tmmm mmm m me N S N S )1 ( )1( ...111 222 111 1 2)1( )1(... )1()()1( 1 1 1 1 1 1 1 tN tN t N ttN N N me e Nmmm mmm m ...)1 ( )1( ...222 333 111 1 3)1( )1(... )1()()1( 2 2 2 1 2 1 1 tN tN t NttN N N me e Nmmm mmm m )1 ( )1( )1 ( )1()()1( )1()()1( 1 1 1 1 11 111 1 mmm mmm mmm mttN m m m t N ttN m m m N me N e e N 1 111 222 111 1 20 )()1( )1(... )1()()1( 1 1 1 1 1...t s stN tN t NttN N N mdW e emmm mmm m ... ...2 1 222 333 111 1 3)()1( )1(... )1()()1( 2 1 2 1 1 t t s stN tN t NttN N N mdW e emmm mmm m t t s stN m t t s st N ttN m N mm mm m m mmm mmm mdW e dW e e1 1 2 111 1)()1( )()1()()1( 1 1 1 Hence, 1 )1( 11)1( )1( 1 1 1 1 1 0 1 111 1 1 mmm m i iii i mtN m m m tN m i N i N m N te N e N S N S 138

PAGE 152

1 1 )1( )1( 1 1)1 ( )1(1 1m i tN tN ii i m ij N jiii m ij iii je e N (6.2.10) 1 1 )()1( )1( 1 1 )()1(1 1 1 1m i t t s stN tN i m ij N j t t s stN mi i iii m ij jjj j m mmdW e e dW e Now, let V be the European option on a stock with respect to hybrid nonlinear model 2. is the value of the option at time t, where is the stock price process defined in (6.2.10). The strike price K and maturity time T are as defined in Section 1.1. r is fixed interest rate. Applying to Theorem 1.1.2, we have (,) VStTS ]|[),()( tT tTrQFCeEtSV, where put SK callKS CT T T},0), max{( },0), max{( For hybrid nonlinear model 2, by knowing and tStT we note that m N T TN S Ym 11 is normal distributed with 1 )1( 1 ][)1( )1( 1 1 1 1 1 011 1 mmm m i iii itN m m m tN m i N i N Te N e N S YE )()1 ( )1(1 1 )1( )1( 1 11 1T e e Nm i tN tN ii i m ij N jiii m ij iii j (6.2.11) and T t s sTN m m i t t s stN tN i m ij N j Tm mm i i iii m ij jjj jdW e dW e e VarYVar1 1 1 1)()1( 1 1 )()1( )1( 1 1)( 1 1 )1(2 )1(2 2 2 1 )1(2)1(2 )1 (1 1m i ii tN tN i m ij N jN e eiii m ij jjj j 139

PAGE 153

)()1 ( )1(22 )()1(2 21T e NmmmtTN mm m (6.2.12) Hence, for European call option, ]|[),()( tT tTrQFCeEtSV K NZTT eEmN m tTrQ 1 1 )()1)()()((, where Z is standard normal random variable. First, we establish for the range of values of Z the integrand is non-zero. 0 )1)()()((1 1 K NZTTmN m is equivalent to d T T N K Zm Nm )( )( 11 where, )(T and )(T are defined in (6.2.11) and (6.2.12) respectively. Now we compute a European call option as d z N m rdz e K NZTTetSVm 2 )1)()()((),(2 1 12 )( 2 )1)()()((2 1 12dKedz e NZTTer d z N m rm (6.2.13) Similarly, the corresponding formula for a European put option is d z N m rdz e NZTTKetSVm 2 )1)()()(( ),(2 1 12 d z N m r rdz e NZTTedKem 2 )1)()()(()(2 1 12. (6.2.14) Illustration 6.2.2 : In the following, we illustrate the usefulness of the above presented result. Suppose the yearly interest rate %5.6 r, by applying (6.2.13) and (6.2.14), the call and put option price of three data sets are computed and recorded in Tables 6.2.3 and 6.2.4 for Hybrid nonlinear models 3.2.1 and 3.2.2, respectively. 140

PAGE 154

Table 6.2.3 Call and Put Option Price of Hybrid Nonlinear Model 3.2.1 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 29.06 33.76 6.01 9.49 3.49 1.45 60 55.40 10.04 8.56 13.13 4.15 2.08 100 74.16 9.68 6.30 11.76 27.22 32.74 200 114.6 9.06 13.57 20.97 53.77 59.20 Table 6.2.4 Call and Put Option Price of Hybrid Nonlinear Model 3.2.2 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 92.56 105.79 17.06 22.34 4.33 0.59 60 50.09 19.55 40.90 44.69 4.98 0.58 100 61.06 22.46 19.68 23.50 5.02 1.20 200 84.54 29.72 16.38 21.94 9.23 14.66 6.2.3 Hybrid Nonlinear Stochastic Model 3 The solution process of the hybrid nonlinear stochastic models 3.3.1 and 3.3.2 is given by, 141

PAGE 155

m mt tt m t t sWs m m WW tt m t tt t t sWs WWtt t sWs Wt ttttS S ds e S e tttSS ds e S e tt SS ds e S e Sm m m m m m ttm m m m tt t1 1 )() 2 ( 1 ) ())( 2 ( 1 2 1 1 )() 2 ( 1 1 )())( 2 ( 1 1 00 0 )() 2 ( 1 0 ) 2 (,lim 1 ... ... ... ,lim 1 0, 11 1 2 1 1 2 1 1 2 2 2 2 1 21 2 2 2 1 2 1 1 1 2 1 1 (6.2.15) Now, let V be the European option on a stock with respect to hybrid nonlinear stochastic model 3. VS is the value of the option at time t, where is the stock price process defined in (6.2.15). The strike price K and maturity time T are as defined in Section 1.1. r is fixed interest rate. Applying to Theorem 1.1.2, we have (, ) tTS]|[),()( tT tTrQFCeEtSV, where (6.2.16) put SK callKS CT T T},0), max{( },0), max{( Illustration 6.2.2 : In the following, we illustrate the usefulness of the above presented result. There is no a simple formula to compute the value of To compute the numerical value of we use equations (6.2.15) to simulate the value of and then compute the expected value in (6.2.16). Suppose the yearly interest rate (,)VStTS %5.6(,) VSt r, by applying (6.2.16), the call and put option price of three data sets are computed a nd recorded in Tables 6.2.5 and 6.2.6 for Hybrid nonlinear models 3.3.1 and 3.3.2, respectively. 142

PAGE 156

Table 6.2.5 Call and Put Option Price of Hybrid Nonlinear Model 3.3.1 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 420.52 0 35.75 0 56.62 0 60 298.64 0 32.59 0 55.61 0 100 279.83 0 30.44 0 54.63 0 200 291.04 0 31.85 0 53.65 0 Table 6.2.6 Call and Put Option Price of Hybrid Nonlinear Model 3.3.2 T Stock X 0S=691.48 K=700 Stock Y 0S=84.84 K=90 S&P 500 Index 0S=903.25 K=910 call put call put call put 20 353.42 0 45.34 0 79.12 0 60 295.16 0 50.79 0 78.17 0 100 338.64 0 53.46 0 77.24 0 200 315.10 0 54.45 0 76.21 0 143

PAGE 157

Chapter 7 Future Research Plan The nonlinear stochastic modeling approach initia ted in this work for solving forecasting and option pricing problems generates several in teresting research problems in the financial engineering. 7.1 Data Smoothing Transformation We note that a stochastic differential equation describes the continuous stock price process. The data sets we apply in our study ar e daily stock prices. In our future research, we want to explore the smoothing functions approach for better prediction and forecasting results. 7.1.1 Nonlinear Stochastic Model 1 In the following, a preliminary study with regard to nonlinear stochastic model 1 is presented. Here, we apply the smoothing function 1,...,2,1, 11 nTjS n Znj ji i j. Table 7.1.1 contains the result of AIC when we use value 3 n, and then apply to the Nonlinear Stochastic Model 1 using overall data set (Section 4.1). The basic sta tistics of the residual errors of fitted model are recorded in Table 7.1.2. Table 7.1.1 AIC of Time Varying Coefficients Nonlinear Model 1 (n=3) of Different Models of Three Dataset s: Stock X, Stock Y and S&P 500 Index Stock X Stock Y S&P500 Index (3, 1, 2) -5925.74 -34022.21 -128409.7 (3, 1, 1) -5667.26 -32847.02 -123984.7 144

PAGE 158

(3, 1, 0) -5600.38 -32401.51 -122989.2 (2, 1, 3) -5925.82 -34016.82 -128410.7 (2, 1, 2) -5927.59 -34019.93 -128409.8 (2, 1, 1) -5575.21 -32161.64 -122653.9 (2, 1, 0) -5566.89 -32090.21 -122485.6 (1, 1, 3) -5927.38 -34019.00 -128404.0 (1, 1, 2) -5929.58 -34019.86 -128373.8 (1, 1, 1) -5549.04 -31995.15 -121906.5 (1, 1, 0) -5528.05 -31892.23 -121206.7 (0, 1, 3) -5929.52 -34019.89 -128377.2 (0, 1, 2) -5926.42 -34021.24 -128345.8 (0, 1, 1) -5335.44 -30982.72 -118845.7 Table 7.1.2 Basic Statistics of Time Varying Coefficients Nonlinear Model 1 (n=3) of Three Data Sets: Stock X, Stock Y and S&P500 Index Model mean variance Standard deviation Stock X (1, 1, 2) 0.590782 59.550900 7.716923 Stock Y (3, 1, 2) 0.012320 0.387723 0.622674 S&P 500 Index (2, 1, 3) 0.045248 47.82333 6.915441 7.1.2 Nonlinear Stochastic Model 2 We repeat the smoothing transformation approach with regard to nonlinear stochastic model 2. The Stochastic Model Identification Procedure 4. 2.1 is applied to obtain the time series model corresponding to the nonlinear stochastic model 2. Table 7.1.3 exhibits the basic statistics of the residual errors of nonlinear model 2 with 3 n. 145

PAGE 159

Table 7.1.3 Basic Statistics of Time Varying Coefficients Nonlinear Model 2 (n=3) of Three Data Sets: Stock X, Stock Y and S&P500 Index Model N mean variance Standard deviation Stock X (1, 1, 2) 0.04 0.632399 57.25794 7.566898 Stock Y (2, 1, 3) 0 0.014074 0.341886 0.58471 S&P 500 Index (2, 1, 2) 0.02 0.067176 46.31976 6.805862 From this preliminary study, comparing the results in Tables 7.1.2, 7.1.3, 4.3.1, 4.3.2 and 4.3.3, we propose to utilize the smoothing function 1,...,2,1, 1 nTjS n Znj ji i j, and also other smoothing linear and nonlinear functions to investigate forecasting problem. 7.2 Forecasting Problem We recall that, in Section 4.3, we studied prediction problem and comparison about the performance of presented and existing models. Th is was based on three overall data sets. We simply attempted to use the formulations of stoc hastic nonlinear Models 4.1.1 and 4.2.1 with time varying coefficient for the data fitting problem. We further note that the performance of these models in the framework of data fitting is s uperior than the existing time series models and nonlinear stochastic models 1 and 2. The forecas ting in the frame work of these models is open research problem. This problem will be also addressed in the future. 7.3 Option Pricing Problem We observe that the parameters in our option pric ing illustrations in Chapter 5 and 6 are estimated from stock price data sets. In practice, these implie d parameters are computed from the historical option pricing data set. In our future research, we attempt to find the historical option pricing data (if available) and then apply to de velop modified option pricing models. 146

PAGE 160

References [1] B. K. ksendal, Stochastic differential Equations: An Introduction with Applications, 5th Edition, Berlin, New York: Springer, (1998). [2] Fred D. Arditti, Derivatives, A Comprehe nsive Resource for Options, Futures, Interest Rates Swaps, and Mortgage Securities, Harvard Business School Press, (1996). [3] Hirotugu Akaike A new look at the statistical model identification, IEEE Transactions on Automatic Control, 19 (6) (1974), pp. 71623. [4] J. Marc Allaire, James B. Bittman, Ri va Aidus Hemond, Elliot Kazt, Harrison Roth, Anthony J. Saliba, Gary L. Trennepohl, C. R. Tucker, and James W. Yates, Jr. Options, Essential Concepts and Trading Strategies, Second Edition, Richard D. IRWIN, Inc. (1995). [5] Turan G. Bali, An Empirical Comparison of Continuous Time Models of The Short Term Interest Rate, The Journal of Futures Markets, Vol.19, No.7, (1999), pp.777-797. [6] Fisher Black, and Piotr Karasinski, Bond and Option Pricing When Short Rates are Lognormal, Financial Analysts Jour nal (July-August) (1991), 52-59. [7] Fischer Black, and Myron Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81 (3), (1973). 637. [8] A. James Boness, Elements of a Theory of Stock Option Value, The Journal of Political Economy, April (1964), pp. 163-175. 147

PAGE 161

[9] George E. P. Box, Gwilym M. Jenkins a nd Gregory C. Reinsel, Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco. (1970). [10] George E. P. Box, Gwilym M. Jenkins and Gregory C. Reinsel, Time Series Analysis, Forecasting and Control, Third Edition, Pr entice-Hall International, Inc. (1994). [11] M. J. Brennan and E. S. Schwartz, Conditional Predictions of Bond Prices and Returns, Journal of Finance 35 (1980), pp. 405-417. [12] P. J. Brockwell and R. A. Davis, Intr oduction to Time Series and Forecasting, Springer, New York, (1996). [13] Robert Goodell Brown, Smoothing, For ecasting and Prediction of Discrete Time Series, Prentice-Hall, Inc. Englewood Cliffs, N.J. (1963). [14] F. Canova and B.E. Hansen, Are seasonal patterns constant over time? A test for seasonal stability. Journal of Business and Economic Statistics 13, (1995), pp. 237-252. [15] J. C. Cox, J. E. Ingersoll and S.A. Ro ss, A Theory of the Term Structure of Interest Rates, Econometrica 53(1985), pp.385407. [16] J. Chandra and G.S.Ladde, Stability Analysis of Stochastic Hybrid Systems, International Journal of Hybrid Systems, Vol.4 (2004), pp. 179-198. [17] K.C. Chan, G. Andrew Karolyi, Francis A. Longstaff, and Anthony B. Sanders, An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, The Journal of Finance, Vol. XLVII, N0.3, (1992) 1209-1227. [18] Uri L. Dothan, On the Term Structure of Interest Rates, Journal of Financial Economics 6(1978), pp. 59-69. [19] Alison Etheridge, A course in Financial Calculus, Cambridge, (2002). 148

PAGE 162

[20] R. J. Elliott, and P. E. Kopp, Mathemat ics of Financial Markets, Springer Verlag, New York, NY, (1998). [21] Kiyoshi It On stochastic differential equations Memoirs, American Mathematical Society 4 (1951), 1. [22] Denis Kwiatkowski, Peter C. B. Phillips, Peter Schmidt and Youngcheol Shin, Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root, Journal of Econometrics 54 (1992) pp.159-178. [23] Masaaki Kijima, Stochastic Processes with Applications to Finance, Chapman & Hall/CRC, (2003). [24] Peter E. Kloeden and Eckhard Platen, Numerical Solution of Stochastic Di_erencial Equations, Springer-Verlag, (1992). [25] A. G. Ladde, and G. S. Ladde, Dynamic Processes Under Random Environment, Bulletin of the Marathwada Mathematical Societ y, Vol.8, No.2 (2007), pp. 96-123. [26] A. G. Ladde, and G. S. Ladde, An Intr oduction to Differential Equations-II: Stochastic Modeling, methods and Analysis, in press. [27] G. S. Ladde, Hybrid Systems: Convergence and Stability Analysis of Stochastic LargeScale Approximation Schemes, Dynamic Systems and Applications, Vol.30 (2004), pp.487-512. [28] G. S. Ladde, and V. Lakshmikantham, Ra ndom Differential Inequaliti es, Academic Press, New York, (1980), MR 84b: 60081. [29] G. S. Ladde, and M. Sambandham, St ochastic Versus Deterministic Systems of Differential Equations, Marcel Dekker, Inc., New York, (2004). 149

PAGE 163

[30] G. S. Ladde and Ling Wu, Developm ent of Modified Geometric Brownian Motion Models by Using Stock Price Data and Basic Statistics, Nonlinear Analysis 71 (2009) e1203e1208. [31] G. S. Ladde and Ling Wu, Development of Nonlinear Stochastic Models by Using Stock Price Data and Basic Statistics, Special Issue, Communications in Applied Analysis (In processing). [32] M. E. Lindelf, Sur l'application de la mthode des approximations successives aux quations diffrentielles ordinaires du premie r ordre, Comptes rendus hebdomadaires des sances de l'Acadmie des sciences, Vol. 114, (189 4), pp. 454. [33] Robert C. Merton, Theory of rational option pricing, The bell Journal of Economics and Management Science, Vol. 4, No. 1, (1973), 141-183. [34] N. Metropolis, and S. Ulam, The Monte Carlo Method. Journal of the American Statistical Association 44 (247) (1949), pp. 335341. [35] Raymond H. Myers, Classical and Mode rn Regression with Applications, Duxbury, (1989). [36] M. F. M. Osborne, Brownian Motion in the Stock Market, Operations Research, March-April (1959), pp. 145-173. [37] Vijay K. Rohatgi and A. K. Md. Ehsan es Saleh, An Introduction to Probability and Statistics, Second Edition, Wiley, New York, (2000). [38] Shou Hsing Shih, Forecasting Models fo r Economic and Environmental Applications, Doctoral dissertation, University of South Florida, (2008). [39] John W. Tukey, Exploratory Data Analysis, Addison-Wesley Publishing Company, (1977). 150

PAGE 164

[40] Y K Tse, Some International Evidence on the Stochastic Behavior of Interest Rates, Journal of International Money and Finance, Vol. 14, No.5 (1995), pp. 721-738. [41] Oldrich Vasicek, An Equilibrium Charact erisation of the Term Structure, journal of Financial Economics 5 (1977), pp. 177-188. [42] W. N. Venables and B. D. Ripley, Modern Applied Statistics with S, Fourth Edition, Springer, (2002). [43] Simon Vine, Options: Trading Strategy and Risk Management, Hoboken, Weiey, N.J. (2005). [44] G. Yin, Q. Zhang, and K. Yin, Constr ained Stochastic Estimation Algorithms for a Class of Hybrid Stock Market Models, Journal of Optimization Theory and Applications, Vol. 118, No.1, (2003) 157-182. 151

PAGE 165

Appendices 152

PAGE 166

Appendix A1: The Estimated Parameters of Stock X Applying Monthly GBM Model 2004 2005 January 0.001161554 0.02942549 February -0.001702899 0.02796643 March -0.001766466 0.01257827 April 0.009640864 0.02096884 May 0.01115012 0.01629551 June 0.002920542 0.02164556 July -0.0009630533 0.01643769 August 0.003216502 0.03775822 -0.0001724213 0.01373443 September 0.01146937 0.02181682 0.00493249 0.01504678 October 0.01924003 0.04152636 0.008160907 0.02977017 November -0.001520906 0.03722648 0.004194011 0.01871986 December 0.002805678 0.01911732 0.001242512 0.01315277 153

PAGE 167

Appendix A1: (Continued) 2006 2007 January 0.002671674 0.03379703 0.004406722 0.01673394 February -0.008647245 0.03598111 -0.005636942 0.01614764 March 0.003500880 0.02592455 0.0009669722 0.01374949 April 0.003836704 0.01975170 0.001483809 0.01109096 May -0.005141313 0.01863313 0.002560355 0.01195841 June 0.005608799 0.01690952 0.002363476 0.009974502 July -0.003991524 0.0119881 -0.001067062 0.01443759 August -0.0008110875 0.01456686 0.0005236452 0.01251902 September 0.003111817 0.01525902 0.005110094 0.009779767 October 0.007971126 0.02201493 0.009694167 0.0155215 November 0.0009373092 0.01435362 -0.0005707263 0.0276292 December -0.002497903 0.01236763 2.591382e-06 0.01499199 154

PAGE 168

Appendix A2: The Estimated Parameters of St ock X Applying Unequal Interval GBM Model Index Index 1 0.006621 0.024521 20 0.010446 0.022202 2 0.013391 0.028731 21 -0.00389 0.02337 3 0.009891 0.055846 22 -0.0026 0.019065 4 0.001149 0.030116 23 0.002168 0.015421 5 0.009663 0.019472 24 -0.00417 0.014351 6 -0.00134 0.026352 25 0.003034 0.012816 7 0.005189 0.038095 26 0.003202 0.014617 8 -0.00261 0.015574 27 0.007015 0.020763 9 0.008553 0.017281 28 -0.00327 0.012388 10 0.00884 0.022777 29 1.37E-05 0.015938 11 -0.00053 0.014418 30 -0.00115 0.017478 12 7.88E-05 0.015385 31 -0.00045 0.014767 13 0.000596 0.015815 32 2.02E-05 0.01084 14 0.016518 0.033196 33 0.007892 0.011226 15 0.009311 0.013909 34 0.0025 0.008967 16 -0.00076 0.016247 35 -0.00407 0.016643 17 0.004541 0.025039 36 0.003471 0.010888 18 -0.01211 0.036582 37 0.008705 0.015797 19 0.000358 0.028831 38 0.010025 0.015063 39 -0.00136 0.022851 155

PAGE 169

Appendix A3: The Estimated Jump Coefficient of Stock X Applying Monthly GBM Model 2004 2005 2006 2007 Jan to Feb N/A 0.9333078 0.8899062 0.9496149 Feb to Mar N/A 1.0128700 1.0826370 1.0392360 Mar to Apr N/A 1.0101639 0.9906786 1.0042266 Apr to May N/A 1.0139198 0.9524189 0.9933268 May to June N/A 1.0276019 1.0794418 1.0103405 June to July N/A 0.9493318 0.9786493 1.0100002 July to Aug N/A 1.0248249 0.9630886 0.9895443 Aug to Sep 0.9751628 0.9868456 1.0288616 1.0153643 Sep to Oct 1.0456927 1.0070887 0.9986621 1.0084123 Oct to Nov 1.0101994 1.0134436 0.9816198 0.9663877 Nov to Dec 0.9644825 1.0051667 1.0108246 0.9845458 Dec to next Jan 1.0668690 1.0269237 1.0249850 N/A 156

PAGE 170

Appendix A4: The Estimated Jump Coefficient of Stock X Applying Unequal Interval GBM Model with Jumps Index Jump Coefficient Index Jump Coefficient 1 N/A 21 1.015838 2 0.938119 22 1.017131 3 0.929323 23 0.994225 4 1.058691 24 1.073491 5 1.040035 25 0.940836 6 0.99704 26 0.994028 7 1.09544 27 0.961578 8 0.995219 28 1.123955 9 0.937131 29 0.996198 10 0.973401 30 0.924911 11 1.090801 31 1.079305 12 0.936869 32 0.936505 13 0.999014 33 0.998823 14 0.92379 34 1.008491 15 1.086789 35 1.07922 16 1.082488 36 0.96916 17 0.908315 37 0.952614 18 1.146248 38 0.988914 19 0.855107 39 1.096968 20 1.000371 157

PAGE 171

About the Author Ling Wu got her undergraduate education at one of the highly respected higher educational institution, namely, DaLian University of Technology in China. In 2004, she came to the United States pursue her further higher education at University of South Florida, Tampa, Florida. Her passion and interest for mathematics and statistics motivated to earn her M.A. Degree in Mathematics with the concentration in statistics in 2006. Thereafter, she continued her Ph.D, and wrote her dissertation, entitled Stochastic Modeling and Statistical Analysis at University of South Florida. In her dissertation, Ling show s her creativity, intellectual independence, and exhibited her both analytical and computational abilities. She is sharp and energetic young lady. As a person, she always extends helping hands to other people.