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PAGE 1 1 AUTO REGRESSIVE ANALYSIS OF STOCK PRICE VARIABLES AND THEIR APPLICATION IN MODELING ARTIFICIAL STOCK MARKET By AJAYDAS KAKI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2012 PAGE 2 2 2012 A jay d as Kaki PAGE 3 3 To my parents, Prof. GS and my friend, Nithya PAGE 4 4 ACKNOWLEDGMENTS This g raduate t hesis has given me a wonderful opportunity to learn new conc epts of simulation modeling and stock markets, which was possible only due to my mentor Dr. Sherman Bai. I would like to thank him for his invaluable inputs ,efforts and for pushing me to explore new ideas within the scope of the research. I would like to thank Prof. Uryasev and his PHD student Constantine Kalinchenko for evaluating my work and giving me inputs on work improvement. I would like to thank all the authors of the literature I reviewed for passively teaching me auto regression. I enjoyed working on this thesis and I hope to get an opportunity to pursue continued research in the field of stock markets and stochastic modeling. I hope that this work would open a new window in the simulation modeling of stock markets. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ ........ 10 ABSTRACT ................................ ................................ ................................ ................... 14 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 15 2 REAL MARKET ANALYSIS ................................ ................................ .................... 18 Market Dynamics ................................ ................................ ................................ .... 19 Volume Analysis ................................ ................................ ................................ ..... 21 3 EMPIRICAL RELATIONS OF STOCK PRICE DYNAMICS ................................ .... 23 Introductory Remarks ................................ ................................ .............................. 23 Return Analysis ................................ ................................ ................................ ....... 23 Test for Auto Correlation in Returns ................................ ................................ 23 Conditional Mean Model ( AR(M) ) ................................ ................................ ... 24 Conditional Variance Model (GARCH(R,Q)) ................................ ..................... 25 Regressive Equations for Volume ................................ ................................ .... 25 4 ESTIMATION ................................ ................................ ................................ .......... 27 Return Analysis ................................ ................................ ................................ ....... 27 Volume Estimates ................................ ................................ ................................ ... 29 5 LITERATURE REVIEW ................................ ................................ .......................... 31 Introductory Remarks ................................ ................................ .............................. 31 Trading in the Real Market ................................ ................................ ...................... 33 6 ARTIFICIAL STOCK MARKETS ................................ ................................ ............. 35 General Framework for ASMs ................................ ................................ ................ 35 A Few ASMs ................................ ................................ ................................ ........... 36 Santa Fe Artificial Market ................................ ................................ ................. 36 Genova Market ................................ ................................ ................................ 36 Simulation of GENOVA Market ................................ ................................ ............... 36 ................................ ................................ ................................ 37 ................................ ................................ ............................... 37 PAGE 6 6 Trading Process ................................ ................................ ............................... 38 GENOVA Discussion ................................ ................................ ........................ 38 Test 1 Initial Conditions ................................ ................................ .................... 38 Drawback Discussions ................................ ................................ ..................... 42 GS Index ................................ ................................ ................................ ................. 43 ................................ ................................ ............................... 45 ................................ ................................ ................................ 45 Trading Process ................................ ................................ ............................... 45 Results and Discussions ................................ ................................ .................. 46 7 CONCLUSION ................................ ................................ ................................ ........ 50 APPENDIX A DOW JONES AVERAGE DATA ................................ ................................ ............. 51 B SIMULATION OUTPUT ................................ ................................ .......................... 79 LIST OF REFERENCES ................................ ................................ ............................. 103 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 104 PAGE 7 7 LIST OF TABLES Table page 3 1 LBQ Test for auto correlation of returns for Dow Jones Industrial average for the year 2011 ................................ ................................ ................................ ...... 2 3 3 2 Correlation coefficients of standard ized volume and stock price returns ............ 25 3 3 LBQ Test for auto correlation of v for Dow Jones Industrial average for the year 2011 ................................ ................................ ................................ ........... 26 4 1 Returns auto regression coefficient values for AR(3) and GARCH (3,3) ............ 27 4 2 Regression coefficients for the empirical expression formulated to calculate v t ................................ ................................ ................................ ......................... 30 6 1 Initial conditions based on Dow jones average for the period January 2012 ...... 38 6 2 historical daily return variances for Dow Jones average ................................ ..... 38 6 3 Sample output data ................................ ................................ ............................ 39 6 4 AR(3) and GARCH(3,3) fit parameters for the simulated outputs of GS index and GENOVA ................................ ................................ ................................ ..... 48 6 5 Kurtosis of returns of ASMs ................................ ................................ ................ 49 A 1 GARCH fit for returns 2006 ................................ ................................ ................. 54 A 2 Regressio n coefficients for the empirical expression formulated to calculate v t for 2006 data ................................ ................................ ................................ ...... 54 A 3 LBQ test for auto correlation in returns ................................ ............................... 54 A 4 LBQ test for auto correlation in v t ................................ ................................ ....... 54 A 5 GARCH fit for returns 2007 ................................ ................................ ................. 58 A 6 Regression coefficients for the empiric al expression formulated to calculate v t for 2007 data ................................ ................................ ................................ ...... 58 A 7 LBQ test for auto correlation in returns ................................ ............................... 58 A 8 LBQ test for auto correlation in v t ................................ ................................ ....... 58 A 9 GARCH fit for returns 2007&2008 ................................ ................................ ...... 62 PAGE 8 8 A 10 Regression coefficients for the empirical expression formulated to calculate v t for 2007&2008 data ................................ ................................ ............................ 62 A 11 LBQ test for auto correlation in returns ................................ ............................... 62 A 12 LBQ test for auto correlation in v t ................................ ................................ ....... 62 A 13 GARCH fit for returns 2008 ................................ ................................ ................. 66 A 14 Regression coefficients for the empirical expression formulated to calculate v t for 2008 data ................................ ................................ ................................ ...... 66 A 15 LBQ test for auto correlation in returns ................................ ............................... 66 A 16 LBQ test for auto correlation in v t ................................ ................................ ....... 66 A 17 GARCH fit for returns 2008&2009 ................................ ................................ ...... 70 A 18 Regression coefficients for the empirical expression formulated to calculate v t for 2008&2009 data ................................ ................................ ............................ 70 A 19 LBQ test for auto correlation in returns ................................ ............................... 70 A 20 LBQ test for auto correlation in v t ................................ ................................ ....... 70 A 21 GARCH fit for returns 2009 ................................ ................................ ................. 74 A 22 Regression coefficients for the empirical expression formulated to calculate v t for 2009 data ................................ ................................ ................................ ...... 74 A 23 LBQ test for auto correlation in returns ................................ ............................... 74 A 24 LBQ test for auto correlation in v t ................................ ................................ ....... 74 A 25 GARCH fit for returns 2010 ................................ ................................ ................. 78 A 26 Regression coefficients for the empirical expression formulated to calculate v t for 2010 data ................................ ................................ ................................ ...... 78 A 27 LBQ test for auto correlation in returns ................................ ............................... 78 A 28 LBQ test for auto correlation in v t ................................ ................................ ....... 78 B 1 Test case 2 Initial co nditions ................................ ................................ ............... 79 B 2 Historical daily return variances calculated using American express(January) Data ................................ ................................ ................................ .................... 79 B 3 Test case 3 Initial con ditions ................................ ................................ ............... 85 PAGE 9 9 B 4 Historical daily return variances calculated using Bank of America(January) Data ................................ ................................ ................................ .................... 85 B 5 Test case 4 Initial condi tions ................................ ................................ ............... 91 B 6 Historical daily return variances calculated using Boeing(January) Data ........... 91 B 7 Test case 5 Initial conditions ................................ ................................ ............... 97 B 8 Randomly generated return variances Data ................................ ....................... 97 PAGE 10 10 LIST OF FIGURES Figure page 2 1 Dow Jo nes average for the years mentioned 2011 ................................ ............ 18 2 2 Returns for the year 2011 ................................ ................................ ................... 20 2 3 Absolute daily returns for Dow Jones average ................................ ................... 20 2 4 Absolute daily returns 2010 ................................ ................................ ................ 22 2 5 v (standardize volume) 2010 ................................ ................................ ............... 22 4 1 2011 GARCH fit ................................ ................................ ................................ .. 28 6 1 ASM lay out. ................................ ................................ ................................ ....... 35 6 2 Price using outputs from simulation 1 ................................ ................................ 40 6 3 Return using outputs from simulation 1 ................................ .............................. 40 6 4 Absolute returns ................................ ................................ ................................ 41 6 5 Garch, return comparison plot ................................ ................................ ............ 42 6 6 Price ................................ ................................ ................................ ................... 46 6 8 Absolute returns ................................ ................................ ................................ 47 6 9 Innovations ................................ ................................ ................................ ......... 47 A 1 2006 Dow Jones average ................................ ................................ ................... 51 A 2 2006 r eturns ................................ ................................ ................................ ...... 51 A 3 2006 Abs olute returns ................................ ................................ ....................... 52 A 4 2006 Traded Volume ................................ ................................ ......................... 52 A 5 2006 sample auto correlation in squared returns ................................ .............. 53 A 6 2006 GARCH fit for returns ................................ ................................ ................ 53 A 7 2007 Dow Jones average ................................ ................................ ................... 55 A 8 2007 Returns ................................ ................................ ................................ ..... 55 A 9 2007 Absolute returns ................................ ................................ ....................... 56 PAGE 11 11 A 10 2007 Traded Volume ................................ ................................ ......................... 56 A 11 2007 S ample auto correlation in squared returns ................................ .............. 57 A 12 2007 GARCH fit for returns ................................ ................................ ................ 57 A 13 2007&2008 Dow Jones average ................................ ................................ ......... 59 A 14 2007&2008 Returns ................................ ................................ ........................... 59 A 15 2007&2008 Absolute returns ................................ ................................ ............. 60 A 16 2007&20 08 Traded Volume ................................ ................................ ............... 60 A 17 2007&2008 sample auto correlation in squared returns ................................ .... 61 A 18 2007&2008 GARCH fit for returns ................................ ................................ ..... 61 A 19 2008 Dow Jones average ................................ ................................ ................... 63 A 20 2008 Returns ................................ ................................ ................................ ..... 63 A 21 2008 Abso lute returns ................................ ................................ ....................... 64 A 22 2008 Traded Volume ................................ ................................ ......................... 64 A 23 2008 sample auto correlation in squared returns ................................ .............. 65 A 24 2008 GARCH fit for returns ................................ ................................ ................ 65 A 25 2008&2009 Dow Jones average ................................ ................................ ......... 67 A 26 2008&2009 ret urns ................................ ................................ ............................ 67 A 27 2008&2009 Absolute returns ................................ ................................ ............. 68 A 28 2007&2008 Traded Volume ................................ ................................ ............... 68 A 29 2008&2009 sample auto correlation in squared returns ................................ .... 69 A 30 2008&2009 GARCH fit for returns ................................ ................................ ..... 69 A 31 2009 Do w Jones average ................................ ................................ ................... 71 A 32 2009 Returns ................................ ................................ ................................ ..... 71 A 33 2009 Absolute returns ................................ ................................ ....................... 72 A 34 2009 Traded Volume ................................ ................................ ......................... 72 PAGE 12 12 A 35 2009 Sample auto correlation in squared returns ................................ .............. 73 A 36 2009 GARCH fit for returns ................................ ................................ ................ 73 A 37 2010 Dow Jones average ................................ ................................ ................... 75 A 38 2010 Returns ................................ ................................ ................................ ..... 75 A 39 2006 Absolute returns ................................ ................................ ....................... 76 A 40 2006 Traded Volume ................................ ................................ ......................... 76 A 41 2006 sample auto correlation in squared returns ................................ .............. 77 A 42 2006 GARCH fit for returns ................................ ................................ ................ 77 B 1 GENOVA price with time for setup case 2 ................................ .......................... 80 B 2 GS index price with time for setup case 2 ................................ .......................... 80 B 3 GENOVA return with time for setup case 2 ................................ ........................ 81 B 4 GS index return with time for setup case 2 ................................ ......................... 81 B 5 GENOVA absolute return with time for setup case 2 ................................ .......... 82 B 6 GS index absolute return with time for setup case 2 ................................ .......... 82 B 7 Sample auto correlation in squared returns of GENOVA for setup case 2 ......... 83 B 8 Sample auto correlation in squared returns of GS index for setup case 2 ......... 83 B 9 GARCH fit for returns of GENOVA output for setup case 2 ................................ 84 B 10 GARCH fit for returns of GS index output for setup case 2 ................................ 84 B 11 GENOVA price with time for setup case 3 ................................ .......................... 86 B 12 GS index price with time for setup case 3 ................................ .......................... 86 B 13 GENOVA return with time for setup case 3 ................................ ........................ 87 B 14 GS index return with time for setup case 3 ................................ ......................... 87 B 15 GENOVA absolute return with time for setup case 3 ................................ .......... 88 B 16 GS index absolute return with time for setup case 3 ................................ .......... 88 B 17 Sample auto correlation in squared returns of GENOVA for setup case 3 ......... 89 PAGE 13 1 3 B 18 Sample auto correlation in squared returns of GS index for setup case 3 ......... 89 B 19 GARCH fit for returns of GENOVA output for setup case 3 ................................ 90 B 20 GARCH fit for returns of GS index output for setup case 3 ................................ 90 B 21 GENOVA price with time for setup case 4 ................................ .......................... 92 B 22 GS index price with time for setup case 4 ................................ .......................... 92 B 23 GENOVA return with time for setup case 4 ................................ ........................ 93 B 24 GS index return with time for setup case 4 ................................ ......................... 93 B 25 GENOVA absolute return with time for setup case 4 ................................ .......... 94 B 26 GS index absolute return with time for setup case 4 ................................ .......... 94 B 27 Sa mple auto correlation in squared returns of GENOVA for setup case 4 ......... 9 5 B 28 Sample auto correlation in squared returns of GS index for setup case 4 ......... 95 B 29 GARCH fit for returns of GENOVA output for setup case 3 ................................ 96 B 30 GARCH fit for returns of GS index output for setup case 4 ................................ 96 B 31 GENOVA price with time for setup case 5 ................................ .......................... 98 B 32 GS index price with time for setup case 5 ................................ .......................... 98 B 33 GENOVA return with time for setup case 5 ................................ ........................ 99 B 34 GS index return with time for setup case 5 ................................ ......................... 99 B 35 GENOVA absolute return with time for setup case 5 ................................ ........ 100 B 36 GS index absolute return with time for setup case 5 ................................ ........ 100 B 37 Sample auto correlation in squared r eturns of GENOVA for setup case 5 ....... 101 B 38 Sample auto correlation in squared returns of GS index for setup case 5 ....... 101 B 39 GARC H fit for returns of GENOVA output for setup case 5 .............................. 102 B 4 0 GARCH fit for returns of GS index output for setup case 5 .............................. 102 PAGE 14 14 Abstract of Thesis Presented to th e Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science AUTO REGRESSIVE ANALYSIS OF STOCK PRICE VARIABLES AND THEIR APPLICATION IN MODELING ARTIFICIAL STOCK MARKET By A jayd as Kaki May 2012 Chair: Sherman X. Bai Major: Industrial and Systems Engineering Autoregressive (AR) methods have been widely used to model the expected value of stock price returns and variance of the stock price returns. In this project we measure the performance of Generalized Autoregressive conditional heteroskedastic (GARCH) methods in modeling the stock price variance coupled with auto regression of the expected value of the stock price returns. We also propose a method to model the trading vol ume under the influence of stock price returns. Most of the ASMs concentrate on the propagation of prices but very few attempts have been made to induce the fluctuations of trading volume under the influence of stock price returns. For this purpose, we exp loit the correlation property between absolute returns and traded volume. We model a sophisticated single stock artificial stock market, which uses GARCH and AR principles to model the trading volume and stock price returns under various market conditions. PAGE 15 15 CHAPTER 1 INTRODUCTION The erratic behavior of the stock market has drawn interests of researchers in dynamics is because of its ultra sensitive nature to various factor s like information Most of the market theories proposed discuss the conditional behavior of the market in a few specialized scenarios but no generalized market theory has been proposed to discuss the market i n every possible scenario. Efficient market hypothesis proposed by Fama (1979),hypothesize that the investors react to new information is zero as the prices follow random walk nullifying the net effect of the information event. Random walk hypothesis descr ibes that the stock price realization is best described by the Brownian motion of particles. Willard (1974) claimed that the stock returns vaguely exhibit log normal behavior. Since no theory proposed so far has succeeded to formulate the market dynamics exactly, a few researchers tried to define the behavior using empirical methods (regression methods) and simulation tools (Artifical stock markets).Robert Engle(1987),proposed a non stationary various model called GARCH to describe the volatility of the s tock price returns for which he was awarded the Noble prize in economics. Campbell and Henstchell (1992) used quadratic GARCH (Q GARCH) to obtain the estimates of the GARCH coefficients and to analyze the volatility of the market. A few researchers focuse d their work on simplistic moving average methods, but their descriptive power of the market proved to be very weak. Conducting studies using Simulation is another most used method to study the market dynamics. PAGE 16 16 Artificial Stock markets (ASMs) are construc ted to conduct studies on the stock market using simulated trading process. ASMs are modeled as replicas to the real stock market so that it would be feasible for the academic researchers to design their experiments to analyze the behavior of the stock. AS Ms offer the advantage of conducting factor wise analysis of the market For example: separating the influence of trading volume on the returns etc. They can also be used to analyze the predictive power of various forecasting methods using the methods of m achine learning. Since the ASMs are used as a proxy to real stock markets and are used to conduct analytical studies, we target to build an ASM which depicts the properties of the real market. The properties of the real market which an ASM should contain a re : Clustering of Volatility. Lepto kurtosic behavior of the returns. Mean reversion of returns Non stationary variance of returns Auto correlation in stock price returns. Positive correlation between absolute stock price returns and trading volume. Man y attempts have been made to model ASMs to match their behavior with the real market and to improve their predictive power over prices. This project is intended to build such prototype of the market which would depict the real market and could be used to a nalyze price dynamics. I discuss the journey of the ASM model building and its validation in five different c hapter s with the Chapter 1 focusing on the back ground of the research. In Chapter 2 I discuss the behavior of the market from the 2006 through 20 11 for the Dow Jones Industrial Average and use the data to prove the existence of lepto kurtosis, mean reversion, auto correlation in stock price returns. In Chapter 4 we discuss the GARCH estimates and also model the expression for trading volume and st ock price returns. PAGE 17 17 The background of artificial stock markets is discussed in Chapter 5. In Chapter 6 we discuss the GENOVA market modeled by Rabertoa.M et al ., (2000) in detail. In Chapter 6 we also provide our model GS Index an improv isation of GENOVA market and the results are also discussed. PAGE 18 18 CHAPTER 2 REAL MARKET ANALYSIS Before the market is analyzed statistically, a brief description of the real market in the recent past would familiarize us with its behavior. The Dow jones Industrial average and its components are used as proxy for the stock market in US in this project. To get an overview of the markets behaviour during the period 2006 2011, the price Vs time are plotted for the year 2006, 2007, the period lasting from 2007 2008, 2008, the perio d lasting from 2008 2009, 2009,2010, 2011. The prices are plotted with various time frame windows to capture the anomalies that would have occurred in the price dynamics during the market crash of 2007 2008, 2008 2009. The 2 year windows are used as cross validation tools for the results produced by their corresponding year wise analysis. The Price Vs time plots are discussed in the Appendix A Figure 2 1 Dow Jones average for the years mentioned 2011 A short look at the F igure 2 1 describes the volatil e nature of the market in the recent past especially in the year 2011. Figure 2 1 also exemplifies the mean reversion PAGE 19 19 process of stock price .Due to the volatile nature of the market, stock price prediction has become increasingly difficult. Though there were many empirical methods proposed to capture the stock price dynamics, their strength in prediction has been limited to a regular market conditions. In order to understand the clear dynamics of the market, various trading processes involved in the marke t have to be studied. In this regard understanding the flow of information about the past price returns and the behavior of the traders in various market conditions should be analyzed. Market Dynamics The history of stock price returns is valuable informat ion and is used by many investors to make their trading decisions. For example : A series of positive returns for a stock is perceived as a positive information about the stock and this information event surges the demand for the stock. Vice versa a serie s of negative returns is perceived as a negative information event contributing to the increase in supply of the stock as the risk averse traders sell their shares realizing the signal sent by the stock price returns. Therefore, the phenomenon of herd beha vior of the traders results in the clustering of volatility and also partially contributes to efficient markets. The returns plot in F igure 2 2 conveys no information about the clustering of returns. But a careful examination at the plot reveals the mean r eversion process of the returns. The clustering of volatility and auto correlation of returns is well understood by studying the absolute stock price returns. The mathematical explanation for the mean reversion process and the clustering of Volatility is e xplained through Generalized Auto regressive heteroschedastic process (GARCH) in the upcoming c hapter s. GARCH expression captures most of the PAGE 20 20 properties of the real market like non stationary variance etc., hence this expression is used as the variance pro pagation rule for our ASM. Figure 2 2 Returns for the year 2011 Figure 2 3 Absolute daily returns for Dow Jones average PAGE 21 21 Volume Analysis It has been widely accepted that volume and absolute returns are correlated. But no empirical results have been established to define their movement with respect to each other. Since the total number of shares differs from company to company, the volume analysis using the actual trading volume becomes very complex. In order to make the analysis easier and appli cable all companies with different total number of tradable shares, we use the standardized trading volume (v) as a proxy for the actual trading volume. v = actual trading volume (V) / maximum number of shares issued (V ). The empirical relations between v and r (rate of return) can be applied to any stock, since the standardized volume and the returns (r) can be generalized for any company. r = (P t P t 1 )/ P t 1 If two variables are positively correlated, their crests fall together and their troughs fall together. Hence their realizations behave alike with time or with a shift in time axis. This descriptive phenomenon of positive correlation between absolute stock price returns and trading volume can be depicted in the F igure 2 4. PAGE 22 22 Figure 2 4 Absolute daily returns 2010 Figure 2 5 v ( standardize volume ) 2010 PAGE 23 23 CHAPTER 3 EMPIRICAL RELATIONS OF STOCK PRICE DYNAM ICS Introductory Remarks In Chapter 2, the properties of the real market were explained descriptively, but a concrete conclusions of the existence of the properties like auto correlation in returns, GARCH returns, positive correlation between absolute returns and trading volume. Statistical tes ts are conducted for the Dow Jones Industrial average for the years 2006 through 2011, to test the existence of the mentioned properties in the real market. Return Analysis Test for Auto Correlation in Returns The existence of auto correlation among stock price returns is tested using Ljung Box Q test (LBQ) whic h is discussed in the appendix A The LBQ test is conducted using the MATLAB which returns the Hypothesis that is true, P value of the test, statistic and critical values. The test returns the value of H as zero if the null hypothesis i.e. no auto correlation in stock price returns and value of one otherwise. A sample test for Dow Jones Industrial average returns(2011) is presented here and the tests for the rest of the periods in discussed in the Ap pendix A Table 3 1. LBQ Test for auto correlation of returns for Dow Jones Industrial average for the year 2011 H P value Statistic Critical value 1 0 125.2668 18.3070 1 0 133.4892 24.9958 1 0 136.2539 31.4104 The same test was conducted f or the years 2006 through 2011 and the two year periods 2007 2008 and 2008 2009. The test proved the existence of strong auto correlation in stock price returns. PAGE 24 24 The existence auto correlation among stock returns is discussed by many researchers. Generali zed autoregressive conditional heteroskedasticity (GARCH, Bollerslev(1986)) model is used as an empirical tool to predict the fluctuations in stock prices in a stable market. GARCH is used to capture the non stationary variance phenomenon among stock price returns. It captures the kurtosis and the skewness properties of the market. GARCH is a conditional variance model whose variance along with which I propose to use auto regressive conditional mean model to model the expected value tion to GARCH, we need a model to analyze the realizations of the mean. In an informed market, the expected value of a stock is dependent on the performance of the stock in its previous periods. Hence it makes logical sense to propose conditional mean and conditional variance models to analyze the stock price realizations and to study the market dynamics. We adopt the auto regressive models to formulate the expected value of the stock price returns and its non stationary variance. The mean and variance mode ls we adopted are described in the section below. Conditional Mean Model ( AR(M) ) The conditional mean model for the return having M lagged auto regressive terms is given by (3 1) Where r t i is the auto regressive PAGE 25 25 Conditional Variance Model (GARCH(R,Q)) The conditional variance model for the returns having R GARCH terms and Q ARCH terms is given by (3 2) 2 t is the error var iance and t is the squared residual of the return at( t)th period. And r t 1 = r t + t Where r t 1 Regressive Equations for Volume Many researchers concurred that the trading volume and absolute stock price ret urns are positively correlated, based on which one can make a logical conclusion that there is auto correlation in trading volume as there is auto correlation in stock price returns. Hence, statistical tests are conducted to check for the auto correlation in trading volume and the trading volume and stock price returns are positively correlated. Table 3 2 Correlation coefficients of standardized volume and stock price returns Period Lag 2011 2009 2008 2009 2007 2008 2006 0 0.158 0.3234 0.3942 0.5842 0.0747 1 0.1074 0.2656 0.3548 0.5423 0.0696 2 0.1556 0.3382 0.3049 0.4689 0.0117 The standardized trading volume (v) and stock price returns are tests for correlation for the periods 2006 through 2011 and tests concur with assumption that they a re positively correlated. The strength of the correlation coefficients are stronger PAGE 26 26 during the periods when the markets are efficient and the correlation strength is weakened when the markets are volatile (The above statement can be understood from the strong correlation coefficients for the periods 2007 2008, the period during which the market crashed. The weak correlation can be observed for the year 2011 when the market is volatile). Table 3 3. LBQ Test for auto correlation of v for Dow Jones Industr ial average for the year 2011 H P value Statistic Critical value 1 0 260.3880 18.3070 1 0 269.8323 24.9958 1 0 273.9217 31.4104 LBQ Test was conducted to test the auto correlation in v and the test was returned positive. Hence, an empirical equation can be formulated exploiting the auto correlation in v and the positive correlation between v and the stock price returns. The tests conducted for the rest of the periods are discussed in Appendix A (3 3) Where v t is the standardized trading vo t  is the absolute stock price PAGE 27 27 CHAPTER 4 ESTIMATION Return Analysis The expressions were built in such away that the traders in the market place their buy rmance. Hence the expected return r t is calculated using AR(M) model and the variance of the returns are modelled using GARCH(R,Q) (refer to Chapter 3). c hapter s), traders observe the short term trend o f the price dynamics and place their decisions. Hence, rather than modeling the returns as Markov process, return propagation among traders was modeled as AR(3) and GARCH(3,3) process. Table 4 1. Returns auto regression coefficient values for AR(3) and G ARCH (3,3) Values Parameter 2011 2010 2009 2008 2009 2008 2008 2007 2007 2006 1 0.6300 0.5317 0.5386 0.4724 0.4054 0.4294 0.4723 0.5124 2 0.3774 0.2623 0.2933 0.3053 0.4064 0.2914 0.1458 0.2614 3 0.1803 0.1104 0.1129 0.1119 0.1291 0.13 30 0.0909 0.1237 1 0.1068 0 0 0 0 0 0 0.8044 2 0.6135 0.2994 0.3325 0.7152 0.3601 0.3404 0.1145 0 3 0 0.3656 0.4463 0 0.2691 0.3082 0.5943 0.0421 1 0 0.1611 0 0.0074 0.0769 0.0727 0 0 2 0.2391 0.1526 0.0053 0.1010 0.2390 0.2253 0.2141 0 3 0.008 0 0 0.1729 0.1468 0.0353 0.0284 0 0.1016 The complete details of the coefficients statistics and their standard errors are presented in the Appendix A influenced by its pr regression is very up on the performance of the stock in the previous periods. In the periods, where the PAGE 28 28 information events arrive into the market successively, the strength of the auto regression coefficients is weakened to a minute extent due to the consideration of the effect of the information in the successive periods. From Table 3, we can also deduce that a period highly dependent on the variance of the return in its preceding months. Though the regressive strength of the forecast errors in returns is weak in predicting the variance of a future periods compared to the past variances strength, it has a significant effect on the variance prediction. Figure 4 1. 2011 GARCH fit The se observations can be used to model the price prediction expressions by adjusting the auto regression coefficients according to the behavior of the mar ket, which PAGE 29 29 could be done by adjusting the weights according to the sample inf ormation deduced from the Table 4 1, after analyzing whether the market is highly volatile or otherwise. The auto regressive expressions when incorporated into an ASM, they facilitate the realistic behavior of the market. The GARCH terms and the ARCH terms in the variance modeling capture the volatility clustering phenomenon. Taking a look at the top plot (innovations ) and the bottom plot(returns), we can clearly observe the capturin g of the volatility phenomenon by GARCH expressions. Volume Estimates The positive correlation between standardized trading volume and the absolute returns are tested and the corresponding results are presented in the Chapter 3 The influence of the corre lation between v and absolute stock price returns are limited to only one period. Similarly, the influence auto correlation in v in calculating the expected standardized trading volume (v t period. These hypotheses were considered to model the empirical expressions to calculate the expected standardized trading volume. The number of regression coefficients were carefully selected to avoid the over fitting phenomenon. Similar to the way the coefficie nts of AR(3) and GARCH(3,3) were calculated, t he parameters of the equation 3 3 corresponding to the standardized volume(v) were calculated. The values of the parameters for variou s periods are presented in the T able 4 2 and the complete information regar ding their corresponding standard errors and statistics are discussed in the Appendix A Since the C and K in equation 3 3 are just regression constants and K is given a negligibly small value used to facilitate regression through GARCHFIT in MATLAB, the influence of K was omitted in calculating the expected v t using equation 3 3. PAGE 30 30 Table 4 2 Regression coefficients for the empirical expression formulated to calculate v t Values Parameter 2011 2010 2009 2008 2009 2008 2008 2007 2007 2006 C 0.0178 0 .0138 0.0130 0.0112 0.0125 0.0075 0.0088 0.0113 Regress1 0.4856 0.6070 0.7026 0.7239 0.6668 0.7399 0.6210 0.4360 Regress2 0.0871 0.3016 0.1381 0.1155 0.1380 0.1698 0.3036 0.1080 The relationships between returns trading volume and the expressions use d to calculate their respective expected values are formulated, literature review of various ASMs modeled so far would make one comfortable with the ASM that was modeled in this research using the equations 3 1 3 2 and 3 3. PAGE 31 31 CHAPTER 5 LITERATURE REVIEW In troductory Remarks In any simulation model of a stock market, agents are divided as buyers and sellers who quote buy and sell prices for a stock. In a few models, traders arrive at discrete time steps which is an unrealistic assumption of the stock market. Therefore, arrival process of traders into the market is an important phenomenon. JOSEPH B. PAPERMAN (1996) modeled the buyer and seller arrivals as non stationary poisson process to study the behavior of liquidity of the market upon information arrival. Poisson process is widely used in the literature to model the trader arrival process, because it gives leverages to control the arrivals without disrupting the continuous trading property of the real market. It also helps generating the influence of opini on propagations among traders. Price relationships are another important aspects of interest in modeling an ASM. The price va riability volume relationships by Mark P itts ( 1983) studied how the joint distributions price and trading volume changes as more a nd more traders enter into the market. Willard (1974) proposed that the stock prices behave log nor mally. Models of stock returns by Stanley J.Kon ( 1984) discovered that the normal distribution models of stock prices fail to replicate them and the stock pr ices have fatter tails compared to the normal distribution. The casual relationship between absolute returns and the trading volume was established by Michael Smirlock (1988) and he suggested that this relationship is stronger in periods surrounding earnin gs announcements. Information plays an important role in the behavior stock price. It causes a drift diffusion process in t he stock price realization. Jump Diffusion Processes are popular PAGE 32 32 in option valuation models. These processes were incorpora ted by Merton (1976) into the theory of option valuation and constitute an important alternative to the lognormal diffusion process assumed in the Black and Scholes (1973) model. The drift in the stock price is stron g after bad news announcements by S.Chan ( 2003) and claims that the reversal in stock price is observed after the extreme movements in the absence of public news. The rate of information arrival influences the realization of stock price and the information arrival is a no n stationary memoryless process. In general, even in the strong efficiency conditions, each information event is perceived differently among different groups of traders and hence causing the trading process. The first artificial stock market model was Sa nta Fe. This model uses the demand supply match process to define the trading process. Though the model is very simple its lacks the real market behavior like asynchronous trading, continuous trading and the agents are cleared after every time step. Consi derable attention has been given to ASMs due to their replicability and control over the parameters (Davis and Holt, 1992). ASMs vary from model to model in their micro structure, trading process, price decision process. A few properties which an artific ial market should possess are : heterogeneous traders, adaptive traders, rich in dynamics and adaption. Agent based simulation has been extensively used to create ASMs that reflect the real market micro structure and real market dynamics. ASMS were class ified by Woolidge (2005) as: fully observable vs partially observable, deterministics vs stochastic, whether the current decisions are dependant on past decisions made, trading in discrete time steps or modeled as a continuous trading process. Kaitalin (2 008) in his dissertation proposed a PAGE 33 33 generic frame for agent based simulation models. According the basic components of an ASM are information source, market makers and traders. Trading in the Real Market In the real market, trading process is a continuou s process. Buyers and sellers arrive into the market placing quotes in the limit order book. The market maker tries to b ) price (P s ). The unmatched quotes wait in the limit order book till they find a matching counterpart in the market. Here we can observe that the trading process is a continuous matching process and the s process. The arrival rate of the buyers and sellers is influenced by the recent returns in the market. In an efficient is less than the fundamental value(P f,t ) of sell the stock if the stock price is greater than P f,t But since the market is not totally efficient and due to the presence of noise traders in the market, stock price is not a true representation of the in formation. Hence, traders engage in bidding at higher prices than the P f,t when the positive trend persists resulting in the speculative bubble. A crash occurs when the bubble ruptures and due to accumulation negative information events which can also be explained by the herding behavior of trading agents. At any time engaging in the trading process. Due to their heterogeneous behavior and memoryless property of the i nformation events, analyzing the market dynamics is very complex. Hence people tried to approach the analysis using ASMs where they have control over PAGE 34 34 the factors that influence the market dynamics and draw conclusions over the sensitivity of the market wit h respect to various factors. PAGE 35 35 CHAPTER 6 ARTIFICIAL STOCK MAR KETS In an artificial stock market,at every discrete time step traders enter into the for a stock at a price less than or equal to Pb,t Similarly, each seller offers his supply at a price greater than or equal to Ps,t In the trading box, buyers are matched with sellers based upon their buying and selling quotes respectively. At the end of the trading process in a time step, traded price (equilibrium price) at which shares were traded, return and traded volume are calculated. With the help of these variables, expected return, expected trading volume, expected return variance and expected stock price are calcu lated, based up on which traders place their quotes in the next time step General Framework for ASMs Figure 6 1 ASM lay out. PAGE 36 36 A Few ASMs Santa Fe Artificial Market This is one of the first ASMs modeled to depict the behavior of real stock markets. It uses the basic concepts of demand and supply matching to compute the new traded price(P t i,t = 1 and desire to sell a stock as a i,t = 1. Th e imbalance between the demand and supply is calculated using the equation And the new price is calculated according to the equation This model fails to capture real dynamics of the market due to it synchronous behavior of the traders and the peri odical clearing of the traders from the market. Genova Market To bring more realism to synchronous models, researchers from Genoa proposed a model called the Genoa artificial stock market in which agents are allowed to emit classical limit orders to the ma rket (see (Raberto, Cincotti, Focardi, and Marchesi 2001), (Cincotti, Focardi, Marchesi, and Raberto 2003) or (Raberto, Cincotti,Focardi, and Marchesi 2003)). This model is able to depict the true properties of the market like 1. Leptokurtosis 2. Serial correlat ion in absolute returns 3. Mean reversion in returns 4. Clustering of volatility Simulation of GENOVA Market This model assumes that the total number of traders (N) in market is fixed and the ed and the total PAGE 37 37 first simulation time step, each trader is assigned A/N amount of assets and T/N number of shares .At each time step, a random proportion of N are se lected as buyers using r b *N = B, where r b is a uniform random number in the interval [0,1] and a random proportion of N r b *N are selected as sellers (S) using r s *(N B)=S, where r s is a uniform random number in the interval [0,1]. S Process At ever th trader issues a sell order for a number of shares of n s (t)= r s N s (t) at a limit price P s,t = p t /N(, recent 20 periods returns. W here r s = uniform random variable in the interval [0,1]. N s (t)= Total number of shares owned by the s th P t is the equilibrium price of the most recent time step. Limit order price of a seller is the price below which his order would not executed. th buyer selects a random proportion of his assets a b (t)= r b A b (t) and places a buyer order for a number of shares n s (t) = a b (t)/ P b,t W here P b,t = p t *N(, ) for a demand R b = uniform random variable in the interval [0,1]. A b (t) = Total assets owned by the b th Limit order price of a buyer is the price above which his order would not executed. PAGE 38 38 Trading Process At every time step, the demand of shares above a certain price p t and the supply of shares below p t are calculated and the demand price curve and the supply price curve are plotted on the same graph. The price p t at which the demand price curve and the supply price curve cross each o ther is set as the equilibrium price at which shares shares of trader are updat ed as per the trading process. The model is built with various initialized scenarios i.e. different total number of shares, different assets, different initial return variance and tried to replicate the properties of the real market. GENOVA Discussion The model is tested for five different initialized scenarios and each simulation is conducted for 3000 time steps with a warm up period of 100 time steps. The results of the simulations are presented in the appendix B The results for a test case are presented Test 1 Initial Conditions Table 6 1 Initial conditions based on Dow jones average for the period January 2012 Total number of shares (T) Total number of traders (N) Total Assets (A) 64056.25 100 2116002.134 Table 6 2. historical daily return varia nces for Dow Jones average Variance of returns 1.64122x10 05 1.64122 x10 05 1.47565 x10 05 PAGE 39 39 Table 6 2. Continued Variance of returns 2.68062 x10 05 3.53283 x10 05 1.50288 x10 05 3.29809 x10 05 3.14786 x10 05 1.46493 x10 05 3.96288 x10 05 2.764 97 x10 05 2.13734 x10 05 3.20945 x10 05 2.18381 x10 05 1.74092 x10 05 2.6467 x10 05 1.77868 x10 05 1.60444 x10 05 Table 6 3. Sample output data Price Volume Return Absolute Return Time step 32.87 6821 0.004574 0.004573876 2 32.43 5201 0.01348 0.013476467 3 31.94 2671 0.01522 0.015224778 4 31.9 10768 0.00125 0.001253133 5 31.37 1741 0.01675 0.016753988 6 30.92 5393 0.01445 0.014448798 7 31.39 1662 0.015086 0.015086147 8 31.29 11661 0.00319 0.003190813 9 31.61 5334 0.010175 0.01017496 9 10 The simulation is conducted for 3000 time steps. At every time step, traded price traded volume, stock price return and return variance were collected as outputs. Since, the output data is huge; it is presented in Appendix B for the refrence. The output data is analyzed to determine if it depicts the properties of the real market. Hence, its tested for the auto correlation of returns, clustering of volitlity, PAGE 40 40 positive correlation between trading volume and returns, lepto kurtosis of returns and mea n reversion of the return phenomenon. Figure 6 2 Price using outputs from simulation 1 From the Figure 5 4 (Absolute returns), we can observe the clustering of volatility, through thick return propagation process. Figure 6 3. Return using outputs f rom simulation 1 PAGE 41 41 Figure 6 4 Absolute returns The kurtosis of a distribution measures the outlier proneness of a distribution. The kurtosis of a normal distribution is three, hence the distributions which exhibit kurtosis greater than three said to be lepto kutosic. When we tested the output data from various simulations of GENOVA market, we noticed a deviation of its kurtosis behavior from the real market, which is lepto kurtosic. The kurtosis details of the GENOVA market are compared with the ASM that is improvised over GENOVA market and presented in Table. A close look at the F igure 5 2 (price Vs time graph), reveals that the price of a stock is deteriorating on a long term scale and is approaching zero. This phenomenon exhibited by the price is undesi rable for a stock market and it is a call for immediate correction in the model.The returns of the GENOVA market is also tested for auto correlation in returns using LBQ test. Though the test returned results that the returns generated through GENOVA marke t are auto correlated, the strength of the parameters PAGE 42 42 of AR(3) and GARCH(3,3) applied to the GENOVA market are very weak compared to that of the real market. The LBQ tests for the returns of the GENOVA market are discussed in the appendix B The authors of the GENOVA market did not also discuss the clear procedure to plot the demand vs price and supply vs price graphs for a time. Improper modeling of this procedure would lead to faulty equilibrium prices and catastrophic realizations of a stock price. Drawb ack Discussions At every time step, GENOVA market selects a random proportion of its traders a buyers and a random proportion of the rest of the traders as sellers and they in turn quote their supply/demand using uniform random numbers. Figure 6 5 Garc h, return comparison plot PAGE 43 43 This procedure does not take into account the influence of stock price returns on trading volume. Hence the positive correlation between trading volume and stock price returns cannot be depicted. In GENOVA market, each buyer quot es his buying price P b,t for his demand of t *N(, ). And each seller quotes his selling price P s,t t /N(, ). Since =1.01 and N(, ) is a normal distribution which is a symmetric distribution, we encounter negative normal random numbers, which in turn contribu te to unrealistic negative buy/sell quotes for shares. Therefore, these negative normal random numbers are ignored which makes the distribution N(, ) asymmetric with more random numbers being greater than 1.01. Precisely, in this case the probability of N (, ) greater than or equal 1.01 is far higher t /N(, )) are always far lesser than the buyer quotes (p t *N(, )). Hence, this procedure would result in a decreasing sequence of equilibrium prices with t ime, which was observed in the F igure 6 2. Since the price quoting process among traders is a Markov process, the influence The GARCH phenomenon which should be observed in low frequency stock market data is very weak in GENOVA. In order to correct the draw backs, GENOVA market is modified with the help of mathematical expressions formulated in Chapter 2 and this new version ASM is called GS index. GS Index GS In Infact, it is an improvisation over GENOVA market. Even this model assumes that the PAGE 44 44 total number of traders (N) in market is fixed and the total assets (total money) present in the m assigned A/N amount of assets and T/N number of shares. And random populations of buyers and sellers are selected from N as done in GENOVA market. GS Index assumes complete efficiency of the market over the returns and the trading volume. Hence, homogeneity among traders is observed when it comes to realization of information regarding traded volume and stock price returns, which is made available through the regression expression developed in Chapter 2. At every time step, equilibrium traded price P t traded volume V t standardized traded volume v t and periodical return r t 1 are collec ted as outputs of the trading process where v t = V t /T. These outputs are used as inputs for the AR(3),GARCH(3,3) and standardized volume regression equation. The equations take the form : r t = 1 r t 1 1 + 2 r t 2 1 + 3 r t 3 1 (6 1) 2 t = 1 2 t 1 + 2 2 t 2 + 3 2 t 3 + 1 t 1 2 + 2 t 2 2 + 3 t 3 2 (6 2) and t = t *z t where z t = N(0,1) which is a standard normal variable and expected standardized tra v t = regress 1 v t 1 + regress 1 * r t 1 1  + C (6 3) where r t 1 t 1  is the absolute value of PAGE 45 45 For all the simula tions of GS index, the parameters were set as: 1 = 0.63001, 2 = 0.378, 3 = 0.1804, 1 = 0.1068, 2 = 0.6135, 3 = 0, 1 = 0, 2 = 0.24, 3 = 0.008,C= 0.018,regre ss1 = 0.49 and regress2 = 0.087. Process th buyer realizes the information about the traded volume through e quation(6 3 ) and places his demand for n s (t) at the limit price P b,t Where P b,t = P t *(1+r t + t ) and n s (t) = v t A b (t)/ P b,t which is followed from the logic that a rational buyer would place his demand proportionate to expected proportion of trading vo lume to the maximum available shares. Process th trader issues a sell order for a number of shares of n s (t) at a limit price P s,t where P s,t = P t *(1+r t + t ) and n s (t)= v t N s (t), which is followed from the logic that a rational seller would place his supply proportionate to the expected proportion of trading volume to the maximum available shares. Trading Process Trading phenomenon is left unaltered from the way it is conducted in GENOVA market except that the sequence o f prices that was used in plotting demand price and supply price graphs are modeled clearly. If P t price and supply price graphs are plotted continuously for every price that falls in 95% confi dence interval. If discrete prices are used in this process, optimal equilibrium price is lost in most of the cases leading to false stock price realizations. PAGE 46 46 Results and Discussions Simulations for GS index are conducted for all the initialization conditi ons for which the simulations for GENOVA market are conducted. Attempts have been made to compare the properties of GS index with the real market and GENOVA market. GS index is able to strongly replicate the most of the real market characteristics. The out put of GS index with respect to the initial ization conditions described for GENOVA is presented here. Figure 6 6 Price Figure 6 7 Returns PAGE 47 47 Figure 6 8 Absolute returns Figure 6 9 Innovations It is obvious from the figures that there is mea n reversion of returns in the GS Index. Volatility clustering is very evident from the absolute returns vs time steps plot. Lepto kurtosic behavior of the returns in GS index is very evident from innovations (forecast errors) vs time steps plot as there ar e evident strong innovations in the plot. PAGE 48 48 The simulated output data from GS index was used to conduct tests for auto correlation of returns using LBQ test. The test confirmed the assumption that the returns are auto correlation. The details of the test for all the simulation conducted are discussed in the Appendix B The data has also shown profound correlation between trading volume and absolute result which is very obvious as GS index is built on the mathematical expressions exploiting positive correlatio n between trading volume and absolute returns. Simulated data from the GENOVA market and GS index are used to fit AR(3) and GARCH (3,3) expressions and the strengths of the parameters are presented here for comparison. Table 6 4. AR(3) and GARCH(3,3) fit parameters for the simulated outputs of GS index and GENOVA Parameter GS 1 GEN 1 GS 2 GEN 2 GS 3 GEN 3 GS 4 GEN 4 GS 5 GEN 5 1 1.222 0.043 1.239 0.019 1.254 0.008 1.249 0.027 1.221 0.017 2 0.753 0.008 0.714 0.007 0.804 0.005 0.791 0.012 0.729 0.003 3 0.380 0.036 0.326 0.033 0.378 0 0.378 0.036 0.340 0.005 1 0 0.263 0 0.105 0 0 0 0.892 0 0 2 0 0.562 0 0.228 0 0 0.270 0 0.409 0.387 3 0.317 0 0.349 0.349 0.091 0.771 0 0 0 0.309 1 0.159 0.034 0.305 0.160 0.181 0.049 0.061 0.026 0.041 0.093 2 0.239 0.085 0.228 0.099 0.283 0.052 0.410 0 0.297 0.064 3 0.167 0.001 0.116 0 0.194 0.045 0.179 0.052 0.140 0.066 In the above T able GS 1 can be interpreted as GS index with initialization parameter set 1 and GEN 1 can be interpreted similarly. For the GS index and GENOVA market, the AR(3) and GARCH(3,3) fit parameters for an initialization set are listed next to ea ch other for easy comparison. The strength of the auto regression parameters is very strong in GS index compared to the real market data while their strength is extremely weak in GENOVA market outputs. The excessive strength of the PAGE 49 49 parameters in GS index i s due to the fact that the ASM itself is built on auto regression principles, which made its output very similar to real data, whereas the GENOVA market merely captures auto correlation of returns. The strength of GS index in exhibiting the real market pro perties is also reflected through its kurtosis of ret urns which is presented in the T able 6 5 below in comparison with the GENOVA market. Table 6 5. Kurtosis of returns of ASMs Simulation set 1 2 3 4 5 GENOVA 8.762 25.938 2.91 2.35 7.62 GS index 11.28 8 .99 6.43 4.03 3.61 From the above table, it can be observed that the kurtosis of the returns of GENOVA market behaves irregularly going as high as 26 for a simulation and as low as 2.35 which is less than the kurtosis of the of normal distribution. GS index consistently exhibits lepto kurtosic behavior of prices which pre dominantly reflects the proximity of GS index PAGE 50 50 CHAPTER 7 CONCLUSION GS index exhibits most of the properties of real stock market. Therefore, it can be extended to the market of mu ltiple company stocks and can be used for the academic research of portfolio problems. It opens scope to factor wise analysis of the stock market. Information based studies using GS index would open scope to analyze the strengths of rumours and news in th e market. It can be used to analyze the circumstances under which GARCH fails in the ASM when it is mixed with heterogeneous traders. The best use of this market would be to use machine learning techniques to analyze how each trader evolves in this market with his own prediction strategies. PAGE 51 51 APPENDIX A DOW JONES AVERAGE DATA Figure A 1. 2006 Dow Jones average Figure A 2. 2006 returns PAGE 52 52 Figure A 3. 2006 Absolute returns Figure A 4. 2006 Traded Volume PAGE 53 53 Figure A 5. 2006 sample auto correl ation in squared returns Figure A 6. 2006 GARCH fit for returns PAGE 54 54 Table A 1. GARCH fit for returns 2006 Parameter Value Standard error T statistic 1 0.512 0.069 7.439 2 0.261 0.080 3.260 3 0.124 0.069 1.779 1 0.804 1.038 0.776 2 0 1.312 0 3 0.0421 0.705 0.059 1 0 0.069 0 2 0 0.090 0 3 0.101 0.103 0.986 Table A 2. Regression coefficients for the empirical expression formulated to calculate v t for 2006 data Parameter Value Standard error T statistic C 0.011 0.001 10.497 Regress1 0.436 0.045 9.665 Regress2 0.108 0.052 2.059 Table A 3 LBQ test for auto correlation in returns H P value Statistic Critical value 1 0 125.2668 1 8.3070 1 0 133.4892 24.9958 1 0 136.2539 31.4104 Table A 4 LBQ test for auto correlation in v t H P value Statistic Critical value 1 0 260.3880 18.3070 1 0 269.8323 24.9958 1 0 273.9217 31.4104 PAGE 55 55 Figure A 7 2007 Dow Jones average Figure A 8 2007 R eturns PAGE 56 56 Figure A 9. 2007 Absolute returns Figure A 10. 2007 Traded Volume PAGE 57 57 Figure A 11. 2007 Sample auto correlation in squared returns Figure A 12. 2007 GARCH fit for returns PAGE 58 58 Table A 5. GARCH fit for returns 2007 Parameter Value Standard error T statistic 1 0.472 0.066 7.112 2 0.145 0.094 1.545 3 0.091 0.069 1.305 1 0 0.279 0 2 0.114 0.165 0.692 3 0.594 0.202 2.941 1 0 0.052 0 2 0.214 0.104 2.052 3 0 0.096 0 Table A 6. Regression coefficients for the empirical expression formulated to calculate v t for 2007 data Parameter Value Standard error T statistic C 0.008 0.001 8.623 Regress1 0.621 0.029 20.964 Regress2 0.303 0.048 6.277 Table A 7. LBQ test for auto correlation in returns H P value Statistic Critical value 1 0 38.493 18. 307 1 0 48.530 24.995 1 0 51.968 31.410 Table A 8 LBQ test for auto correlation in v t H P value Statistic Critical value 1 0 604.654 18.307 1 0 658.419 24.995 1 0 661.528 31.410 PAGE 59 59 Figure A 13. 2007&2008 Dow Jones average Figure A 14 2007&2008 R eturns PAGE 60 60 Figure A 15 2007&2008 Absolute returns Figure A 16 2007&2008 Traded Volume PAGE 61 61 Figure A 17 2007&2008 sample auto correlation in squared returns Figure A 18 2007&2008 GARCH fit for returns PAGE 62 62 Table A 9. GARCH fit for r eturns 2007&2008 Parameter Value Standard error T statistic 1 0.429 0.047 9.088 2 0.291 0.060 4.807 3 0.133 0.050 2.653 1 0 1.046 0 2 0.340 0.336 1.011 3 0.308 0.513 0.600 1 0.072 0.063 1.151 2 0.225 0.084 2.672 3 0.028 0.234 0.121 Table A 10. Regression coefficients for the empirical expression formulated to calculate v t for 2007&2008 data Parameter Value Standard error T statistic C 0.007 0 8.866 Regress1 0.74 0.019 37.220 Regress2 0.169 0.016 10.657 Table A 11. LBQ test for auto correlation in returns H P value Statistic Critical value 1 0 361.192 18.307 1 0 560.359 24.995 1 0 691.352 31.410 Table A 12. LBQ test for auto correlation in v t H P value Statistic Critical value 1 0 2371.9 18.307 1 0 3117.8 24.995 1 0 3777.2 31.410 PAGE 63 63 Figure A 19 2008 Dow Jones average Figure A 20 2008 R eturns PAGE 64 64 Figure A 21 2008 Absolute returns Figure A 22 2008 Traded Volume PAGE 65 65 Figure A 23 2008 sample auto correlation in squared returns Figure A 24 2008 GARCH fit for returns PAGE 66 66 T able A 13. GARCH fit for returns 2008 Parameter Value Standard error T statistic 1 0.405 0.067 6.018 2 0.406 0.084 4.814 3 0.129 0.070 1.826 1 0 1.828 0 2 0.360 0.543 0.662 3 0.269 0.857 0.313 1 0.076 0.098 0.784 2 0.239 0. 145 1.6 37 3 0.035 0.424 0.083 Table A 14. Regression coefficients for the empirical expression formulated to calculate v t for 2008 data Parameter Value Standard error T statistic C 0.012 0.001 7.996 R egress1 0.666 0.032 20.649 Regress2 0.138 0.025 5.466 Table A 15 LBQ test for auto correlation in returns H P value Statistic Critical value 1 0 131.241 18.307 1 0 210.706 24.995 1 0 256.303 31.410 Table A 16 LBQ test for auto correlation in v t H P value Statistic Critical value 1 0 703.1 18.307 1 0 854.6 24.995 1 0 1007.8 31.410 PAGE 67 67 Figure A 25 2008&2009 Dow Jones average Figure A 26 2008&2009 returns PAGE 68 68 Figure A 27 2008&2009 Absolute returns Figure A 28 2007&2008 Traded Volume PAGE 69 69 Figure A 29 2008&2009 sample auto correlation in squared returns Figure A 30 2008&2009 GARCH fit for returns PAGE 70 70 Table A 17. GARCH fit for returns 2008&2009 Parameter Value Standard error T statistic 1 0.472 0.044 10.670 2 0.305 0.051 5.905 3 0.112 0.048 2.311 1 0 0.393 0 2 0.715 0.212 3.364 3 0 0.337 0 1 0.007 0.051 0.146 2 0.101 0.065 1.542 3 0.146 0.076 1.917 Table A 18. Regression coefficients for the empirical expressio n formulated to calculate v t for 2008& 2009 data Parameter Value Standard error T statistic C 0.011 0.001 9.378 Regress1 0.723 0.025 28.890 Regress2 0.115 0.019 6.067 Table A 19. LBQ test for auto correlation in returns H P value Statistic Critical value 1 0 280.114 18.307 1 0 440.687 24.995 1 0 533.879 31.410 Table A 20. LBQ test for auto correlation in v t H P value Statistic Critical value 1 0 1463.6 18.307 1 0 1806.9 24.995 1 0 2119.9 31.410 PAGE 71 71 Figure A 31 2009 Dow Jones average Figure A 32 2009 R eturns PAGE 72 72 Figure A 3 3 2009 Absolute returns Figure A 34 2009 Traded Volume PAGE 73 73 Figure A 35 2009 S ample auto correlation in squared returns Figure A 36 2009 GARCH fit for returns PAGE 74 74 Table A 21. GARCH fit for return s 2009 Parameter Value Standard error T statistic 1 0.538 0.068 7.822 2 0.293 0.069 4.195 3 0.112 0.067 1.675 1 0 0.372 0 2 0.332 0.251 1.323 3 0.446 0.363 1.226 1 0 0.099 0 2 0.005 0.057 0.093 3 0.172 0.099 1.736 Table A 22. Regression coeffici ents for the empirical expression formulated to calculate v t for 20 09 data Parameter Value Standard error T statistic C 0.013 0.001 7.222 Regress1 0.607 0.043 13.848 Regress2 0.301 0.051 5.820 Table A 23. LBQ test for auto correlation in returns H P value Statistic Critical value 1 0 56.253 18.307 1 0 100.682 24.995 1 0 128.129 31.410 Table A 24. LBQ test for auto correlation in v t H P value Statistic Critical value 1 0 769.6 18.307 1 0 1013.6 24.995 1 0 1159.5 31.410 PAGE 75 75 Figure A 37 2010 Dow Jones average Figure A 38 2010 R eturns PAGE 76 76 Figure A 3 9 2006 Absolute returns Figure A 40 2006 Traded Volume PAGE 77 77 Figure A 41 2006 sample auto correlation in squared returns Figure A 42 2006 GARCH fit for returns PAGE 78 78 Table A 25 GARCH fit for returns 2010 Parameter Value Standard error T statistic 1 0.531 0.0957 5.550 2 0.262 0.102 2.564 3 0.110 0.087 1.269 1 0 1.242 0 2 0.299 0.710 0.421 3 0.365 0.886 0.412 1 0.161 0.107 1.494 2 0.152 0.176 0.866 3 0 0.246 0 Table A 26. Regression c oefficients for the empirical expression formulated to calculate v t for 20 10 data Parameter Value Standard error T statistic C 0.013 0.001 7.222 Regress1 0.607 0.043 13.848 Regress2 0.301 0.051 5.820 Table A 27 LBQ test for auto correlation in ret urns H P value Statistic Critical value 1 0 34.959 18.307 1 0 40.072 24.995 1 0 50.439 31.410 Table A 2 8 LBQ test for auto correlation in v t H P value Statistic Critical value 1 0 507.526 18.307 1 0 672.598 24.995 1 0 749.338 31.410 PAGE 79 79 APPENDIX B SIMULATION OUTPUT The data for the simulation is presented in the article. Hence, the data related to the rest of the articles is presented here. LBQ test is not presented as the details of the test cases are similar and the auto correlation can be easily detected in sample auto correlation pictures Table B 1 Test case 2 Initial conditions Total number of shares (T) Total number of traders (N) Total Assets (A) 13462250 100 864956293.62 Table B 2 Historical daily return variances calculated using American express (January) Data Variance of returns 1.64122x 10 05 1.64122X 10 05 4.83071X 10 06 2.64565X 10 05 1.57275X 10 05 1.23917X 10 05 3.78324X 10 05 4.45549X 10 05 2.86645X 10 05 7.80265X 10 05 6.05916X 10 05 4.85433X 10 05 8.190 67X 10 05 5.82015X 10 05 4.79373X 10 05 0.000128876 5.56439X 10 05 8.31089X 10 05 PAGE 80 80 Figure B 1. GENOVA price with time for setup case 2 Figure B 2 GS index price with time for setup case 2 PAGE 81 81 Figure B 3 GENOVA return with time for setup ca se 2 Figure B 4 GS index return with time for setup case 2 PAGE 82 82 Figure B 5. GENOVA absolute return with time for setup case 2 Figure B 6. GS index absolute return with time for setup case 2 PAGE 83 83 Figure B 7. Sample auto correlation in s quared returns of GENOVA for setup case 2 Figure B 8. Sample auto correlation in squared returns of GS index for setup case 2 PAGE 84 84 Figure B 9 GARCH fit for returns o f GENOVA output for setup case 2 Figure B 10. GARCH fit for returns of GS index outpu t for setup case 2 PAGE 85 85 Table B 3 Test case 3 Initial conditions Total number of shares (T) Total number of traders (N) Total Assets (A) 3 786713 00 100 2837762722.00 Table B 4 Historical daily return variances calculated using B ank of A merica (January) Da ta Variance of returns 1.64122 x10 5 1.64122 x10 5 7.57759 x10 6 3.15015 x10 5 0.000612335 7.25329 x10 5 0.000804558 0.00056676 0.00074968 0.001628211 0.001049233 0.002119121 0.004534573 0.00238161 0.004859028 0.004868979 0.002803701 0.004 462068 PAGE 86 86 Figure B 11. GENOVA price with time for setup case 3 Figure B 12. GS index price with time for setup case 3 PAGE 87 87 Figure B 13. GENOVA return with time for setup case 3 Figure B 1 4. GS index r eturn with time for setup case 3 PAGE 88 88 Figure B 15 GENOVA absolute return with time for setup case 3 Figure B 16. GS index absolute return with time for setup case 3 PAGE 89 89 Figure B 17. Sample auto correlation in squared returns of GENOVA for setup case 3 Figure B 18. Sample auto correlation in square d returns of GS index for setup case 3 PAGE 90 90 Figure B 19. GARCH fit for returns of GENOVA output for setup case 3 Figure B 20. GARCH fit for returns of GS index output for setup case 3 PAGE 91 91 Table B 5 Test case 4 Initial conditions Total number of shares ( T) Total number of traders (N) Total Assets (A) 5971518.75 100 1136371060.84 Table B 6 Historical daily return variances calculated using Boeing(January) Data Variance of returns 1.64122 x10 05 1.64122 x10 05 6.43762 x10 06 2.76273 x10 05 1.80892 x10 05 1.72471 x10 05 3.34239 x10 05 5.85157 x10 05 5.97852 x10 05 0.000115108 0.000112643 0.000112022 0.000167812 0.000136664 0.000151558 0.000157405 0.000126696 0.000122828 PAGE 92 92 Figure B 21. GENOVA price with time for setup case 4 Fig ure B 22. GS index price with time for setup case 4 PAGE 93 93 Figure B 23. GENOVA return with time for setup case 4 Figure B 24. GS index return with time for setup case 4 PAGE 94 94 Figure B 25. GENOVA absolute return with time for setup case 4 Figure B 26. GS i ndex absolute return with time for setup case 4 PAGE 95 95 Figure B 27. Sample auto correlation in squared returns of GENOVA for setup case 4 Figure B 28. Sample auto correlation in squared returns of GS index for setup case 4 PAGE 96 96 Figure B 29. GARCH fit fo r returns of GENOVA output for setup case 3 Figure B 30. GARCH fit for returns of GS index output for setup case 4 PAGE 97 97 Table B 7 Test case 5 Initial conditions Total number of shares (T) Total number of traders (N) Total Assets (A) 6074137.5 100 4 5654128 5 70 Table B 8 Randomly generated return variances Data Variance of returns 1.64122 x10 05 1.64122 x10 05 9.45257 x10 05 2.82943 x10 05 1.94583 x10 05 4.63883 x10 05 2.86269 x10 05 4.37487 x10 05 4.16191 x10 05 2.31417 x10 05 4.11259 x10 05 3.4848 x10 05 2.3873 x10 05 3.39638 x10 05 2.99189 x10 05 2.72016 x10 05 6.00624 x10 05 0.000104401 PAGE 98 98 Figure B 31. GENOVA price with time for setup case 5 Figure B 32. GS index price with time for setup case 5 PAGE 99 99 Figure B 33. GENOVA return wit h time for setup case 5 Figure B 34. GS index return with time for setup case 5 PAGE 100 100 Figure B 35. GENOVA absolute return with time for setup case 5 Figure B 36. GS index absolute return with time for setup case 5 PAGE 101 101 Figure B 37. Sample auto correlation in squared returns of GENOVA for setup case 5 Figure B 38. Sample auto correlation in squared returns of GS index for setup case 5 PAGE 102 102 Figure B 39. GARCH fit for returns of GENOVA output for setup case 5 Figure B 4 0. GARCH fit for returns of GS inde x output for setup case 5 PAGE 103 103 LIST OF REFERENCES Ajinkya, Bipin B. and Jain, Prem C. ( 1989 ) The behavior of daily stock market trading volume. Journal of accounting and economics 11:331 359. Binder, John J. ( 1985 ) On the use of the multivariate regression model in event studies Journal of accounting research 23(1): 370 383. Campbell, John Y., Grossman, Sanford J. and Wang,J. ( 1993 ) Trading volume and serial correlation in stock r eturns. 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Raberto, M.,Cincotti, S., Focardi, Sergio M. and Marchesi, M.( 2001 ) Agent based simulation of a financial market Physica A 299:319 327 Tauchen, George E. and Pitts, M. ( 1983 ) Price variability v olume relationships on speculative markets Econometrica 51(2):485 505 PAGE 104 104 BIOGRAP HICAL SKETCH Ajayd as attended Indian Institute of Technology Madras (IIT M), to pursue his undergraduate studies in c iv il engineering with a minor in operations r esearch. After undergraduate studies, he worked as a research assistant with Prof. G.Srinivasan at IITM to prepare for graduate studies in the field of operations research H e graduated with a Master of Science in industrial and systems e ngineering a t University of Flo rida, where he worked on auto regression analysis of stock prices and their application in simulated stock markets. This project was his first stochastic modeling problem which gave him scope to explore various mathematical problems in relation to stock ma rkets. After his graduate studies, he went on to pursue a research career in the field of mathematical modeling and optimization. Ap art from mathematics, Ajay likes meta physics and music. 