Scaling, clustering and dynamics of volatility in financial time series

SCALING, CLUSTERING AND DYNAMICS OF VOLATILITY IN FINANCIAL TIME SERIES BAOSHENG YUAN (M.Sc., B.Sc.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF COMPUTATIONAL SCIENCE NATIONAL UNIVERSITY OF SINGAPORE 2005 ii In memory of my parents Acknowledgments First and foremost, I would like to thank Kan Chen, my advisor, for providing me the opportunity to work on this project. His sharp foresight and insight, great enthusiasm, kind encouragement and full support are the most important sources of inspiration and driving force for me to excel. He was always available to help and I benefited greatly from the frequent discussions with him on multitudes of problems during all these years. He also instilled in me the essential discipline to tackle the challenging problems systematically and scientifically. I am very fortunate to have him as my advisor and am very grateful to him for the success of this thesis. My second deep gratitude goes to Professor Bing-Hong Wang of the University of Science and Technology of China, who introduced me to the research field of Minority Game and Complex Networks. These research practices in the early years helped equip me with some essential research skills and gave me some fresh insights regarding the problem from a different perspective. This constitutes an important part of my research capability. iii Acknowledgments iv I am indebted to A/Prof. Baowen Li of the Department of Physics for introducing me to my advisor when I decided to pursue a Ph.D. Without his recommendation, it may have been impossible for me to start this great endeavor. There are many other individuals I would like to thank: Dr. Lou Jiann Hua and Dr. Liu Xiaoqing, of the Department of Mathematics, from whom I learned most of the knowledge on graduate level financial mathematics; Prof. Jian-Sheng Wang of our department, from whom I enhanced my knowledge of Monte Carlo; the members of our department have been supportive and all of them, faculty and administrators, deserve a note of gratitude. I also owe my gratitude to many of my colleagues and friends: Liwen Qian, Yibao Zhao, Yunpeng Lu, Yanzhi Zhang, Nan Zen, Honghuang Lin, Hu Li, Lianyi Han and Jie Sun, just name a few. Their help and friendship made my study a more pleasant experience. Another individual who deserves a special note is Anand Raghavan, one of my best friends; his careful corrections of the final manuscript and some suggestions made this thesis easier to understand. I am obliged to the National University of Singapore for the financial support. Last but not least, I would like to thank my daughter Jessica Yuan Xi for her emotional support. Her love and support are additional sources of my motivation to push myself for excellence. Her careful reading and corrections of the first manuscript helped make it a better written thesis. I am very proud of that! Baosheng Yuan Dec. 2005 Contents Acknowledgments iii Summary xii List of Tables xv List of Figures xvi Introduction 1.1 Volatility Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Volatility clustering and its characteristics . . . . . . . . . . . 1.1.2 Direct measure of volatility clustering . . . . . . . . . . . . . . 1.1.3 Universal curve of volatility clustering . . . . . . . . . . . . . 1.2 Time Series Modeling of Volatility Clustering . . . . . . . . . . . . . 1.2.1 Modeling volatility clustering with GARCH model . . . . . . . 1.2.2 A phenomenological volatility clustering model . . . . . . . . . v Contents 1.2.3 vi Property of the phenomenological model . . . . . . . . . . . . 1.3 Agent-Based Modeling of Volatility Clustering . . . . . . . . . . . . . 1.3.1 What is the underlying mechanism of volatility clustering? . . 1.3.2 Consumption-based asset pricing with agent-based modeling . 1.3.3 Agent-based model with heterogeneous and dynamic risk aver- 1.3.4 sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic risk aversion: a key factor of volatility clustering . . 1.4 Evolution of Strategies in a Stylized Agent Based Models —Minority Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 Population distribution of agents’ probabilistic trend-based strategies in EMG . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.2 Dynamics and phase structure of EMG with adaptive and deterministic strategies . . . . . . . . . . . . . . . . . . . . . . 12 1.4.3 EMG with agent-agent interaction . . . . . . . . . . . . . . . 13 1.5 Summary and Dissertation Outline . . . . . . . . . . . . . . . . . . . 14 Literature Review 17 2.1 Financial Markets and Financial Assets . . . . . . . . . . . . . . . . . 21 2.1.1 Financial markets . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Financial assets . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.3 Market uncertainty and price fluctuation . . . . . . . . . . . . 22 2.1.4 Key properties of FTS . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Features of Financial Assets . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Prices and frequency of observations . . . . . . . . . . . . . . 26 2.2.2 Price regularities . . . . . . . . . . . . . . . . . . . . . . . . . 26 Contents 2.2.3 vii Returns and time scales . . . . . . . . . . . . . . . . . . . . . 27 2.3 Financial Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.1 The first principle . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Consumption-based model . . . . . . . . . . . . . . . . . . . . 29 2.3.3 Utility function and risk preferences . . . . . . . . . . . . . . . 30 2.3.4 An alternative risk preference: Prospect Theory . . . . . . . . 31 2.4 Modeling Financial Time Series . . . . . . . . . . . . . . . . . . . . . 33 2.4.1 Statistics of returns . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4.2 Stationarity and white noise . . . . . . . . . . . . . . . . . . . 35 2.4.3 Ergodicity and estimation of expectation . . . . . . . . . . . . 36 2.4.4 Conditional statistical analysis . . . . . . . . . . . . . . . . . . 37 2.4.5 Measures of volatility . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.6 Measures of excess volatility and volatility clustering . . . . . 39 2.4.7 Summary of stylized statistical facts . . . . . . . . . . . . . . 40 2.5 Agent-Based Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.1 Why agent-based modeling? . . . . . . . . . . . . . . . . . . . 41 2.5.2 The challenges of the agent-based modeling . . . . . . . . . . 42 2.5.3 The current status of ABM . . . . . . . . . . . . . . . . . . . 43 2.6 Modeling Interactive Agents with Evolutionary Minority Game . . . . 46 2.6.1 Population distribution of agents’ probabilistic trend-based strategies in EMG . . . . . . . . . . . . . . . . . . . . . . . . 47 2.6.2 Dynamics and phase structure of EMG with deterministic and adaptive strategies . . . . . . . . . . . . . . . . . . . . . . . . 48 2.6.3 Network based EMG with adaptive strategies . . . . . . . . . 48 Contents viii 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Volatility Clustering in Financial Time Series 52 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Excess Volatility and Associated Clustering . . . . . . . . . . . . . . 54 3.2.1 Excess volatility of financial asset series . . . . . . . . . . . . . 55 3.2.2 Volatility clustering and its measures . . . . . . . . . . . . . . 55 3.3 Conditional Probability Distribution of Asset Returns as a Measure of Volatility Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.1 Construction of the CPD . . . . . . . . . . . . . . . . . . . . . 57 3.3.2 The choice of the number of conditional returns . . . . . . . . 58 3.3.3 Determination of the widths of the bins . . . . . . . . . . . . . 58 3.3.4 Estimation of the probability distribution . . . . . . . . . . . . 59 3.3.5 Measuring volatility clustering with the CPD . . . . . . . . . 59 3.4 Analyzing Asset Returns using CPD Measure . . . . . . . . . . . . . 60 3.4.1 The CPDs for the returns of various financial asset series . . . 61 3.4.2 Quantitative measure of volatility clustering with the CPDs . 68 3.4.3 Universal curves of the CPDs of asset returns . . . . . . . . . 69 3.4.4 Super universal curves of the CPDs of asset returns . . . . . . 75 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Time Series Modeling of Financial Assets 79 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.1 The model description . . . . . . . . . . . . . . . . . . . . . . 82 Contents ix 4.2.2 Major impact and properties of ARCH/GARCH model . . . . 83 4.2.3 Property of time duration dependence of kurtosis for GARCH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.4 Time duration dependence of kurtosis of real FTS and GARCH simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3 A Phenomenological Model of Volatility Clustering . . . . . . . . . . 92 4.3.1 The model dynamics . . . . . . . . . . . . . . . . . . . . . . . 92 4.3.2 Analytical properties of the model . . . . . . . . . . . . . . . . 93 4.3.3 Simulation results of the model . . . . . . . . . . . . . . . . . 97 4.3.4 Duration of Volatility Clustering 4.3.5 Continuous-time model . . . . . . . . . . . . . . . . . . . . . . 100 4.3.6 Time duration dependence of excess volatility . . . . . . . . . 101 4.3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 . . . . . . . . . . . . . . . . 99 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Modeling Volatility Clustering with an Agent-Based Model 109 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2 Demand and Price Setting under the Power Utility Function . . . . . 113 5.2.1 Demand and price setting with consumption-based model . . . 113 5.2.2 Derivation of demand and price equations . . . . . . . . . . . 115 5.3 The Baseline Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3.1 Price prediction . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3.2 Dividend process . . . . . . . . . . . . . . . . . . . . . . . . . 118 Contents 5.3.3 x The price setting equation of the baseline model . . . . . . . . 119 5.4 Model with Dynamic Risk Aversion . . . . . . . . . . . . . . . . . . . 119 5.4.1 Heterogeneous and dynamic risk averse agents . . . . . . . . . 119 5.4.2 Price equation with dynamic risk aversion . . . . . . . . . . . 120 5.4.3 The range of DRA indices . . . . . . . . . . . . . . . . . . . . 121 5.5 The Simulation Results and Analysis . . . . . . . . . . . . . . . . . . 121 5.5.1 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.5.2 Simulation price and trading volume . . . . . . . . . . . . . . 122 5.5.3 Excess volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.5.4 Volatility clustering . . . . . . . . . . . . . . . . . . . . . . . . 124 5.6 The SFI Market Model with Dynamic Risk Aversion . . . . . . . . . 125 5.6.1 Brief introduction to SFI market model . . . . . . . . . . . . . 125 5.6.2 Numerical results of SFI market model with DRA . . . . . . . 127 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Modeling the Market with Evolutionary Minority Game 133 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.2 Population Distribution of Agents’ Probabilistic Trend-Based Strategies in EMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.2.1 The description of the model . . . . . . . . . . . . . . . . . . . 138 6.2.2 Numerical results of phase 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List of Publications 1) Chen, K., B.-H. Wang and B. Yuan, “Adiabatic theory for the population distribution in the evolutionary minority game”, Phys. Rev. E. (Rapid Communication), 69, 025102(R) (2004). 2) Chen, K., B.-H. Wang and B. Yuan, “Theory of the three-group minority game”, International Journal of Modern Physics B. Vol.18, Nos. 17-19, 2387-93. (2004). 3) Yuan, B. and K. Chen, “Evolutionary dynamics and the phase structure of the minority game”, Phys. Rev. E., 69, 067106 (2004). 4) Yuan, B. and K. Chen, “Growing Directed Networks: Organization and Dynamics”, arXiv:cond-mat/0408391 5) Yuan, B. and K Chen, “Evolutionary Dynamics in Complex Networks of Competing Boolean Agents”, arXiv:cond-mat/0411664. 6) Yuan, B. and K. Chen, “Characteristics of Non-Gaussian Fluctuations in Financial Time Series”, 22nd International Conference on Statistical Physics July 4-9, 2004. 7) Yuan, B. and K. Chen, “Impact of Investor’s Varying Risk Aversion on the 199 List of Publications 200 Dynamics of Asset Price Fluctuations”, SSRN Working Paper Series, July, 2005. 8) Chen, K., C. Jayaprakash and B. Yuan, “Conditional probability as a measure of volatility clustering in finncial time series”, arXiv: physics/0503157. Name: Baosheng YUAN Degree: Doctor of Philosophy Department: Computational Science Thesis Title: Scaling, Clustering and Dynamics of Volatility in Financial Time Series Abstract This thesis investigates volatility clustering, scaling and dynamics in financial series of asset returns and studies the underlying mechanism. We propose a direct measure of volatility clustering (VC) based on the conditional probability distribution (CPD) of the returns given the return in the previous time interval. We found that the CPDs of returns in real financial time series exhibits universal scaling, characterized by a collapse of the CPDs (of different time lags and of different returns in the previous interval) into to a universal curve exhibiting a power-law tail with an exponent of −4. We construct a simple phenomenological model to explain the emergence of VC and the associated volatility scaling. We also study agent-based models of financial markets, and explore the impact of dynamical risk aversion (DRA) of heterogeneous agents on the price fluctuations. We found that the DRA is the primary driving force responsible for excess price fluctuations and the associated VC. Both our models (phenomenological model and agent-based model) are able to generate time series that reproduces stylized facts of the market data on different time scales. We have also studied general herding behavior often exhibited in financial markets in the context of an evolutionary Minority Game. We discovered a general mechanism for the transition from segregation into opposing groups to clustering towards cautious behavior. List of Publications 202 Keywords: Complex system, Scaling, Universality, Modeling, Financial Time Series, Risky Asset, Risk-free Asset, Financial Asset Pricing, Fluctuation, Dynamics, Volatility, Excess Volatility, Stochastic Volatility, Volatility Clustering, Heteroskedasticity, GARCH, Heterogeneous Agent, Dynamic Risk Aversion, Conditional Probability, Power Law, Fat Tail, Utility, Behavioral Finance, Uncertainty. SCALING, CLUSTERING AND DYNAMICS OF VOLATILITY IN FINANCIAL TIME SERIES BAOSHENG YUAN NATIONAL UNIVERSITY OF SINGAPORE 2005 SCALING, CLUSTERING AND DYNAMICS OF VOLATILITY IN FINANCIAL TIME SERIES BAOSHENG YUAN 2005 [...]... wide range of time intervals of the returns, but also for different asset series, suggesting that the CPD may serve as another measure characterizing volatility in FTS 3 1.2 Time Series Modeling of Volatility Clustering 1.2 1.2.1 Time Series Modeling of Volatility Clustering Modeling volatility clustering with GARCH model Engle’s Autoregressive Conditional Heteroskedasticity (ARCH) model and its generalization... worth noting that all these problems are interrelated and the central focus is the dynamics of volatility fluctuations 1 1.1 Volatility Clustering In the next section, we will discuss volatility clustering and its measure Section 2 describes how to model volatility clustering Section 3 investigates the underlying mechanism for the emergence of volatility clustering and introduces heterogeneous and dynamic... model volatility clustering with an agent-based model In section 4, we summarize the study of the financial market from the viewpoint of interactive game and reveal some key general properties of the market in the context of an evolutionary minority game The last section gives an overall summary and the outline of this thesis 1.1 1.1.1 Volatility Clustering Volatility clustering and its characteristics Volatility. .. critical dynamics evolves to the highest level of global coordination among its agents, leading to the best performance All these results may give some new insight in understanding the scaling, clustering and dynamics of volatility in FTS and their underlying mechanism of the financial market List of Tables 3.1 Four moments of the daily returns for some S&P component stocks 55 4.1 The parameters and kurtosis... any time scales of real FTS, making it easy to use in practice 1.3 Agent-Based Modeling of Volatility Clustering 1.3.1 What is the underlying mechanism of volatility clustering? Although there have been considerable efforts in exploring statistical models of volatility clustering, the underlying economic causality remains very much an unsolved puzzle [19] There is a long-established literature which links... Harris and Raviv [82], Shalen [147], Kurz 6 1.3 Agent-Based Modeling of Volatility Clustering 7 and Motolese [111], and Brock and Lebaron [23]) Although investment sentiment has been an important issue in behavioral finance [151], the literature is scarce on work in understanding price impacts (represented as the excess price fluctuation and in particular volatility clustering) of investor’s changing risk... in an evolutionary minority game (EMG) and study the mechanism for such behavior with possible interpretation in the context of financial markets (market impact and market inefficiency) The aim is to gain better understanding of investors’ behavior, such as herding, clustering of strategies, trend-following and segregation, in financial markets This can be done through modeling of evolution mechanism and. .. FTS, such as scaling and clustering of volatility The organization of the thesis is as follows The next chapter gives a review on the fundamental features of assets and recent developments on analyzing and modeling of financial time series Chapter 3 describes a direct and quantitative measure of volatility clustering which is used to analyze real FTS and reveals a new universal property of real FTS Chapter... the volatility, which embodies essential information on the dynamics of financial time series (FTS) Since it is first observed, volatility clustering is found to be ubiquitous in the FTS of asset returns from different markets, for different assets and in different time periods It is also observed that the strength of volatility clustering in FTS strongly depends on (among other variables) the sampling... in the thesis will be: how to quantify the key concept of volatility clustering and construct a direct and quantitative measure for it 1.1.2 Direct measure of volatility clustering A direct measure of volatility clustering is a necessity for a quantitative analysis of volatility dynamics in FTS We introduce a conditional probability measure (CPM) of financial asset return distribution as a direct measure . SCALING, CLUSTERING AND DYNAMICS OF VOLATILITY IN FINANCIAL TIME SERIES BAOSHENG YUAN (M.Sc., B.Sc.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF COMPUTATIONAL. 3 1.2 Time Series Modeling of Volatility Clustering . . . . . . . . . . . . . 4 1.2.1 Modeling volatility clustering with GARCH model . . . . . . . 4 1.2.2 A phenomenological volatility clustering. 180 Bibliography 181 List of Publications 199 Summary This thesis investigates volatility clustering (VC), scaling and dynamics in financial time series (FTS) of asset returns and their underlying mechanism. To