经济物理学导论：金融中的相关性与复杂性

本书讨论如何运用统计物理学中的概念来描述金融系统。具体来说，作者首先对在概率论、临界现象物理学和完全扰动湍流（fullydevelopedturbulent）理论中被广泛使用的标度（scaling）等概念进行了说明，然后将这些概念运用于对金融时间序列的分析，以获得对金融市场行为的新理解。作者还提供了一个新的随机模型，以描述在经验数据中观察到的几项统计特征。.通常，在研究经济系统时，用不同的尺度来考察经济系统是完全可能的。但要获得描述特定系统中经济体（economicentity）交互作用的精确方程却往往不可能。一些统计物理学概念，如随机动力学、短程与长程相关、自相似性和标度等，在不需要对所研究的经济系统事先做出详细与精微描述的前提下，就能提供对该经济系统全局行为的一种理解。本书符合物理学家与经济学家的兴趣。由于经济系统是我们可以研究的最具吸引力的复杂系统之一，以至物理学家在运用统计物理学概念于经济系统时会发现兴趣与挑战。经济学家与金融领域的工作人员将发现本书所提供的实证分析方法和表述清晰的理论工具是非常有用的，它们将有助于描述那些由大量交互作用的子系统所组成的复杂系统。

Preface viii
1 Introduction 1
1.1 Motivation 1
1.2 Pioneering approaches 2
1.3 The chaos approach 4
1.4 The present focus 5
2 Efficient market hypothesis 8
2.1 Concepts, paradigms, and variables 8
2.2 Arbitrage 8
2.3 Efficient market hypothesis 9
2.4 Algorithmic complexity theory 11
2.5 Amount of information in a financial time series 12
2.6 Idealized systems in physics and finance 12
3 Random walk 14
3.1 One-dimensional discrete case 14
3.2 The continuous limit 15
3.3 Central limit theorem 17
3.4 The speed of convergence 19
3.4.1 Berry-Esseen Theorem 1 20
3.4.2 Berry-Esseen Theorem 2 20
3.5 Basin of attraction 21
4 Levy stochastic processes and limit theorems 23
4.1 Stable distributions 23
4.2 Scaling and self-similarity 26
4.3 Limit theorem for stable distributions 27
4.4 Power-law distributions 28
4.4.1 The St Petersburg paradox 28
4.4.2 Power laws in finite systems 29

4.5 Price change statistics 29
4.6 Infinitely divisible random processes 31
4.6.1 Stable processes 31
4.6.2 Poisson process 31
4.6.3 Gamma distributed random variables 32
4.6.4 Uniformly distributed random variables 32
4.7 Summary 33
5 Scales in financial data 34
5.1 Price scales in financial markets 35
5.2 Time scales in financial markets 39
5.3 Summary 43
6 Stationarity and time correlation 44
6.1 Stationary stochastic processes 44
6.2 Correlation 45
6.3 Short-range correlated random processes 49
6.4 Long-range correlated random processes 49
6.5 Short-range compared with long-range
correlated noise 51
7 Time correlation in financial time series 53
7.1 Autocorrelation function and spectral density 53
7.2 Higher-order correlations: The volatility 57
7.3 Stationarity of price changes 58
7.4 Summary 59
8 Stochastic models of price dynamics 60
8.1 Levy stable non-Gaussian model 61
8.2 Student's t-distribution 63
8.3 Mixture of Gaussian distributions 63
8.4 Truncated Levy flight 64
9 Scaling and its breakdown 68
9.1 Empirical analysis of the S&P 500 index 68
9.2 Comparison with the TLF distribution 73
9.3 Statistical properties of rare events 74
10 ARCH and GARCH processes 76
10.1 ARCH processes 77
10.2 GARCH processes 80
10.3 Statistical properties of ARCH/GARCH
processes 81
10.4 The GARCH(1,1) and empirical observations 85
10.5 Summary 87
11 Financial markets and turbulence 88
11.1 Turbulence 89
11.2 Parallel analysis of price dynamics and fluid velocity 90

11.3 Scaling in turbulence and in financial markets 94
11.4 Discussion 96
12 Correlation and anticorrelation between stocks 98
12.1 Simultaneous dynamics of pairs of stocks 98
12.1.1 Dow-Jones Industrial Average portfolio 99
12.1.2 S&P 500 portfolio 101
12.2 Statistical properties of correlation matrices 103
12.3 Discussion 103
13 Taxonomy of a stock portfolio 105
13.1 Distance between stocks 105
13.2 Ultrametric spaces 106
13.3 Subdominant ultrametric space of a portfolio of stocks 111
13.4 Summary 112
14 Options in idealized markets 113
14.1 Forward contracts 113
14.2 Futures 114
14.3 Options 114
14.4 Speculating and hedging 115
14.4.1 Speculation: An example 116
14.4.2 Hedging: A form of insurance 116
14.4.3 Hedging: The concept of a riskless portfolio 116
14.5 Option pricing in idealized markets 118
14.6 The Black & Scholes formula 120
14.7 The complex structure of financial markets 121
14.8 Another option-pricing approach 121
14.9 Discussion 122
15 Options in real markets 123
15.1 Discontinuous stock returns 123
15.2 Volatility in real markets 124
15.2.1 Historical volatility 124
15.2.2 Implied volatility 125
15.3 Hedging in real markets 127
15.4 Extension of the Black & Scholes model 127
15.5 Summary 128
Appendix A: Martingales 136
References 137