This book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully-developed turbulent fluids. These concepts are then applied to financial time series. The authors also present a new stochastic model that displays several of the statistical properties observed in empirical data. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behavior of economic systems without first having to work out a detailed microscopic description of the system. Physicists will find the application of statistical physics concepts to economic systems interesting. Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.

Dr. H. Eugene Stanley has served for 30 years on the physics faculties of MIT and Boston University.

Preface

viii

Introduction

1

(7)

Motivation

1

(1)

Pioneering approaches

2

(2)

The chaos approach

4

(1)

The present focus

5

(3)

Efficient market hypothesis

8

(6)

Concepts, paradigms, and variables

8

(1)

Arbitrage

8

(1)

Efficient market hypothesis

9

(2)

Algorithmic complexity theory

11

(1)

Amount of information in a financial time series

12

(1)

Idealized systems in physics and finance

12

(2)

Random walk

14

(9)

One-dimensional discrete case

14

(1)

The continuous limit

15

(2)

Central limit theorem

17

(2)

The speed of convergence

19

(2)

Berry--Esseen Theorem 1

20

(1)

Berry--Esseen Theorem 2

20

(1)

Basin of attraction

21

(2)

Levy stochastic processes and limit theorems

23

(11)

Stable distributions

23

(3)

Scaling and self-similarity

26

(1)

Limit theorem for stable distributions

27

(1)

Power-law distributions

28

(1)

The St Petersburg paradox

28

(1)

Power laws in finite systems

29

(1)

Price change statistics

29

(2)

Infinitely divisible random processes

31

(2)

Stable processes

31

(1)

Poisson process

31

(1)

Gamma distributed random variables

32

(1)

Uniformly distributed random variables

32

(1)

Summary

33

(1)

Scales in financial data

34

(10)

Price scales in financial markets

35

(4)

Time scales in financial markets

39

(4)

Summary

43

(1)

Stationarity and time correlation

44

(9)

Stationary stochastic processes

44

(1)

Correlation

45

(4)

Short-range correlated random processes

49

(1)

Long-range correlated random processes

49

(2)

Short-range compared with long-range correlated noise

51

(2)

Time correlation in financial time series

53

(7)

Autocorrelation function and spectral density

53

(4)

Higher-order correlations: The volatility

57

(1)

Stationarity of price changes

58

(1)

Summary

59

(1)

Stochastic models of price dynamics

60

(8)

Levy stable non-Gaussian model

61

(1)

Student's t-distribution

62

(1)

Mixture of Gaussian distributions

63

(1)

Truncated Levy flight

64

(4)

Scaling and its breakdown

68

(8)

Empirical analysis of the S&P 500 index

68

(4)

Comparison with the TLF distribution

72

(2)

Statistical properties of rare events

74

(2)

ARCH and GARCH processes

76

(12)

ARCH processes

77

(3)

GARCH processes

80

(1)

Statistical properties of ARCH/GARCH processes

81

(4)

The GARCH (1,1) and empirical observations

85

(2)

Summary

87

(1)

Financial markets and turbulence

88

(10)

Turbulence

89

(1)

Parallel analysis of price dynamics and fluid velocity

90

(4)

Scaling in turbulence and in financial markets

94

(2)

Discussion

96

(2)

Correlation and anticorrelation between stocks

98

(7)

Simultaneous dynamics of pairs of stocks

98

(5)

Dow--Jones Industrial Average portfolio

99

(2)

S&P 500 portfolio

101

(2)

Statistical properties of correlation matrices

103

(1)

Discussion

103

(2)

Taxonomy of a stock portfolio

105

(8)

Distance between stocks

105

(1)

Ultrametric spaces

106

(5)

Subdominant ultrametric space of a portfolio of stocks

111

(1)

Summary

112

(1)

Options in idealized markets

113

(10)

Forward contracts

113

(1)

Futures

114

(1)

Options

114

(1)

Speculating and hedging

115

(3)

Speculation: An example

116

(1)

Hedging: A form of insurance

116

(1)

Hedging: The concept of a riskless portfolio

116

(2)

Option pricing in idealized markets

118

(2)

The Black & Scholes formula

120

(1)

The complex structure of financial markets

121

(1)

Another option-pricing approach

121

(1)

Discussion

122

(1)

Options in real markets

123

(7)

Discontinuous stock returns

123

(1)

Volatility in real markets

124

(3)

Historical volatility

124

(1)

Implied volatility

125

(2)

Hedging in real markets

127

(1)

Extension of the Black & Scholes model

127

(1)

Summary

128

(2)

Appendix A: Notation guide

130

(6)

Appendix B: Martingales

136

(1)

References

137

(8)

Index

145

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