Preface
1 Introduction
1.1 Motivation
1.2 Pioneering approaches
1.3 The chaos approach
1.4 The present focus
2 Efficient market hypothesis
2.1 Concepts， paradigms， and variables
2.2 Arbitrage
2.3 Efficient market hypothesis
2.4 Algorithmic complexity theory
2.5 Amount ofinformation in a financial time series
2.6 Idealized systems in physics and finance
3 Random walk
3.1 One-dimensional discrete case
3.2 The continuous limit
3.3 Central limit theorem
3.4 The speed of convergence
3.4.1 Berry-Esseen Theorem 1
3.4.2 Berry-Esseen Theorem 2
3.5 Basin of attraction
4 Levy stochastic processes and limit theorems
4.1 Stable distributions
4.2 Scaling and self-similarity
4.3 Limit theorem for stable distributions
4.4 Power-law distributions
4.4.1 The St Petersburg paradox
4.4.2 Power laws in finite systems
4.5 Price change statistics
4.6 Infinitely divisible random processes
4.6.1 Stable processes
4.6.2 Poisson process
4.6.3 Gamma distributed random variables
4.6.4 Uniformly distributed random variables
4.7 Summary
5 Scales in financial data
5.1 Price scales in financial markets
5.2 Time scales in financial markets
5.3 Summary
6 Stationarity and time correlation
6.1 Stationary stochastic processes
6.2 Correlation
6.3 Short-range correlated random processes
6.4 Long-range correlated random processes
6.5 Short-range compared with long-range correlated noise
7 Time correlation in financial time series
7.1 Autocorrelation function and spectral density
7.2 Higher-order correlations: The volatility
7.3 Stationarity of price changes
7.4 Summary
8 Stochastic models of price dynamics
8.1 Levy stable non-Gaussian model
8.2 Student's t-distribution
8.3 Mixture of Gaussian distributions
8.4 Truncated Levy fiight
9 Scaling and its breakdown
9.1 Empirical analysis of the S&P 500 index
9.2 Comparison with the TLF distribution
9.3 Statistical properties of rare events
10 ARCH and GARCH processes
10.1 ARCH processes
10.2 GARCH processes
10.3 Statistical properties of ARCH／GARCH processes
10.4 The GARCH（1，1） and empirical observations
10.5 Summary
11 Financial markets and turbulence
11.1 Turbulence
11.2 Parallel analysis of price dynamics and fiuld velocity
11.3 Scaling in turbulence and in financial markets
11.4 Discussion
12 Correlation and anticorrelation between stocks
12.1 Simultaneous dynamics of pairs of stocks
12.1.1 Dow-Jones Industrial Average portfolio
12.1.2 S&P 500 portfolio
12.2 Statistical properties of correlation matrices
12.3 Discussion
13 Taxonomy of a stock portfolio
13.1 Distance between stocks
13.2 Ultrametric spaces
13.3 Subdominant ultrametric space of a portfolio of stocks
13.4 Summary
14 Options in idealized markets
14.1 Forward contracts
14.2 Futures
14.3 Options
14.4 Speculating and hedging
14.4.1 Speculation: An example
14.4.2 Hedging: A form ofinsurance
14.4.3 Hedging: The concept of a riskless portfolio
14.5 Option pricing in idealized markets
14.6 The Black & Scholes formula
14.7 The complex structure of financial markets
14.8 Another option-pricing approach
14.9 Discussion
15 Options in real markets
15.1 Discontinuous stock returns
15.2 Volatility in real markets
15.2.1 Historical volatility
15.2.2 Implied volatility
15.3 Hedging in real markets
15.4 Extension of the Black & Scholes model
15.5 Summary
Appendix A: Notation guide
Appendix B: Martingales
References
Index
