Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- Dedication -- 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 of information 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-Esséen Theorem 1 -- 3.4.2 Berry-Esséen Theorem 2 -- 3.5 Basin of attraction -- 4 Lévy 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 Lévy stable non-Gaussian model.

8.2 Student's t-distribution -- 8.3 Mixture of Gaussian distributions -- 8.4 Truncated Lévy flight -- 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 fluid 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 of insurance -- 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.