1 Probability theory:basic notions
1.1 Introduction
1.2 Probability distributions
1.3 Typical values and deviations
1.4 Moments and characteristic function
1.5 Divergence of moments-asymptotic behaviour
1.6 Gaussian distribution
1.7 Log-normal distribution
1.8 Levy distributions and Paretian tails
1.9 Other distributions（*）
1.10 Summary
2 Maximum and addition of random variables
2.1 Maximum of random variables
2.2 Sums of random variables
2.2.1 Convolutions
2.2.2 Additivity of cumulants and of tail amplitudes
2.2.3 Stable distributions and self-similarity
2.3 Central limit theorem
2.3.1 Convergence to a Gaussian
2.3.2 Convergence to a Levy distribution
2.3.3 Large deviations
2.3.4 Steepest descent method and Cramer function（*）
2.3.5 The CLT at work on simple cases
2.3.6 Truncated Levy distributions
2.3.7 Conclusion:survival and vanishing of tails
2.4 From sum to max:progressive dominance of extremes（*）
2.5 Linear correlations and fractional Brownian motion
2.6 Summary
3 Continuous time limit, Ito calculus and path integrals
3.1 Divisibility and the continuous time limit
3.1.1 Divisibility
3.1.2 Infinite divisibility
3.1.3 Poisson jump processes
3.2 Functions of the Brownian motion and Ito calculus
3.2.1 Ito's lemma
3.2.2 Novikov's formula
3.2.3 Stratonovich's prescription
3.3 Other techniques
3.3.1 Path integrals
3.3.2 Girsanov's formula and the Martin-Siggia-Rose trick（*）
3.4 Summary
4 Analysis of empirical data
4.1 Estimating probability distributions
4.1.1 Cumulative distribution and densities-rank histogram
4.1.2 Kolmogorov-Smirnov test
4.1.3 Maximum likelihood
4.1.4 Relative likelihood
4.1.5 A general caveat
4.2 Empirical moments:estimation and error
4.2.1 Empirical mean
4.2.2 Empirical variance and MAD
4.2.3 Empirical kurtosis
4.2.4 Error on the volatility
4.3 Correlograms and variograms
4.3.1 Variogram
4.3.2 Correlogram
4.3.3 Hurst exponent
4.3.4 Correlations across different time zones
4.4 Data with heterogeneous volatilities
4.5 Summary
5 Financial products and financial markets
5.1 Introduction
5.2 Financial products
5.2.1 Cash（Interbank market）
5.2.2 Stocks
5.2.3 Stock indices
5.2.4 Bonds
5.2.5 Commodities
5.2.6 Derivatives
5.3 Financial markets
5.3.1 Market participants
5.3.2 Market mechanisms
5.3.3 Discreteness
5.3.4 The order book
5.3.5 The bid-ask spread
5.3.6 Transaction costs
5.3.7 Time zones, overnight, seasonalities
5.4 Summary
6 Statistics of real prices:basic results
6.1 Aim of the chapter
6.2 Second-order statistics
6.2.1 Price increments vs. returns
6.2.2 Autocorrelation and power spectrum
6.3 Distribution of returns over different time scales
6.3.1 Presentation of the data
6.3.2 The distribution of returns
6.3.3 Convolutions
6.4 Tails,what tails?
6.5 Extreme markets
6.6 Discussion
6.7 Summary
7 Non-linear correlations and volatility fluctuations
7.1 Non-linear correlations and dependence
7.1.1 Non identical variables
7.1.2 A stochastic volatility model
7.1.3 GARCH(1,1)
7.1.4 Anomalous kurtosis
7.1.5 The case of infinite kurtosis
7.2 Non-linear correlations in financial markets:empirical results
7.2.1 Anomalous decay of the cumulants
7.2.2 Volatility correlations and variogram
7.3 Models and mechanisms
7.3.1 Multifractality and multifractal models（*）
7.3.2 The microstructure of volatility
7.4 Summary
8 Skewness and price-volatility correlations
8.1 Theoretical considerations
8.1.1 Anomalous skewness of sums of random variables
8.1.2 Absolute vs. relative price changes
8.1.3 The additive-multiplicative crossover and the q-transformation
8.2 A retarded model
8.2.1 Definition and basic properties
8.2.2 Skewness in the retarded model
8.3 Price-volatility correlations:empirical evidence
8.3.1 Leverage effect for stocks and the retarded model
8.3.2 Leverage effect for indices
8.3.3 Return-volume correlations
8.4 The Heston model:a model with volatility fluctuations and skew
8.5 Summary
9 Cross-correlations
9.1 Correlation matrices and principal component analysis
9.1.1 Introduction
9.1.2 Gaussian correlated variables
9.1.3 Empirical correlation matrices
9.2 Non-Gaussian correlated variables
9.2.1 Sums of non Gaussian variables
9.2.2 Non-linear transformation of correlated Gaussian variables
9.2.3 Copulas
9.2.4 Comparison of the two models
9.2.5 Multivariate Student distributions
9.2.6 Multivariate Levy variables（*）
9.2.7 Weakly non Gaussian correlated variables（*）
9.3 Factors and clusters
9.3.1 One factor models
9.3.2 Multi-factor models
9.3.3 Partition around medoids
9.3.4 Eigenvector clustering
9.3.5 Maximum spanning tree
9.4 Summary
9.5 Appendix A:central limit theorem for random matrices
9.6 Appendix B: density of eigenvalues for random correlation matrices
10 Risk measures
10.1 Risk measurement and diversification
10.2 Risk and volatility
10.3 Risk of loss, value at risk'(VaR) and expected shortfall
10.3.1 Introduction
10.3.2 Value-at-risk
10.3.3 Expected shortfall
10.4 Temporal aspects: drawdown and cumulated loss
10.5 Diversification and utility-satisfaction thresholds
10.6 Summary
11 Extreme correlations and variety
11.1 Extreme event correlations
11.1.1 Correlations conditioned on large market moves
11.1.2 Real data and surrogate data
11.1.3 Conditioning on large individual stock returns:exceedance correlations
11.1.4 Tail dependence
11.1.5 Tail covariance（*）
11.2 Variety and conditional statistics of the residuals
11.2.1 The variety
11.2.2 The variety in the one-factor model
11.2.3 Conditional variety of the residuals
11.2.4 Conditional skewness of the residuals
11.3 Summary
11.4 Appendix C:some useful results on power-law variables
12 Optimal portfolios
12.1 Portfolios of uncorrelated assets
12.1.1 Uncorrelated Gaussian assets
12.1.2 Uncorrelated'power-law' assets
12.1.3 'Exponential' assets
12.1.4 General case: optimal portfolio and VaR（*）
12.2 Portfolios of correlated assets
12.2.1 Correlated Gaussian fluctuations
12.2.2 Optimal portfolios with non-linear constraints（*）
12.2.3 'Power-law' fluctuations-linear model（*）
12.2.4 'Power-law' fluctuations-Student model（*）
12.3 Optimized trading
12.4 Value-at-risk-general non-linear portfolios（*）
12.4.1 Outline of the method: identifying worst cases
12.4.2 Numerical test of the method
12.5 Summary
13 Futures and options: fundamental concepts
13.1 Introduction
13.1.1 Aim of the chapter
13.1.2 Strategies in uncertain conditions
13.1.3 Trading strategies and efficient markets
13.2 Futures and forwards
13.2.1 Setting the stage
13.2.2 Global financial balance
13.2.3 Riskless hedge
13.2.4 Conclusion:global balance and arbitrage
13.3 Options:definition and valuation
13.3.1 Setting the stage
13.3.2 Orders of magnitude
13.3.3 Quantitative analysis-option price
13.3.4 Real option prices, volatility smile and ‘implied’kurtosis
13.3.5 The case of an infinite kurtosis
13.4 Summary
14 Options:hedging and residual risk
14.1 Introduction
14.2 Optimal hedging strategies
14.2.1 A simple case: static hedging
14.2.2 The general case and‘△’hedging
14.2.3 Global hedging vs. instantaneous hedging
14.3 Residual risk
14.3.1 The Black-Scholes miracle
14.3.2 The‘stop-loss’strategy does not work
14.3.3 Instantaneous residual risk and kurtosis risk
14.3.4 Stochastic volatility models
14.4 Hedging errors. A variational point of view
14.5 Other measures of risk-hedging and VaR （*）
14.6 Conclusion of the chapter
14.7 Summary
14.8 Appendix D
15 Options: the role of drift and correlations
15.1 Influence of drift on optimally hedged option
15.1.1 A perturbative expansion
15.1.2 ‘Risk neutral’probability and martingales
15.2 Drift risk and delta-hedged options
15.2.1 Hedging the drift risk
15.2.2 The price of delta-hedged options
15.2.3 A general option pricing formula
15.3 Pricing and hedging in the presence of temporal correlations（*）
15.3.1 A general model of correlations
15.3.2 Derivative pricing with small correlations
15.3.3 The case of delta-hedging
15.4 Conclusion
15.4.1 Is the price of an option unique?
15.4.2 Should one always optimally hedge?
15.5 Summary
15.6 Appendix E
16 Options:the Black and Scholes model
16.1 Ito calculus and the Black-Scholes equation
16.1.1 The Gaussian Bachelier model
16.1.2 Solution and Martingale
16.1.3 Time value and the cost of hedging
16.1.4 The Log-normal Black-Scholes model
16.1.5 General pricing and hedging in a Brownian world
16.1.6 The Greeks
16.2 Drift and hedge in the Gaussian model（*）
16.2.1 Constant drift
16.2.2 Price dependent drift and the Ornstein-Uhlenbeck paradox
16.3 The binomial model
16.4 Summary
17 Options:some more specific problems
17.1 Other elements of the balance sheet
17.1.1 Interest rate and continuous dividends
17.1.2 Interest rate corrections to the hedging strategy
17.1.3 Discrete dividends
17.1.4 Transaction costs
17.2 Other types of options
17.2.1 ‘Put-call’parity
17.2.2 ‘Digital’options
17.2.3 ‘Asian’options
17.2.4 ‘American’options
17.2.5 ‘Barrier’options（*）
17.2.6 Other types of options
17.3 The ‘Greeks’and risk control
17.4 Risk diversification（*）
17.5 Summary
18 Options:minimum variance Monte-Carlo
18.1 Plain Monte-Carlo
18.1.1 Motivation and basic principle
18.1.2 Pricing the forward exactly
18.1.3 Calculating the Greeks
18.1.4 Drawbacks of the method
18.2 An ‘hedged’Monte-Carlo method
18.2.1 Basic principle of the method
18.2.2 A linear parameterization of the price and hedge
18.2.3 The Black-Scholes limit
18.3 Non Gaussian models and purely historical option pricing
18.4 Discussion and extensions. Calibration
18.5 Summary
18.6 Appendix F:generating some random variables
19 The yield curve
19.1 Introduction
19.2 The bond market
19.3 Hedging bonds with other bonds
19.3.1 The general problem
19.3.2 The continuous time Gaussian limit
19.4 The equation for bond pricing
19.4.1 A general solution
19.4.2 The Vasicek model
19.4.3 Forward rates
19.4.4 More general models
19.5 Empirical study of the forward rate curve
19.5.1 Data and notations
19.5.2 Quantities of interest and data analysis
19.6 Theoretical considerations（*）
19.6.1 Comparison with the Vasicek model
19.6.2 Market price of risk
19.6.3 Risk-premium and the θ law
19.7 Summary
19.8 Appendix G:optimal portfolio of bonds
20 Simple mechanisms for anomalous price statistics
20.1 Introduction
20.2 Simple models for herding and mimicry
20.2.1 Herding and percolation
20.2.2 Avalanches of opinion changes
20.3 Models of feedback effects on price fluctuations
20.3.1 Risk-aversion induced crashes
20.3.2 A simple model with volatility correlations and tails
20.3.3 Mechanisms for long ranged volatility correlations
20.4 The Minority Game
20.5 Summary
Index of most important symbols
