Cover -- Half-title -- Title -- Copyright -- Contents -- Foreword -- Preface -- Acknowledgements -- 1 Probability theory: basic notions -- 1.1 Introduction -- 1.2 Probabilities -- 1.2.1 Probability distributions -- 1.2.2 Typical values and deviations -- 1.2.3 Moments and characteristic function -- 1.2.4 Divergence of moments-asymptotic behaviour -- 1.3 Some useful distributions -- 1.3.1 Gaussian distribution -- 1.3.2 Log-normal distribution -- 1.3.3 Lévy distributions and Paretian tails -- 1.3.4 Other distributions (*) -- 1.4 Maximum of random variables-statistics of extremes -- 1.5 Sums of random variables -- 1.5.1 Convolutions -- 1.5.2 Additivity of cumulants and of tail amplitudes -- 1.5.3 Stable distributions and self-similarity -- 1.6 Central limit theorem -- 1.6.1 Convergence to a Gaussian -- 1.6.2 Convergence to a Lévy distribution -- 1.6.3 Large deviations -- 1.6.4 The CLT at work on a simple case -- 1.6.5 Truncated Lévy distributions -- 1.6.6 Conclusion: survival and vanishing of tails -- 1.7 Correlations, dependence and non-stationary models (*) -- 1.7.1 Correlations -- 1.7.2 Non-stationary models and dependence -- 1.8 Central limit theorem for random matrices (*) -- 1.9 Appendix A: non-stationarity and anomalous kurtosis -- 1.10 Appendix B: density of eigenvalues for random correlation matrices -- 1.11 References -- 2 Statistics of real prices -- 2.1 Aim of the chapter -- 2.2 Second-order statistics -- 2.2.1 Variance, volatility and the additive-multiplicative crossover -- 2.2.2 Autocorrelation and power spectrum -- Power spectrum -- 2.3 Temporal evolution of fluctuations -- 2.3.1 Temporal evolution of probability distributions -- The elementary distribution P -- Maximum likelihood -- Convolutions -- Tails, what tails? -- 2.3.2 Multiscaling-Hurst exponent (*) -- 2.4 Anomalous kurtosis and scale fluctuations.

2.5 Volatile markets and volatility markets -- 2.6 Statistical analysis of the forward rate curve (*) -- 2.6.1 Presentation of the data and notations -- 2.6.2 Quantities of interest and data analysis -- 2.6.3 Comparison with the Vasicek model -- 2.6.4 Risk-premium and the… -- The average FRC and value-at-risk pricing -- The anticipated trend and the volatility hump -- 2.7 Correlation matrices (*) -- 2.8 A simple mechanism for anomalous price statistics (*) -- 2.9 A simple model with volatility correlations and tails (*) -- 2.10 Conclusion -- 2.11 References -- Scaling and Fractals in Financial Markets -- The interest rate curve -- Percolation, collective models and self organized criticality -- Other recent market models -- 3 Extreme risks and optimal portfolios -- 3.1 Risk measurement and diversification -- 3.1.1 Risk and volatility -- 3.1.2 Risk of loss and 'Value at Risk' (VaR) -- 3.1.3 Temporal aspects: drawdown and cumulated loss -- Worst low -- Cumulated losses -- Drawdowns -- 3.1.4 Diversification and utility-satisfaction thresholds -- 3.1.5 Conclusion -- 3.2 Portfolios of uncorrelated assets -- 3.2.1 Uncorrelated Gaussian assets -- Effective asset number in a portfolio -- 3.2.2 Uncorrelated 'power-law' assets -- 3.2.3 'Exponential' assets -- 3.2.4 General case: optimal portfolio and VaR (*) -- 3.3 Portfolios of correlated assets -- 3.3.1 Correlated Gaussian fluctuations -- The CAPM and its limitations -- 3.3.2 'Power-law' fluctuations (*) -- 'Tail covariance' -- Optimal portfolio -- 3.4 Optimized trading (*) -- 3.5 Conclusion of the chapter -- 3.6 Appendix C: some useful results -- 3.7 References -- Statistics of drawdowns and extremes -- Portfolio theory and CAPM -- Optimal portfolios in a Lévy world -- Generalization of the covariance to Lévy variables -- 4 Futures and options: fundamental concepts -- 4.1 Introduction.

4.1.1 Aim of the chapter -- 4.1.2 Trading strategies and efficient markets -- 4.2 Futures and forwards -- 4.2.1 Setting the stage -- 4.2.2 Global financial balance -- 4.2.3 Riskless hedge -- Dividends -- Variable interest rates -- 4.2.4 Conclusion: global balance and arbitrage -- 4.3 Options: definition and valuation -- 4.3.1 Setting the stage -- 4.3.2 Orders of magnitude -- 4.3.3 Quantitative analysis-option price -- Bachelier's Gaussian limit -- Dynamic equation for the option price -- 4.3.4 Real option prices, volatility smile and 'implied' kurtosis -- Stationary distributions and the smile curve -- Non-stationarity and 'implied' kurtosis -- 4.4 Optimal strategy and residual risk -- 4.4.1 Introduction -- 4.4.2 A simple case -- 4.4.3 General case… -- Cumulant corrections to… -- 4.4.4 Global hedging/instantaneous hedging -- 4.4.5 Residual risk: the Black-Scholes miracle -- The 'stop-loss' strategy does not work -- Residual risk to first order in kurtosis -- Stochastic volatility models -- 4.4.6 Other measures of risk-hedging and VaR (*) -- 4.4.7 Hedging errors -- 4.4.8 Summary -- 4.5 Does the price of an option depend on the mean return? -- 4.5.1 The case of non-zero excess return -- 'Risk neutral' probability -- Optimal strategy in the presence of a bias -- 4.5.2 The Gaussian case and the Black-Scholes limit -- Ito calculus -- 4.5.3 Conclusion. Is the price of an option unique? -- 4.6 Conclusion of the chapter: the pitfalls of zero-risk -- 4.7 Appendix D: computation of the conditional mean -- 4.8 Appendix E: binomial model -- 4.9 Appendix F: option price for (suboptimal)… -- 4.10 References -- Some classics -- Market efficiency -- Optimal filters -- Options and futures -- Stochastic differential calculus and derivative pricing -- Option pricing in the presence of residual risk -- Kurtosis and implied cumulants -- Stochastic volatility models.

5 Options: some more specific problems -- 5.1 Other elements of the balance sheet -- 5.1.1 Interest rate and continuous dividends -- Systematic drift of the price -- Independence between price increments and interest rates-dividends -- Multiplicative model -- 5.1.2 Interest rates corrections to the hedging strategy -- 5.1.3 Discrete dividends -- 5.1.4 Transaction costs -- 5.2 Other types of options: 'Puts' and 'exotic options' -- 5.2.1 'Put-call' parity -- 5.2.2 'Digital' options -- 5.2.3 'Asian' options -- 5.2.4 'American' options -- American puts -- 5.2.5 'Barrier' options -- Other types of option -- 5.3 The 'Greeks' and risk control -- 5.4 Value-at-risk for general non-linear portfolios (*) -- 5.5 Risk diversification (*) -- 'Portfolio' options and 'exogenous' hedging -- Option portfolio -- 5.6 References -- More on options, exotic options -- Stochastic volatility models and volatility hedging -- Short glossary of financial terms -- Index of symbols -- Index.