金融风险理论--从统计物理到风险管理特色及评论 在上一个世纪五十、七十年代的两个时间段，有一些智者提出了“风险的处理和效益的优化”两个现代金融学的中心议题。从此，几乎所有数理金融的理论也都围红绕着这两个基本问题而展开。 金融风险理论--从统计物理到风险管理内容简介 本书的重点是金融风险的控制和管理，为此必须要有可管、可控的指标，有了这些指标，就可以对风险定价，给出合理的模式和方法，所以本书的最后一章，广泛讨论了各种期权的定价和风险管理。这是一本视角、方法都很有特点的书，自始至终贯穿着用实际的证券市场的数据来说明、验证相应的分析结论，用股票市场的指数、外汇市场的交易和国债市场的行情作为实例，因此是有数据支持，令人不感到枯燥的分析。各种不同观点的人，从这本书的分析中都会有所收获。 ~Preface 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 Cram~~r function (*) 2.3.5 The CLT at work on simple cases 2.3.6 Truncated L6vy 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. I 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(I,I) 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 L~~vy 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 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 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 'A' 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 martingale s 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 Omstein-Uhlenbeck paradox 16.3 The binomial model 16.4 Summary 17 Options: some more specific 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 Ganssian 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 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 Index~
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