Copula methods in finance
Copula methods in finance
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发表时间：
2004
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通讯作者：
Umberto Cherubini;E. Luciano;Walter Vecchiato
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作者：
Umberto Cherubini;E. Luciano;Walter Vecchiato
前言。常用符号和符号列表。1衍生工具定价、套期保值和风险管理：最新进展。1.1简介。1.2衍生工具定价基础：二项模型。1.2.1复制投资组合。1.2.2无套利与风险中性概率测度。1.2.3无套利与客观概率测度。1.2.4不同概率测度下的贴现。1.2.5世界的多重状态。1.3 Black-Scholes模型。1.3.1 Ito引理。1.3.2 Girsanov定理。1.3.3鞅性质。1.3.4数字期权。1.4利率衍生品。1.4.1仿射因子模型。1.4.2远期鞅测度。1.4.3 LIBOR市场模型。1.5 Smile1.5.1随机波动率模型。1.5.2局部波动率模型。1.5.3隐含概率。1.6不完全市场。1.6.1回到效用理论。1.6.2超级对冲策略。1.7信用风险。1.7.1结构模型。1.7.2简化形式模型。1.7.3隐含违约概率。1.7.4交易对手风险。1.8金融中的Copula方法：1.8.1联合概率、边际概率和联结函数。1.8.2联结函数的对偶性。1.8.3联结函数的例子。1.8.4联结函数和市场变动。1.8.5尾部依赖性。1.8.6股票挂钩产品。1.8.7信用挂钩产品二元Copula函数。2.1定义和性质。2.2 Frechet界和协序。2.3 Sklar定理和Copula的概率解释。2.3.1 Sklar定理。2.3.2 Sklar定理中的次Copula。2.3.3建模结果。2.3.4 Sklar定理在金融应用中的应用：走向非black - scholes世界。2.4作为依赖函数的Copula：基本事实。2.4.1独立性。2.4.2共性。2.4.3单调变换和Copula不变性。2.4.4应用：2.5.1一个应用：带有外生冲击的违约概率。2.6密度与典型表示。2.7 r.v.s和分布函数的边界。2.7.1一个应用：VaR界。2.8附录.3市场运动与Copula族。3.1关联测度。3.1.1一致性。3.1.2 Kendall's τ。3.1.3 Spearman's rors .3.1.4线性相关。3.1.5尾部依赖。3.1.6正象限依赖。3.2二元Copula的参数族。3.2.1二元高斯Copula .3.2.2二元Student's t Copula .3.2.3 fri -echet家族。3.2.4阿基米德Copula .3.2.5 Marshall-Olkin Copula .44.1定义和基本性质。4.2 Frechet界和协调顺序：多维情况。4.3 Sklar定理和基本概率解释。多维情况。4.3.1建模结果。4.4生存联结和联合生存函数。4.5多维联结的密度和正则表示。4.6 n个随机变量和的分布函数的界。4.7多变量依赖。4.8 n维联结的参数族。4.8.1多变量高斯联结。4.8.2多变量Student’st联结。4.8.3多变量离散联结。4.8.4阿基米德联结市场数据的估计和校准。5.1 copula的统计推断。5.2精确最大似然法。5.2.1示例。5.3 IFM方法。5.3.1应用：市场数据参数copula的估计。5.4 CML方法。5.4.1应用：高斯copula相关矩阵的估计。5.5非参数估计。5.5.1经验copula.5.5.2核copula.5.6使用样本相关性度量的校准方法。5.7应用。5.8 copula的评价准则。5.9条件copula.5.9.1在股票投资组合中的应用市场场景的模拟。6.1蒙特卡罗在copuls中的应用。6.2椭圆copuls的模拟方法。6.3条件抽样。6.3.1 Clayton n-copula.6.3.2 Gumbel n-copula.6.3.3 Frank n-copula.6.4 Marshall和Olkin的方法。6.5模拟实例信用风险应用。7.1信用衍生产品。7.2部分信用衍生产品概述。7.2.1信用违约互换。7.2.2一揽子违约互换。7.2.3其他信用衍生产品。7.2.4债务抵押债券（CDO）。7.3 Copula方法。7.3.1单一生存时间建模与校准的回顾。7.3.2多重生存时间：建模。7.3.3多重违约：7.3.5损失分布和同质一篮子违约掉期定价。7.4应用：cdo定价和风险监控。7.4.1道琼斯EuroStoxx50 cdo。7.4.2应用：一篮子违约掉期。7.4.3 EuroStoxx50 cdo的实证应用。7.4.4 EuroStoxx50定价与风险监测。7.4.5一篮子违约掉期的定价与风险监测。7.5技术附录。7.5.1多元Clayton copula密度的推导。7.5.2四变量Frank copula密度的推导。7.5.3相关违约时间。7.5.4方差协方差稳健估计。7.5.5分析中的利率与汇率基于Copula的期权定价。8.1简介。8.2完全市场中二元期权的定价。8.2.1 Copula定价内核。8.2.2可选定价技术。8.3不完全市场中二元期权的定价。8.3.1定价：8.3.2 Copula定价核。8.4易损期权定价。8.4.1易损数字期权定价。8.4.2易损看涨期权定价。8.4.3易损看跌期权定价。8.4.4易损期权的实际定价。8.5彩虹双色期权定价。8.5.1两个资产中最小值的看涨期权。8.5.2两个资产中最大值的看涨期权。8.5.3两个资产中最大值的看跌期权。8.5.4两个资产中最小值的看跌期权。8.5.5期权to8.6.2看跌期权定价：一般框架8.6.3指定触发事件8.6.4校准依赖结构8.6.5反射copula 8.7多变量期权定价：蒙特卡罗方法8.7.1应用：篮子期权。
Preface.List of Common Symbols and Notations.1 Derivatives Pricing, Hedging and Risk Management: The State of the Art.1.1 Introduction.1.2 Derivative pricing basics: the binomial model.1.2.1 Replicating portfolios.1.2.2 No-arbitrage and the risk-neutral probability measure.1.2.3 No-arbitrage and the objective probability measure.1.2.4 Discounting under different probability measures.1.2.5 Multiple states of the world.1.3 The Black-Scholes model.1.3.1 Ito's lemma.1.3.2 Girsanov theorem.1.3.3 The martingale property.1.3.4 Digital options.1.4 Interest rate derivatives.1.4.1 Affine factor models.1.4.2 Forward martingale measure.1.4.3 LIBOR market model.1.5 Smile and term structure effects of volatility.1.5.1 Stochastic volatility models.1.5.2 Local volatility models.1.5.3 Implied probability.1.6 Incomplete markets.1.6.1 Back to utility theory.1.6.2 Super-hedging strategies.1.7 Credit risk.1.7.1 Structural models.1.7.2 Reduced form models.1.7.3 Implied default probabilities.1.7.4 Counterparty risk.1.8 Copula methods in finance: a primer.1.8.1 Joint probabilities, marginal probabilities and copula functions.1.8.2 Copula functions duality.1.8.3 Examples of copula functions.1.8.4 Copula functions and market comovements.1.8.5 Tail dependence.1.8.6 Equity-linked products.1.8.7 Credit-linked products.2 Bivariate Copula Functions.2.1 Definition and properties.2.2 Frechet bounds and concordance order.2.3 Sklar's theorem and the probabilistic interpretation of copulas.2.3.1 Sklar's theorem.2.3.2 The subcopula in Sklar's theorem.2.3.3 Modeling consequences.2.3.4 Sklar's theorem in financial applications: toward a non-Black-Scholes world.2.4 Copulas as dependence functions: basic facts.2.4.1 Independence.2.4.2 Comonotonicity.2.4.3 Monotone transforms and copula invariance.2.4.4 An application: VaR trade-off.2.5 Survival copula and joint survival function.2.5.1 An application: default probability with exogenous shocks.2.6 Density and canonical representation.2.7 Bounds for the distribution functions of sum of r.v.s.2.7.1 An application: VaR bounds.2.8 Appendix.3 Market Comovements and Copula Families.3.1 Measures of association.3.1.1 Concordance.3.1.2 Kendall's tau.3.1.3 Spearman's rhoS.3.1.4 Linear correlation.3.1.5 Tail dependence.3.1.6 Positive quadrant dependency.3.2 Parametric families of bivariate copula.3.2.1 The bivariate Gaussian copula.3.2.2 The bivariate Student's t copula.3.2.3 The Fr-echet family.3.2.4 Archimedean copulas.3.2.5 The Marshall-Olkin copula.4 Multivariate Copulas.4.1 Definition and basic properties.4.2 Frechet bounds and concordance order: the multidimensional case.4.3 Sklar's theorem and the basic probabilistic interpretation: the multidimensional case.4.3.1 Modeling consequences.4.4 Survival copula and joint survival function.4.5 Density and canonical representation of a multidimensional copula.4.6 Bounds for distribution functions of sums of n random variables.4.7 Multivariate dependence.4.8 Parametric families of n-dimensional copulas.4.8.1 The multivariate Gaussian copula.4.8.2 The multivariate Student's t copula.4.8.3 The multivariate dispersion copula.4.8.4 Archimedean copulas.5 Estimation and Calibration from Market Data.5.1 Statistical inference for copulas.5.2 Exact maximum likelihood method.5.2.1 Examples.5.3 IFM method.5.3.1 Application: estimation of the parametric copula for market data.5.4 CML method.5.4.1 Application: estimation of the correlation matrix for a Gaussian copula.5.5 Non-parametric estimation.5.5.1 The empirical copula.5.5.2 Kernel copula.5.6 Calibration method by using sample dependence measures.5.7 Application.5.8 Evaluation criteria for copulas.5.9 Conditional copula.5.9.1 Application to an equity portfolio.6 Simulation of Market Scenarios.6.1 Monte Carlo application with copulas.6.2 Simulation methods for elliptical copulas.6.3 Conditional sampling.6.3.1 Clayton n-copula.6.3.2 Gumbel n-copula.6.3.3 Frank n-copula.6.4 Marshall and Olkin's method.6.5 Examples of simulations.7 Credit Risk Applications.7.1 Credit derivatives.7.2 Overview of some credit derivatives products.7.2.1 Credit default swap.7.2.2 Basket default swap.7.2.3 Other credit derivatives products.7.2.4 Collateralized debt obligation (CDO).7.3 Copula approach.7.3.1 Review of single survival time modeling and calibration.7.3.2 Multiple survival times: modeling.7.3.3 Multiple defaults: calibration.7.3.4 Loss distribution and the pricing of CDOs.7.3.5 Loss distribution and the pricing of homogeneous basket default swaps.7.4 Application: pricing and risk monitoring a CDO.7.4.1 Dow Jones EuroStoxx50 CDO.7.4.2 Application: basket default swap.7.4.3 Empirical application for the EuroStoxx50 CDO.7.4.4 EuroStoxx50 pricing and risk monitoring.7.4.5 Pricing and risk monitoring of the basket default swaps.7.5 Technical appendix.7.5.1 Derivation of a multivariate Clayton copula density.7.5.2 Derivation of a 4-variate Frank copula density.7.5.3 Correlated default times.7.5.4 Variance-covariance robust estimation.7.5.5 Interest rates and foreign exchange rates in the analysis.8 Option Pricing with Copulas.8.1 Introduction.8.2 Pricing bivariate options in complete markets.8.2.1 Copula pricing kernels.8.2.2 Alternative pricing techniques.8.3 Pricing bivariate options in incomplete markets.8.3.1 Frcicing: super-replication in two dimensions.8.3.2 Copula pricing kernel.8.4 Pricing vulnerable options.8.4.1 Vulnerable digital options.8.4.2 Pricing vulnerable call options.8.4.3 Pricing vulnerable put options.8.4.4 Pricing vulnerable options in practice.8.5 Pricing rainbow two-color options.8.5.1 Call option on the minimum of two assets.8.5.2 Call option on the maximum of two assets.8.5.3 Put option on the maximum of two assets.8.5.4 Put option on the minimum of two assets.8.5.5 Option to exchange.8.5.6 Pricing and hedging rainbows with smiles: Everest notes.8.6 Pricing barrier options.8.6.1 Pricing call barrier options with copulas: the general framework.8.6.2 Pricing put barrier option: the general framework.8.6.3 Specifying the trigger event.8.6.4 Calibrating the dependence structure.8.6.5 The reflection copula.8.7 Pricing multivariate options: Monte Carlo methods.8.7.1 Application: basket option.Bibliography.Index.