Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method for performing inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. Our approach represents the first step towards the development of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.

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Pair-copula constructions of multiple dependence

Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book’s methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives.

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Quantitative Risk Management

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Introduction To Robust Estimation And Hypothesis Testing

Preface. Acknowledgments. 1 Complex Electricity Markets. 1.1 Liberalization. 1.2 The Marketplace. 1.2.1 Power Pools and Power Exchanges. 1.2.2 Nodal and Zonal Pricing. 1.2.3 Market Structure. 1.2.4 Traded Products. 1.3 Europe. 1.3.1 The England and Wales Electricity Market. 1.3.2 The Nordic Market. 1.3.3 Price Setting at Nord Pool. 1.3.4 Continental Europe 13. 1.4 North America. 1.4.1 PJM Interconnection. 1.4.2 California and the Electricity Crisis. 1.4.3 Alberta and Ontario. 1.5 Australia and New Zealand. 1.6 Summary. 1.7 Further Reading. 2 Stylized Facts of Electricity Loads and Prices. 2.1 Introduction. 2.2 Price Spikes. 2.2.1 Case Study: The June 1998 Cinergy Price Spike. 2.2.2 When Supply Meets Demand. 2.2.3 What is Causing the Spikes?. 2.2.4 The Definition. 2.3 Seasonality. 2.3.1 Measuring Serial Correlation. 2.3.2 Spectral Analysis and the Periodogram. 2.3.3 Case Study: Seasonal Behavior of Electricity Prices and Loads. 2.4 Seasonal Decomposition. 2.4.1 Differencing. 2.4.2 Mean or Median Week. 2.4.3 Moving Average Technique. 2.4.4 Annual Seasonality and Spectral Decomposition. 2.4.5 Rolling Volatility Technique. 2.4.6 Case Study: Rolling Volatility in Practice. 2.4.7 Wavelet Decomposition. 2.4.8 Case Study: Wavelet Filtering of Nord Pool Hourly System Prices. 2.5 Mean Reversion. 2.5.1 R/S Analysis. 2.5.2 Detrended Fluctuation Analysis. 2.5.3 Periodogram Regression. 2.5.4 Average Wavelet Coefficient. 2.5.5 Case Study: Anti-persistence of Electricity Prices. 2.6 Distributions of Electricity Prices. 2.6.1 Stable Distributions. 2.6.2 Hyperbolic Distributions. 2.6.3 Case Study: Distribution of EEX Spot Prices. 2.6.4 Further Empirical Evidence and Possible Applications. 2.7 Summary. 2.8 Further Reading. 3 Modeling and Forecasting Electricity Loads. 3.1 Introduction. 3.2 Factors Affecting Load Patterns. 3.2.1 Case Study: Dealing with Missing Values and Outliers. 3.2.2 Time Factors. 3.2.3 Weather Conditions. 3.2.4 Case Study: California Weather vs Load. 3.2.5 Other Factors. 3.3 Overview of Artificial Intelligence-Based Methods. 3.4 Statistical Methods. 3.4.1 Similar-Day Method. 3.4.2 Exponential Smoothing. 3.4.3 Regression Methods. 3.4.4 Autoregressive Model. 3.4.5 Autoregressive Moving Average Model. 3.4.6 ARMA Model Identification. 3.4.7 Case Study: Modeling Daily Loads in California. 3.4.8 Autoregressive Integrated Moving Average Model. 3.4.9 Time Series Models with Exogenous Variables. 3.4.10 Case Study: Modeling Daily Loads in California with Exogenous Variables. 3.5 Summary. 3.6 Further Reading. 4 Modeling and Forecasting Electricity Prices. 4.1 Introduction. 4.2 Overview of Modeling Approaches. 4.3 Statistical Methods and Price Forecasting. 4.3.1 Exogenous Factors. 4.3.2 Spike Preprocessing. 4.3.3 How to Assess the Quality of Price Forecasts. 4.3.4 ARMA-type Models. 4.3.5 Time Series Models with Exogenous Variables. 4.3.6 Autoregressive GARCH Models. 4.3.7 Case Study: Forecasting Hourly CalPX Spot Prices with Linear Models. 4.3.8 Case Study: Is Spike Preprocessing Advantageous?. 4.3.9 Regime-Switching Models. 4.3.10 Calibration of Regime-Switching Models. 4.3.11 Case Study: Forecasting Hourly CalPX Spot Prices with Regime-Switching Models. 4.3.12 Interval Forecasts. 4.4 Quantitative Models and Derivatives Valuation. 4.4.1 Jump-Diffusion Models. 4.4.2 Calibration of Jump-Diffusion Models. 4.4.3 Case Study: A Mean-Reverting Jump-Diffusion Model for Nord Pool Spot Prices. 4.4.4 Hybrid Models. 4.4.5 Case Study: Regime-Switching Models for Nord Pool Spot Prices. 4.4.6 Hedging and the Use of Derivatives. 4.4.7 Derivatives Pricing and the Market Price of Risk. 4.4.8 Case Study: Asian-Style Electricity Options. 4.5 Summary. 4.6 Further Reading. Bibliography. Index.

Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach

ural and Political Observations Made upon the Bills of Mortality (1662) stands as the first great example of modern statistical data analysis, Bernstein also spends some time telling us about the genesis of Lloyd’s of London, the famous insurance firm. This weaving of topics that are standard in the history of probability and statistics with many that are not is one of the strengths and attractions of the book. Another is the obvious zest with which Bernstein describes the many intriguing people and tantalizing mysteries so peculiar to the mathematics of chance.

Against the Gods: The Remarkable Story of Risk

This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation

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Rdiger Frey

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Quantitative Risk Management: Concepts, Techniques, and Tools