Stochastic Modelling of Electricity and Related Markets

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Author :
Publisher : World Scientific
ISBN 13 : 981281230X
Total Pages : 352 pages
Book Rating : 4.8/5 (128 download)

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Book Synopsis Stochastic Modelling of Electricity and Related Markets by : Fred Espen Benth

Download or read book Stochastic Modelling of Electricity and Related Markets written by Fred Espen Benth and published by World Scientific. This book was released on 2008 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: The markets for electricity, gas and temperature have distinctive features, which provide the focus for countless studies. For instance, electricity and gas prices may soar several magnitudes above their normal levels within a short time due to imbalances in supply and demand, yielding what is known as spikes in the spot prices. The markets are also largely influenced by seasons, since power demand for heating and cooling varies over the year. The incompleteness of the markets, due to nonstorability of electricity and temperature as well as limited storage capacity of gas, makes spot-forward hedging impossible. Moreover, futures contracts are typically settled over a time period rather than at a fixed date. All these aspects of the markets create new challenges when analyzing price dynamics of spot, futures and other derivatives.This book provides a concise and rigorous treatment on the stochastic modeling of energy markets. Ornstein?Uhlenbeck processes are described as the basic modeling tool for spot price dynamics, where innovations are driven by time-inhomogeneous jump processes. Temperature futures are studied based on a continuous higher-order autoregressive model for the temperature dynamics. The theory presented here pays special attention to the seasonality of volatility and the Samuelson effect. Empirical studies using data from electricity, temperature and gas markets are given to link theory to practice.

Stochastic Modeling Of Electricity And Related Markets

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Author :
Publisher : World Scientific
ISBN 13 : 9814471313
Total Pages : 352 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Stochastic Modeling Of Electricity And Related Markets by : Fred Espen Benth

Download or read book Stochastic Modeling Of Electricity And Related Markets written by Fred Espen Benth and published by World Scientific. This book was released on 2008-04-14 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: The markets for electricity, gas and temperature have distinctive features, which provide the focus for countless studies. For instance, electricity and gas prices may soar several magnitudes above their normal levels within a short time due to imbalances in supply and demand, yielding what is known as spikes in the spot prices. The markets are also largely influenced by seasons, since power demand for heating and cooling varies over the year. The incompleteness of the markets, due to nonstorability of electricity and temperature as well as limited storage capacity of gas, makes spot-forward hedging impossible. Moreover, futures contracts are typically settled over a time period rather than at a fixed date. All these aspects of the markets create new challenges when analyzing price dynamics of spot, futures and other derivatives.This book provides a concise and rigorous treatment on the stochastic modeling of energy markets. Ornstein-Uhlenbeck processes are described as the basic modeling tool for spot price dynamics, where innovations are driven by time-inhomogeneous jump processes. Temperature futures are studied based on a continuous higher-order autoregressive model for the temperature dynamics. The theory presented here pays special attention to the seasonality of volatility and the Samuelson effect. Empirical studies using data from electricity, temperature and gas markets are given to link theory to practice.

Stochastic Modelling of Electricity and Related Markets

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Author :
Publisher : World Scientific
ISBN 13 : 9812812318
Total Pages : 352 pages
Book Rating : 4.8/5 (128 download)

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Book Synopsis Stochastic Modelling of Electricity and Related Markets by : Fred Espen Benth

Download or read book Stochastic Modelling of Electricity and Related Markets written by Fred Espen Benth and published by World Scientific. This book was released on 2008 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: The markets for electricity, gas and temperature have distinctive features, which provide the focus for countless studies. For instance, electricity and gas prices may soar several magnitudes above their normal levels within a short time due to imbalances in supply and demand, yielding what is known as spikes in the spot prices. The markets are also largely influenced by seasons, since power demand for heating and cooling varies over the year. The incompleteness of the markets, due to nonstorability of electricity and temperature as well as limited storage capacity of gas, makes spot-forward hedging impossible. Moreover, futures contracts are typically settled over a time period rather than at a fixed date. All these aspects of the markets create new challenges when analyzing price dynamics of spot, futures and other derivatives. This book provides a concise and rigorous treatment on the stochastic modeling of energy markets. OrnsteinOCoUhlenbeck processes are described as the basic modeling tool for spot price dynamics, where innovations are driven by time-inhomogeneous jump processes. Temperature futures are studied based on a continuous higher-order autoregressive model for the temperature dynamics. The theory presented here pays special attention to the seasonality of volatility and the Samuelson effect. Empirical studies using data from electricity, temperature and gas markets are given to link theory to practice. Sample Chapter(s). A Survey of Electricity and Related Markets (331 KB). Contents: A Survey of Electricity and Related Markets; Stochastic Analysis for Independent Increment Processes; Stochastic Models for the Energy Spot Price Dynamics; Pricing of Forwards and Swaps Based on the Spot Price; Applications to the Gas Markets; Modeling Forwards and Swaps Using the HeathOCoJarrowOCoMorton Approach; Constructing Smooth Forward Curves in Electricity Markets; Modeling of the Electricity Futures Market; Pricing and Hedging of Energy Options; Analysis of Temperature Derivatives. Readership: Researchers in energy and commodity markets, and mathematical finance.

Mathematical Modelling of Contemporary Electricity Markets

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Publisher : Academic Press
ISBN 13 : 0128218398
Total Pages : 444 pages
Book Rating : 4.1/5 (282 download)

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Book Synopsis Mathematical Modelling of Contemporary Electricity Markets by : Athanasios Dagoumas

Download or read book Mathematical Modelling of Contemporary Electricity Markets written by Athanasios Dagoumas and published by Academic Press. This book was released on 2021-01-30 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical Modelling of Contemporary Electricity Markets reviews major methodologies and tools to accurately analyze and forecast contemporary electricity markets in a ways that is ideal for practitioner and academic audiences. Approaches include optimization, neural networks, genetic algorithms, co-optimization, econometrics, E3 models and energy system models. The work examines how new challenges affect power market modeling, including discussions of stochastic renewables, price volatility, dynamic participation of demand, integration of storage and electric vehicles, interdependence with other commodity markets and the evolution of policy developments (market coupling processes, security of supply). Coverage addresses all major forms of electricity markets: day-ahead, forward, intraday, balancing, and capacity. Provides a diverse body of established techniques suitable for modeling any major aspect of electricity markets Familiarizes energy experts with the quantitative skills needed in competitive electricity markets Reviews market risk for energy investment decisions by stressing the multi-dimensionality of electricity markets

Modeling and Forecasting Electricity Loads and Prices

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Publisher : John Wiley & Sons
ISBN 13 : 0470059990
Total Pages : 192 pages
Book Rating : 4.4/5 (7 download)

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Book Synopsis Modeling and Forecasting Electricity Loads and Prices by : Rafal Weron

Download or read book Modeling and Forecasting Electricity Loads and Prices written by Rafal Weron and published by John Wiley & Sons. This book was released on 2007-01-30 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an in-depth and up-to-date review of different statistical tools that can be used to analyze and forecast the dynamics of two crucial for every energy company processes—electricity prices and loads. It provides coverage of seasonal decomposition, mean reversion, heavy-tailed distributions, exponential smoothing, spike preprocessing, autoregressive time series including models with exogenous variables and heteroskedastic (GARCH) components, regime-switching models, interval forecasts, jump-diffusion models, derivatives pricing and the market price of risk. Modeling and Forecasting Electricity Loads and Prices is packaged with a CD containing both the data and detailed examples of implementation of different techniques in Matlab, with additional examples in SAS. A reader can retrace all the intermediate steps of a practical implementation of a model and test his understanding of the method and correctness of the computer code using the same input data. The book will be of particular interest to the quants employed by the utilities, independent power generators and marketers, energy trading desks of the hedge funds and financial institutions, and the executives attending courses designed to help them to brush up on their technical skills. The text will be also of use to graduate students in electrical engineering, econometrics and finance wanting to get a grip on advanced statistical tools applied in this hot area. In fact, there are sixteen Case Studies in the book making it a self-contained tutorial to electricity load and price modeling and forecasting.

Decision Making Under Uncertainty in Electricity Markets

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Publisher : Springer Science & Business Media
ISBN 13 : 1441974210
Total Pages : 549 pages
Book Rating : 4.4/5 (419 download)

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Book Synopsis Decision Making Under Uncertainty in Electricity Markets by : Antonio J. Conejo

Download or read book Decision Making Under Uncertainty in Electricity Markets written by Antonio J. Conejo and published by Springer Science & Business Media. This book was released on 2010-09-08 with total page 549 pages. Available in PDF, EPUB and Kindle. Book excerpt: Decision Making Under Uncertainty in Electricity Markets provides models and procedures to be used by electricity market agents to make informed decisions under uncertainty. These procedures rely on well established stochastic programming models, which make them efficient and robust. Particularly, these techniques allow electricity producers to derive offering strategies for the pool and contracting decisions in the futures market. Retailers use these techniques to derive selling prices to clients and energy procurement strategies through the pool, the futures market and bilateral contracting. Using the proposed models, consumers can derive the best energy procurement strategies using the available trading floors. The market operator can use the techniques proposed in this book to clear simultaneously energy and reserve markets promoting efficiency and equity. The techniques described in this book are of interest for professionals working on energy markets, and for graduate students in power engineering, applied mathematics, applied economics, and operations research.

An Introduction to Stochastic Modeling

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Publisher : Academic Press
ISBN 13 : 1483269272
Total Pages : 410 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis An Introduction to Stochastic Modeling by : Howard M. Taylor

Download or read book An Introduction to Stochastic Modeling written by Howard M. Taylor and published by Academic Press. This book was released on 2014-05-10 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: An Introduction to Stochastic Modeling provides information pertinent to the standard concepts and methods of stochastic modeling. This book presents the rich diversity of applications of stochastic processes in the sciences. Organized into nine chapters, this book begins with an overview of diverse types of stochastic models, which predicts a set of possible outcomes weighed by their likelihoods or probabilities. This text then provides exercises in the applications of simple stochastic analysis to appropriate problems. Other chapters consider the study of general functions of independent, identically distributed, nonnegative random variables representing the successive intervals between renewals. This book discusses as well the numerous examples of Markov branching processes that arise naturally in various scientific disciplines. The final chapter deals with queueing models, which aid the design process by predicting system performance. This book is a valuable resource for students of engineering and management science. Engineers will also find this book useful.

Valuation, Hedging and Speculation in Competitive Electricity Markets

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Publisher : Springer Science & Business Media
ISBN 13 : 146151701X
Total Pages : 220 pages
Book Rating : 4.4/5 (615 download)

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Book Synopsis Valuation, Hedging and Speculation in Competitive Electricity Markets by : Petter L. Skantze

Download or read book Valuation, Hedging and Speculation in Competitive Electricity Markets written by Petter L. Skantze and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: The challenges currently facing particIpants m competitive electricity markets are unique and staggering: unprecedented price volatility, a crippling lack of historical market data on which to test new modeling approaches, and a continuously changing regulatory structure. Meeting these challenges will require the knowledge and experience of both the engineering and finance communities. Yet the two communities continue to largely ignore each other. The finance community believes that engineering models are too detailed and complex to be practically applicable in the fast changing market environment. Engineers counter that the finance models are merely statistical regressions, lacking the necessary structure to capture the true dynamic properties of complex power systems. While both views have merit, neither group has by themselves been able to produce effective tools for meeting industry challenges. The goal of this book is to convey the fundamental differences between electricity and other traded commodities, and the impact these differences have on valuation, hedging and operational decisions made by market participants. The optimization problems associated with these decisions are formulated in the context of the market realities of today's power industry, including a lack of liquidity on forward and options markets, limited availability of historical data, and constantly changing regulatory structures.

Integrating Renewables in Electricity Markets

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Publisher : Springer Science & Business Media
ISBN 13 : 1461494117
Total Pages : 434 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Integrating Renewables in Electricity Markets by : Juan M. Morales

Download or read book Integrating Renewables in Electricity Markets written by Juan M. Morales and published by Springer Science & Business Media. This book was released on 2013-12-03 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: This addition to the ISOR series addresses the analytics of the operations of electric energy systems with increasing penetration of stochastic renewable production facilities, such as wind- and solar-based generation units. As stochastic renewable production units become ubiquitous throughout electric energy systems, an increasing level of flexible backup provided by non-stochastic units and other system agents is needed if supply security and quality are to be maintained. Within the context above, this book provides up-to-date analytical tools to address challenging operational problems such as: • The modeling and forecasting of stochastic renewable power production. • The characterization of the impact of renewable production on market outcomes. • The clearing of electricity markets with high penetration of stochastic renewable units. • The development of mechanisms to counteract the variability and unpredictability of stochastic renewable units so that supply security is not at risk. • The trading of the electric energy produced by stochastic renewable producers. • The association of a number of electricity production facilities, stochastic and others, to increase their competitive edge in the electricity market. • The development of procedures to enable demand response and to facilitate the integration of stochastic renewable units. This book is written in a modular and tutorial manner and includes many illustrative examples to facilitate its comprehension. It is intended for advanced undergraduate and graduate students in the fields of electric energy systems, applied mathematics and economics. Practitioners in the electric energy sector will benefit as well from the concepts and techniques explained in this book.

Energy Markets and Responsive Grids

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Publisher : Springer
ISBN 13 : 1493978225
Total Pages : 518 pages
Book Rating : 4.4/5 (939 download)

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Book Synopsis Energy Markets and Responsive Grids by : Sean Meyn

Download or read book Energy Markets and Responsive Grids written by Sean Meyn and published by Springer. This book was released on 2018-06-09 with total page 518 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consists of selected essays by participants of the workshop Control at Large Scales: Energy Markets and Responsive Grids held at the Institute for Mathematics and its Applications, Minneapolis, Minnesota, U.S.A. from May 9-13, 2016. The workshop brought together a diverse group of experts to discuss current and future challenges in energy markets and controls, along with potential solutions. The volume includes chapters on significant challenges in the design of markets and incentives, integration of renewable energy and energy storage, risk management and resilience, and distributed and multi-scale optimization and control. Contributors include leading experts from academia and industry in power systems and markets as well as control science and engineering. This volume will be of use to experts and newcomers interested in all aspects of the challenges facing the creation of a more sustainable electricity infrastructure, in areas such as distributed and stochastic optimization and control, stability theory, economics, policy, and financial mathematics, as well as in all aspects of power system operation.

Stochastic Models for Prices Dynamics in Energy and Commodity Markets

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Author :
Publisher : Springer Nature
ISBN 13 : 3031403673
Total Pages : 250 pages
Book Rating : 4.0/5 (314 download)

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Book Synopsis Stochastic Models for Prices Dynamics in Energy and Commodity Markets by : Fred Espen Benth

Download or read book Stochastic Models for Prices Dynamics in Energy and Commodity Markets written by Fred Espen Benth and published by Springer Nature. This book was released on 2023-11-16 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents a theory for random field models in time and space, viewed as stochastic processes with values in a Hilbert space, to model the stochastic dynamics of forward and futures prices in energy, power, and commodity markets. In this book, the well-known Heath–Jarrow–Morton approach from interest rate theory is adopted and extended into an infinite-dimensional framework, allowing for flexible modeling of price stochasticity across time and along the term structure curve. Various models are introduced based on stochastic partial differential equations with infinite-dimensional Lévy processes as noise drivers, emphasizing random fields described by low-dimensional parametric covariance functions instead of classical high-dimensional factor models. The Filipović space, a separable Hilbert space of Sobolev type, is found to be a convenient state space for the dynamics of forward and futures term structures. The monograph provides a classification of important operators in this space, covering covariance operators and the stochastic modeling of volatility term structures, including the Samuelson effect. Fourier methods are employed to price many derivatives of interest in energy, power, and commodity markets, and sensitivity 'delta' expressions can be derived. Additionally, the monograph covers forward curve smoothing, the connection between forwards with fixed delivery and delivery period, as well as the classical theory of forward and futures pricing. This monograph will appeal to researchers and graduate students interested in mathematical finance and stochastic analysis applied in the challenging markets of energy, power, and commodities. Practitioners seeking sophisticated yet flexible and analytically tractable risk models will also find it valuable.

Forecasting Models of Electricity Prices

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Author :
Publisher : MDPI
ISBN 13 : 3038424153
Total Pages : 259 pages
Book Rating : 4.0/5 (384 download)

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Book Synopsis Forecasting Models of Electricity Prices by : Javier Contreras

Download or read book Forecasting Models of Electricity Prices written by Javier Contreras and published by MDPI. This book was released on 2018-04-06 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a printed edition of the Special Issue "Forecasting Models of Electricity Prices" that was published in Energies

Competitive Electricity Markets

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Publisher : Elsevier
ISBN 13 : 9780080557717
Total Pages : 624 pages
Book Rating : 4.5/5 (577 download)

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Book Synopsis Competitive Electricity Markets by : Fereidoon P. Sioshansi

Download or read book Competitive Electricity Markets written by Fereidoon P. Sioshansi and published by Elsevier. This book was released on 2011-10-10 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: After 2 decades, policymakers and regulators agree that electricity market reform, liberalization and privatization remains partly art. Moreover, the international experience suggests that in nearly all cases, initial market reform leads to unintended consequences or introduces new risks, which must be addressed in subsequent “reform of the reforms. Competitive Electricity Markets describes the evolution of the market reform process including a number of challenging issues such as infrastructure investment, resource adequacy, capacity and demand participation, market power, distributed generation, renewable energy and global climate change. Sequel to Electricity Market Reform: An International Perspective in the same series published in 2006 Contributions from renowned scholars and practitioners on significant electricity market design and implementation issues Covers timely topics on the evolution of electricity market liberalization worldwide

Financial Modelling in Commodity Markets

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Publisher : CRC Press
ISBN 13 : 1351730959
Total Pages : 145 pages
Book Rating : 4.3/5 (517 download)

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Book Synopsis Financial Modelling in Commodity Markets by : Viviana Fanelli

Download or read book Financial Modelling in Commodity Markets written by Viviana Fanelli and published by CRC Press. This book was released on 2020-01-14 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt: Financial Modelling in Commodity Markets provides a basic and self-contained introduction to the ideas underpinning financial modelling of products in commodity markets. The book offers a concise and operational vision of the main models used to represent, assess and simulate real assets and financial positions related to the commodity markets. It discusses statistical and mathematical tools important for estimating, implementing and calibrating quantitative models used for pricing and trading commodity-linked products and for managing basic and complex portfolio risks. Key features: Provides a step-by-step guide to the construction of pricing models, and for the applications of such models for the analysis of real data Written for scholars from a wide range of scientific fields, including economics and finance, mathematics, engineering and statistics, as well as for practitioners Illustrates some important pricing models using real data sets that will be commonly used in financial markets

Mathematical Modeling, Simulation and Optimization for Power Engineering and Management

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Author :
Publisher : Springer Nature
ISBN 13 : 3030627322
Total Pages : 333 pages
Book Rating : 4.0/5 (36 download)

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Book Synopsis Mathematical Modeling, Simulation and Optimization for Power Engineering and Management by : Simone Göttlich

Download or read book Mathematical Modeling, Simulation and Optimization for Power Engineering and Management written by Simone Göttlich and published by Springer Nature. This book was released on 2021-02-02 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edited monograph offers a summary of future mathematical methods supporting the recent energy sector transformation. It collects current contributions on innovative methods and algorithms. Advances in mathematical techniques and scientific computing methods are presented centering around economic aspects, technical realization and large-scale networks. Over twenty authors focus on the mathematical modeling of such future systems with careful analysis of desired properties and arising scales. Numerical investigations include efficient methods for the simulation of possibly large-scale interconnected energy systems and modern techniques for optimization purposes to guarantee stable and reliable future operations. The target audience comprises research scientists, researchers in the R&D field, and practitioners. Since the book highlights possible future research directions, graduate students in the field of mathematical modeling or electrical engineering may also benefit strongly.

Uncertainty in the Electric Power Industry

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Publisher : Springer Science & Business Media
ISBN 13 : 0387230475
Total Pages : 325 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Uncertainty in the Electric Power Industry by : Christoph Weber

Download or read book Uncertainty in the Electric Power Industry written by Christoph Weber and published by Springer Science & Business Media. This book was released on 2005 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: Around the world, liberalization and privatization in the electricity industry have lead to increased competition among utilities. At the same time, utilities are now exposed more than ever to risk and uncertainties, which they cannot pass on to their customers through price increases as in a regulated environment. Especially electricity-generating companies have to face volatile wholesale prices, fuel price uncertainty, limited long-term hedging possibilities and huge, to a large extent, sunk investments. In this context, Uncertainty in the Electric Power Industry: Methods and Models for Decision Support aims at an integrative view on the decision problems that power companies have to tackle. It systematically examines the uncertainties power companies are facing and develops models to describe them – including an innovative approach combining fundamental and finance models for price modeling. The optimization of generation and trading portfolios under uncertainty is discussed with particular focus on CHP and is linked to risk management. Here the concept of integral earnings at risk is developed to provide a theoretically sound combination of value at risk and profit at risk approaches, adapted to real market structures and market liquidity. Also methods for supporting long-term investment decisions are presented: technology assessment based on experience curves and operation simulation for fuel cells and a real options approach with endogenous electricity prices.

Handbook of Multi-Commodity Markets and Products

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Author :
Publisher : John Wiley & Sons
ISBN 13 : 0470661836
Total Pages : 1064 pages
Book Rating : 4.4/5 (76 download)

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Book Synopsis Handbook of Multi-Commodity Markets and Products by : Gianluca Fusai

Download or read book Handbook of Multi-Commodity Markets and Products written by Gianluca Fusai and published by John Wiley & Sons. This book was released on 2015-02-19 with total page 1064 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Multi-Commodity Markets and ProductsOver recent decades, the marketplace has seen an increasing integration, not only among different types of commodity markets such as energy, agricultural, and metals, but also with financial markets. This trend raises important questions about how to identify and analyse opportunities in and manage risks of commodity products. The Handbook of Multi-Commodity Markets and Products offers traders, commodity brokers, and other professionals a practical and comprehensive manual that covers market structure and functioning, as well as the practice of trading across a wide range of commodity markets and products. Written in non-technical language, this important resource includes the information needed to begin to master the complexities of and to operate successfully in today’s challenging and fluctuating commodity marketplace. Designed as a practical practitioner-orientated resource, the book includes a detailed overview of key markets – oil, coal, electricity, emissions, weather, industrial metals, freight, agricultural and foreign exchange – and contains a set of tools for analysing, pricing and managing risk for the individual markets. Market features and the main functioning rules of the markets in question are pres- ented, along with the structure of basic financial products and standardised deals. A range of vital topics such as stochastic and econometric modelling, market structure analysis, contract engineering, as well as risk assessment and management are presented and discussed in detail with illustrative examples to commodity markets. The authors showcase how to structure and manage both simple and more complex multi-commodity deals. Addressing the issues of profit-making and risk management, the book reveals how to exploit pay-off profiles and trading strategies on a diversified set of commodity prices. In addition, the book explores how to price energy products and other commodities belonging to markets segmented across specific structural features. The Handbook of Multi-Commodity Markets and Products includes a wealth of proven methods and useful models that can be selected and developed in order to make appropriate estimations of the future evolution of prices and appropriate valuations of products. The authors additionally explore market risk issues and what measures of risk should be adopted for the purpose of accurately assessing exposure from multi-commodity portfolios. This vital resource offers the models, tools, strategies and general information commodity brokers and other professionals need to succeed in today’s highly competitive marketplace.