Quantitative Methods for Electricity Trading and Risk Management: Advanced Mathematical and Statistical Methods for Energy Finance (Finance and Capital Markets)

Quantitative Methods for Electricity Trading and Risk Management Advanced Mathematical and Statistical Methods for Energy Finance

Stefano Fiorenzani

QUANTITATIVE METHODS FOR ELECTRICITY TRADING AND RISK MANAGEMENT

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Quantitative Methods for Electricity Trading and Risk Management Advanced Mathematical and Statistical Methods for Energy Finance

STEFANO FIORENZANI

© Stefano Fiorenzani 2006 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2006 by PALGRAVE MACMILLAN Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world PALGRAVE MACMILLAN is the global academic imprint of the Palgrave Macmillan division of St. Martin’s Press, LLC and of Palgrave Macmillan Ltd. Macmillan® is a registered trademark in the United States, United Kingdom and other countries. Palgrave is a registered trademark in the European Union and other countries. ISBN-13: 978–1–4039–4357–6 ISBN-10: 1–4039–4357–5 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress. 10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10 09 08 07 06 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham and Eastbourne

Contents

List of Tables

viii

List of Figures

ix

Introduction

xi

Part I Distributional and Dynamic Features of Electricity Spot Prices 1 Liberalized Electricity Markets Organization 1.1 1.2 1.3

3

The liberalization process Spot electricity exchanges organization Electricity derivatives markets: organized exchanges and OTC markets

3 4 6

2 Electricity Price Driving Factors 2.1 2.2 2.3

Price determination in a liberalized context Electricity demand driving factors Electricity supply driving factors

3 Electricity Spot Price Dynamics and Statistical Features 3.1 3.2 3.3

Preliminary data definitions Detecting periodic components in electricity prices Statistical properties of electricity prices

Part II

8 12 14

19 19 22 30

Electricity Spot Price Stochastic Models

4 Electricity Modeling: General Features 4.1 4.2 4.3

8

Scope of a financial model Econometric models versus purely probabilistic models Characteristics of an ideal model and state of the art

39 39 40 41 v

vi

CONTENTS

5 Econometric Modeling of Electricity Prices 5.1 5.2 5.3 5.4

Traditional dynamic regression models Transfer function models Capturing volatility effects: GARCH models Capturing long-memory effects in electricity price level and volatility: fractionally integrated models

6 Probabilistic Modeling of Electricity Prices 6.1 Traditional stochastic models 6.2 More advanced and realistic models Appendix Semimartingales in financial modeling

43 44 46 48

49

51 52 57 63

Part III Electricity Derivatives: Main Typologies and Evaluation Problems 7 Electricity Derivatives: Main Typologies 7.1 7.2 7.3

Exchange-traded derivatives and OTC derivatives Exotic options Options typically embedded in electricity physical contracts

8 Electricity Derivatives: Valuation Problems 8.1 8.2

Derivative pricing: the traditional approach The spot-forward price relationship in traditional and electricity markets 8.3 Non-storability and market incompleteness 8.4 Pricing and hedging in incomplete markets: basic principles 8.5 Calibrating the pricing measure Appendix An equilibrium principle for pricing electricity assets in incomplete markets

9 Electricity Derivatives: Numerical Methods for Derivatives Pricing 9.1 Monte Carlo simulations 9.2 The lattice approach Appendix A Pricing electricity swaptions by means of Monte Carlo simulations Appendix B Pricing swing options by means of trinomial tree forests

71 71 75 79

83 83 85 88 89 92 93

95 95 98 104 106

CONTENTS

vii

Part IV Real Asset Modeling and Real Options: Theoretical Framework and Numerical Methods 10

Financial Optimization of Power Generation Activity 10.1 10.2 10.3

Optimization problems and the real option approach Generation asset modeling: the spark spread method Generation asset modeling: the stochastic dynamic optimization approach Appendix Discrete time stochastic dynamic programing

11

111 111 116 117 123

Framing and Solving the Optimization Problem

127

11.1 Optimization problems in a deterministic environment 11.2 Naïve application of Monte Carlo methods 11.3 Solving Bellman’s problem 11.4 Alternative solution methods: ordinal optimization Appendix Generation asset modeling: numerical results

127 129 130 134 136

Part V Electricity Risk Management: Risk Control Principles and Risk Measurement Techniques 12

Risk Definition and Mapping 12.1 12.2 12.3

13

Market risk definition and basic principles Different risk factors and their mapping onto the company value-creation chain Risk and opportunity (enterprise risk attitude)

Risk Measurement Methods 13.1 Risk measures for financial trading portfolios 13.2 Risk measures for physical trading portfolios Appendix On the coherence of risk measures

14

Risk-Adjusted Planning in the Electricity Industry 14.1 14.2

Production value and risk–return measures Survival performance level and extreme market events 14.3 A practical application

145 145 146 150

152 152 161 163

165 166 170 172

Bibliography

176

Index

179

List of Tables

3.1 3.2 3.3 5.1 5.2 9A.1 9A.2 11A.1 11A.2 11A.3 11A.4 14.1 14.2 14.3

viii

Powernext de-structured time series correlogram analysis Powernext de-structured time series unit root test Powernext squared de-structured price time series correlogram analysis Dynamic regression model estimation results Transfer function model estimation results Simulation experiment results Sensitivity experiment result Summary of test case characteristics Production value and risk measure versus spark spread Power production value versus volatility Worst case value versus volatility Power plant physical characteristics used in the example Price model characteristics used in the example Empirical experiments results

34 34 36 45 47 105 106 136 137 140 140 168 168 173

List of Figures

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 3.1

3.2 3.3 3.4 3.5 3.6 3.7

Alberta power exchange merit order, 12 November 1999, 13.06 Demand/supply schematic representation Demand shock in normal load regime Demand shock in high load regime Supply local shock Italian national load (April 04–April 05) Typical daily max load and daily max temperature Mean hourly consumption of two European countries (Italy and France) Mean monthly consumption of two European countries (Italy and France) Powernext and EEX base load daily prices (546 observations 2003–04) Daily base-load prices from major European and American electricity exchanges (546 observations from 11-01-02 to 16-03-04) IPEX (Italian Power Exchange) hourly electricity spot price matrix (1 Jan. 2005–16 Mar. 2005) EEX (German market) historical hourly prices, Oct. 2000–Oct. 2003 EEX periodogram calculated with hourly prices, Oct. 2000–Oct. 2003 EEX seasonal component y(t) EEX intra-week shaping component w(t) New South Wales electricity spot prices (Jan. 1998– Aug. 1998): original series denoised by means of wavelets (two examples)

9 10 11 11 12 13 14 15 16 18

20 22 23 26 27 28

30

ix

x

LIST OF FIGURES

3.8

Powernext hourly spot prices distributional analysis (8,784 observations from 1 Jan. 2004–31 Dec. 2004) 3.9 Powernext hourly spot prices structural and unpredictable components 3.10 Powernext hourly spot de-structured prices distributional analysis 5.1 Dynamic regression model graphical representation 5.2 Powernext daily price (base load) 5.3 Transfer function model graphical representation 7.1 Tolling contract scheme 9.1 Lattice approach schematic representation 9.2 State price aggregation scheme 9A.1 Barlow process simulated price paths 9B.1 Trinomial tree forest scheme 10.1 Power generation/marketing decisional process 10.2 Diagrammatic representation of the dynamic equation (10.14) 11.1 Trinomial lattice scheme 11A.1 Production value versus spark spread (case 1) 11A.2 Production value versus spark spread (case 2) 11A.3 Production value versus spark spread (case 3) 11A.4 Production value versus spark spread (relative differences from case 1) 11A.5 Production value versus volatility 11A.6 Worst case versus volatility 12.1 Company risk map 13.1 Thick loss function vs normal 13.2 Non-monotonic loss function vs normal 13.3 Discontinuous loss function vs normal 13.4 Graphical representation of VaR, ES and CVaR 14.1 Asset optimization scheme 14.2 Power generation economic performance distribution 14.3 Power generation performance risk measure 14.4 Portfolio’s economic performance distribution 14.5 A portfolio’s economic performance survival risk measure 14.6 Distributional results from first experiment 14.7 Distributional results from second experiment

32 32 33 45 47 48 81 99 103 104 107 115 122 133 138 138 139 139 140 141 147 158 159 159 161 166 168 169 171 171 174 175

Introduction

The electricity sector is traditionally a business sector with a strong industrial connotation. Due to regimes of strong national or regional protection under which the power sector in many countries has been born and developed, not much effort has been traditionally put into developing methods to increase the level of competitiveness of operators. Among such ‘competitiveness weapons’ we may also consider all those merely financial issues related to the management and control of economic risks. The progressive liberalization process which has characterized many countries in recent years, and is developing in many others now, has led to the fragmentation of big national power operators and the entrance of new players into the different sectors of the electricity market. The increase in the number of players has obviously been accompanied by an increase in the level of competitiveness in all branches of the power sector: generation, marketing and distribution. Competition has also stimulated many players to look at and experiment with new business developments, and attention towards financial trading and risk-management issues has registered a significant increase such that today they play a central role in business practice. The same persons who, until few years ago, used to deal only with problems related to generation efficiency, transmission loads or power tariffs, are today asked to manage complex problems typically belonging to the financial area such as: financial trading, risk management, derivative products and portfolio optimization. The need to manage the problems of a market in rapid and strong evolution brings with it the need to integrate standard business and analysis tools with new methods more suitable to tackle this new set of problems. Hence we have observed massive efforts towards applying traditional methods of financial analysis to the new exigencies of the electricity market. Quantitative finance methods are now used not only for xi

xii

INTRODUCTION

financial risk assessment or for derivatives trading; real asset valuation with real options methodology and commercial deal structuring are also potential fields of application. A first problem is that of inducting non-experts to the application of mathematical and statistical tools for energy finance problems. This has been the focus of the biggest part of the specific scientific literature in the last few years. Initially we did not observe the development of specific models or methodologies for the power market, but rather the straightforward transposition to the electricity sector of instruments traditionally used in more mature financial sectors such as fixed-income or foreign-exchange markets. Often, this straightforward transposition has not given balanced attention to the potential dangers which may arise from a limited consideration of the physical peculiarities which characterize electricity markets. This book has been conceived and written as a natural continuation of the above approach. Following the traditional scientific literature of the sector, I also introduce some elements of discontinuity with the past tradition in the hope that they may help to inaugurate a new and independent field of research in the area of mathematical finance. I shall review the existing literature with a critical eye, emphasizing the limits and dangers of a straightforward application of traditional mathematical models in the peculiar environment represented by power markets. Advanced modeling approaches and mathematical tools are introduced to enhance the cultural and technical background of the reader. The necessary increase in complexity of the whole analysis is the price we have to pay in order to develop realistic and ad hoc analysis tools for the electricity world. Economics concepts such as that of market incompleteness or mathematical methods such as stochastic dynamic programing are introduced in this spirit. In addition, I want to give space to some extremely interesting and innovative research initiatives that have characterized the specialist literature in very recent years. Mainly I am referring to mathematical or statistical methods developed by researchers working in the field, with the precise scope of solving typical valuation or analysis problems of the electricity business. The reason for this choice lies in the need to stimulate the academic world also, too often concentrated on theoretical issues, towards real business problems. The book is more inclined towards practical issues than theoretical ones, although theoretical issues are considered if and when they are functional to business practice. The book is essentially dedicated to two groups of people:
electricity market operators willing to enlarge their cultural and technical backgrounds concerning quantitative methods; and
quantitative analysts actively working in traditional financial sectors curious to approach the fantastic world of power markets.

INTRODUCTION

xiii

Nevertheless, I strongly recommend the use of the book to all postgraduate students who are considering research in the energy finance sector. The contents of the book are essentially the fruits of my personal professional and academic experience, but I do not want to forget the contributions in terms of ideas and professional experiences that many people I have worked with or had the pleasure to meet in recent years have been able to transfer to me. STEFANO FIORENZANI

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PART I

Distributional and Dynamic Features of Electricity Spot Prices

1

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CHAPTER 1

Liberalized Electricity Markets Organization

1.1 THE LIBERALIZATION PROCESS Traditionally, economics sectors of public utility such as power and gas have been developed and run under strict monopolistic regimes. We may find examples of state monopoly, especially in south-east Europe, regional or municipal monopoly in central and north Europe or somewhere in the United States, and, finally, natural monopoly (privately owned). There have been two main reasons which justify the success of monopolistic regimes. The first is the Keynesian economics beliefs which influenced a lot of the economics policies of many industrialized countries just after the Second War World. The Keynesian economics vision suggests a significant intervention of the state in economic affairs, and the period we are talking about was characterized by the greatest development of utility sectors all over the world. The second important reason is related to the huge initial investment necessary to start up any industrial initiatives in the utility sector itself. Only big industrial subjects or national governments have the resources necessary to face such investments. From the beginning of the 1990s a growing number of countries worldwide, including most of the developed ones such as the USA and the UK, have chosen a substantial restructuring of their utility sectors, starting from their electric power sectors, massively oriented towards liberalization of the whole sector. With different speeds and approaches, many countries have progressively opened their power sectors to competition. In general, the liberalization process has involved all the different areas of the electricity industrial sector, such as generation, retailing and distribution activities. However, these different areas were often not opened at the same time and not always 3

4

DISTRIBUTIONAL AND DYNAMIC FEATURES

in the same order. In many countries liberalization started from generation, as is logically plausible, but we may find examples where retailing activities were opened to competition before generation, or eventually also in the presence of monopolistic regimes for generation. The general scope of a liberalization process in any economy sector is that of supporting a more efficient allocation of economic resources, especially when those resources are scarce. The free interaction of demand and supply which characterizes competitive markets should produce an efficient allocation of scarce resources, whatever is the nature of those resources: goods, capital or labour. Efficient allocation means that resources are allocated to those subjects which are able to make the best economic use of them. This notion of economic efficiency should reflect a situation of a higher degree of satisfaction for all market participants, especially final consumers. However, we all know that perfect competition is a theoretical situation and we often observe that liberalization processes have produced transitions from state-owned monopolies to natural monopolies or oligopolies. Hence, the improvement of final consumers’ utility is not guaranteed. The physical peculiarities of a commodity such as electric power greatly influence the processes of market liberalization. In fact, electricity is a commodity hardly storable or transportable. Electricity delivered at different times and at different places is considered by users as a non-fungible commodity. In this situation, in order to make the liberalization process effective, it is necessary to guarantee all market participants a non-discriminatory access to the grid. To this extent, we usually have a situation where the transmission network is owned and managed by an independent subject, which acts as a transmission system operator under the regulatory control and authority of the central government. Deregulating processes are usually accompanied by the introduction of competitive wholesale markets and often by the introduction of electricity derivatives contracts. These markets are often organized in real exchanges with rules and operational processes similar to those traditionally present in financial markets. Obviously, the internal organization of electricity spot and derivatives exchanges cannot ignore the physical characteristics of electric power. Hence, peculiar rules have been studied and adopted in order to make electricity transactions feasible.

1.2 SPOT ELECTRICITY EXCHANGES ORGANIZATION This book is essentially dedicated to the analysis of quantitative methodologies applied to electricity trading and risk-management issues. Hence, a detailed discussion of spot and derivatives electricity markets organization is beyond our scope. However, the intuition of the working mechanisms of these markets is essential in order to fully understand electricity price

LIBERALIZED ELECTRICITY MARKETS ORGANIZATION

5

behavior and the impact that market organizational rules have on it. For this reason, in this section we briefly introduce the basic concepts of the organization and functioning of electricity physical spot markets. Electricity spot markets are day-ahead markets, in the sense that physical quantities expressed in MWh, and prices negotiated during a market session for each of the 24 hours of the day, will physically deliver on the specific hour of the following day.1 The first example of an electricity spot exchange was that of Nordpool (Nordic Power Exchange, Norway) in 1993. Nowadays, in almost all the EU countries, in the USA and in Australia many exchanges are actively operating short-term physical transactions. Ideally, the main goal of organized electricity exchanges should be that of facilitating electricity short-term physical transactions by improving market information, competition and liquidity. Power exchanges also represent neutral marketplaces, where deals can take place reducing transaction costs and counterparty risk. The price reference may also represent an important benchmark for over-the-counter (OTC) transactions (derivative contracts or bilateral physical ones). Competitive electric power markets are usually organized around one or more auctions, but the way these auctions take place may be different from market to market. However, in every situation, suppliers’ and demanders’ bids are combined together by the market-maker to obtain the production rule that minimizes the costs of aggregated demand, given the physical constraints imposed by the transmission grid. Different market models diverge substantially because of issues related to the type of auction used and the participation rule adopted. Regarding the participation rule, we can have mandatory or non-mandatory auctions, which means that all the market participants may be forced, or not, to submit their bids to the market participating in the day-head auction. Bidding sides represents a first criterion for auction-type classification. If only power generators are asked to submit their bids, the market is called one-sided,2 while if both buyers and sellers participate the market is called double-sided. A second criterion for auction-type classification is represented by the price-formation rule. We can have a uniform pricing rule, where all participants get the same price independently on their price bid, or a pay-as-bid pricing rule, where the bidder pays or receives the price of his offer (if the offer is accepted by the market). Usually, the uniform pricing rule is associated with the classical System Marginal Price mechanism, which means that the uniform price that market participants pay or receive is the price of the last accepted bid in order of economic merit. 1

Typically, different hours of the day are grouped together into homogeneous groups in order to form standardized products such as peak and off-peak hours or blocks. 2 One-sided auctions are present in early stage phases of market evolution, and their scope is essentially that of creating a market model which can be simply understood by producers.

6

DISTRIBUTIONAL AND DYNAMIC FEATURES

As we have said, economic efficiency is not the only scope of electricity markets. Physical sustainability and grid-balancing are essential issues that have to be ensured by a certain spot market model. For this reason the simple day-ahead auction is not sufficient for a satisfactory allocation of installed and available power generation capacity. Despatching services auctions, transmission rights auctions or even intra-day electricity auctions (also known as the hour-ahead market or adjustment market) are essential elements of an efficient electricity spot exchange. The detailed description of all possible market models is beyond our scope, but the interested reader can refer to Madlener and Kaufmann (2002) for an extensive discussion on this topic and a detailed description of most important European spot power exchanges.

1.3 ELECTRICITY DERIVATIVES MARKETS: ORGANIZED EXCHANGES AND OTC MARKETS Even if the electricity sector liberalization process usually leads to a reduction in the average electricity price, the same process is accompanied by a necessary and sometimes pronounced randomness about this average value. In regulated regimes, the monopoly or the state (when they do not coincide) establishes the tariff, which may be fixed for a predetermined time period or may barely fluctuate according to some index reflecting inflation, average production costs or other drivers. In liberalized regimes, the electricity spot price determined in organized exchanges varies a lot according to the interaction of power supply and demand. In the following chapters we will study this erratic behavior, here suffice it to say that the uncertainties which characterize future spot price realizations may be really huge. As in any other market (financial or not) price uncertainty means economic risk, and in our field electricity price volatility means economic risk for power generators, retailers and final consumers. The need for electricity derivatives contracts and electricity derivatives markets comes from the exigency of hedging-out this risk. Electricity derivatives represent claims on the future delivery of electricity, either physical delivery or purely financial settlements made with reference to the physical electricity price. Such contracts are usually traded up to some days before the payoff fixing, but theoretically they can be traded up to some seconds before it. Typical derivative instruments are forwards, futures and options. However, as we will see, derivative instrument structuring is an extremely dynamic discipline. Hence, the range of derivative products changes rapidly according to market exigencies, especially in over-the-counter (OTC) markets. To date, exchange-based electricity derivatives markets have been established both by spot power exchanges that usually operate in the

LIBERALIZED ELECTRICITY MARKETS ORGANIZATION

7

physical market, such as Nordpool, or by existing derivative exchanges which already operated in traditional commodity derivative contracts, such as Nymex (New York Mercantile Exchange) or IPE (International Petroleum Exchange, London). Electricity derivatives can be either physically or financially settled. This feature, which is not usually shared by traditional commodity derivative products, makes electricity derivatives particularly interesting for financial market participants. Hence, the potential market liquidity is significantly enhanced. Market liquidity is also sensibly increased by instrument standardization. Liquidity is normally maximized when the number of contract clauses is limited through standardization. Regarding the clearing mechanism and the daily settlement procedure, the electricity derivatives markets are not that different from traditional derivatives markets. Hence we usually find a clearing house which works as counterparty for every single trade, and a margin call system3 that prevents credit risk being embedded by any transaction. Once we have an overview of the reasons for electricity market liberalization processes and the structure of electricity spot and derivative markets, we shall be in a position to examine the quantitative methods necessary for optimizing trading or risk-management activities in this field.

3 In some electricity derivatives markets the margin account system is structured in such a way that the additional risk related to the potential impact of unexpected physical outages or congestions on the electricity price is considered.

CHAPTER 2

Electricity Price Driving Factors

2.1 PRICE DETERMINATION IN A LIBERALIZED CONTEXT Classical economic theory teaches us that in a free and competitive market the price of any traded good or service is completely determined by the interaction between aggregated demand and aggregated supply. Hence, at least in theory, the price of electricity (the commodity) and ancillary service related to it is determined, in a liberalized market, by this kind of dynamic equilibrium. Because of that, before considering electricity price behavior it is fundamental to understand and analyze the shape and dynamics of aggregated demand and supply. In the previous chapter we briefly discussed the main features of electricity spot market models. We have seen that in ‘pay-as-bid’ markets the concept of System Marginal Price has a fundamental importance. The System Marginal Price is exactly determined, hour by hour, by the interception of the system merit order curve and the system aggregated demand. Figure 2.1 shows the characteristic shape of merit order and aggregated demand curves for the Alberta electricity day-ahead spot exchange. The merit order curve is a map of the ability of the productive system to offer different quantities of electricity at different prices, in a given time. Consequently, it provides us with information about the marginal cost of production of the power generation units operating in the system and about the bidding strategy of their managers. In summary we can say that merit order curves can be interpreted as short-term aggregated supply functions. Very efficient, but not extremely flexible plants contribute to the left bottom side of the curve, while less efficient or very flexible generation 8

ELECTRICITY PRICE DRIVING FACTORS

Bid and offer prices ($/MWh)

Fri Nov 12 13:06:25 1999 1000

9

Supply/demand plot

Merit order curve

100 $39.6

Demand curve 10

1 4,000

4,500

5,000

5,500

6,000

6,500

7,000

7,500

MW

Figure 2.1 Alberta power exchange merit order, 12 November 1999, 13.06

units act in the upper right corner. The supply function defined is always upward-sloping, but its shape will obviously depend on the inner physical characteristics of the productive system. In particular, if the productive system is composed of a heterogeneous set of small units characterized by different technologies (hence, different marginal costs and different efficiency), the shape of the merit order curve should appear smoother. On the other hand, if the system is composed of big and technologically homogeneous generation units the curve will look more discontinuous. The shape of the merit order curve has a drastic impact on the price behavior, as we will see later. Aggregated electricity demand is typically price-inelastic, at least in the short run. This characteristic of the aggregated demand function can be graphically inferred from its steepness. In fact, the curve is almost vertical. In addition, in the case of aggregated demand the peculiarities of the system, in terms of the electricity consumption structure, significantly affect the slope and the shape of the demand curve. In fact, electricity demand can typically be divided into industrial demand and domestic demand, which typically have different behaviors and mostly

10

DISTRIBUTIONAL AND DYNAMIC FEATURES

Price

Demand

Supply

Load

Figure 2.2 Demand/supply schematic representation

display different price elasticities. Hence, the slope and shape of the aggregated demand curve of a certain country or geographic area is influenced by the proportions within it of aggregated demand of industrial and domestic consumption. Once we have seen the static shape of price determinants, aggregated demand and aggregated supply, and before analysing their dynamic behavior, we are in a position to study their impact on electricity prices by means of the classic tools of microeconomic analysis such as comparative static analysis.1 For simplicity let us represent the demand curve as an almost vertical line and the supply curve as a hyperbolic curve with a vertical asymptote positioned in correspondence to the maximum available capacity of this hypothetical system. In Figure 2.2 it is easy to imagine that if the demand fluctuates around a ‘normal’ load level, this will cause price fluctuations around its normal regime and the ratio of the demand fluctuation amplitude and the price fluctuation amplitude depends upon the slope of the supply curve in the normal load area. However, if the demand fluctuates in an area close to the maximum available capacity, also a small demand shock can potentially determine significant price spikes. Figures 2.3 and 2.4 show us the situations described above. Supply curve movements can also cause similar price effects. The supply curve could have a parallel upward or downward movement caused by 1

The examination of a change in outcome in response to a change in underlying economic parameters is typically known as comparative static analysis.

11

Price

ELECTRICITY PRICE DRIVING FACTORS

P⬘ P*

Load

Figure 2.3 Demand shock in normal load regime

P⬘

Price

P*

Load

Figure 2.4 Demand shock in high load regime

a diffuse increase in the production cost (fuel costs, taxes or other costs), but more frequently we observe breaks in the supply curve due to plant outages or maintenance operations, during which the plant is unable to offer electricity for a period (Figure 2.5).

DISTRIBUTIONAL AND DYNAMIC FEATURES

Price

12

P⬘ P*

Load

Figure 2.5 Supply local shock

In addition to physical reasons that may shock the supply function, the bidding strategies of unit managers can also affect the short-term shape of the aggregate supply. Moreover, if the production system of a certain country or geographic area is intrinsically not sufficient to match the consumption exigencies of the same area (the system is intrinsically in a situation of undercapacity) then other factors such as congestion of transmission lines used to import electricity can play an important role in the determination of aggregate electricity supply. However, since our scope is that of understanding electricity price dynamics, the analysis of the static relation between demand and supply is not sufficient even if it is important. We need to analyze the demand supply equilibrium in a dynamic perspective focusing on the economic and physical factors which drive their evolution and consequently affect the electricity price evolution.

2.2 ELECTRICITY DEMAND DRIVING FACTORS We said before that electricity aggregated demand is mainly composed of two principal components: industrial demand and domestic demand. These two components may display very different dynamics and consequently their proportional contributions to the global demand may determine

ELECTRICITY PRICE DRIVING FACTORS

13

52,000 LOAD 48,000 44,000

Central August and Christmas holidays

MW

40,000 36,000 32,000 28,000 24,000 20,000 16,000 1,000 2,000 3,000 4,000 5,000 6,000 7,000 Hours

Figure 2.6 Italian national load (April 04–April 05)

different behaviors. Electricity is an important productive input for many industrial sectors, and hence industrial electricity consumption is driven by different economic factors typical of different industrial sectors. Moreover, industrial electricity demand may reflect the different cycles and seasonalities which may characterize different industrial sectors. If the industrial structure of a certain country or geographic area is sufficiently diversified then all the idiosyncratic and seasonal components of each single sector tend to compensate themselves within the whole consumption basket. Obviously, not all the components may be diversified away because some factor may jointly affect all the industrial sectors. This is, for example, the case in common periods of inactivity of industrial processes such as night-time inactivity. In many European countries it is also quite common to interrupt firms’ activity for the traditional vacation periods (Christmas, Easter or central summer period), and these common phenomena are sometimes evident just by looking at the load path during the year (Figure 2.6). The main characteristic of industrial electricity demand, in general, is its relative insensitivity to the electricity price in the short term. The explanation is quite evident and related to the fact that industrial programs and production schedules, especially in heavy industry, are not extremely flexible and cannot be changed instantaneously to react to price changes. Obviously, this consideration sensibly affects electricity procurement contracts and hedging strategies of big industrial players.

DISTRIBUTIONAL AND DYNAMIC FEATURES

Load

14

Temperature

Figure 2.7 Typical daily max load and daily max temperature

Domestic demand may be a significant and sometimes predominant component of aggregate electricity demand. Typical domestic consumption is concentrated on specific hours of the day when people use electricity for heating, freezing or cooking. This generates the typical intraday load shape. Domestic electricity consumption related to heating or air conditioning is essentially related to weather conditions. In particular, many empirical studies demonstrate that temperature and domestic consumption are extremely related, especially in developed countries. In particular, once we have adjusted load data in order to consider the non-temperature-related periodical components, we often come out with a parabolic relation between maximum daily load and maximum daily temperature, which may be approximately represented as in Figure 2.7. The two main components of electricity aggregated demand, industrial and domestic demand, merge together in different proportions, in different countries, and contribute to the typical shape of aggregated demand, with its typical hourly and monthly behavior. Figures 2.8 and 2.9 represent micro and macro electricity load dynamic behavior.

2.3 ELECTRICITY SUPPLY DRIVING FACTORS Given the fact that electricity is not, at least now, a storable commodity, when we analyze electricity aggregated supply dynamics drivers we mainly analyze the determinants of electricity capacity available, more than that physically or theoretically installed. As we mentioned before, the static shape of the electricity supply, the merit order curve, depends upon the

ELECTRICITY PRICE DRIVING FACTORS

15

Italy 43,000

MW

38,000

33,000

28,000

23,000

18,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hours France

59,000 54,000

MW

49,000 44,000 39,000 34,000 29,000 24,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hours

Figure 2.8 Mean hourly consumption of two European countries (Italy and France)

characteristics of the plants which comprise the completely productive system. Thermoelectric power plants have obviously different characteristics in terms of efficiency and flexibilities in comparison to hydro power plants or nuclear plants. Hence, the structure of the production system of a country

16

DISTRIBUTIONAL AND DYNAMIC FEATURES

Average load (MW)

40,000

Average load (MW)

Italy

80,000

30,000 20,000 10,000

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4

0

Figure 2.9 Mean monthly consumption of two European countries (Italy and France) strongly affects the dynamics of its electricity supply capability. Also, we distinguish between the long-term behavior of electricity supply from short-term behavior, because they have different dynamic determinants. In the end, electricity supply dynamics is obviously influenced by the installed capacity, which is determined mainly by the number and the typology of operating power plants. New entrant plants, characterized presumably by more modern technology, have the effect of decreasing the price of offered electricity for a given quantity. This effect may be described as a rightwards parallel (or almost parallel) shift of the merit curve. Nevertheless, the entrance of new productive units is a very long-term and predictable event. Within the year, which for our purposes is still a long-term horizon, given the frequency of operating decisions or trading transactions that can be done in such a period of time, we can consider as fixed the productive system installed capacity. Within the same time horizon, fuel costs have a strong impact on electricity supply. Fuel costs influence the marginal opportunity cost of production, and presumably impact the electricity price according to the short-term bidding strategy of electricity producers. The study of the relation between electricity price dynamics and fuel-cost dynamics is not a simple issue since the loading of fuel costs in final electricity prices also

ELECTRICITY PRICE DRIVING FACTORS

17

depends on fuel procurement and hedging strategies established by producers. Of course, where a liquid market for the fuel exists, the opportunity costs and the real production costs tend to converge with the consequence of a more direct impact of fuel costs on final prices. Unfortunately, this is a realistic example only for those countries where a liquid natural-gas spot market exists, such as the UK or the USA. Another important long-term determinant of electricity supply is the dependence structure of different fuel prices. In fact, even in welltechnologically-diversified productive systems, fuel costs have a deeper impact on final prices if different fuel prices tend to move in the same direction (highly dependent fuel prices). The degree of diversification of generation systems in terms of plant technology, efficiency and fuel burned (at least for thermo power plants which often represent the most diffuse technology) could mitigate the impact of fuel costs on final prices. In a shorter time horizon, other factors play an important role. For example, transmission constraints (international and regional interconnectors and merchant lines) play a very important role. This is particularly true in countries or areas that are intrinsically undersupplied by installed capacity, because there, power imports are determinant for the full satisfaction of internal aggregated demand. The impact that transmission constraints have on prices (regional or national) can be fully understood by analyzing price spreads between neighbor countries or regions from the same country. If two neighbor areas are well interconnected, electricity trading is more fluid between them and prices tend to be similar without arbitrage arguments. In the opposite situation, prices can potentially move in a very different way, both in the long, medium and short term. Figure 2.10 shows prices from two well interconnected European countries such as France and Germany, over the same period. Over very short time horizons plant outages, unpredicted maintenance operations and, mainly, plant-manager strategic behavior determine the shape of the supply function, since they determine the proportion of installed capacity (producible or importable) which is effectively available in that specific market moment. In addition to the list of supply and demand dynamics determinants we have described so far there are some economic drivers which jointly influence electricity demand and supply in the medium and long term. We mainly refer to interest-rate dynamics and country production and income trends. Our scope here is not that of describing the impact of macroeconomic variables on electricity demand and supply, since that would require an in-depth economic discussion. However, it is quite simple to understand that the impact that macroeconomic variables such as interest rates and production level have on electricity market behavior is not dissimilar from their impact on other, more traditional, financial markets.

DISTRIBUTIONAL AND DYNAMIC FEATURES

PN 100

€/MWh

80 60 40 20 0 100

200 300 400 Daily observations

500

EEX 200 160 €/MWh

18

120 80 40 0 100

200 300 400 Daily observations

500

Figure 2.10 Powernext and EEX base load daily prices (546 observations 2003–04)

CHAPTER 3

Electricity Spot Price Dynamics and Statistical Features

In the previous chapter we saw that electricity price dynamics are influenced by many economic and physical factors most of which are strongly related to the specific system and market structure of the country or geographic area. The natural conclusion is that the dynamics of electricity prices for a certain area cannot be fully generalized. However, there are common features which characterize the time evolution of electricity prices over heterogeneous areas, which, mostly, can be analyzed and studied with common tools. The determination of the principal common features of electricity price dynamics and the specification of the most advanced methodologies for their detection is the main focus of this chapter. The topics discussed will provide the essential basis for the price modeling methodologies to be discussed in the next part of the book.

3.1 PRELIMINARY DATA DEFINITIONS The first step in any analysis is the definition of the object of analysis. In our case, the object of the analysis is obviously the electricity spot price,1 and we 1 All the methodologies proposed here may also be applied to analyze the forward price behavior. Our approach, here and in the rest of the book, will be that of deriving forward price properties from spot price ones.

19

20

DISTRIBUTIONAL AND DYNAMIC FEATURES

EEX – German market

ECAR – USA market 140

200

$/MWh

€/MWh

120 150 100

100 80 60 40

50

20 0

0 600

700

800

900 1000

600 700

Daily observations

POWERNEXT – French market 100 80

800 €/MWh

NOK/MWh

900 1000

Daily observations

NORDPOOL – North Europe market 1000

600 400

60 40 20

200

0

0 600 700

800

900 1000

600

700

800

900 1000

Daily observations

Daily observations OMEL – Spanish market

UKPX – UK market 100 POUNDS/MWh

10 € CENTS/KWh

800

8 6 4 2

80 60 40 20 0

0 600

700

800 900 1000

Daily observations

600 700

800 900 1000

Daily observations

Figure 3.1 Daily base-load prices from major European and American electricity exchanges (546 observations from 11-01-02 to 16-03-04) are interested in studying its time evolution in a general context. However, it is not that obvious how to prepare the empirical dataset necessary to perform the analysis. In fact, as we saw in the first chapter, electricity spot markets have a peculiar organization since they are day-ahead markets.2 This implies 2

Real time spot markets represent a marginal part of electricity spot markets.

SPOT PRICE DYNAMICS AND STATISTICAL FEATURES

21

that for every single daily auction a string of 24 electricity hourly prices is determined. Hence, when we collect historical data from electricity spot exchanges we have alternative ways of studying historical prices:
as multivariate vectors of hourly prices with daily granularity (frequency of observations); or
as univariate vectors of hourly prices with hourly granularity (frequency of observations). When we analyze financial data, especially financial asset prices, we effectively try to extract from the available dataset all the relevant information useful for a full understanding of the phenomenon we are studying. If the market is efficient, according to the classical Fama (1970) definition, prices contain all the relevant information. The natural approach to studying electricity spot-price time series should be the first method suggested, that is as a multivariate vector of hourly prices, since hourly prices within the same trading day are simultaneously determined and should reflect the same set of relevant information. However, the study of the time evolution of a multidimensional phenomenon is not a simple task because it is not only important to analyze the time evolution of each single component of the price vector, but also its dependence structure. In traditional quantitative finance applications we face similar problems when we try to study the time evolution of the interest rate forward curve, with the difference that the number of components here is much higher (24 hourly components most of which exhibit independent behavior) and the dynamic behavior is much more complex as we can see from Figure 3.2. Here the Italian power price is represented in such a way to emphasize its multidimensional evolution. The complexity of power price time evolution is evident in both the daily and intraday behavior. Hence, the matrix of empirical prices is usually transformed and studied as a vector of hourly prices with an hourly frequency of observations, in order to reduce the complexity of data analysis. This does not mean that the alternative, and most correct, approach has to be discarded a priori. It may be used for specific purposes. We said before that it is possible to identify common features in electricity price dynamics. In particular, in a very general setting we can write the electricity price dynamics as the sum of different dynamical components, as follows: Et = α(t) + µ(t) + W(t) + J(t)
Deterministic behavior

 Random behavior

(3.1)

22

DISTRIBUTIONAL AND DYNAMIC FEATURES

200

100

€/MWh

150

50 Day

s

s of Hour y a e th d

0

Figure 3.2 IPEX (Italian Power Exchange) hourly electricity spot price matrix (1 Jan. 2005–16 Mar. 2005) where: α(t) is a linear drift component; µ(t) is a periodic component; W(t) is a probabilistic noise component; and J(t) is a pure spike component. Each of the generic components listed above is representative of a typical electricity price behavior: a linear tendency, a periodic component for the micro and macro price frequency, a component for the description of the price variability (unpredictable behavior) in the ‘normal regime’, and a component for the description of price unpredictable behavior in extreme situations (spiky behavior). Quite often in traditional financial data analysis, a lot of emphasis has been concentrated on the study of random components of financial asset prices. This is because the deterministic behavior (especially the periodic behavior) of financial asset prices is usually not significant. On the contrary, in electricity markets, spot price dynamics are essentially characterized by a strong and complex periodic component, which has to be analyzed with the appropriate instruments and filtered out from the dataset before concentrating attention on the study of its statistical properties.

3.2 DETECTING PERIODIC COMPONENTS IN ELECTRICITY PRICES Let us concentrate, first, on the detection of periodic components present in electricity spot prices. Typically, it is assumed that an intraday periodic component and an annual or seasonal periodic component characterize the electricity spot price. However, this assumption is quite simplistic and not supported by empirical evidence. In Figure 3.3 we put in evidence that hourly price evolution may be deeply different year to year. These differences may be imputed to structural

SPOT PRICE DYNAMICS AND STATISTICAL FEATURES

Euro

400

2000–01

200 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

400 Euro

23

0.9

1

2001–02

200 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

200 Euro

2002–03 100 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Years

Figure 3.3 EEX (German market) historical hourly prices, Oct. 2000–Oct. 2003 or contingent factors. It is necessary to develop analysis tools which enable us clearly to separate structure from randomness. In practice, electricity price dynamics are characterized by a very complex periodic structure made up of many different overlapping oscillators with different characteristic periods, which quite often cannot be detected simply by observing price time series. Of course, daily period and yearly periods are the predominant ones, but not the only relevant ones. The most appropriate mathematical tool for a correct detection of difference frequency components of electricity price periodic behavior is Fourier analysis.

3.2.1 Fourier analysis and fast Fourier transform (FFT) Fourier analysis is a traditional tool of signal analysis and has many similarities with spectral analysis in economic data analysis applications. The main scope of Fourier analysis is that of expressing a certain signal,3 originally expressed in two-dimensional space (time, price), into its frequency domain, in order to reveal its predominant periods. 3

We will use the word signal as synonymous with a time series or empirical observations.

24

DISTRIBUTIONAL AND DYNAMIC FEATURES

According to standard Fourier theory, every periodic function or waveform can be decomposed by means of its Fourier transform into sinusoid functions of different frequency which sum to the original function or waveform. The Fourier transform effectively detects and distinguishes the different frequencies and respective amplitudes of the periodic components which characterize a certain signal and express them in terms of sinusoid functions. Without going into the details and the properties of Fourier transforms,4 we may say that the Fourier transform of a certain function f (t) is defined as:
+∞ F(s) = f (t) exp(−i2πts) dt −∞

and its inverse
+∞ f (t) = F(s) exp(i2πts) dt

(3.2)

−∞

so that

⎡ ⎤
+∞
+∞ ⎣ f (t) = f (t) exp(−i2πts) dt⎦ exp(i2πts) dt −∞

−∞

As we mentioned before, the most common use of the Fourier transform is that of finding the frequency components of an empirical signal originally buried in a noisy time domain. Since sample observations coming from a data-generating process (price formation, in our case) are discrete time observations, Fourier analysis has to be performed using the discrete time representation of the Fourier transform, the so-called discrete Fourier transform. In order to define the discrete Fourier transform F(r) of the continuous time function f (t) let us define f (k) as the discrete sample values of the same continuous time function, or signal. If f (kT) and F(rs0 ) are the k-th and r-th samples of f (t) and F(s) respectively, and N0 is the samples in the signal in one period T0 , then: f (k) = T f (kT) = and

T0 f (kT) N0 (3.3)

F(r) = F(rs0 ) where s0 = 2π/T0 . 4

For more details on Fourier theory see M.A. Pinsky, Introduction to Fourier Analysis and Wavelets, Pacific Grove (2002).

SPOT PRICE DYNAMICS AND STATISTICAL FEATURES

25

The discrete Fourier transform (DFT) and its inverse can be defined as follows: F(r) =

N 0 −1 k=0

  2π f (k) exp −ir k (DTF) N0

  N0 −1 1  2π f (k) = F(r) exp ir k (inverse DTF) N0 N0

(3.4)

r=0

In the case that f (t) is a real function, and consequently f (k) ∈ R ∀ k, we can rewrite the above equation in terms of a summation of sine and cosine functions with real coefficients: f (k) =

    N0 2πrk 2πrk 1  + β(r) sin α(r) cos N0 N0 N0

(3.5)

r=0

where α(r) = real(F(r)) and β(r) = −imag(F(r)). The fast Fourier transform (FFT) is an efficient algorithm for computing the DFT of a sequence. This methodology was developed by Turky and Cooley in 1965 in order to sensibly reduce the number of computations necessary for DFT calculations. Many mathematical software programs include routines for FFT calculations. For a length N input sequence x, the DFT is a length N complex vector X. The magnitude of X squared is usually called the power spectrum of x. The plot of the power spectrum of x against the vector of detectable frequencies is called the ‘periodogram’ of x. The periodogram is one of the most useful tools for the analysis of the periodic component of a certain signal since it presents strong peaks in correspondence to the frequencies of the relevant periodic components. Figure 3.4 shows the periodogram of the same EEX signal (historical spot prices) used for the plot of Figure 3.3. Figure 3.4 demonstrates that many different periodic components can potentially characterize electricity spot price dynamics, and not all of them are trivially detectable form the price plot (Figure 3.3). Again, we can see that daily and seasonal components are not the only relevant periodic ones, at least in the EEX dataset we used for the analysis. The fact that the periodic structure of power price dynamic is complex is evidenced here by the number of relevant peaks present in the periodogram. Fourier analysis is not only important for the full comprehension of the periodic behavior of electricity spot prices, but can also be used successfully to filter out from our dataset the component itself and also to give it an analytic representation. In fact, once we have identified the right frequencies at which it is better to study the empirical signal, it is possible to use a truncated Fourier series to fit the predictable component of an empirical

26

DISTRIBUTIONAL AND DYNAMIC FEATURES

3

⫻109

‘weekly’ peaks

‘daily’ peaks

2.5

2

1.5 ‘seasonal’ 1 component

0.5

0

0.03

0.1

0.2 0.3 0.4 0.5

1

2

3

4 5 6

Days⫺1

Figure 3.4 EEX periodogram calculated with hourly prices, Oct. 2000–Oct. 2003 signal, considering the fit residuals as purely random. In practice, we can fit the following equation: E(t) = f (t) + ε(t)

(3.6)

where f (t) = α + ni=1 βi cos(nωt) + λi sin(nωt) is the periodic component and ε(t) is a pure random component. Eventually external components such as Gaussian or exponential functions can be added to the Fourier series in order to capture ‘abnormal’ price behavior which, nevertheless, is considered structural. As an example here we report a fitting experiment done on the EEX dataset used throughout this section, in order to give an idea of the complexity of the functional representation that a realistic modeling approach to the predictable properties of spot electricity price dynamics may require. We can think, for example, of modeling the predictable component of the signal as the product of two building blocks: f (t) = w(t) × y(t)

(3.7)

where w(t) denotes the ‘intra-week’ shaping factor, and y(t) denotes the seasonal component.

27

SPOT PRICE DYNAMICS AND STATISTICAL FEATURES

40 35

Euro/MWh

30 25 20 15 10

EEX week average Periodic fit

5 0

0

0.1

0.2

0.3

0.4

0.5 Years

0.6

0.7

0.8

0.9

1

Figure 3.5 EEX seasonal component y(t) Obviously, the level of goodness of fit with the truncated Fourier method is highly dependent on the length of the series selected. We used additional Gaussian components, according to the empirical evidence, in order to increase the goodness of fit. The model used can be generalized in the following formulation: βi sin(tπωi ) + β2i+1 cos(tπω2i+1 ) α0 + i=0   w(t) = abs λ0 + γi sin(tπψi ) + γ2i+1 cos(tπψ2i+1 ) i=0

(Weekly component)

(3.8a)

and y(t) = ∂0 +



i sin(tπhi ) + 2i+1 cos(tπh2i+1 ) +

i=0

(Seasonal component)

 j=0

   t−µ 2 ξj exp − η (3.8b)

Figures 3.5 and 3.6 graphically represent the capacity of the proposed analytical form to capture deterministic behavior of the series used. In any case it is necessary to note that the time horizon and the number of observations of the dataset size are essential elements for the significance of any kind of analysis aimed to separate structural price phenomena from

28

DISTRIBUTIONAL AND DYNAMIC FEATURES

2.5

EEX week Fit 1 Fit 2

Euro/MWh

2

1.5

1

0.5

0

0.002 0.004 0.006 0.008

0.01 Years

0.012 0.014 0.016 0.018

Figure 3.6 EEX intra-week shaping component w(t) idiosyncratic ones. In fact, working with only a few years of data, which if they are hourly data already represent a huge quantity of observations, we cannot really say anything certain about structural phenomena which are supposed to influence price behavior a few times a year. Hence, we may run the risk of interpreting as structural a certain phenomenon, which in fact is occasional, and vice versa.

3.2.2 Introduction to wavelet analysis The basic idea behind Fourier analysis is to represent data from a certain data-generating process by means of a superposition of sine and cosine functions. Scientists have used this approach, proposed by Joseph Fourier, to approximate choppy signals since the early 1800s. However, by construction, sine and cosine functions are non-local functions and extend to infinity. This implies that standard Fourier analysis and serial representation are not extremely accurate for the analysis and the approximation of a signal which presents sharp spikes or discontinuities. The fundamental idea behind wavelet analysis is not very different from that of traditional Fourier analysis, with the difference that the basic building blocks of wavelet analysis are not sine and cosine functions but particular mathematical functions that are contained neatly in finite domains (wavelet functions). Wavelets analyze the signal according to scale, hence the scale on which we wish to examine the data plays an important role. This means that

SPOT PRICE DYNAMICS AND STATISTICAL FEATURES

29

if we are interested in studying the signal within a large window, wavelet analysis will emphasize gross features of the signal, while if we restrict the window we will be able to notice fine features. Wavelet analysis is particularly suitable to study empirical signals which display a significantly spiky behavior, like power price dynamics. Wavelet analysis adopts a wavelet prototype function, called the mother function, in order to create an orthogonal basis for the scaling function we want to analyze, making possible its representation as a linear combination of basis components (an orthogonal wavelet series). In particular, dilations and translations of the mother function ϕ(x), selected among the family of wavelet functions, define our wavelet basis: −s

ϕ{s,l} (x) = 2 2 ϕ(2−s x − l)

(3.9)

The parameters s and l are integer parameters, which have the respective function of scaling and dilating the mother function ϕ(x). The scale index s determines the wavelet’s width, while the location index l determines its position. The mathematical properties of the mother function fully determine the properties of the basis. To span the data domain at different resolutions, the wavelet has to be used in a scaling equation as follows: W(x) =

N−2 

(−1)k ck+1 ϕ(2x + k)

(3.10)

k=−1

where W(x) is the scaling function; ϕ(x) the mother function; and ck the wavelet coefficients. Wavelet coefficients must satisfy the following linear and quadratic constraints: N−1  k=0

ck = 2,

N−1 

ck ck+2l = 2δl,0

(3.11)

k=0

where δ is the delta function and l is the location index. A full and comprehensive discussion on wavelet analysis is beyond the scope of this book;5 here it is sufficient to realize that wavelet analysis and the related filtering tool is a useful and advanced instrument for denoising both electricity spot price and electricity demand time series (see Figure 3.7). On the contrary, the use of wavelet tools for the analytic representation of spot price periodic behavior is not possible since wavelet series functions usually do not have an explicit functional representation. A good application of wavelet analysis to the electricity sector has been recently proposed by Stevenson (2002). 5

The interested reader can refer again to Pinsky (2002) for a more detailed discussion on wavelets.

30

AUD/MWh

DISTRIBUTIONAL AND DYNAMIC FEATURES

350 300 250 200 150 100 50 0 2,500

5,000 Hours Original spot prices

7,500

2,500

7,500

2,500

7,500

300 AUD/MWh

250 200 150 100 50 0 5,000 Hours Denoised spot prices (1)

AUD/MWh

300 250 200 150 100 50 0 5,000 Hours Denoised spot prices (2)

Figure 3.7 New South Wales electricity spot prices (Jan. 1998–Aug. 1998): original series are denoised by means of wavelets (two examples) Source: Stevenson (2002)

3.3 STATISTICAL PROPERTIES OF ELECTRICITY PRICES In the previous section we faced the problem of detecting and eventually modeling the predictable components of electricity price dynamics, if present. We started our research on this topic having in mind the fact that predictable (structural) and unpredictable price components were linked together by means of a linear (multiplicative or additive formulation) functional relationship. If the assumptions of this model are true and if we are able to completely detect all structural components from price empirical

SPOT PRICE DYNAMICS AND STATISTICAL FEATURES

31

data, then the unpredictable components of price dynamics will simply be a white-noise process with zero mean and constant variance:6 E(t) = f (t) + ε(t) (additive formulation) E(t) = f (t) × ε(t) (multiplicative formulation)

(3.12)

where E(t) is the electricity price; f (t) is the price structural component; and ε(t) is the white-noise random component. Unfortunately, the complete detection of all the periodic or structural components of electricity spot prices is not a simple task. As we have already discussed, all the structural components of demand and supply interact together in a nonlinear way and consequently they download their effects on prices in a non-trivial way. The result of this complex interaction is that not all the structural components of price dynamics can be modeled and filtered out completely from the data by means of simple representations such as those used so far. However, we can try to capture additional information about power price structural behavior by means of a sophisticated statistical analysis. After the application of all the detecting and filtering methodologies discussed so far we may be left with a ‘de-structured’ signal (original signal minus its structural component), which is not exactly a white-noise process but a stochastic process with much more complex characteristics than white noise. Accurate analysis of the statistical properties of the de-structured signal can significantly improve our understanding of the price behavior. In this section we will see how electricity price statistical properties can be analyzed and explained by means of traditional statistical tools. A detailed discussion of all the relevant statistical techniques is beyond the scope of this book, and hence we will concentrate on only the most important ones.

3.3.1 Correlogram analysis and unit-root presence The importance of a correct, even if partial, detection of electricity price structural components (especially periodic ones) can be noted from a standard distributional analysis of the electricity price marginal distribution (Figure 3.8). In fact, if we analyze a rough hourly price time series distribution we can observe strange distribution shapes due to undetected periodic components. Once we filter out periodic components7 from the same dataset, we obtain a time series with very different statistical properties (Figures 3.9 and 3.10). 6

This process typically represents the situation of a random noise signal when no additional information can be extracted from the data. 7 In the example presented here the filtering process has been performed by subtracting price structural components. However, multiplicative filters can be used as well.

32

DISTRIBUTIONAL AND DYNAMIC FEATURES

2,000

Series: PRICE Sample 1 8784 Observations 8,784

Frequency

1,600 1,200 800 400

Mean Median Maximum Minimum Std. dev. Skewness Kurtosis

28.18322 28.01200 100.0040 0.000000 10.47256 0.208797 3.551207

Jarque-Bera Probability

170.7225 0.000000

0 0

20

40 60 €/ MWh

80

100

Figure 3.8 Powernext hourly spot prices distributional analysis (8,784 observations from 1 Jan. 2004–31 Dec. 2004)

80

Price (€/MWh)

Residual Actual Fitted

60 40 20

80

0 ⫺20

40

⫺40 0 ⫺40 ⫺80 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Hours

Figure 3.9 Powernext hourly spot prices structural and unpredictable components From Figures 3.9 and 3.10 it can be easily seen that after the removal of structural components, the price marginal distribution appears smoother and displays a more standard shape. Note that here we studied absolute price behavior, however, the same analysis can be applied to the logarithm of the price. The logarithmic representation may be useful, especially for

SPOT PRICE DYNAMICS AND STATISTICAL FEATURES

2,400

Series: Residuals Sample 25 8568 Observations 8,544

2,000 Frequency

33

800

Mean ⫺6.43e-13 Median ⫺0.281747 Maximum 59.66402 Minimum ⫺39.44432 Std. dev. 5.510964 Skewness 1.023309 Kurtosis 10.01567

400

Jarque-Bera Probability

1,600 1,200

19013.35 0.000000

0 ⫺25

0 25 €/MWh

50

Figure 3.10 Powernext hourly spot de-structured prices distributional analysis modeling purposes, as we will see in Part 2, in order to avoid the possibility of negative prices. In both situations (absolute prices or logarithms) the distributional properties of de-structured price time series do not tend to support standard assumptions of normal or lognormal price behavior. Pronounced kurtosis and asymmetry typically characterize de-structured price marginal distributions. A static distributional analysis is a useful tool but is not sufficient for a full understanding of price statistical properties. In fact, dynamical properties such as serial dependence may have much more importance than marginal distribution ones. In this case correlogram analysis is the basic statistical tool for the study of time series serial dependence properties.8 Table 3.1 shows us that de-structured Powernext time series still contain extractable information. In fact, autocorrelation and partial autocorrelation functions reported here display the typical shape of a first-order autoregressive process. A test for unit-root presence is a fundamental step in electricity price statistical analysis, since the presence of a unit root would imply nonstationary behavior and consequently the absence of mean reversion of electricity de-structured prices. The Augmented Dickey–Fuller test performed in our example excludes this possibility in favour of the hypothesis of a strong mean reversion, at least for the case of Powernext 2004 hourly prices. However, it is important to stress the fact that, typically, unit-root 8

For a detailed discussion about traditional correlogram analysis, applied to financial time series analysis, see Campbell, Lo and MacKinlay (1997).

34

DISTRIBUTIONAL AND DYNAMIC FEATURES

Table 3.1 Powernext de-structured time series correlogram analysis Autocorrelation

Partial Correlation

AC

PAC

Q-Stat.

Prob.

|****** |

|****** |

1

0.824

0.824

5806.9

|***** |

|*

2

0.704

0.078

10049.

|***** |

|

|

3

0.604

0.015

13173.

0.000

|**** |

| |

4

0.517

−0.004

15455.

0.000

|

0.000

|*** |

|

|

5

0.452

0.032

17202.

0.000

|*** |

|

|

6

0.390

−0.012

18501.

0.000

|*** |

|

|

7

0.347

0.034

19533.

0.000

|** |

|

|

8

0.314

0.022

20377.

0.000

|** |

|

|

9

0.282

0.002

21059.

0.000

|** |

|

|

10

0.252

−0.004

21603.

0.000

|** |

|

|

11

0.223

−0.005

22029.

0.000

|* |

|

|

12

0.192

−0.016

22346.

0.000

Table 3.2 Powernext de-structured time series unit root test Null hypothesis: DESTRUCT has a unit root Exogenous: constant Lag length: 27 (automatic based on SIC, MAXLAG = 36) t-statistic

Prob.*

Augmented Dickey–Fuller test statistic

−15.97956

0.0000

Test critical values:

1% level

−3.430942

5% level

−2.861686

10% level

−2.566889

*MacKinnon one-sided p-values.

tests are low-power statistical tests, hence accurate analysis (for example on undetected regime changes in the data) are recommended in order to minimize potentially biased conclusions regarding price process stationarity. Table 3.2 reports results obtained performing a unit root test to a Powernext de-structured signal. The question of whether the electricity price process has stationary or non-stationary behavior is extremely delicate as it has a tremendous impact on modeling assumptions, as we will see in Part 2. To give an example of its importance and the implication of stationarity, note that the variance of a non-stationary process tends to increase linearly with time, while for

SPOT PRICE DYNAMICS AND STATISTICAL FEATURES

35

stationary processes this does not happen since unconditional variance is time independent.9 Xt = βXt−1 + εt β = 1 ⇒ Xt

autoregressive process, with εt IID(0, σε2 ) non-stationary (unit-root presence)

→ σ2 (XT ) = T · σε2 → 0 < β < 1 ⇒ Xt

lim σ2 (XT ) = ∞

T→∞

stationary (mean reversion) → σ2 (XT) =

(3.13) σε2 ,∀T 1 − β2

Unless there is strong contrary evidence, we shall assume that the electricity spot price is stationary.

3.3.2 Heteroskedasticity and long memory Electricity spot-price time series may also display another statistical feature, quite common in traditional financial time series, heteroskedasticity. Heteroskedasticity means that even if the unconditional variance of the electricity price is finite and independent of time, as a consequence of stationarity conditional variance may not be constant but may display a structured behavior. In Part 2 we will see how to model heteroskedasticity by means of econometric models, here we are interested in simply detecting it and understanding the reasons behind it. The application of correlogram analysis to squared de-structured price time series may reveal dynamic variance effects. Going back to our example, we can see from Table 3.3 that Powernext price time series present significant heteroskedastic effects. The de-structured price process is a zero-mean reverting process; hence, the series obtained by squaring de-structured prices may be interpreted as a proxy for the variance process. Detecting an autoregressive structure in this squared series may be strong evidence in favour of the presence of conditional autoregressive heteroskedastic effects. In the next Part, we will see how these effects can be captured and scientifically tested by means of ARCH and GARCH modeling support. In general, we can say that conditional heteroskedasticity is the result of an improper assumption that price determinants (periodic and statistical components) are linearly related. The problem is that of selecting a nonlinear assumption which is particularly suitable for our purposes. ARCH and GARCH modeling approaches do not represent the only available and realistic non-linear modeling approach, but they represent the optimal compromise between effectiveness and simplicity. 9 See Verbeek (2000) for a detailed discussion on autoregressive and moving average univariate time series models.

36

DISTRIBUTIONAL AND DYNAMIC FEATURES

Table 3.3 Powernext squared de-structured price time series correlogram analysis Autocorrelation |***

|

Partial Correlation |***

|

AC

PAC

Q-Stat.

1

0.452

0.452

1748.7

Prob.

|

|

**|

|

2

0.040

−0.207

1762.2

0.000

|

|

|*

|

3

0.027

0.131

1768.5

0.000

|

|

*|

|

4

0.018

−0.061

1771.3

0.000

|

|

|

|

5

0.015

0.045

1773.2

0.000

|

|

|

|

6

0.009

−0.021

1773.9

0.000

|

|

|

|

7

0.001

0.009

1773.9

0.000

|

|

|

|

8

0.001

−0.003

1773.9

0.000

|

|

|

|

9 −0.003

−0.004

1774.0

0.000

|

|

|

|

10 −0.002

0.002

1774.0

0.000

|

|

|

|

11

0.003

0.003

1774.1

0.000

|

|

|

|

12

0.001

−0.003

1774.1

0.000

Another common effect which may also be recovered in electricity price time series is the so-called long-memory effect. Many empirically observed financial time series, although they seem to satisfy the assumption of stationarity, as the electricity price, seem to exhibit a non-negligible, even if small, dependence between very distant observations. This phenomenon, quite common also in geological branches such as hydrology, is usually associated with the presence of undetected cycles whose length (period) is not exactly measurable given the actual sample size, because it is too long. The discovery of long-memory effects (also known as long-run persistence effects) in electricity time series is not infrequent, and has been documented in some recent studies.10 Often there are effects strictly related to the fact that electricity spot markets have been introduced only recently, and consequently it is quite common to have a dataset insufficiently long to consistently measure long-term periodic components. However, even if this phenomenon is present in electricity spot price dynamics, its effect has to be considered marginal compared to those listed and discussed so far. We will discuss long-memory effect modeling approaches in Part 2.

10

See for example Atkins and Chen (2002) or Haldrup and Nielsen (2004).

P A R T II

Electricity Spot Price Stochastic Models

37

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CHAPTER 4

Electricity Modeling: General Features

4.1 SCOPE OF A FINANCIAL MODEL During the first part of the book we have seen that electricity price dynamics, in organized markets, display peculiarities which partly have to be attributed to the characteristics of the particular market we are looking at, as well as more general features of markets. It is therefore necessary to study analytically the peculiarities of electricity price dynamics in order to identify the main economic drivers. The scope of this study will be that of describing and forecasting electricity price behavior for the evaluation of financial and real assets, which depend upon it. This kind of exigency is satisfied by the construction of a descriptive mathematical model. Of course, the perfect mathematical model, able to describe and forecast price behavior and useful for the valuation of financial and real assets related to electricity prices, is difficult or even impossible to construct. For this reason, the basis for building a good financial model lies also in analyzing the scope of the model itself and the selection of its characteristics. As we have already discussed, the principal exigencies of a financial model for the description of electricity price dynamics are:
a description and comprehension of the main economic drivers;
forecasting; and
valuation of the related financial and real assets. The first two are highly correlated topics since they are both typical applications of an economic model. In particular, in the electricity industry a good 39

40

ELECTRICITY SPOT PRICE STOCHASTIC MODELS

comprehension of electricity price economic drivers and a good forecasting capacity are fundamental elements for efficient bidding, scheduling and production optimization policies. On the other hand, the need to evaluate financial and real assets whose value is sensitive to the price of electricity is something which cannot be fully solved within a purely economic theoretical framework; a higher degree of mathematical sophistication is necessary to face and successfully solve this kind of problem. Traditionally, complex mathematical modeling approaches have been borrowed from physics and natural sciences.

4.2 ECONOMETRIC MODELS VERSUS PURELY PROBABILISTIC MODELS In the energy finance field we also encounter the traditional clash between econometric modeling approaches and probabilistic approaches. Without going into the details of this philosophical dispute, in this section we try to identify the general features and aims of the traditional econometric modeling approach in contrast to those typical of a purely probabilistic one. Both approaches have their theoretical strengths and fields of application, and both can be successfully used for the analysis and solution of electricity finance problems. Generally speaking, an econometric model is conceived and developed in order to describe and measure the interaction between economic or financial variables, which need to be linked in some way. Hence, the principal scope of an econometric model is to test the empirical consistency of a certain economic theory. Obviously, this test function is conditioned by the building hypothesis upon which the model itself is based, and sometimes those hypotheses can be too stringent or simplistic with respect to the complexity of the phenomenon that the model is intended to describe or to test. This is exactly the case of traditional econometric models such as linear autoregressive models. Such models may be considered too simplistic for a realistic description of complex dynamics such as those of electricity spot and forward prices. However, some recent developments in theoretical and applied econometrics, especially in the field of nonlinear, multiple regimes and jump models can be considered as strong and extremely useful tools for the quantitative analysis of electricity price dynamics. Moreover, another very important application of econometric modeling is forecasting. Forecasting with an econometric model is the natural outcome of the data analysis and model-testing phases. From a mathematical point of view, econometric models are typically discrete time-stochastic models. Usually, but not necessarily, they are linear models. The time dimension of the model is partitioned in relation to the frequency of empirical observation of the phenomenon under study.

E L E C T R I C I T Y M O D E L I N G: G E N E R A L F E A T U R E S

41

Purely probabilistic models are conceived and developed to describe the probabilistic properties (trajectorial and distributional) that a certain measurable phenomenon displays. They have traditionally been extensively used to model physical phenomena, but by the early 1970s their use had become common also in financial analysis, especially in the field of derivatives pricing or for real asset valuation purposes. They are continuous time-stochastic models and the mathematical complexity involved is usually superior to that associated with econometric models. Typically, probabilistic models are reduced form models, in the sense that the stochastic variables involved do not have a particular and exact economic or financial meaning, only a descriptive role. Generally speaking, it is possible to state that every kind of modeling approach has its own strength if it is rigorous and well-founded. It is also true, however, that certain modeling approaches are more indicated than others for particular applications. This implies that the choice of the right approach should not be ideological but consistent with the contingency of the application. This book is mainly oriented towards pricing and risk-management problems, and hence it is natural that more space and attention will be given to models which derive form the field of applied probability. However, we will also try to provide the reader with a comprehensive overview of the most used and useful econometric instruments.

4.3 CHARACTERISTICS OF AN IDEAL MODEL AND STATE OF THE ART Apart from the intrinsic nature of the model, the ideal model should have some general characteristics. Obviously, the ideal model does not exist, but it is nevertheless possible to define the characteristics it should have:
The ideal model should be able to reproduce the trajectorial and statistical properties of price and returns dynamics. In the first part of the book we have seen that those characteristics can be summarized as: high volatility, asymmetry, the presence of spikes, fat tails, and so on. Traditional models are not always able to reproduce all these characteristics and, clearly, the realism of the model is strongly affected by this fact.
The ideal model should incorporate in an explicit way (econometric models) or in an implicit way (reduced-form probabilistic models) all the risk factors which have an impact on price dynamics.
The ideal model should be flexible enough to adapt to all the different characteristics displayed by different electricity markets. We have already seen that the peculiarities which make markets different can sometimes

42

ELECTRICITY SPOT PRICE STOCHASTIC MODELS

be non-negligible. The capacity of the model to simply adapt to different environments is also evidence of the fact that the model is able to capture the intrinsic drivers of the phenomenon, not only describing its dynamics in a mechanic way.
The ideal model should be parsimonious in the number of parameters or degrees of freedom which it incorporates. A limited number of parameters allow a higher statistical robustness in estimating the parameters themselves. Moreover, a limited number of parameters implies a greater mathematical simplicity of the model itself, and hence a more simple interpretation and a higher facility in its utilization.
The ideal model should be the simplest possible, mathematically speaking. Mathematical complexity, when not justified by a higher descriptive capacity, has to be considered a defect of the model itself. Sometimes it could happen that unnecessary mathematical complexity is considered as a positive feature of a model, but for practical purposes the mathematical elegance of a model is not as relevant as its effectiveness. Moreover, as we will see later, there are some mathematical properties that we hope a model presents. In particular, properties of Markov processes or semimartingales may sometimes be welcome because they reduce the complexity of the whole analysis, whilst at other times they may be simply necessary for the application of mathematical results useful in constructing the model itself. As we have noted, the ideal model does not exist in any field of application, especially in the energy field. After almost ten years from the publication of the first specific paper concerning electricity prices (see Kaminski and Gibner, 1997), few of the models developed can really be considered as realistic models currently used in trading and risk management in daily activities. We can say that in the energy finance field standard benchmarks for quantitative analysis do not exist. This situation can partly be attributed to a very limited specific literature (in 2004 I know of about 10 papers, only five published in refereed journals), but also to power markets that are not always liquid and mature enough to allow a serious implementation and test program. However, in the last few years, the growing number of electricity marketplaces and market participants is improving the situation. In the following chapters we will try to select and present the most interesting and useful modeling approaches provided by the recent literature, emphasizing for each of their positive and negative features.

CHAPTER 5

Econometric Modeling of Electricity Prices

In the previous chapter we noted that the main scope of an econometric model is that of describing in the best possible way the interrelation existing among different economic variables. Applying this idea to the study of electricity price dynamics we have the possibility to investigate the relation that links electricity prices (dependent variable of the model) to other economic drivers such as, for example, electricity consumption. This relationship can be expressed in mathematical terms by a simple and generic functional representation of the following type: Et = f (Xit )

(5.1)

where Xi represents the value of the i-th generic explanatory variable. The simplest possible explicit formulation for this generic functional relationship is obviously the linear one. In its simplicity, the linear hypothesis can also be interpreted as a first-order approximation for every generic functional relation. The mathematical representation for the linear relation is the following: Et = α +

N


βi Xit + εt

(with εt i.i.d. process)

(5.2)

i=1

where i.i.d. stands for Independent and Identically Distributed Process. The main issue related to this kind of model representation is not exclusively that of the correct selection of the set of explanatory variables Xi , but is mainly due to the fact that a simple linear regression model is intrinsically not able to capture the strong serial dependence effects which characterize electricity price time series, that we documented in Part 1. Hence, in order to 43

44

ELECTRICITY SPOT PRICE STOCHASTIC MODELS

have a realistic representation of the phenomenon it is necessary to construct models able to reproduce non-trivial serial dependence effects. Moreover, the presence of such effects can be interpreted as proof that the functional relationship between prices and economic drivers is much more complex than the simple linear one. Such complex relations can be studied much better by dynamic models rather than simple static ones.

5.1 TRADITIONAL DYNAMIC REGRESSION MODELS The simplest econometric modeling technique able to capture serial dependence phenomena is represented by classical dynamic regression models. Those models are nothing more than simple multivariable extensions of simple ARMA or ARIMA models, well-known in classical econometric analysis. If we want to use dynamic regression models for the study of electricity price dynamics we may assume a linear relationship between the level of electricity price at time t(Et ), its lagged values and the (Et−1 , Et−2 , . . ., Et−n ) and the values (lagged and not) of a set of explanatory variables (Xit , Xi(t−1) , . . ., Xi(t−k) ). More formally: Et = α +

n
i=1

βi Et−i +

k m



γlj Xl(t−j) + εt

(5.3)

l=1 j=1

Typically, electricity price time series do not contain unit roots, and hence a dynamical regression analysis can be conducted on price levels (or on their logarithmic transformation) without differencing and without incurring the risk of spurious results. Obviously, a preliminary test for the presence of unit roots, conducted considering all the plausible components of the datagenerating process (intercept, linear trends, seasonal components, regime shifts…), is always useful and informative. A simple autoregressive model for electricity prices can represent the simplest and most concrete application of traditional dynamic regression models in the electricity field, where demand quantity acts as the explanatory variable. In this class of models, proposed for short-term forecasting purposes by Nogales, Contreras, Conejo and Espinola (2002), the price of electricity at time t is related to the past values of prices (Et ) and demands (Dt ) at times t − 1, t − 2, . . ., t − n. This is done in order to obtain a model that has uncorrelated error terms. Formally, this class of models can be generalized in the following formulation: Et = c + ωd (B)Dt + ωe (B)Et + εt

(5.4)

ECONOMETRIC MODELING OF ELECTRICITY PRICES

45

 K d k e e k where the functions ωd (B) = K k=0 ωk B and ω (B) = k=1 ωk B are polynol mial functions of the lag operator B(B : B xt = xt−l ). The efficiency of this approach depends on the capability of the selected parameters to achieve an uncorrelated set of errors εt . The correct selection of the appropriate number of lags can be performed by means of classical correlogram analysis as suggested in many econometrics textbooks for generalized ARMA models (see for example Hamilton, 1994). In addition, concerning the estimation methodologies for this class of models, both ordinary least squares (OLS) and maximum likelihood (ML) methods are available. In Table 5.1 and Figure 5.1 we present the results

Table 5.1 Dynamic regression model estimation results Variable

Coefficient

Std. Error

T-Statistic

Prob.

31.14958

5.137272

6.063448

0.0000

C QUANTITY

−0.002425

0.001285

−1.886387

0.0610

PRICE (−1)

0.920817

0.029821

30.87808

0.0000

R-squared

0.861828

Mean dependent var.

26.25526

Adjusted R-squared

0.860143

S.D. dependent var.

12.75797

S.E. of regression

4.771162

Akaike info criterion

5.980858

Sum squared resid.

3733.294 −496.4016

Log likelihood Durbin–Watson stat.

1.466219

Schwarz criterion

6.036869

F -statistic

511.4616

Prob (F -statistic)

0.000000

80 Residual Actual

€/MWh

Fitted 20

60 40 20

10

0

0 ⫺10 ⫺20 20

40

60 80 100 120 Hours (of the week)

140

160

Figure 5.1 Dynamic regression model graphical representation

46

ELECTRICITY SPOT PRICE STOCHASTIC MODELS

coming from the application of this simple modeling approach to a sample of one week of hourly prices and quantities from the French market (Powernext). It is simple to infer from the estimation results reported in Table 5.1 that this simple regression model provides a good fit to the original dataset (R2 about 90%). However, the regression residuals appear to be not completely cleaned from relevant information. From Figure 5.1 we can clearly observe regression residual structural behavior. This means that models that are more complex can be appropriately used to extract this additional information.

5.2 TRANSFER FUNCTION MODELS A simple way to include serial correlation in a regression model is through a serially correlated error term. Such an approach is referred as a transfer function model; it was first applied in the electricity sector again by Nogales, Contreras Conejo and Espinola (2002). Specifically, it assumes that prices and demands are both stationary variables modeled through a general transfer function in the following way: Et = c + ωd (B)Dt + Nt

(5.5)

 d k where ωd (B) = K k=0 ωk B is a polynomial function of the lag operator θ(B) εt with B and Nt is an ARMA disturbance term of the type: Nt = φ(B) 
  l l θ(B) = 1 − l=0 θl B and φ(B) = 1 − l=0 φl B and εt = i.i.d. error term. In the transfer function model the actual electricity price is related to the actual and lagged demand through the function ωd (B) in the same way as in the standard dynamic regression model and to lagged prices through the function φ(B). The parameters of the model can be estimated recursively, first by a direct selection and estimation of the parameters of the polynomial function ωd (B), and then by using the residuals of the first regression it is possible to select and estimate the parameters of the ARMA component. The following results come from the application of the transfer function model to a sample dataset of daily base load prices from the French market (Figure 5.2 and Table 5.2). In these results the behavior displayed by the series appears much more complicated than the simple hourly series analyzed previously. As we can see both from Table 5.2 and Figure 5.3 the results in this second experiment are not comparable to those obtained for the hourly time series but are not so bad in consideration of the fact that the biggest deviations from the model are represented by the few spiky values. Those outliers do not affect the

47

400

€/MWh price

300

200

100

04 g. ’ Au 23

Fe 5

20

Ju

ly

b. ’

’0

04

3

3 Ja

n.

’0

2 1

27

15

N

Ju

ov

n.

.’

’0

01

0

Figure 5.2 Powernext daily price (base load)

Table 5.2 Transfer function model estimation results Variable

Coefficient

Std. Error

T-Statistic

Prob.

C

22.98863

1.296460

17.73184

0.0000

QUANTITY

0.000311

8.60E–05

3.614820

0.0003

−0.000146

8.60E–05

−1.701995

0.0891

MA(1)

0.413144

0.031512

13.11086

0.0000

MA(2)

0.221703

0.034079

6.505523

0.0000

MA(3)

0.115793

0.034554

3.351066

0.0008

MA(4)

0.061808

0.034078

1.813731

0.0700

MA(5)

−0.026673

0.031593

−0.844259

0.3987

QUANTITY(−1)

R-squared

0.208945

Mean dependent var.

26.14291

Adjusted R-squared

0.203425

S.D. dependent var.

14.65342 7.987673

S.E. of regression

13.07834

Akaike info criterion

Sum squared resid.

171556.1

Schwarz criterion

8.026594

F -statistic

37.84679

Prob (F -statistic)

0.000000

Log likelihood Durbin–Watson stat.

−4029.769 2.003887

Note: Here the model has been estimated in its moving average representation.

48

ELECTRICITY SPOT PRICE STOCHASTIC MODELS

€/MWh

400 Residual Actual

300

Fitted

200 100

300

0

200 100 0

04 g. ’ Au 24

6

21

Ju

ly

Fe b. ’

’0

04

3

3 2

Ja

n.

’0 16

Ju

n.

.’ ov N 28

’0

2

01

⫺100

Figure 5.3 Transfer function model graphical representation residuals’ linear dependence1 but they have a significant impact on the squared residuals’ dependence structure. The significant persistence in squared residuals, which is usually observed, can be considered as strong evidence in favor of the hypothesis of embedded non-linear relations among the variables. The transfer function model is not able to capture this type of non-linear effect. The presence of outliers, which cause large deviations from normality in the regression residuals, introduces us to another more advanced category of models which have the scope to explain the observed volatility persistence. In the next section we will introduce GARCH models.

5.3 CAPTURING VOLATILITY EFFECTS: GARCH MODELS Up to this point we have analyzed models where the disturbance term εt is an i.i.d. process, that is a process with zero mean and constant conditional and unconditional variance. However, as in many financial time series, electricity price time series also support the hypothesis of models with heteroskedastic error terms. This new feature significantly improves both the goodness of fit and the out-of-sample forecasting ability of standard ARMA models. 1

We do not report here LM test results on linear correlation.

ECONOMETRIC MODELING OF ELECTRICITY PRICES

49

The GARCH (generalized autoregressive conditionally heteroskedasticity) models, proposed by Bollerslev in 1986, represent a parsimonious and efficient way of capturing the autocorrelation effects displayed by squared residuals time series. Their generalized representation (GARCH (n, m)) can be formulated in the following way: Et = α +

N
i=1

βi Et−i +

M


γj εt−j + et

(5.6)

j=1

  with E(ε2t |
t−1 ) = σt2 ∀t and σt2 = a + ni=1 bi σt−i + m j=1 cj ηt−j + ηt and ηt = i.i.d. GARCH (1,1) represents the most popular formulation of this modeling approach. GARCH modeling is well-known in standard financial applications and analysis, and for this reason we do not describe further its theoretical background; the interested reader can refer to Mills (2000). In the electricity field, the GARCH modeling approach has been proposed by Garcia (2003) for short-term forecasting of electricity spot prices. One of the main problems with the standard GARCH representation is that parameter restrictions have to be imposed in order to guarantee the non-negativity of the variance process. There are also other data peculiarities which cannot be incorporated by standard GARCH models; in particular volatility persistence effects, asymmetry effects and other non-linearities in the volatility process. To deal with these effects many refinements have been proposed in the financial literature such as: exponential GARCH (EGARCH), power GARCH (PGARCH), integrated GARCH (IGARCH) and component GARCH, and others. We mention these GARCH refinements without going into further detail of their particular formulations because their usefulness in the electricity field has not yet been proved. Again, the interested reader can refer to Mills (2000) for a comprehensive discussion on all the GARCH methodologies.

5.4 CAPTURING LONG-MEMORY EFFECTS IN ELECTRICITY PRICE LEVEL AND VOLATILITY: FRACTIONALLY INTEGRATED MODELS Many empirically observed time series seem to exhibit a significant dependence between empirical observations very distant in time. This phenomenon, particularly common in hydrologic time series, is known as long memory, and in economic time series can affect both price and volatility. Recently, a few papers have analyzed the long-memory phenomenon in electricity markets. Atkins and Chen (2002) analyzed long-memory behavior in the electricity prices of the Alberta deregulated market, while Haldrup and Nielsen (2004) focused their research on the Nordpool market.

50

ELECTRICITY SPOT PRICE STOCHASTIC MODELS

Long-memory effects are commonly characterized by an extremely slow decay of the empirical autocorrelation between subsequent observations. This characteristic is not compatible with the behavior displayed by both standard stationary processes (I(0) processes) and non-stationary processes (I(1) processes). An efficient way of capturing this effect is through fractionally integrated processes. An integrated process of order d (I(d) process) is defined in such a way that it becomes stationary after d differentiations. Traditional econometric modeling usually deals with processes which are integer-integrated, in the sense that the integration order d is an integer and a positive number. Most frequently, in financial time series analysis a lot of attention is placed on I(0) or I(1) processes. Fractionally integrated processes are processes whose integration order is not an integer, and in particular we are interested in introducing a sub-class such that 0 < d < 1. ARFIMA (p, d, q) processes represent a general category of fractionally integrated processes, whose general formulation can be expressed in a more compact form by means of a classical lag operator representation in the following way: Xt = ω(B)(1 − B)d εt

(5.7)

where εt is a white-noise process. Baillie (1996) provides a self-comprehensive and detailed presentation of ARFIMA processes and their economics applications. It is not infrequent in financial time series analysis to find long-memory effects not only in price level dynamics, but also in volatility time evolution. This feature can also be observed in the electricity price time series, even if at the present time it has not been documented in any scientific paper. However, long-memory effects in volatility time evolution are a well-documented phenomenon for more traditional financial time series. In order to capture the volatility long-memory feature, Baillie, Bollerslev and Mikkelson (1996) proposed a class of processes called fractionally integrated GARCH processes (FIGARCH). In analogy with what we said about standard GARCH processes, we may state that FIGARCH processes are processes whose conditional variance behaves like an ARFIMA process:   E ε2t |
t−1 = σt2 σt2 = ω(B)(1 − B)d νt

(5.8)

where νt is a white-noise process. Many other even more complicated modeling approaches, such as non-linear dynamic models or multiple regime models, can be usefully considered for application in electricity price econometric modeling. However, it is always important to remember that an increase in model complexity should be compensated by a sensible increase in the model’s explanatory power.

CHAPTER 6

Probabilistic Modeling of Electricity Prices

The use of continuous time probabilistic modeling tools for financial applications can be dated back from the beginning of the 1960s, with Samuelson and Mandelbroat, or even at the beginning of the twentieth century with Bachelier. However, we can observe its massive diffusion after the publication of the famous Black–Scholes and Merton articles (1973) on option pricing and the subsequent growing importance of derivative pricing problems in financial economics. In addition, in the energy finance field such modeling tools have become important as the energy markets have been progressively deregulated and price dynamics behavior has become similar to that of other purely financial assets and markets of derivative products linked to energy prices started. As we discussed in Chapter 4, the main scope of a probabilistic model for the electricity price is not that of the explaining or forecasting its dynamics, but that of a realistic description of its distributional and trajectorial features. Typically, probabilistic models are reduced-form models in the sense that economic determinants of price fluctuations are not directly observable in the model, but they are indirectly represented by continuous time stochastic processes (traditionally Brownian motion). They are mainly used for derivative pricing or generation assets modeling. The class of stochastic processes which has been used until now for electricity price modeling can be divided into two main categories: traditional and more advanced models. Traditional modeling approaches have been inspired and developed for financial modeling applications other than electricity. For this reason, often, they do not provide a realistic description of the typical features of electricity prices. On the contrary, by advanced models we mean all those models which have been developed explicitly 51

52

ELECTRICITY SPOT PRICE STOCHASTIC MODELS

to model electricity prices and for this reason are able to provide a more realistic description, even if their mathematical complexity is a bit higher.

6.1 TRADITIONAL STOCHASTIC MODELS 6.1.1 Classical mean reverting processes The class of mean reverting processes is a class of processes often considered as the benchmark for the evaluation of commodity derivatives. Their success is mainly because they are simple models and able to produce closed-form formulas for a wide class of derivative products. Their dynamics can be represented by the following stochastic differential equation: dEt = −λ(Et − m(t))dt + σ(t)dWt

(6.1)

where λ is the mean reversion parameter; m(t) is the periodic attractor; and σ(t)dW is the martingale stochastic component. A martingale process is a process with zero mean conditional increments or in formal terms we can say that Xt is a martingale process if E(XT /ft ) = Xt ∀ t < T. In expression (6.1) the martingale component is made up of the product of a standard Brownian motion Wt and a deterministic volatility process σ(t). According to this representation, electricity prices diffuse around a longterm attractor, which likely has a periodic behavior, and the speed of the reversion is determined by the constant parameter lambda. A generalized Brownian motion with a constant volatility parameter generally represents the martingale stochastic component. But similar representation is available for processes with a deterministic, locally deterministic and stochastic volatility factor.1 This class of stochastic processes has the nice feature of being very simple to calibrate both on historical prices, by means of the ML estimator, and forward market data, by means of least-squares fitting of the analytical forward curve (forward curve calibration). The analytic functional form of the forward curve that the model is able to produce can be written as follows: F(t, T) = exp[(r − y)(T − t)]m(T)(1 − exp(−λ(T − t))) + Et exp[(−λ(T − t))] with y convenience yield

(6.2)

One of the main reasons which make this class of processes not completely adequate for modeling electricity price dynamics is the fact that they can potentially generate negative prices.2 To overcome this problem, some 1

However, stochastic volatility introduces some more complexities in the model, which will be analyzed in detail in the following paragraphs. 2 Traditionally, economic theory imposes positive prices for both financial asset prices and commodity prices.

PROBABILISTIC MODELING OF ELECTRICITY PRICES

53

refinements have been introduced to the traditional formulation, and the Lucia and Schwartz model (2002) is certainly the nicest and easiest among such refinements. In their approach, they propose a formulation which deals directly with the natural logarithm of the spot price instead of the spot price itself: dEt = k(b(t) − ln Et )Et dt + σEt dWt

(6.3)

Obviously, in this case the price process has a lognormal conditional distribution. Even if we correct the process to overcome the problem of potential negative prices, the class of mean reverting processes does not seem to be adequate for electricity spot price modeling because the linear diffusive dynamics is not able to replicate spiky behavior. In the limited scientific literature of recent years regarding the problem of electricity price modeling, multivariate generalizations of standard mean reverting models have also been presented (see for example Pilipovic, 1997). However, these generalizations have not met with much success since they do not completely solve the main problems we have discussed for the univariate case. Moreover, they introduce additional mathematical complexity.

6.1.2 Mean reverting with jumps processes This class of processes represents the natural extension of the previous case. A jump component is introduced in the standard mean reverting process in order to reproduce the spiky behavior. In its simplest formulation, the jump component is a standard compound Poisson process similar to that proposed by Merton (1976).3 The associated stochastic differential equation can be written as follows: dEt = −λ( ln Et − l(t))Et dt + σ(t)Et dWt + dNt

(6.4)

where N(t) is a classical compound Poisson process. This class of processes is considered by many people working in the electricity trading sector as a good model, since it merges together the two main features of electricity spot price dynamics: mean reversion and jumps. Unfortunately, the ways in which this class of models merge mean that reversion and jumps cannot be considered very realistic. In fact, the reversion intensity is constant for both normal and spike regimes, while empirically we observe that the spike reversion intensity is much more significant than the standard mean reversion. This implies that when a 3 A jump arrival process, which is a Poisson process with constant intensity, and the jump amplitude process make up this simple jump process, which is an R+ random variable (typically exponential).

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ELECTRICITY SPOT PRICE STOCHASTIC MODELS

positive jump occurs reversion towards the normal regime is slower than that observed. Moreover, the simple compound Poisson process N(t) is characterized by a constant jump frequency while we know also that the probability of a spike occurrence is not constant over time but is often cyclical, since it depends on some price determinants which are themselves periodical. Another major disadvantage with this class of processes is related to parameter estimation. In fact, filtering methods have to be applied in order to separate, in the original empirical time series, jump components from diffusive components, and this operation has a massive effect on the estimate we get of the whole set of parameters. Hence, the model specification is extremely sensitive to the filtering method we decide to adopt and this could be dangerous. Another problem we face in using mean reversion with jump processes in derivative valuation models is related to market incompleteness. The jump component we add to the standard mean reversion process introduces an additional source of uncertainty in the model, which cannot be diversified away within the standard derivative evaluation theoretical frameworks. This important problem is common to all the models that make use of discontinuous stochastic components. The implications and the possible solutions to market incompleteness will be discussed in Part 3.

6.1.3 Markov switching regime models In the previous sections we have mentioned the fact that electricity spot price dynamics is characterized by normal and spiky behaviors or regimes. Hence, a natural way of mathematically representing this feature is through the class of multiple regime processes. According to this modeling approach, the electricity spot price is assumed to follow two different and independent regimes. The first one, let us call it the ‘mean reverting regime’ is intended to describe the non-spiky behavior of the dynamics, while the second is intended to replicate the ‘spiky regime’ of the process. The spikes in the second regime are modeled with a simple lognormal behavior whose mean and standard deviation are much higher than those of the mean reverting regime process. Formally, we have the following specification: d log Et = −λ( log Et − m(t))dt + σ(t) dWt

(stable regime)

d log Et = µ(t)dt + σ(t) dWt

(spiky regime)

(6.5)

The switching from one regime to the other one is governed by a two-state Markov process Mt for each t and T > 0 with T > t, characterized by the

PROBABILISTIC MODELING OF ELECTRICITY PRICES

55

following 2 × 2 transition matrix P(t, T):
P(t, T) =



Pss (t, T) Psr (t, T)

Prs (t, T) Prr (t, T)

(6.6)

The role of the transition probability matrix is very important since it determines the likelihood to jump from one regime to the other in every single instant of time. Conditional on the regime state, the parameters of the two processes can be easily estimated by means of ML estimators based on the normal distribution of the stochastic terms of the two processes. However, in practice, parameter estimation is not that easy since we do not know what regime reigns in each single instant of time (regime is a latent variable). The Kalman filtering methodology4 helps us to solve estimation problems, but again we stress the fact that filtering procedures usually have a big impact on estimation results. Another source of complexity is represented by the fact that the regime is not the only latent variable, prices are as well. In fact, prices in the mean reverting regime continue as a latent process during a spike regime, and thus are unobservable there. From the mathematical point of view, the Markov switching process as presented is not a one-dimensional Markov process, and this creates many serious problems, especially related to parameter estimation. Deng (1999) proposed a methodology to circumvent estimation problems for Markov switching processes. Excluding estimation problems, however, this class of processes has the important advantage of producing closed formulae both for forward and plain-vanilla options prices, as linear combinations of prices under the two regimes and regime probabilities.5 In addition, though, for this class of models the problem of market incompleteness arises since the source of uncertainty is multiple and not completely eliminable.

6.1.4 Stochastic volatility and subordination Up to this point we have discussed in depth the problems we can incur in using standard Brownian motions as stochastic driving factors in our model. Brownian motions are, in general, inadequate to realistically describe the stochastic dynamics of financial asset prices or returns. In the electricity field this is particularly true given the peculiarities of electricity price dynamics. We saw in previous sections that the introduction of discontinuities in the 4

For an introduction to the Kalman filtering approach, see Welch and Bishop (2004). See De Jong and Huisman (2002) for a more detailed discussion of the model and as a reference for closed form formulae. 5

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ELECTRICITY SPOT PRICE STOCHASTIC MODELS

stochastic driving factor may represent a remedy to this problem, but it is not the only one. In the literature, a different way of improving the traditional financial Brownian-based models is represented by stochastic volatility models. The main feature of stochastic volatility processes is, clearly, the fact that their unconditional volatility moves stochastically over time. This effect is traditionally obtained assuming that the volatility parameter in a Black– Scholes-type model is no longer a known constant but is itself a stochastic process. Black himself commenting on the limits of the modeling approach proposed with Scholes suggested adopting a stochastically time-varying volatility parameter; however, the first application of stochastic volatility for derivative pricing is due to Hull and White (1987). A huge number of different stochastic volatility models have been developed since 1987, but the majority of them can be represented by the following generic stochastic differential equation: dEt = µ(t, Et ) dt + σt Et dB1t dσt2 = α(t, σt2 ) dt + σt2 ξ dB2t

(6.7)

where µ and α are real functions and ξ ∈ R+ and B1t and B2t are correlated Brownian motions. Jump processes and stochastic volatility processes had an independent origin and evolution but it is not difficult to show that they are closely related. The probabilistic concept of subordination allows us to reconcile these two apparently different approaches. Subordination is a transformation of a stochastic process into a new one through a random time change by an increasing process, called the subordinator, independent of the original stochastic process. The process obtained with this procedure is described as subordinated to the original one. In a traditional stochastic volatility model the log-return process can be represented as follows: dZt = µ dt + σ(t) dBt

(6.8)

where σ(t) is a stochastic process independent of Bt . The marginal distribution of Z(t), conditional on σ(t), will be normal with t mean µt and variance σt2 = 0 σ2 (u) du. On the other hand, we can think of a generalized simple Brownian subordinated model as follows: ¯ t = µdt + σ¯ dB(Nt ) dZ

(6.9)

where N(t) is a subordinator process independent of Bt . In addition, for the subordinated process (6.9) the marginal distribution, conditional on the subordinator process, is normal with mean µt and variance σ¯ 2 Nt . Hence, it is plausible to state that the stochastic volatility

PROBABILISTIC MODELING OF ELECTRICITY PRICES

57

process (6.8) and the Brownian subordinated process (6.9) are equivalent in t law if the subordinator process Nt = σ¯12 0 σ2 (u) du. t By construction, the integrated process σt2 = 0 σ2 (u) du possesses the characteristics of a subordinator process since it is right-continuous and increasing with σ2 (0) = 0. In other words, it will always be possible to express a volatility which varies in time as accelerations and decelerations of a stochastic time scale. Moreover, it may be formally shown that a huge class of jump processes can be written as time-changed Brownian motion.6 Hence, as mentioned before, the concept of subordination allows us to reconcile the stochastic volatility approach with those modeling approaches related to the use of jump-diffusion or generalized discontinuous stochastic processes.

6.2 MORE ADVANCED AND REALISTIC MODELS In the previous section we saw that the use of a traditional financial modeling approach for the electricity sector is not a simple task, and the results obtained may often be not very satisfactory. Electricity prices display peculiarities different from those usually displayed by other ‘financial’ assets such as stocks, interest rates, foreign exchange rates and so on. For this reason, for a realistic modeling of electricity price dynamics it seems to be necessary to develop ad hoc instruments. In the present discussion, we will refer to as advanced or realistic models all those models developed in the literature precisely for electricity price modeling. In the last few years academic and industrial research has produced some extremely interesting modeling approaches which correct, or attempt to correct, the bad features displayed by traditional models. Since we cannot discuss in detail all the interesting contributions which have been provided in this field over the last few years, we concentrate our attention on two recent models which, for different reasons, can be considered serious candidates to become evaluation benchmarks in electricity modeling:
The Barlow model; and the
Geman–Roncoroni model.

6.2.1 The Barlow model The original paper by Barlow was published in 1999, at the early stages of the electricity sector liberalization process worldwide. Here we briefly 6

See Fiorenzani (2003) for a detailed discussion on subordination and jump processes.

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ELECTRICITY SPOT PRICE STOCHASTIC MODELS

analyze its characteristics. The Barlow model is a simple but attractive supply/demand model, able to reproduce spikes without introducing jumps as an external and independent source of randomness. The model assumes that supply is non-random and independent of time and that demand is very inelastic with respect to the price level. Assuming that demand can be expressed as a function of the price by means of the following exponential functional representation: D(t) = g(Et )

(6.10)

where g(x) = a0 − b0 xα (g−1 (y) = ((a0 − y)/b0 )−1/α , y < a0 ) then we will obtain an equilibrium price function, obtained by equating supply and demand, given by the following relation:   a0 − Dt 1/α if Dt < a0 − ε0 b0 Et = b0 (6.11) 1/α

Et = ε0

if Dt ≥ a0 − ε0 b0

where ε0 is a cutting value for the price level. The demand function D(t) can be modeled as an Ornstein–Uhlenbeck process where (λ, a1 , σ1 ) are respectively the reversion speed, long-term mean and volatility parameters: D(t) = λ(a1 − D(t)) dt + σ1 Wt

(6.12)

Alternatively, the demand process can be represented as follows: D(t) = a1 + σ1 Yt

(6.13)

where Y(t) is an OU process (λ, 0, 1). By inversion, the price process E(t) can be written as a non-linear Ornstein–Uhlenbeck process (NLOU process) of the following type:  Et =

a0 − a1 σ1 Yt + b0 b0

1/α

= (1 + αXt )1/α

where X(t) is itself an OU process of the following type:   a0 − a1 − b0 σ1 Yt Xt = + αb0 αb0

(6.14)

(6.15)

Finally, we will have: Et = (1 + αXt )1/α 1/α

Et = ε0

if 1 + αXt > ε0 otherwise

where dXt = −λ(Xt − a) dt + σ dWt .

(6.16)

PROBABILISTIC MODELING OF ELECTRICITY PRICES

59

The Barlow model is an attractive model for several reasons. It displays a high degree of mathematical tractability since it is a diffusive semimartingale, and it can be easily calibrated on market data by means of the ML estimation procedure, since its marginal density is a non-linear transformation of a Gaussian density.7 In fact, if the process Y is a non-linear transformation of X, such that Y = f (X), and q(x, y) is the transition density of X, then the transition density of Y, p(x, y) can be calculated according to the following rule:     p(x, y) = q( f −1 (x), f −1 (y)) ·  f −1 (y) (6.17) where f −1 denotes the inverse of f . Moreover, it is able to reproduce in a realistic way the typical spiky behavior of electricity price dynamics and the mechanism for spiky behavior replication is economically consistent, even if, maybe, it is too simplistic. However, there are also negative aspects of the model. For example, if we try to modify one or more of the simplistic economic and mathematical assumptions of the original model in order to make it more realistic, for example with the introduction of a deterministic periodic component in the demand process, or by introducing the possibility of unexpected supply shocks, the mathematical complexity of the model becomes much higher and also the parameter estimation process becomes more difficult. Moreover, even in its simplest version the Barlow model doesn’t produce closed-form solutions for the calculation of conditional expectations, and this represents a very significant problem in derivative pricing, since it prevents the possibility of having closed formulae for the pricing of plain-vanilla derivatives such as forwards or plain-vanilla calls and puts. On the other hand, the Barlow stochastic differential equation (6.16) can be easily discretized by means of the standard Euler scheme, and this make it extremely simple to use this approach within a Monte Carlo simulation routine. In summary, we can state that the Barlow approach represents in many ways the best compromise between all the different exigencies that a stochastic model for electricity prices should provide. It is very well-appreciated also by business practitioners who are not specialists in probability or stochastic calculus because of its economic consistency with the physical market, which stands behind electricity price determination.

6.2.2 A class of NLOU processes As we have seen, the Barlow model effectively belongs to the class of non-linear OU processes (NLOU). In particular, it is a NLOU process of 7

See the original paper Barlow (2002) for a detailed description of the ML estimation procedure.

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ELECTRICITY SPOT PRICE STOCHASTIC MODELS

exponential type. The exponential transformation is justified by the simple supply/demand approach, and by the shape that we want to assume for the inverse supply function. However, from many points of view the exponential transformation is not the best possible solution since it may pose some practical problems, some important ones being: 1 The exponential transformation is not sufficiently flexible to cope with all the price time series observed in electricity markets. Many European market prices display a high degree of non-spiky volatility, which cannot be reproduced by the classical Barlow process. 2 The robustness of ML estimates of the model’s parameters is extremely sensitive to the shape of the non-linear transformation. The exponential transformation is very flat (first derivative almost null) for low-demand values and this may produce an increase in the estimation errors of those parameters which determine the normal regime of the price dynamics. For these reasons we may consider using a more general and flexible class of non-linear transformations, which avoids these problems without introducing an additional source of complexity. An example of an alternative non-linear transformation is the hyperbolic sine transformation. The hyperbolic sine function sinh(z) =

exp(z) − exp(−z) 2

(z = x + iy)

(6.18)

can be generalized to capture all the features of the demand/supply relationship that a particular market can display. Moreover, the hyperbolic sine function is invertible in its entire domain, its inverse function has a known functional representation, and this is extremely important in the construction of the maximum likelihood function of the process. The generic representation of the hyperbolic sine NLOU process can be the following: Et = max(0, α + β sinh(Xt ))

(6.19)

where dXt = −λ(Xt − a) dt + σ dWt . Cutting values for the price level can be easily introduced.

6.2.3 The Geman–Roncoroni model In a recent paper, Geman and Roncoroni (2003) studied in very precise detail the dynamic and distributional features of electricity prices. They considered all the specific features that a realistic and complete probabilistic model for electricity prices needs to have and they proposed an advanced and complex

PROBABILISTIC MODELING OF ELECTRICITY PRICES

61

modeling approach. Here we present only the main features of their model, and we suggest the interested reader refer to the original paper for a more detailed analysis. The Geman–Roncoroni model is a pure mathematical model without any economics derivation. The probabilistic structure used to describe the electricity spot price dynamics is particularly suitable to realistically model spiky behavior. In their paper they proposed to model the log-price process in order to avoid the possibility of negative price outcomes. The electricity log-price process is represented by the unique solution of the following stochastic differential equation: d ln(Et ) = µ (t) dt + θ1 [µ(t) − ln(Et− )] dt + σ dWt + h(t− ) dJt

(6.20)

where µ(t) is a periodic function of time and h(t) is a sign function, which determines the jump direction; θ1 and σ are constant parameters; W(t) is a standard Wiener process; and J(t) is the jump component. The interesting and original part of the completely stochastic process adopted by Geman and Roncoroni is represented by the jump component J(t). J(t) is a special semimartingale process8 called a ‘marked point process’. It is a purely discontinuous process characterized by a jump time process N(t) and a jump amplitude process Ji (t)(R+ process; only positive jumps) such that  t Nt  Ji = x µ(dx, dt) (6.21) Jt = i=1

0




where µ(dx, dt) is the random measure of jumps of J. The intensity process which drives the jump time process N(t) can be modeled both as a deterministic (periodic) function of time or as a level dependent stochastic process. On the other hand, the jump amplitude process is modeled as an exponential process. The exponential process is not the unique alternative we can use to model the jump amplitude process; fact the only restriction we need to impose on the jump amplitude process is that, in any case, it has to be an R+ process. Because of the assumption of independence between the jump time process and the jump amplitude process it is possible to write the compensator process v(dx, dt) as follows: N

t  v(dx, dt) = v(dx, t)dt = E Ji dt 

i=1



= s(t) [0,φ] 8

x1{x\0} p(x, λ) dx dt

(6.22)

See the Appendix to this chapter for more details on semimartingale theory, random measures of jumps and compensators.

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ELECTRICITY SPOT PRICE STOCHASTIC MODELS

where s(t) is the intensity function, and p(x, λ) is the density function of the jump amplitude process. The jump direction function h(t), which has the effect of determining the sign of the jump size process, is modeled by Geman and Roncoroni as a simple level dependent function of the following type: h(t− ) = 1

if Et < Tt

h(t− ) = −1

elsewhere

(6.23)

The Geman–Roncoroni process is a marked point process, hence it is a semimartingale and a Markov process. Note that it can also be thought of as a mean reverting process with positive jumps and a level dependent mean/jumps reversion attitude. In the original formulation the jump reversion rate depends upon the threshold level T(t) of the sign function (6.23). According to the formulation proposed, this class of model is able to potentially capture all the dynamical and statistical features of electricity prices we have analyzed so far. We can model the mean reversion calibrating the parameter θ1 ; we can capture periodicity by means of the µ function; and, finally, we can replicate spike behavior by means of the market point process J(t). Theoretically, another interesting feature of the Geman–Roncoroni model is the calibration procedure they propose in their original paper. In fact, they derive the log-likelihood function of the process by means of its Radon–Nikodym derivative process, which is always known for every semimartingale. Unfortunately, this elegant way of calibrating the process has some bad features which may interfere with its practical implementation. In fact, the likelihood function obtained by means of this methodology depends explicitly on continuous and discontinuous components of the empirical signal. Hence, filtering methods have to be applied previously to separate the two components of the price process. Since the result of the calibration procedure turns to be highly dependent on the cutting levels and filters adopted, as we have seen for other jump processes, the choice of an appropriate filtering method is fundamental. Another inconvenience, which prevents a straightforward application of the Geman–Roncoroni model, is due to the impossibility of deriving closed formulae for the pricing of plain-vanilla derivatives. As we have seen, this inconvenience is common to many modeling approaches in the energy field. Although we have mentioned some significant problems in the practical implementation of this modeling approach, the Geman–Roncoroni model certainly represents the most realistic model developed specifically for electricity spot price modeling. The mathematical intuitions which support the model’s theoretical framework and its flexibility in reproducing typical features of electricity price dynamics lead us to seriously consider this model as a benchmark in electricity price modeling.

PROBABILISTIC MODELING OF ELECTRICITY PRICES

APPENDIX

63

SEMIMARTINGALES IN FINANCIAL MODELING

Traditionally, financial modeling has made strong use of sophisticated mathematical tools belonging to the area of stochastic calculus. Stochastic processes such as Brownian motions, Ito processes or Ornstein–Uhlenbeck processes are commonly used for financial applications. Furthermore, many important theoretical probability results such as stochastic integration, Ito’s lemma and probability measure change theorems are, nowadays, well-known concepts also to non-mathematicians working in the financial sector. However, the growing complexity of the problems typically faced in finance and affine economic sectors calls for a continuous update and upgrade of the mathematical toolsets. The energy finance sector is a typical example where the absence of concrete modeling benchmarks makes the use of basic tools of stochastic calculus and probability insufficient for the construction of realistic models. For this reason we believe it is necessary to raise the level of understanding of financial operators as to stochastic processes, stochastic calculus and probability. More advanced mathematical concepts can help in the construction of more developed and realistic models and can facilitate the full understanding of the basic assumptions and implications of traditional ones. In this Appendix we will try to describe in simple but non-trivial ways the basic foundations and results of semimartingales theory. The class of stochastic processes called semimartingales is the widest and most general class of stochastic processes for which probability theory provides an articulated set of strong results. Brownian motions, Ito processes and Poisson processes are classical examples of semimartingales. This series of structured and strong theoretical results facilitates the use of these stochastic processes in practical applications. Moreover, financial modeling, particularly derivative pricing modeling, is based on two important theorems related to semimartingales: stochastic integration and probability measure changes. This justifies a more in-depth study of semimartingales theory.

Semimartingales and stochastic integrals In general, naïve stochastic integration is impossible. This means that it is impossible to define stochastic integrals as limits of sums as is usual in a Lesbegue–Stieljes theoretical framework. However, according to Ito (1944), it is possible to construct the integral of the function h with respect to the integrator process x, as a limit of sums, if we are able to see the trajectory of x on (ti , ti+1 ). This implies that if we want to apply Ito stochastic integration results, we are forced to restrict our attention to stochastic integrals, whose integrators are represented by an appropriate class of adapted processes.9 Definition: A stochastic process X is called a semimartingale if, for each t ∈ R+
Xt is a process right continuous and left limited (Cadlag);
Xt is an adapted process; and
the linear mapping IX : S → L0 such that:

IX (H) = H0 X0 +

n 

Hi (XT(i+1) − XT(i) )

(6A.1)

i=1 9 A stochastic

process X is said to be adapted to the filtration F if Xt is Ft -measurable for every t belonging to R+ .

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ELECTRICITY SPOT PRICE STOCHASTIC MODELS

where H ∈ S (the class of simple predictable processes) has the following representation Ht = H0 1{0} +

n 

Hi 1(T(i),T(i+1))

i=1

is continuous. Now we are in the position to define formally the Ito’s stochastic integral. Definition: Let X be a semimartingale. The continuous linear mapping JX (H) obtained as the ucp10 extension of the linear mapping, JX (H) = H0 X0 +

n 

Hi (XT(i+1) − XT(i) )

(6A.2)

i=1

is called the stochastic integral of H with respect to X. Using an alternative formulation,

 JX (H) =

Hs dXs = ucp lim H0 X0 +

n 

 Hi (XT(i+1) − XT(i) )

(6A.3)

i=1

Ito’s lemma and stochastic exponentials for general semimartingales Ito’s lemma is maybe the most important result of probability theory used in financial mathematics applications. It allows a determination of the stochastic differential equation describing the dynamics of a certain function of a stochastic process, whose stochastic differential equation is known. In general, the Ito’s lemma version for generalized Brownian motions and Ito processes is widely known in the financial community. This result is essential for the derivation of the classical Black–Scholes partial differential equation. In fact, if we assume, as they suggest in their original paper on stock option pricing, that the stock price process is represented by a classical geometric Brownian motion, dSt = µSt dt + σSt dWt

(6A.4)

and that f is a generic C2 function representing the price of a generic derivative product written on S, we will have that:
df =

 df df df 1 d2 f 2 2 σ S dt + µS + + σS dWt dS dt 2 dS2 dS

(6A.5)

Unfortunately, this nice and easy representation is still not valid when we try to enlarge the class of stochastic processes from geometric or generalized Brownian motions to semimartingales. 10

ucp stands for ‘uniformly on compacts in probability’.

PROBABILISTIC MODELING OF ELECTRICITY PRICES

65

Theorem (Ito’s formula for generalized semimartingales): Let X be a semimartingale and f a C2 real function. Then f (X) is again a semimartingale and the following formula holds:   t 1 t  f  (Xs− )dXs + f (Xs− )d[X, X]cs f (Xt ) = f (X0 ) + 2 0+ 0+     

+

continuous part

f (Xs ) − f (Xs− ) − f  (Xs− )Xs

0<s≤t







(6A.6)



discontinuous part

where Xt− = lim Xs (left lim); and [X, X]cs is the continuous part of the quadratic variation s→t

process of X. Hence, compared with the classical representation we have a jump component of the formula, which was not present before and that considerably complicates Ito’s formula representation and applications. A simple, but very important application of Ito’s formula is the derivation of the so-called stochastic exponential of a semimartingale, also known as the Doleans–Dade exponential. Theorem: Let X be a semimartingale, such that X(0) = 0. Then there exists a unique t semimartingale Z that satisfies the equation: Zt = 1 + 0 Zs− dXs . Z is called the stochastic exponential of X(ε(X)) and is given by:   1 (1 + Xs ) exp(−Xs ) Zt = exp Xt − [X, X]ct (6A.7) 2 s≤t

where the infinite product converges.

Canonical decomposition and representation of semimartingales Sometimes, semimartingales processes are defined through their canonical decomposition property. In fact, semimartingales have the property of being uniquely decomposable in the sum of a canonical martingale process M and a finite variation process A. Hence, if the process X is a semimartingale there exists a martingale process M and a finite variation process A such that: X = X0 + M + A

(6A.8)

This decomposition is unique and is called the canonical decomposition of X. This property is extremely important since the semimartingale process space has the nice property of remaining stable under a variety of transformations such as localization, change of time and change of probability measure (absolutely continuous). Hence, the previous result guarantees that a canonical decomposition always exists and often there are theorems to derive it explicitly. The most famous example is the Girsanov theorem for transformations of the probability measure (see next section). The canonical decomposition formula can be expanded further in order to specify in a more detailed manner the properties of the processes M and A. In particular it is possible to express every semimartingale process as a function of its characteristic triplet. A detailed exposition of semimartingale theory is beyond the scope of the present discussion,11 but 11 For a complete and detailed discussion on semimartingale theory, see Jacod and Shiryaev (2003).

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ELECTRICITY SPOT PRICE STOCHASTIC MODELS

it is in any case important to introduce the formalism implicit in the so-called canonical representation of a semimartingale. Canonical representation: Let X be a generic semimartingale and h a truncation function such that h(x) = x in the neighborhood of zero, then we can represent X as follows:  X = X0 + X C +

 h(x) d(µX − v) +

(x − h(x)) dµX + B

(6A.9)

where X C is the continuous part of X such that [X C , X C ] = C; µX is the random measure of jumps of X; v is the compensator of the random measure of jumps of X; and B is a finite variation process. (B, C, v) is called the characteristic triplet of X and it uniquely identifies a semimartingale through its canonical representation.

Probability measure changes: the Girsanov theorem As is known from standard derivative valuation theory, which will be considered in the next chapter, of particular importance in financial applications of stochastic processes is probability measure changes. The Girsanov theorem is a strong result which allows us to analytically define the transformation of a certain stochastic process, a semimartingale in particular, by means of a change in the probability measure it was originally defined on. This theorem is generally known in the financial field in a simplified version which is applicable to Brownian-based stochastic processes, traditionally used for financial applications. Here we present a general version of the Girsanov theorem, without proving it, as we believe it is important to understand how matters become complicated when we pass from traditional Brownian-based processes to general semimartingales. Girsanov theorem: Let X be a semimartingale in probability space (, , (t )t≥0 , P) satisfying the usual hypothesis, hence X has the canonical decomposition X = M + A under the P-measure. Let Q be a probability measure equivalent to P, then X is a classical semimartingale also under the Q-measure and has a canonical decomposition X = N + C where:  Nt = Mt −

t 0

1 d [Z, M]s Zs

(6A.10)

  P where Zt = EP dQ dP | t s.t. E (Z) = 1 is a uniformly integrable martingale called the Radon–Nykodin derivative process. An alternative and useful representation of the same theorem allows us to analytically express the changes that the probability transformation implies on the characteristic triplet of a semimartingale. In fact, given the assumptions stated above let us assume that (B, C, v) is the characteristic triplet of X (assume X is a d-dimensional semimartingale) under the P measure, then there exists a measurable function Y and a predictable process β = (βi )i≤d satisfying:       ij j   c β   · At < ∞ and  j≤d 

⎛ ⎝

 j,k≤d

⎞ βj cjk βk ⎠· At < ∞ Q as for t ∈ R+

(6A.11)

PROBABILISTIC MODELING OF ELECTRICITY PRICES

67

where A is an increasing predictable process s.t. Cij = cij A, such that the characteristic triplet of X with respect to Q will be: BiQ

⎞ ⎛   ij j =B +⎝ c β ⎠ · A + hi (x) (Y − 1) dv i

j≤d

CQ = C

(6A.12)

vQ = Y · v Hence, for a general semimartingale process the probability measure change has an impact not only on the drift component as in the case of Brownian-based processes, but it also drastically influences its jump structure.

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P A R T III

Electricity Derivatives: Main Typologies and Evaluation Problems

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CHAPTER 7

Electricity Derivatives: Main Typologies

Liberalized electricity markets are characterized by high volatility levels, which mean high risk for both electricity producers and consumers. This high-risk level is not always compatible with agents’ risk attitudes, and hence derivative instruments represent a necessary tool to reconcile agents’ economic exigencies with electricity markets’ natural characteristics. The world of electricity derivatives is relatively young but is extremely dynamic. In fact, the typologies of derivative instruments traded on organized and over-the-counter (OTC) markets change continuously in order to match the dynamic exigencies of electricity producers, traders and final consumers. In this chapter we try to list and describe the most important kinds of electricity derivatives actually traded or typically embedded in electricity physical contracts.

7.1 EXCHANGE-TRADED DERIVATIVES AND OTC DERIVATIVES As in every traditional financial market, derivatives products can be divided into two main categories: exchange-traded derivatives and over-the-counter (OTC) derivatives. The difference between these two groups is mainly due to standardization of the instrument itself. In fact, an organized exchange market, in order to guarantee a regular and liquid trading activity for a certain derivative instrument, first has to ensure that the instrument possesses characteristics general enough to match the exigencies of a large slice of market participants. On the other hand, the typical feature of OTC derivatives is that of being flexible with respect to the economic and financial needs of both 71

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parties involved in the agreement. Another well-known difference between the two groups is the counterparty risk involved in the deal. In organized derivative markets, the clearinghouse is the counterparty of every transaction concluded. Hence its financial solidity is a warranty for a good outcome of the deal itself. In OTC derivative agreements, counterparty credit risk is necessarily a relevant variable, which should always be considered in deal evaluations and conclusions. Most electricity derivatives transactions are concluded over the counter. However, an increasing number of exchanges worldwide, such as Nymex, EEX, Powernext, Neta and Nordpool, quote electricity derivative products. The products traded are mainly plain-vanilla instruments such as futures, forwards, swaps and plain-vanilla options. Let us, here, try to identify the main peculiarities which characterize derivative instruments traded in these markets.

7.1.1 Electricity forward/futures contracts Forward and futures contracts are the simplest examples of derivative products. They differ only in the settlement procedure, which for futures is ‘marketed to the market’ on a daily basis, while for forward contracts it occurs at the natural maturity of the contract. Almost all exchanges worldwide actively trade forward or futures contacts on base load and peak load electricity prices. The contracts actually traded in different markets display different characteristics related to:
delivery type; and
payoff type. Forward/futures contracts can have a physical or a purely financial delivery type. The Powernext future contract, for example, is a contract with physical delivery, while the Nordpool one is a standard purely financial contract. The future contract traded at the German electricity market EEX is a mixed physical/financial contract. In fact, the contract buyer can decide to have a physical or financial delivery some days before the maturity of the contract itself. The standard payoff of a forward contract maturing at T is (E(T) − F), where F is the forward price established at contract birth and E(T) is the electricity spot price at the contract’s maturity. However, we frequently find forward contracts traded in organized exchanges with Asian-type payoffs. This feature implies that the contract payoff depends in some way on the average spot price. The French futures contract traded at Powernext is an

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example of such a type of payoff formula. This payoff feature makes the forward contract practically indistinguishable from the plain-vanilla electricity swap we will now analyze.

7.1.2 Electricity swap Energy swaps are typical hedging instruments for the oil and gas markets (for example, a Brent swap), and the electricity market is following the example of these two more mature markets. Plain-vanilla swaps are also known as ‘contracts for differences’ (CFDs); they are contracts to exchange a floating electricity price for a fixed one. Typically, the floating leg of the contract is linked to the average spot electricity price calculated over a certain (quite long) period such as a month, a quarter or the year (calendar swap). The unitary payoff of such an instrument is then given by the following formula:  
 N 1  (7.1) Ei − K Swap payoff = N i=1

Since they are simple linear derivative instruments, plain-vanilla electricity swaps can be hedged and priced by means of a simple portfolio of electricity forward contracts: Swap value =

N 1  (Fi − K)βi N

(7.2)

i=1

with βi a unitary discount factor. In addition to plain-vanilla swaps, which are usually traded in both organized exchanges and over-the-counter markets, there is a wide set of ‘exotic swap contracts’ such as variable volume swaps (swaps with swing rights), differential swaps (time-spread swaps), spark spread swaps (commodityadjusted spread swaps) and many others, which are traded exclusively in OTC markets.

7.1.3 Plain vanilla options Not many electricity exchanges regularly trade options. In Europe, only the Nordpool market trades European-style options written on season-forward contracts. In the USA, plain-vanilla options on futures of American style are traded at the Nymex. In general, exchange-traded options are purely financial options written on forward or futures contracts with a quite long maturity (a season

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or longer), which makes the pricing and the standardization much easier. However, the traded volume for such kinds of instruments is very low, while the traded volume of OTC electricity options is much higher. In over-thecounter markets, not only plain-vanilla options on futures are traded but also plain-vanilla options whose payoff depends more directly on the spot electricity price. Some of them could be:
Block options. These are OTC options whose underlying asset is a block of hours (for example 17–20) of a certain day or of a certain group of days. They can display typical option or swaption features in the sense that the opportunity to exercise can be limited to one period or to a multi-period. Sometimes, electricity physical grid-owners or transmission managers use this kind of contract for medium-term regulation and optimization activities.
Hourly options. These are OTC options whose underlying asset is a single hour of a single day or of a group of days. They display more or less the same characteristics of block options. They are important because they represent the maximum flexibility tradable in electricity markets, although they are not intensively traded since they are extremely risky instruments. Plain-vanilla options can be combined together in order to create derivative structures such as caps, floors or collars. Actually, there are no evaluation models for electricity plain-vanilla options which could be considered benchmarks such as the traditional Black–Scholes model. This is mainly due to the peculiar stochastic dynamics that electricity spot and forward prices display, as we have seen in previous chapters, and to some difficulties in the straightforward application of standard derivative pricing approaches in electricity markets. The next chapter will be focused on this last topic. However, traditional pricing models such as the Black–Scholes or the Black model can be used at least to have a proxy of the true fair price, especially when the underlying asset is an electricity forward contract or an average of spot prices. Numerical methodologies can be applied for a more exact pricing.

7.1.4 Electricity swaptions In general, an electricity swaption is a European option on an electricity swap, that is an option to enter into a swap contract at option maturity. The option is called American if the exercise opportunity is not limited to the maturity time but extended to the whole option lifetime. When the option is a call option, then the swaption is called a payer swaption, since the owner has

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the right to enter into an electricity swap where he pays the fixed price K and receives the floating electricity price. On the contrary, when the option is a put option, the swaption contract is called a receiver swaption and the owner has the right to enter into the same swap but with an opposite position. The payoff function of a payer swaption can be written as follows:   N 1  Swaption payoff = max 0, (Fi − K) (7.3) N i=1

Generally, swaptions contracts are traded exclusively in OTC markets even if they are quite standardized and not extremely exotic in their payoff. We have already explained that a portfolio of forward contracts can replicate a swap contract; hence, a swaption contract can be seen as an option on a portfolio of forwards. No analytical closed-form formulas are available to directly price this kind of instrument, and hence numerical techniques should be used for the evaluation.1 Similarly to swaptions, instruments such as captions and floptions can also be defined as plain-vanilla options written on a portfolio of call options (cap) or put option (floor).

7.2 EXOTIC OPTIONS Usually a derivative product is defined as exotic when its payoff function is more complex than that of standard plain-vanilla instruments, such as those described in the previous section. Exotic derivatives are not very much used in electricity markets, but it is important to provide here a brief description of their main typologies as they may become important electricity trading instruments in future years. The high degree of complexity in the payoff function of an exotic option may be because the maturity value of the derivative depends on the terminal value of the underlying asset (the electricity spot or forward price), eventually, jointly with the terminal value of other underlying assets, through a complex non-linear function: Exotic derivative payoff function = g(ET , FT , XTi )

(7.4)

However, it is often the case that an option is defined exotic because its payoff is a function of the path followed by the underlying asset price during the life of the option itself (g(E˜ t,T )). In this case, the exotic option is defined as a ‘path-dependent option’: Path-dependent derivative payoff function = g(E˜ t,T ) 1 A review

on numerical techniques for derivatives valuation is given in Chapter 9.

(7.5)

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ELECTRICITY DERIVATIVES

It is difficult to give an exhaustive presentation of all the exotic derivative typologies traded in energy and purely financial markets. In fact, since they are typical OTC instruments they are subject to rapid change to match the needs of market players. However, we shall try to delineate the main features of the most common exotic derivatives actually traded.

7.2.1 Time-spread options Time-spread options are exotic options on futures. In particular, they are options on the spread between two futures contracts written on the same underlying asset, but with a different maturity (option on time-spread futures). The owner of such a type of option can efficiently hedge his position against changes in the shape (mainly the slope) of the forward curve. Of course, we can have time-spread calls and puts of European or American exercise style.

7.2.2 Commodity-spread options Commodity-spread options are typical options on futures contracts. Here, the futures contracts involved have the same maturity (not necessarily), but are written on two different commodities such as, for example, electricity and coal. The payoff at maturity of the option depends on the differential (spread) between the prices of futures contracts on the two different commodities. The use of such a kind of derivative instrument is a natural solution for all those industrial companies which have a natural exposure to a certain commodity price-spread fluctuation, because, for example, one commodity represents a productive input (cost) and the other a productive output (revenue). Power generation or oil refinery activities provide useful examples of this kind of risk exposure. The famous spark spread option, mentioned more than once in this book, is a subclass of the commodity-spread options particularly important in the world of electricity. (See Section 7.3.2, p. 81.)

7.2.3 Performance-spread options For these instruments, the spread is calculated on the relative performance of two different energy products, although the two energy products involved could both be electric, for example peak and off-peak electricity prices. For a standard call option, the payoff function might display the following functional form:     E1T E2T Performance spread call payoff = max 0, − 2 −K (7.6) E1t Et

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7.2.4 Asian options Asian options are typical instruments in commodity markets. They are simple path-dependent options whose final payoff depends in some way on the average price of the underlying asset over a certain period called the averaging period. In energy markets, these kinds of options are extensively2 used to hedge exposure since many oil, coal and gas contracts are related in some way to the average commodity price over the delivery period. In addition, in electricity markets Asian options are commonly used instruments because the averaging procedure effectively reduces the volatility incorporated by spot electricity price fluctuations. For this reason they usually cost less than standard plain-vanilla options. It is possible to divide the Asian option class into two main categories: average price options and average strike options. The average price option final payoff depends upon the average price of the underlying asset in the following way: 

 N 1  Average price call payoff = max 0, Ei − K N i=1   N 1  Ei Average price put payoff = max 0, K − N

(7.7)

i=1

On the other hand, average strike options depend both upon the average underlying asset price, over the averaging period, and on the final underlying asset price: 

 N 1  Average strike call payoff = max 0, ET − Ei N i=1   N 1  Average strike put payoff = max 0, Ei − ET N

(7.8)

i=1

Asian options characteristics can vary as regards the averaging period, which could coincide or be smaller than the time to maturity of the option, or regarding the exercise rule of the contract which could be European or American according to the exigencies of the contract parties. Within the Black–Scholes–Merton theoretical framework, an evaluation benchmark is represented by the Lévy–Vorst model which, however, cannot be considered as a realistically applicable model in the electricity field. Hence, numerical 2

In the oil market, Asian options are considered the true plain-vanilla options and are usually much more liquid instruments than standard calls or puts.

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procedures such as Monte Carlo simulations have to be used to price such kinds of instruments.3

7.2.5 Barrier options Barrier options are particular types of path-dependent options that have assumed a particular importance both in standard financial sectors and in the energy finance field. Their main characteristic is that the option starts or ceases to exist if the underlying asset price reaches a barrier, at any instant during the option lifetime.4 They are of interest particularly due to the fact that they are cheaper than standard options and consequently the cost involved in a hedging strategy made up of barrier options is lower. Unfortunately, a lower cost implies a much higher risk, especially when the underlying asset value is close to the barrier. There are many kinds of traded barrier options which can be divided into the following four groups: up and out; up and in; down and out; down and in. Obviously, both calls and puts can display barrier-option features. For a complete description of all kinds of barrier options and their evaluation models, within the Black–Scholes framework, the interested reader can refer to Musiela and Rutkowsky (1996). In the electricity field the barrier option feature is not very much used. In fact, since electricity spot and forward markets are characterized by high volatility and spikes, the fact that a hedge may disappear after a strong and fast price move can represent a non-sustainable risk.

7.2.6 Options on the maximum or minimum (lookback options) Lookback options are particular path-dependent options whose payoff depends in some way on the maximum or minimum level reached by the underlying asset price during the lifetime of the option itself. The range of lookback options to be found traded in financial markets or described in the literature is extremely wide, and we refer again to Musiela and Rutkowsky (1996) for a more detailed discussion. Lookback options are considered important derivative instruments in electricity markets because they can effectively be used to hedge sudden and large price spikes. In fact, they can be structured in such a way as to cap the biggest n price spikes which occurs in a certain period of time. Lookback options’ payoffs can be thought of and structured in a very exotic way, but obviously this implies a high degree of complexity in the pricing techniques. 3

See Chapter 9. can eventually be present in the option.

4 A rebate

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7.3 OPTIONS TYPICALLY EMBEDDED IN ELECTRICITY PHYSICAL CONTRACTS In the previous section we saw how in over-the-counter electricity markets an extremely wide range of exotic derivative instruments can be traded for hedging and trading purposes. However, financial derivatives’ complexity is nothing compared to the complexity involved in options typically embedded in physical electricity contracts. Here we deal only with the two main types of physical embedded options: swing rights (swing options), and tolling rights.

7.3.1 Swing options Swing rights are typical options embedded in physical gas and electricity contracts. According to such a type of clause the contract buyer agrees to purchase up to a maximum volume of the underlying commodity (gas or electricity) in a given period of time and at a fixed price. The contract is often characterized by a minimum volume that the buyer has to take in the same period. Long-term agreements can be equipped with more than one swing opportunity during the global duration of the contract, and often the global maximum and minimum volumes which can be taken are smaller and greater, respectively, than the simple sum of the period maximum and minimum volumes (non-trivial volume constraints). Penalty payments can be imposed if the volume constraints are exceeded in order to stimulate the buyer to respect the volumetric limits imposed. Contract schemes are normally defined dividing the total delivery period [0,T] into N sub-periods,5 by means of the time partition Ti : 0 ≤ T1 ≤ T2 ≤ · · · ≤ TN−1 ≤ T

(7.9)

Over each one of the N sub-periods, and over all the duration period of the contract, minimum and maximum delivery quantities are established:
SQ = sub-period quantity
LSQ = lower sub-period quantity
MSQ = maximum sub-period quantity
LGQ = lower global quantity
MGQ = maximum global quantity 5

Typically the day, but it could also be the week or the month.

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ELECTRICITY DERIVATIVES

The contract buyer should respect the following relations: LSQ ≤ SQ ≤ MSQ

(7.10)

and LGQ ≤

N 

SQ ≤ MGQ

(7.11)

i=1

The sub-period volume constraints are non-trivial with respect to the global volume constraints if LGQ > N ∗ LSQ

and

MGQ < N ∗ MSQ

(7.12)

The difference between swing contracts with trivial and non-trivial volume constraints is extremely important in the pricing and hedging of the contract itself. The aim of such a kind of contractual clause is to guarantee the buyer a certain degree of flexibility in the volume which can be taken, since, typically, the gas and electricity consumer is not always in the situation to know exactly, a priori, the quantity he is going to consume period-by-period and in total. Since the commodity, the object of the contract, is usually costly to store,6 the flexibility in the taken volume that the swing right guarantees is very important and extremely valuable. The pricing and hedging issues related to swing options are fundamental challenges in energy markets. Unfortunately, in general, this is not a simple topic, especially for electricity swing options. Numerical procedures have to be used in order to determine the exact value of a non-trivial swing option.7 In fact, if the volume constraints imposed by the contract are trivial, the swing contract is intuitively decomposable in a portfolio of simple forwards and plain-vanilla options. On the contrary, in the general case of non-trivial volume constraints, the swing option contract can be seen as a portfolio of American call options with co-dependent exercise, which cannot be priced by means of standard financial techniques. Stochastic dynamic programming or stochastic control techniques can provide numerical solutions to the swing option-pricing problem. We will deal with such a problem in the Appendix to Chapter 9. The pricing and hedging schemes are not the only features we would like to know about a swing contract. In addition, the optimal exercise policy is fundamental information which should be obtained from a complete and realistic evaluation model. 6 In the case of electricity, storage is impossible, and hence the storage cost can be though of as infinite. 7 A swing option-pricing example is proposed in Chapter 9.

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7.3.2 Tolling rights and spark spread options Tolling agreements are contracts which usually involve a power producer and a power marketer (toller). In general, according to this kind of contract, the toller has the right (the option) to use the power plant of the producer in order to transform a certain quantity of fuel into electricity. On the other side, for this service the power producer is entitled to receive a fixed tolling fee which is supposed to cover its fixed production costs and the energy transformation service price (Figure 7.1). The whole duration of the tolling agreement is divided into sub-periods, and at the beginning of each sub-period the toller informs the power producer about the fuel quantity he wants to transform into electricity for that period. Minimum and maximum period quantities, or also swing rights, can be established in order to match the exigencies of both parties to the contract. Tolling contracts are typically physical contracts, hence they are usually equipped with ancillary clauses which regulate the rights and dues of the parties in the case of asset default or outages, fuel misprocurement, transportation problems, and so forth. However, it could also be the case that the tolling agreement is purely financial. In this case the toller does not physically take the electricity, he just receives the financial equivalent coming from selling the electricity in the spot market. By subscribing to a tolling agreement, the power producer remains responsible only for those operational risks related to the generation activity; while the market risk originated by the fluctuation of fuel and electricity prices belongs to the toller. The financial valuation of tolling contracts is related to the so-called spark spread of the generation asset involved (at least virtually) in the tolling. The spark-spread is a synthetic indicator of the financial efficiency of a certain power plant. In fact, it combines the financial convenience to transform fuel into electricity, given the market conditions, and the physical efficiency of the plant measured by its heat rate (which is the quantity of fuel necessary

Power producer Fuel

Tolling fee

Electricity

Toller

Figure 7.1 Tolling contract scheme

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to produce 1MWh of electricity). Formally, the spark spread is defined as follows: spark − spread = (Ppower − HR ∗ Pfuel )

(7.13)

From a financial point of view, tolling agreements can be evaluated as strings of options on the spark spread.8 Spark spread options are nothing other than options to exchange one asset for another (modified using a conversion rate), and their payoff can be expressed as follows: payoff = max((Ppower − HR ∗ Pfuel ), 0)

(7.14)

These kinds of derivatives are actively traded in OTC electricity markets because they represent the natural hedge to a power generation asset exposure. The benchmark model for the valuation of these kinds of options is the Margrabe model (1978), which is a classical BSM model modified to take into account the correlation between the two assets. Synthetically, the Margrabe formula applied to the spark-spread option pricing problem is given by the following expression:  √  power N(d) − HR · Ptfuel N(d − σ T) (7.15) SSoption = exp(−rT) · Pt where T is the option maturity; r is the risk-free rate; N(.) is the cumulative density function of the normal distribution: √ /HR · Ptfuel ) + 0.5 · σ T √ σ T

power

d= σ=

log(Pt 

2 2 σpower + σfuel − 2ρσpower σfuel

where ρ is the correlation factor.

8

This representation is very general and does not consider all the possible features which can be incorporated into a tolling contract.

CHAPTER 8

Electricity Derivatives: Valuation Problems

The Black–Scholes–Merton theoretical framework represents the benchmark for derivative pricing in standard financial markets. However, in the electricity field some of the basic assumptions of this famous approach are violated, and consequently the traditional derivative pricing approach has to be revised in order to work properly for electricity derivatives. In this chapter we will briefly review the traditional pricing approach focusing attention on those assumptions which cannot be considered realistic for electricity derivatives.

8.1 DERIVATIVE PRICING: THE TRADITIONAL APPROACH The traditional derivative pricing valuation theory, initiated by Black, Scholes and Merton at the beginning of the 1970s and further developed some years later by Harrison and Pliska (1981) and by Delbean and Schachermayer (1994) is not a standard microeconomic valuation theory, but it can be defined as a ‘relative pricing theory’. In fact, while in the classical microeconomic approach a certain asset equilibrium price is determined by the interaction of its demand and supply curve, in contingent claim valuation we usually refer to the concept of a fair price. The fair price of a contingent claim is the price which prevents possible arbitrages between the contingent itself, its underlying asset and all the other contingent claims written on the same underlying asset. In the classical Black–Scholes–Merton theoretical framework the fair price of a generic derivative contract1 is determined by a dynamic self-financing 1

Here we use the terms contingent claim and derivative product synonymously. 83

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trading strategy, made up of a certain quantity of the underlying asset and of the risk-free one. This dynamic trading strategy is intended to perfectly replicate the continuous-time cash flow generated by the derivative product. By no-arbitrage arguments, they conclude that since the replicating portfolio and the derivative produce exactly the same cash flow they should have the same price at the evaluation date. Hence, the derivative price will result as a function of the underlying asset price and of the risk-free asset value (interest rate level). This theoretical paradigm is based on the assumption that any traded derivative is a redundant security and hence its equilibrium price necessarily depends on the underlying asset equilibrium price. Obviously, efficient and frictionless financial markets guarantee that the underlying asset price is the equilibrium price and consequently the derivative fair price notion is consistent with that of an equilibrium price. A market model which assumes that all traded contingent claims are replicable by means of a dynamic self-financing trading strategy made up of a certain quantity of the underlying asset and of the risk-free one is called a complete market model. It is usually said that in a complete market model all contingent claims are ‘attainable’. Within the same theoretical market model considered by Black, Scholes and Merton, Harrison and Pliska have obtained a very important result, known as the first fundamental theorem of asset pricing. This theorem relates the economics notion of no-arbitrage and the mathematical notion of a martingale process. In particular, they have shown that if the market is complete and does not allow arbitrage opportunities, then there exists a unique equivalent martingale probability measure Q such that the fair price of a generic contingent claim, whose payoff at maturity is a function, g(ST ), of the underlying asset termination price, can be obtained by means of the following relationship: g(St ) = EQ [β g(ST )]

(8.1)

where β is a discount factor. Again, assuming the non-admissibility of successful arbitrage opportunities but not necessarily market completeness, Delbean and Schachermayer, in a series of highly technical papers (synthetically known as the second fundamental theorem of asset pricing), define the conditions necessary for the previous relation to still remain valid. However, in this enlarged theoretical environment it is no longer possible to talk about a unique fair price, but we will refer to a set of admissible no-arbitrage prices. Mathematically, this means that under the conditions established by Delbean and Schachermayer the equivalent martingale measure Q is not unique, but there exists a set of admissible martingale measures Me such that the pricing measure Q belongs to Me .

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85

Non-storability of the underlying asset is one of the main causes for market incompleteness for electricity derivatives. As a consequence, in this situation the straightforward application of the classical no-arbitrage argument is not always possible. For this reason it is necessary to describe in a detailed way pricing techniques in a situation of market incompleteness. Let us start by describing the implications that non-storability has on the simplest derivative product, the forward contract.

8.2 THE SPOT-FORWARD PRICE RELATIONSHIP IN TRADITIONAL AND ELECTRICITY MARKETS The simplest derivative instrument available on the market is the standard forward contract. In traditional financial markets, the simplicity of such a kind of derivative instrument allows for the construction of very nice and easy trading strategies, which permit the exploitation of potential arbitrage opportunities between spot and forward market products (cash and carry trading strategies). This possibility, together with the assumption of an extremely liquid and frictionless market, can generate exact spot-forward price no-arbitrage relationships.2 In commodity markets, this relationship is weaker and sometimes ceases to be valid, which is exactly the case for non-storable commodities such as electricity. In this section we analyze the relationship between spot and forward prices, how the no-arbitrage pricing axiom impacts on the modeling of forward price dynamics, and finally the relevance and the limitation of general results in the case of non-storable commodities. The logic path we will follow will be that of examining the spot-forward relation in three different markets: equity, storable commodities and non-storable commodities such as electricity.

8.2.1 The spot-forward relationship in stockmarkets It is a widely known fact that in an efficient and frictionless stockmarket, where participants can borrow and lend at a unique risk-free rate (r), the forward price of a stock is linked to the spot price by the following no-arbitrage relation: F(t, T) = St exp(r(T − t))

(8.2)

The straightforward proof of this comes from the simple application of the famous ‘cash and carry’ trading strategy associated with the principle of no arbitrage. 2

See Hull (2000) for further details on standard spot-forward relationships.

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Two important conclusions can be inferred from the above simple relationship:
The forward price is a deterministic function of the spot price and of the time to maturity (T − t) of the forward itself.
The forward price converges smoothly to the spot price as the time to maturity goes to zero. lim F(t, T) = St T→t

(8.3)

The last conclusion is particularly important since it allows derivation of the forward price stochastic dynamics moving from the spot price forward and vice versa.

8.2.2 The spot-forward relationship in storable commodity markets When the underlying asset of the forward contract is a storable commodity the relationship between forward and spot prices as described in the previous section becomes more complicated, but does not disappear. In fact, if we define as U the total discounted storage cost over the time period (t, T), a trading strategy known as ‘cash and stock’ can be used, together with the classical no-arbitrage assumption, to show that: F(t, T) > (St + U) exp(r(T − t))

(8.4)

Unfortunately, this is not an exact relation but is just a lower-bound relation, although the fact that the lower bound converges to the spot price as the forward contract reaches its maturity remains valid. The cash and stock trading strategy effectively induces a dynamic relationship between spot and forward prices, which is obviously influenced by limited inventory facilities and production capacity. For example, we could consider the effect that, typically, news of a future drop or rise in production of oil immediately has on today’s spot price. The need to model this kind of dynamic relationship between spot and forward prices in storable commodities markets has led to the notion of a convenience yield y representing the additional benefit coming from the ownership of the physical commodity, instead of a forward contract. With the introduction of the notion of convenience yield, the spot forward relationship becomes:
 (8.5) F(t, T) = (St + U) exp (r − y)(T − t) The convenience yield can be thought of and modeled as a deterministic or as a stochastic function. This obviously has a deep impact of the forward prices dynamics.

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87

8.2.3 The spot-forward relationship in the electricity market It is quite straightforward to infer from our previous discussion that the cash and stock trading strategy is not available for non-storable commodities such as electricity. Because of this fact, the dynamic relationship between spot and forward prices that we have described does not hold for electricity and the smooth convergence vanishes. This fact has an important impact on the way we model electricity spot and forward prices and on their relation. From a mathematical point of view it is important to emphasize that since the smooth convergence is no longer valid it becomes impossible to derive the spot price stochastic dynamics from forward prices one and vice versa: lim F(t, T)  = St T→t

(8.6)

The failure of a the spot-forward smooth convergence is extremely important, and has an important effect on traditional derivatives pricing arguments based on the notion of no arbitrage and perfect replication. From the mathematical point of view, this fact is essentially related to the discontinuities in the electricity spot price process induced by nonstorability (electricity spot price spiky behavior). A formal proof of the lack of convergence between spot and forward electricity prices can be constructed by exploiting the results of generalized stochastic integrals limit theory. However, rather than a formal mathematical proof, we are more interested in showing the validity of relation (8.6) by means of trading and risk arguments. To this end, let us consider the risk involved in a long (or short) position in a generic forward contract. The risk of the position is essentially related to the final payoff uncertainty characterizing the position. In fact, the position payoff is: forward payoff = (S(T) − F(t, T))

(8.7)

We know from previous paragraphs that the higher the uncertainty about the future spot price realization S(T), the higher the risk embedded in the position since, once established, the forward quote F(t, T) is fixed. We also know that since forward contracts are zero premium contracts, the forward quote F(t, T) should not simply reflect the expectation of a future spot price realization but also fair remuneration for the position market risk. If we assume that the underlying spot price is not characterized by discontinuities we will have that: lim S(t) = S(T)

t→T

(8.8)

Hence, we can state that, given the assumption of continuity for the underlying spot price, the spot forward convergence holds because the risk

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embedded in a forward position, the risk premium, tends to vanish as the forward contract maturity becomes shorter. Formally: lim Risk[(S(T) − F(t, T))] = 0 ⇒ lim F(t, T) = S(T)

t→T

t→T

(8.9)

Here, we refer with the generic term ‘Risk’ any kind of risk measure we believe is suitable to describe the risk embedded in the trading position. The result (8.9) remains valid for any kind of reasonable risk measure. For example, it is valid if we use a percentile measure like value-at-risk but also if we simply use the statistical variance of the position value’s potential drop. If we no longer assume continuity of the underlying price process expression (8.8) is no longer valid, and consequently the risk embedded in the forward contract position does not vanish as the forward contract maturity vanishes. As a consequence, the forward price quote F(T − ε(τ), T), of the forward contract with the shortest possible time to maturity ε(τ), should contain the non-zero price of this non-vanishing market risk, related to the underlying price discontinuity. Hence, smooth convergence is prevented. Finally, we see here that non-storability and spikes lead us into a world where the pricing of a simple forward contract is not a simple task; this is the world of incomplete markets.

8.3 NON-STORABILITY AND MARKET INCOMPLETENESS In the first section of this chapter we noted that the Black, Scholes and Merton theoretical environment is based on the assumption of an efficient, extremely liquid and frictionless market model. In other words, a perfect market model. When we introduce ‘market imperfections’ to this traditional theoretical framework, such as market incompleteness, restrictions on short-selling, different borrowing and lending rates and so on, the standard no-arbitrage arguments cannot generally be applied. Hence, the classical notion of no arbitrage (or fair) price of a derivative product is no longer well-defined. In this section we deal, in particular, with the notion of market incompleteness as a particular kind of market imperfection. As we have mentioned briefly in previous sections, a market model is defined as incomplete when not all the contingent claims traded are attainable. The financial concept of non-attainability has many economic and mathematical implications that we are not going to discuss in full here. However, let us note that the definition we gave of market incompleteness is not the only one possible, even if, maybe, it is the most appropriate in the actual context. When we introduce a market model which is intrinsically incomplete, the risk exposure associated with derivative products trading cannot be diversified away totally by means of an appropriate continuous time trading strategy, not even in theory. This implies that the determination of a

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unique fair price for the contingent claim is no longer possible even if the no-arbitrage assumption still holds. Thus, the contingent claim price that a certain market participant is willing to pay will depend, within a certain range, on assessment of future market behavior and on the position he is willing to take on the deal (bid/ask spread). The most important situation that causes market incompleteness is when the underlying asset is not a traded asset. This could also be the case for derivative products written on non-storable commodities such as electricity (but also weather variables). In fact, if we consider an electricity derivative or a weather derivative it is intuitively simple to understand that a replicating trading strategy made up of the underlying asset and the risk-free one is not available. In fact, it is not possible to hold a non-storable commodity such as electricity or a weather variable such as the rain level or temperature. This makes the electricity derivative or the weather derivative non-attainable, and, therefore, the market model incomplete. Moreover, in the case of electricity, market incompleteness may also be generated by the fact that the electricity spot price dynamics is characterized by discontinuities (spikes) which represent additional sources of non-eliminable risk. In conclusion, we can state that when the market model is incomplete, the no-arbitrage assumption is no longer a sufficient criterion for the identification of a unique fair price for a derivative product. It is, then, necessary to invoke supplementary economic criteria, which support no arbitrage and allow the determination of a contingent claim price, which is no longer unique even if free of arbitrage. In the following section we present the basic results on derivatives pricing and hedging techniques in incomplete market conditions.

8.4 PRICING AND HEDGING IN INCOMPLETE MARKETS: BASIC PRINCIPLES There are two alternative approaches to the problem of pricing derivatives in incomplete markets. The first is a utility-based approach (or equilibrium approach) which is considered the most traditional and economically more intuitive. The second is a purely mathematical pricing approach and which we will refer to as the minimal distance pricing approach. In general, however, these two approaches do not have to be considered in opposition to each other. In fact, recently, it has been shown that they can be reconciled in a unique vision and we will see how.

8.4.1 The utility-based approach When the market model is incomplete not all the contingent claims traded are attainable, hence a unique no-arbitrage price does not exist. However,

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the investor’s overall objective will still be that of trading the derivative instruments if and until it increases its utility to do so. Consequently, in order to evaluate contingent claims, it is in general possible to apply the traditional theory of asset valuation based on optimal consumption or wealth principles. An excellent exposition of this theory is given in Foldes (1990). Here it is sufficient to remind ourselves of some basic issues. If we assume that T is the maturity date of the derivative and that B(XT ) is its exercise value at time T, with XT as the underlying asset value at T, then the investor’s objective function will be that of maximizing the expected utility, E[U(HT )] coming from the investment portfolio value at maturity H(T), which is made up of the derivative, of the underlying asset and of the risk-free one. Obviously, the composition of the optimal portfolio of tradable assets which maximizes the investor’s objective function, will depend on the investor’s initial endowments and preference structure. We can remind ourselves that, basically, the utility function shape determines the investor’s risk attitude. Closed form solutions to the utility maximization problem are given, under specific assumptions, by Davis (1997). The main problems in the application of a utility-based pricing approach are mainly related to the difficulty in the specification of the inputs required for optimization. In fact, rarely is a corporation or a single investor able to exactly specify the current endowment, the joint stochastic dynamics of the relevant asset values, or the utility function over all wealth levels. Moreover, even if these variables are in some way inferred by past decisions, it is frequently observed that they are inconsistent over time and across assets. The famous Allais paradox represents an example of this last statement. All these facts make the standard equilibrium-based pricing approach difficult to be used for practical applications, even if it is extremely useful for the full economic comprehension of the implications of market incompleteness. We need to find alternative solutions to the problem.

8.4.2 The minimal distance pricing approach From a mathematical point of view, the fact that a dynamic investment strategy which exactly replicates the contingent claim does not exist, implies that the equivalent martingale measure Q is no longer unique, and that, potentially, there exists an infinite number of them. In general, we denote with Me the set of equivalent martingale measures such that Q belongs to Me . All the elements of Me satisfy the no-arbitrage condition.3 Hence, in this situation it will not be possible to talk about the no-arbitrage price of a certain contingent claim, but we will deal, necessarily, with the set of no-arbitrage prices of the contingent. This set is specified by 3

Here we assume that Me is a non-empty class of equivalent martingale measures.

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the following formulation:
= {EQ [βC] : Q ∈ Me }

(8.10)

where β is the discount factor and C is the derivative final payoff function. In all the situations of market incompleteness, in order to price contingent claims we need to find a criterion which supports and integrates the no-arbitrage condition and which allows us to select a probability measure, among all the measures which belong to Me . From a strictly mathematical point of view, given a certain space V it is possible to define a distance function. In our context, the space V represents the space of probability measures Me which satisfy some notion of the no-arbitrage condition and are equivalent to the natural probability measure Q. In particular we want to define a function I(Q, Q ) which measures, in some sense, the distance between two equivalent measures belonging to Me . Generally, the function I(Q, Q ) is defined as follows:    dQ  Q with Q equivalent to Q I(Q, Q ) = E  (8.11) dQ where (x) is a real function of x ∈ (0, +∞) such that (0) = 0. A probability measure Q0 belonging to Me can be defined as the ‘minimal distance martingale measure’ if it satisfies the following condition:    dQi I(Q0 , Q) = min I(Qi , Q) = min EQ  (8.12) Qi ∈Me Qi ∈Me dQ The functional  has to be continuous, strictly convex and differentiable in its domain such that I(Qi , Q) > 0, for each Qi belonging to Me , and I(Qi , Q) = 0 only if Qi ≡ Q. Particular choices of the functional form of  induce different choices of the equivalent pricing measure. In the literature some cases have been studied in particular: 1 the ‘minimal variance measure’ when (x) = x2 ; 2 the ‘minimal entropy measure’ when (x) = x ln(x); √ 3 the ‘minimal Hellinger measure’ when (x) = − x; and 4 the ‘minimal reverse entropy measure’ when (x) = −ln(x). Concerning the existence and the uniqueness of the minimal distance measure, general results do not exist, but some conditions are specified when the distance function assumes particular forms. For example, Delbean and Schachermayer (1994) delineate the existence conditions for the minimal variance measure, while Frittelli (2000) considers the case of the minimal entropy measure.

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Generally, the two different pricing approaches presented in this section, the utility-based approach and the minimal distance approach, are not in total opposition. In fact, the choice of an equivalent martingale measure which minimizes a certain distance with the natural measure, can be justified using arguments related to the problem of an agent’s utility maximization. This is shown in Frittelli (2000) for the minimal entropy martingale measure and in Bellini and Frittelli (2002) for the general case. Bellini and Frittelli (2002) show that, in general, the distance-minimizing problem for the two equivalent measures can be seen as the counterpart of the agent’s utility maximization problem. Clearly, a different functional form of the utility function that the agent maximizes corresponds to each particular choice of the ‘distance function’.

8.5 CALIBRATING THE PRICING MEASURE As we have seen in previous sections, the criteria for the identification of an equivalent martingale measure, which is ‘minimal’ in some sense, are numerous and most of them possess a strong financial justification. However, practically speaking, there are no strong features which allow us to select the best criterion, a priori. We are referring, mainly, to a criterion which could produce theoretical prices for the contingent claims that are not very biased. With this in mind, Keller (1997) suggests not using a theoretical criterion for the selection of the best equivalent martingale measure, but to directly ask the market which is the best measure to use as the pricing functional. Practically, this method consists of choosing the equivalent probability measure Qi , belonging to Me , which produces the minimum bias between theoretical and empirical contingent claims prices. Keller defines this approach as the ‘statistical martingale measure’; however, practitioners prefer to refer to this approach as the ‘calibration approach’. Obviously, the statistical martingale measure is determined by assuming the goodness of the theoretical model, which we use to derive the theoretical derivative price. The statistical martingale measure can be considered optimal in the sense that it minimizes the distance (quadratic or whatever) between theoretical prices of the model and prices actually observed in the market. The main difficulty in the application of the calibration approach to the electricity derivatives market is related to the relative illiquidity of those markets. In fact, as we saw in the previous chapter, the majority of electricity derivatives are traded in OTC markets and the standard volume of the transactions is not usually very high. This implies that in traded derivative prices, when they are available, there is insufficient information to calibrate an electricity derivative pricing model. This problem is particularly important for such models because, as we have seen, they are usually complex and have

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a high number of parameters whose values have to be calibrated on the derivatives market prices. A potential alternative to a pure calibration approach, even if not theoretically fully unbiased, is a mixed ‘estimation/calibration’ approach of the type developed in Cabibbo and Fiorenzani (2004) for the forward curve-shaping problem.

APPENDIX AN EQUILIBRIUM PRINCIPLE FOR PRICING ELECTRICITY ASSETS IN INCOMPLETE MARKETS In this chapter we have seen that ‘no-arbitrage’ arguments can almost be applied also when the market model is incomplete, even if some assumptions have to be imposed and some strong implications on pricing arguments follow as a result. We also emphasized the close relationship between the notion of equilibrium pricing and that of no arbitrage, but we did not go into the details of this interesting relation for the specific case of electricity markets and electricity assets valuation. This Appendix is dedicated precisely to this argument and is mainly based on a recent, very technical and mathematically complex, work of Hinz (2003). We will try to present the topic in a simplified but non-trivial manner. The valuation principle proposed by Hinz in 2003 is based on the hypothesis that electricity cannot be stored, with all the implications related to market incompleteness we have analyzed so far. Although electricity cannot be stored, it can be produced and so the underlying basis of any electricity contract is effectively the ‘physical ability to produce power’. In this Appendix we want to show that, given some conditions and assumptions to reduce the complexity of the analysis, there exists a probability measure Q equivalent to P such that the equilibrium price of any electricity asset (physical or financial) can be obtained as the discounted expectation of the future revenues of the asset itself, calculated under the Q measure: ⎛ ⎞

⎜ ⎟ g(Et ) = EQ ⎝ (8A.1) Rˆ u (e)| t ⎠ u∈
(0,T) R where R is the revenue process; Rˆ = N ; N is the numeraire asset;
(0, T) is a discrete phys partition of (0, T); and e ∈ E = E ∪ E fin . Let us describe the electricity market as a discrete time market over the time partition t = 0, 1, . . ., T on the filtrated probability space (, , P, ( t )Tt=0 ). Let us also assume that E = E phys ∪ Efin is the finite set of tradable assets and that (Rt (e))Tt=1 is the RE-valued process describing the revenue process coming from holding the asset e ∈ E within [t − 1, t]. Each of the I-th agents of the economy is univocally determined by its initial endowments xi and its utility function Ui. Let us assume that for every agent i   (8A.2) Ui ∈ U ∈ C1 ]0, ∞[ : U  > 0 is strictly decreasing with lim Ui (z) = 0 z→∞

At each interval of time over the discrete partition [t = 0, 1, . . ., T], agents trade assets (which are assumed to be arbitrarily divisible), obtain their part of the revenues and reallocate their wealth. Given the market model defined above let us now define the notion of market equilibrium we are going to use for pricing purposes.

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Let Et = (Et (e))e∈E denote the price process of all physical and financial assets belonging to E. A trading strategy can be represented by the pair (θt , ϑt (e))Tt=0 , which identifies the proportions of the initial endowment which have to be invested, respectively, in the saving security N and in the electric asset e ∈ E for the time period [t, t + 1]. A trading strategy (θt , ϑt (e))Tt=0 is called self-financing if Xt = θt Nt + ϑt ⊗ Et

(trading strategy value)

and

Xt+1 = Xt + θt (Nt+1 ) + ϑt ⊗ (Et+1 + Rt+1 )

(8A.3)

A self-financing trading strategy (θt , ϑt (e))Tt=0 is uniquely determined by the initial wealth x = θ0 N0 + ϑ0 ⊗ E0 and by the assets position ϑt since the saving account position can be constructed recursively by difference. For a given initial endowment x ∈ ]0, ∞[, asset prices Et and utility function U it is possible to introduce the set of admissible positions (trading strategies) as follows: A(x, E, U) =

⎧ ⎨ ⎩

 (ϑt )T−1 t=0 :

ˆ x,V,E X T

=

Xtx,V,E Nt

T ≥ 0;

 E

ˆ x,V,E )− U(X T

t=0

⎫ ⎬ <∞ ⎭

(8A.4)

If we assume that each agent behaves rationally, which means that given prices E he chooses trading strategies belonging to A such that its expected final utility is maximized,    ˆ x,V,E max E U X T

(8A.5)

ϑ∈A

then the equilibrium of an electricity market characterized by agents (xi , Ui )Ii=1 consists of a price process E∗t and agent’s positions ϑt∗ such that: I  i=1 I  i=1

ϑti∗ (e) = 1

∀ e ∈ Ephy

ϑti∗ (e)

∀e ∈

(8A.6) =1

Efin

and ϑt∗ maximizes the expected final utility for all the agents. In order to ensure the existence of the equilibrium we have to enforce the analysis with additional assumptions:
the revenue process is integrable and bounded from below;
all contracts lose their value at the final date; and
all the agents are equal for initial endowments and utility functions xi = x and Ui = U

for all i = 1, . . ., I.

Hinz (2003) provides the formal proof that, given the assumptions stated, the existence of the equilibrium is guaranteed and this implies that it and the equivalent probability measure Q exists such that expression (8A.1) is true.

CHAPTER 9

Electricity Derivatives: Numerical Methods for Derivatives Pricing

In the previous chapter we have seen that, for various reasons, no closedform pricing formulas are available for the pricing of electricity derivatives, at least if we concentrate on realistic pricing approaches. Not even the simplest plain-vanilla instruments can be priced realistically by means of a traditional Black–Scholes-type model, because of the particular features which characterize the stochastic dynamics of electricity spot and forward prices. Hence, numerical procedures have to be used in order to determine electricity derivative prices and analytical risk measures, such as Greeks. In this chapter we will discuss general numerical procedures for derivative pricing such as Monte Carlo simulations and Lattice methods, presenting also some practical pricing applications related to derivative instruments introduced in previous chapters.

9.1 MONTE CARLO SIMULATIONS Previously we learned that, even when the market model we are considering is incomplete, a fundamental result of derivative pricing still holds. In particular, it will always be possible to obtain the derivative ‘fair price’ as the discounted value of its expected payoff calculated under an appropriate risk-adjusted probability measure: 
(9.1)
= EQ [βC] : Q ∈ Me where β is the discount factor, and C is the derivative final payoff function. 95

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If we define with g(ET ) the final payoff function of the derivative at maturity it is possible to write its value in t, g(Et ) as follows: g(Et ) = exp(−r(T − t))EQ [g(ET ) | t ]

(9.2)

When the derivative payoff function does not depend on a particular optimal exercise policy, as in the case of American options, it is possible to apply straightforward Monte Carlo methods to obtain a quite exact estimate of the derivative fair price. If we indicate with E˜ t,T a sample trajectory of the electricity price in the time interval [t, T] an estimate of the conditional expectation can be obtained by simulating many paths. In fact, we will have: g(Et ) = exp(−r(T − t))EQ [g(ET ) | t ] 1  ˜j g(Et,T ) = exp(−r(T − t)) lim m→∞ m m

(9.3)

j=1

It is evident that the main challenge in the application of Monte Carlo methodology in electricity derivative pricing, for those instruments types for which such a method is allowed, is the simulation of underlying asset price sample trajectories. In the following we will present discretization and simulation schemes for very simple and progressively more complex stochastic processes, which can be considered for the description of the electricity price stochastic dynamics.

9.1.1 Ito processes simulations Geometric Brownian motion is the stochastic process traditionally used in financial derivatives analysis, and the stochastic differential equation which characterizes it is: dXt = αXt dt + σXt dWt

(9.4)

This stochastic differential equation can be efficiently discretized by means of the well-known Euler’s scheme, which uses the standard properties of Brownian motions and can be represented in the following way: √ Xt = Xt−1 + αXt−1 t + σXt−1 t εt

(9.5)

where εt is i.i.d. Process N(0, 1) distributed. Geometric Brownian motions are not suitable processes to describe electricity prices, mainly because of mean reversion and spikes; however, Euler’s

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scheme can be easily adapted to a wide class of Ito processes, such as classical mean-reverting processes or non-linear Ornstein–Uhlenbeck (NLOU) processes. In fact, for simple mean-reverting processes we will have: √ dXt = λ(µ − Xt )dt + σ dWt → Xt = Xt−1 + λ(µ − Xt−1 )t + σ t εt (9.6) Concerning NLOU processes, a similar rationale can be applied. NLOU processes are Ito processes since they are non-linear transformations of OU processes, which are Ito processes themselves. Hence, they can be simulated simply by applying the non-linear transformation to the discretized OU process: Yt = f (Xt ) ⇒ dXt = λ(µ − Xt )dt + σ dWt

√ ⇒ Xt = Xt−1 + λ(µ − Xt−1 )t + σ t εt

(9.7)

In general, this kind of approximation and simulation scheme can also be applied without any particular problems when the drift and diffusion parameters of the Ito process are not constants but deterministic or locally stochastic functions of time. Moreover, similar schemes can also be adopted for stochastic processes other than Ito processes, such as Lévy processes.

9.1.2 Jump processes simulation In Chapter 6 we saw that, for a realistic description of the stochastic dynamics of electricity prices, the jump component plays an important role. We had occasion to analyze two examples of jump processes: the classical jump diffusion process and the Geman–Roncoroni process. A general approach for the simulation of general semimartingale processes does not exist, and hence it is necessary to face the simulation problem case by case. For simple jump diffusion processes (with mean reversion) the relative stochastic differential equation can be generally written as: dXt = λ(α − Xt )dt + σ(t)dWt + dNt

(9.8)

where Nt is the classical compound Poisson process also used by Merton (1976). This simple jump process is built up using a simple Poisson process as the jump intensity process, and a standard normal distribution to model the jump amplitude.1 Hence, the discretized version of the stochastic dynamic equation becomes: √ (9.9) Xt = Xt−1 + λ(α − Xt−1 )t + σ t εt + Jt · ηt 1

The normal and the Poisson variable implied by the process are assumed to be independent.

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This process has a constant intensity of jumps, which is not a good feature in the electricity field as we have discussed earlier in the book. The Geman– Roncoroni model tries to overcome this problem, and the continuous time stochastic differential equation related to this model is: dXt = µ (t)dt + θ1 [µ(t) − Xt− ]dt + σ dWt + h(t− )dJt

(9.10)

The jump process in (9.10) is a purely discontinuous semimartingale with a time-dependent intensity process.2 Denoting the intensity function as I(t) the discretization scheme becomes: Xt = Xt−1 + µ (t)t + θ1 [µ(t − 1) − Xt−1 ]t √ + σ t εt + h(t − 1) · P(it ) · et

(9.11)

where P(It ) is a Poisson random variable with time-dependent intensity, and et is an exponential random variable. In addition, for such a class of stochastic processes, many modifications can be imposed onto the model to make it more realistic, without affecting the efficiency of the discretization scheme.

9.2 THE LATTICE APPROACH A quite famous and useful approach to pricing derivative products is based on the construction of a discrete time and space lattice to describe the underlying asset price dynamics. The lattice represents all the possible trajectories that the underlying asset price may follow during the derivative lifetime (Figure 9.1). Risk-neutral arguments can be used to determine the derivative price over the lattice. In fact, if we consider a simple binomial tree case we have that: G(Et ) = βEQ [G(Et+1 ) | t ] ⇒ β[Prob(up)G(Et+1 (up)) + Prob(down)G(Et+1 (down))]

(9.12)

The previous recursive relationship can be applied backwards in time to derive the derivative fair price at the evaluation date, starting from the derivative maturity value which is a known function of the underlying asset price. Cox, Ross and Rubinstein (1979) first applied the binomial tree approach for the pricing of financial plain-vanilla options. The success of the lattice approach is mainly due to its flexibility, since it may be used to price a wide range of derivative products. In fact, this method provides the derivative value in every node of the lattice, the derivative optimal exercise policy can 2

See Chapter 6 for a complete description of the Geman–Roncoroni model.

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Et
1(up) Prob(up)

G(Et
1(up))

Et G(Et) Et
1(down)

Prob(down)

G(Et
1(down)) t

t
1

Figure 9.1 Lattice approach schematic representation be determined when necessary as for the case of American-style options or swing options. For example, for American-style options the lattice approach always gives us the possibility of comparing the early exercise value of the option with its continuation value, and hence to consider as the theoretical option value the maximum of the two.3 The main problem in building up a lattice pricing framework is the definition of the discretization scheme which allows us to pass from a specific continuous time-stochastic hypothesis for the underlying asset price dynamics, which is assumed to be realistic, to the corresponding lattice representation. A rigorous general solution to this kind of problem does not exist, but different solutions have been proposed in the last few years for specific classes of stochastic processes. However, a recombining lattice scheme is only available for specific sub-classes of diffusive processes, while for jump processes numerical approximations are necessary for their lattice scheme construction.

9.2.1 The classical CRR binomial model The simple binomial model developed by Cox, Ross and Rubinstein (CRR) can be thought of as the lattice approximation of the classical geometric Brownian motion (GBM) used within the Black–Scholes and Merton (BSM) model. It can also be proved that, under some general conditions,4 the binomial process converges, in distribution, to the GBM as the time step 3

See Cox, Ross and Rubinstein (1979) for a more detailed discussion on the standard binomial tree derivatives pricing approach. 4 In particular, the convergence holds when up and down probabilities are not extremely asymmetric. In this case, the convergence is towards a Poisson process.

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goes to zero. Hence in practice, when we have to construct a CRR binomial tree model, the first problem to solve is how to determine the up up (u → Et+1 = u · Et ) and down (d → Edown t+1 = d · Et ) parameters in order to be consistent with the continuous time risk-neutral dynamics of the underlying asset price. In particular, within the BSM theoretical framework we have to determine u and d as functions of the volatility parameter σ and the risk free rate r. Skipping all the details about the derivation, let us propose the most popular way of realizing the match, which consists in setting: √ √ √ exp(r t) − d u = exp(σ t) d = exp(−σ t) ⇒ Prob(up) = u−d (9.13) This binomial calibration approach is based on the idea that if the GBM and binomial process have to converge in distribution, then the distribution moments also have to converge. Since the marginal distribution of a GBM is lognormal, the relevant distribution moments which have to match are the first two: the mean and variance. Unfortunately, as the GBM process is not suitable to describe the stochastic dynamics of electricity prices, the CRR binomial approach is also not a realistic lattice model to price electricity derivatives; more advanced lattice schemes have to be applied.

9.2.2 Trinomial trees for electricity spot prices As we have seen in the previous chapter, one of the most important requirements for a derivative pricing model is that of being consistent with the observed derivative prices (forward or option prices). This basic principle is effectively the conceptual base that justifies all the model-calibration approaches. Clewlow and Strickland (2000) have presented a trinomial tree-modeling approach for electricity, which has the good feature of being consistent with the observed electricity forward curve. They assume a continuous time stochastic behavior for the electricity spot price described by the single-factor Schwartz process: dEt = λ[µ(t) − ln Et ]dt + σ dWt (9.14) Et Consistency with the observed forward curve F(0, T) is obtained by imposing:5 µ(t) = 5

σ2 d ln F(0, t) + ln F(0, t) + (1 − exp(−2λt)) dt 4

Details about the Clewlow and Strickland model can be found in the original paper.

(9.15)

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The procedure suggested by Clewlow and Strickland to derive the trinomial tree representation of the spot price process described above is similar to that used for the interest rate model by Hull and White (1994). If we denote by xt the natural logarithm of the electricity spot price, we have that: xt = ln(Et ) dxt = [θ(t) − λxt ]dt + σ dWt   d ln F(0, t) σ2 1 2 θ(t) = + λ ln F(0, t) + (1 − exp(−2λt)) − σ dt 4 2

(9.16)

The tree-building procedure which Clewlow and Strickland suggest is made up of two stages. First, a preliminary tree has to be constructed for xt under the assumption that θ(t) = 0 for every t and the initial value for that tree is zero. The time dimension of the tree is equally partitioned such that 0 < t + t < · · · < t + i t < · · · < T, and each node of the tree is identified by the pair of time and price state indices (I, j). An appropriate choice for the √relation between the price state step and the time step is given by: x = σ 3t. The trinomial branching process and the associated probabilities are chosen in order to be consistent with the drift and volatility parameters of the continuous time process for x. The transition probabilities associated with an up, middle and down movement from node (I, j) can then be written as follows:  1 σ2 t + λ2 x2 (i, j)t2 + (k − j)2 pup (i, j) = 2 x2  λx(i, j)t (1 + 2(k − j)) + (k − j) − x  (9.17) 1 σ2 t + λ2 x2 (i, j)t2 pdown (i, j) = + (k − j)2 2 2 x  λx(i, j)t + (1 + 2(k − j)) + (k − j) x pmiddle (i, j) = 1 − pup (i, j) − pdown (i, j) The three nodes which can be reached starting from (I, j) are (I + 1, k − 1), (I + 1, k) and (I + 1, k + 1) where the parameter k is chosen in such a way that the value of the x middle branch is as close as possible to the expected value of x at t(I + 1). This procedure applies to the process xt once we have imposed that θ(t) = 0 and x0 = 0. Let us denote it as x¯ t . The second step in the construction of the tree consists in displacing the nodes with a drift component, which makes the tree consistent with the

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observed forward curve. Let us denote by the displacement term ai . The value of x in the node (I, j) will be that of x¯ , obtained before, imposing θ(t) = 0 and x0 = 0, plus the displacement term ai , while the transition probabilities remain unchanged. The displacement term is obtained as a function of the Arrow–Debreu securities prices Q(I, j) that are the prices at time zero of those simple risky securities which pay 1 monetary unit if the node (I, j) is reached. The Arrow–Debreu securities prices can be calculated through forward induction according to the following relationship: Q(0, 0) = 1 Q(i + 1, j ) =



Q(i, j ) p (i, j, j )B(i t, (i + 1)t)

(9.18)

where p(I, j, j ) is the probability of moving from node (I, j) to node (I + 1, j ) while B(i t, (i + 1)t) is the price in Iτ of a pure discount bond maturing at (I + 1)t. Clewlow and Strickland show that the displacement term that ensures the forward curve consistency of the tree has the following functional form: 

B(0, i t)F(0, i t) ai = ln  xi,j ) j Qi,j exp(¯

 (9.19)

The main advantage of this lattice modeling approach with respect to the classical CRR model consists in the greater flexibility that the higher number of parameters ensures for the model. In fact, with this approach many forward curve shapes can be captured and represented. The main limit of the approach is the impossibility of incorporating jumps into the model representation, and we know that in the electricity field jumps play a fundamental role.

9.2.3 The lattice approximation of jump processes Up to this point we have analyzed exact lattice schemes obtained by starting from a specific continuous time stochastic hypothesis and a mathematically rigorous discretization method. Unfortunately, only for a few classes of processes are exact lattice schemes available, and many derivative valuation problems can be solved only through backward induction methods. To an extent, state price aggregation methods can be adopted to map the price dimension spanned by a non-standard stochastic process onto a manageable discrete grid. State price aggregation consists in partitioning the price space into discrete cells and then considering the price cells, and the corresponding transition probabilities, for the valuation of the derivative contract over those cells. In other words, it can be said that state price

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103

Price

aggregation schemes help us to separate, at the price space when analytical schemes are not available. Lattice schemes, with recombining and non-recombining branches, have been used in traditional financial analysis but always with a tractable number of total nodes, which can be explored. In a similar way, in aggregation schemes the state partition should be properly chosen in such a way that the numerical problem remains manageable (the number of reached nodes is not exploding) and the approximate exercise strategy for the derivative is as close as possible to the actual optimal one. State aggregation is a classical approximation technique for the numerical solution of stochastic optimal control problems, particularly useful for high-dimensional valuation problems. Here we do not want to go into the mathematical details on aggregation techniques,6 simply to emphasize that aggregation schemes can be applied for the pricing of simple derivative products when a non-standard stochastic hypothesis is assumed for the underlying asset price. Numerical and consistent approximations of lattice transition probabilities can be obtained through Monte Carlo simulations once the continuous time stochastic dynamics have been specified and the state aggregation scheme has been decided. Figure 9.2 gives an approximate representation of the state aggregation methodology. Once the price aggregation scheme has been built up and the transition probabilities have been estimated by means of Monte Carlo simulations, classical backward induction arguments can be used over the lattice nodes to obtain derivative prices.

0

T1

T2

T3

T

Time

Figure 9.2 State price aggregation scheme

6

The interested reader can refer to Kushner and Dupuis (1992).

104

ELECTRICITY DERIVATIVES

APPENDIX A PRICING ELECTRICITY SWAPTIONS BY MEANS OF MONTE CARLO SIMULATIONS In this Appendix we present numerical results coming from the application of Monte Carlo simulation methods for the pricing of a generic electricity swaption, under the assumption that the underlying asset price dynamics is described by means of a Barlow model. The swaption contract considered is a one-month call option on a three-month electricity monthly swap. The owner of the swaption contract has the right to enter, at option maturity, into a three-months electricity swap. Hence, it is possible to synthesize the contract features and payoff as follows:
t is the evaluation date;
T is the option maturity;
K is the option strike;
(T1 , T2 , T3 ) with T < T1 are the swap settlement dates;
S(t, T3 ) is the electricity swap curve in t;
S(T, T3 ) is the electricity swap curve in T;
P(T) = max(I δI (S(T, TI ) − K); 0) is the payoff function of the option at maturity.

Here we are going to evaluate this kind of swaption contract by MC simulation, assuming that electricity spot price (daily averaged) is generated by NLOU process of the type proposed by Barlow: Et = (1 + αXt )1/α

(9A.1)

where dXt = λ(µ − Xt )dt + σ dWt . The discretization schemes presented in the previous section have been applied to generate 500 sample trajectories for the underlying asset price. Each sample trajectory represents the price evolution over the period of interest, of four months, with a daily time step (120 days). Figure 9A.1 represents some of the trajectories generated. The swaption price was then calculated by averaging and discounting the instrument payoff function over each sample trajectory. Table 9A.1 is an example of the calculations.

400

200 100

Figure 9A.1 Barlow process simulated price paths

113

97

89

81

105

Days

73

65

57

49

41

33

25

17

9

0 1

Price

300

Table 9A.1 Simulation experiment results Option maturity Swap maturity Swap settlement Strike Risk-free rate

30 days 90 days 30 days 50 5%

Model parameters

Swaption value

Trajectory

lamda 90

mu 5

alpha 0.2

2

3

4

delta 0.002778

initial 50

7.71701

Time in days 1

sigma 25

Option payoff „„,

119

120

Monthly average month 1

month 2

month 3 70.77699

1

20.2810061

29.99934

67.80934

57.16278812

„„,

28.66344

14.9026

13.43344134

51.75558

40.90088

2

21.35061071

13.71343

32.40294

13.93155011

„„,

12.43883

7.59738

0

47.19012

37.19827

52.44437

3

60.159085

122.0981

105.3159

134.9853243

„„,

8.698105

24.21162

9.647829589

47.20937

51.20758

61.23088

„„,

„„,

„„,

„„,

„„,

„„,

„„,

„„,

„„,

„„,

„„,

„„,

499

78.74801155

72.91397

48.08012

140.1382048

„„,

53.53163

14.00179

32.57774677

61.47708

88.6548

32.44587

500

28.57798955

77.22919

85.47402

64.25137518

„„,

68.58487

61.78112

0

39.41567

21.95998

24.3566

105

106

ELECTRICITY DERIVATIVES

Table 9A.2 Sensitivity experiment result Lambda

70

80

90

100

110

Swaption price

20.18

11.75

7.71

4.881

3.02

Sigma

15

20

25

30

35

Swaption price

0.4801

2.21

7.71

21.39

51.44

Alpha

0.1

0.15

0.2

0.25

0.3

Swaption price

192.18

50.04

7.71

0.098

0

It is also interesting to note how, under the assumption of the Barlow model, the swaption value is jointly influenced by all the parameters of the NLOU underlying process. In this model, particular importance is played by the joint effect that the alpha and sigma parameters have on the spot price spike behavior and frequency. As we can see from Table 9A.2, the alpha parameter, which influences the curvature of the non-linear transformation of the underlying OU process has a strong impact on the swaption value. A general consideration in Monte Carlo pricing approaches is about the price convergence rate. In fact, as can be seen for simpler and analytically tractable models, the convergence of the approximated price to the ‘true’ price can be highly dependent on the type of derivative we are trying to evaluate. In fact, the pricing of instruments with very complex payoffs necessitates a higher number of trials for a satisfactory level of approximation. In such situations, variance reduction techniques can be applied to reduce the number of trials necessary.

APPENDIX B PRICING SWING OPTIONS BY MEANS OF TRINOMIAL TREE FORESTS In Chapter 7 we discussed swing option contracts, which are extremely important in the energy field due to the volume flexibility they guarantee. In fact, energy suppliers and consumers are exposed to volumetric risk given the uncertainty about the energy they are going to sell or consume. Swing options are thought of as allowing a partial hedge to this kind of volumetric risk. As we noted, a non-trivial swing option can be thought as a string of American call options with co-dependent exercise. This particular feature causes the evaluation problem of a swing option to be strictly related to the problem of determining the optimal exercise policy.7 The valuation of the swing contracts and the determination of the associated exercise policy can be performed by constructing a trinomial tree forest. Using the notation introduced in Chapter 7 we have that each node of the forest will be identified by the triplet (i, j, k) where i denotes the time step dimension, j the different price levels and k the number of swings which have been exercised up to time i. Not all the nodes of the forest are valid. In fact, if the swing contract starts at 0 and terminates at time n the only valid nodes will be those for which: k = 0, . . ., kmax 7

i = k, . . ., min(n − kmin + i + k, n)

(9B.1)

We mean the optimal quantity of energy to sell or purchase at each exercise date. This optimal quantity should be consistent with global and local volume constraints.

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107

K2

K1

K0

Figure 9B.1 Trinomial tree forest scheme where kmin and kmax denote the maximum and minimum number of swings admissible in order to respect maximum and minimum global constraints (MGQ, LGQ): kmax =

MGQ − n ∗ LSQ MSQ − LSQ

kmin =

LGQ − n ∗ LSQ MSQ − LSQ

(9B.2)

Hence, we have that:
if k ≤ kmin − 1 → i = (k, . . ., n − kmin + k + 1)
if k ≥ kmin − 1 → i = (k, . . ., n)

The valuation of the swing contract is done backwards in time and vertically for the various levels of swing admissible, assigning at each node of the process the maximum value of the contract. Figure 9B.1 represents a simplified example of a swing contract with three exercise periods and only two swings available.8 In general terms, it is possible to arrive at a valuation of the contract just by performing the following steps: 1 Calculate the value of the contract at the node (i = n, j, k = kmax ). Since all the swings available for the contract have been exercised, then the value of the contract in that node will be obtained by multiplying its mark to market (MTM) by the minimum sub-period quantity LSQ: Vi,j,k = (Ei,j,k − K) ∗ LSQ ∀ j 8

Dotted lines represent non-valid branches.

(9B.3)

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ELECTRICITY DERIVATIVES

2 Calculate the value of the contract for the remaining instants backwards in time (i = kmax , n − 1) since last valid instant of time. In those nodes the value of the contract will be given by the sum of the MTM multiplied by the minimum sub-period quantity and the so-called continuation value, which is the discounted expected value of the contract in the attainable following braches:  pj,s Vi+1,s,k ∀ j (9B.4) Vi,j,k = (Ei,j,k − K) ∗ LSQ + exp(−r t) s

where pj,s is the transition probability to move from node (i, j) to (i + 1, s). 3 Calculate, with the same methodology, the value of the contract for the remaining trees of the forest (k = kmax −1, . . ., 0). For each tree the procedure can be synthesized as follows:
For each remaining tree calculate the value of the contract for the last valid instant

of time i = min(n − kmin + k + 1, n). If k > kmin − 1 then the minimum amount of swings necessary to reach the minimum global constraint has already been done, hence the last valid instant of time will be n and the value will be obtained as max (MTM ∗ LSQ; MTM ∗ MSQ). If, otherwise, k ≤ kmin − 1 the minimum number of swings has not yet been exercised, then the last valid instant of time does not coincide with n, but will be n − kmin + k + 1. In this case, the only strategy available is the exercise of the swing. The value of the contract will then be given by:

Vi,j,k = max (Ei,j,k − K)LSQ; (Ei,j,k − K)MSQ ∀ j if k > kmin − 1  Vi,j,k = (Ei,j,k − K)MSQ + exp(−rt) pj,s Vi+1,s,k+1 ∀ j if k ≤ kmin − 1

(9B.5)

s


Calculate the value of the contract for remaining instants of time i = k, …, min

(n − kmin + k − 1, n) − 1. The procedure is re-iterated backward in time until the last valid instant is reached. Independently on the value of k, the value of the contract will be obtained as the sum of max(MTM ∗ LSQ; MTM ∗ MSQ) and the continuation value:  ⎡ ⎤ pj,s Vi+1,s,k+1 ; (Ei,j,k − K)LSQ + exp(−rt) ⎢ ⎥ s  Vi,j,k = max⎣ (9B.6) ⎦ ∀j (Ei,j,k − K)MSQ + exp(−rt) pj,s Vi+1,s,k+1 s

The value of the swing contract will be obtained as the value of the forest in the node (0, 0, 0). In a second moment the trinomial tree forest can be examined to identify for each node the lowest price level ‘j_low’ over which the swing right is exercised. The highest price level will be associated to those nodes that have been defined as non-attainable. The optimal exercise surface can also be illustrated by means of a bidimensional representation, where the generic node (i, j) is substituted by the index t defined as follows: t = k ∗ (n − kmin + 2) + i − k + 1

if k ≤ kmin

1 t = kmin ∗ (n − kmin + 2) + ∗ (k − kmin ) 2 ∗ [2n + 1 − k − kmin + 2] + i − k + 1 if k > kmin

(9B.7)

Once the optimal strategy is identified, the maximum sub-period quantity MSQ will be associated to those nodes where the swing is exercised and the minimum LSQ where it is not. Through transition probabilities, it is also possible to determine the trinomial tree forest of the expected taken quantity.

P A R T IV

Real Asset Modeling and Real Options: Theoretical Framework and Numerical Methods

109

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C H A P T E R 10

Financial Optimization of Power Generation Activity

10.1 OPTIMIZATION PROBLEMS AND THE REAL OPTION APPROACH Optimization problems have always been an important issue in any industry, particularly in the power generation industrial sector. Before the liberalization process of the electricity markets started, the electricity market was monopolistic; the government authority fixed power tariffs and the only uncertainty faced was exogenous demand and mechanical failures giving operational and demand risk top priority. Under a regulated regime, optimization meant cost-minimization. Electricity producers were forced to meet demand by forecasting future demand preferences along with efficiently managing their generation assets, reaching demand targets at the lowest cost. Liberalization, however, shed new light on the optimization argument. The presence of the free market brought equilibrium in the supply and demand of electricity along with price volatility and uncertainty. In this new and uncertain environment, optimization problems related to power production have become increasingly complex and cover different decisional processes. The marginal-cost-determination problem, the traditional optimization problem in regulated environments, still persists and remains important as a basis for the construction of a bidding strategy, but is no longer central in the unit commitment optimal decision process. When a liberalized electricity market is present the determination of optimal bidding strategies and optimal unit commitment and power production 111

112

REAL ASSET MODELING AND REAL OPTIONS

strategies are crucial issues for economic margin maximization.1 However, in a context where demand is stochastic and power producers are price-takers, the determination of the optimal production level is the true challenge to maximize margin in a risk-reward perspective. In other words, the important optimization problem for the power producer is that of determining how to manage his assets in such a way that the expected margin is maximized and the risk involved can be handled. In this chapter we will concentrate on that last problem. In particular, our scope here is that of emphasizing the close relationship between operations management problems and risk-management problems involved in power generation. In fact, as much as a certain generation asset is structurally different for efficiency and operational flexibility from the marginal plant of the economy, the more it can be considered as a potential risk source. Hence, a correct financial modeling of a generation power plant is essential for understanding and measuring a firm’s natural risk exposure. Moreover, asset modeling is also an essential tool for optimization and investment decision purposes. In order to present power generation modeling issues we will make an extensive use of real options theory and stochastic optimization techniques. The real options valuation approach has attained significant importance in modern financial theory since it combines the powerful tools of option valuation theory with the critical importance of valuation and hedging of real assets. In particular, in the electricity field the existence of developed derivative securities markets allows a fuller exploitation of real options tools and provide a more accurate representation of asset flexibilities. Compared to traditional valuation approaches, such as discounted cash flow (DCF) and net present value (NPV), the real options methodology seeks to capture as much value as possible in uncertain environments. It is possible to identify two important areas for real options analysis:
Investment decisions
Operating decisions Traditional valuation approaches have their origins in bond valuation; hence, they are not well-equipped to deal with uncertainty. In fact, a positive net present value may not be sufficient to start an investment project. In a context of economic and technical uncertainty, it is normally necessary that a project be substantially in the money to exercise the option to invest. 1A

bidding strategy is defined optimal when the quantity of power to offer in the exchange, for each level of an admissible bidding price, is determined in such a way that the potential gain of being selected to produce is bigger or equal, at least, to the possible loss of being idle (see Hintz, 2003).

FINANCIAL OPTIMIZATION OF POWER GENERATION

113

DCF and NPV methods are simple to use and understand, but they do not consider in the correct way the flexibilities embedded in a project/asset. Consequently, they are not able to correctly price those flexibilities in an uncertain environment. However, the real options approach also has some disadvantages, especially when evaluation methods are concerned. In fact, a real option is different from a standard financial option because its underlying basis is a non-traded asset (highly illiquid). Hence, the classical replicating portfolio approach for the valuation is no longer valid. In order to evaluate real options we have to refer to incomplete markets pricing techniques (equilibrium approaches, stochastic optimization, and so on) with all the related difficulties. In a very simplistic vision, power generation assets are ‘black boxes’ useful to transform fuel in electricity. In this simplistic world, the owner of the power plant has the option to exchange one asset, the fuel, for another, the power. He will exercise this option every time he believes it is convenient to do so, but even if he does not he faces a fixed cost representative of his initial investment. Using financial terminology, he holds a string of spark spread options. As explained by Tseng and Barz (2001), the payoff of the generator, and hence the spark spread, has been modeled as a linear function of the market prices of the two commodities: Payoff = PE − HPF

(10.1)

where PE is the electricity price, PF is the fuel price, and H is the heat rate parameter. Thereby, a spark spread call option’s payoff is: Payoff = max(PE − HPF , 0)

(10.2)

Hence, in order to maximize profits, the electricity producer in market conditions which favour the price of electricity, that is when PE > HPF, should purchase fuel and sell electricity in the market resulting in a payoff similar to a call option. Following studies by Hsu (1998a, 1998b) and Deng, Johnson and Sogomonian (1998), the power plant can be valued as the sum of a string of spark spread call options: Payoff =

T


[max(PtE − HPtF , 0)]

(10.3)

t=1

Competitive markets have made it possible for electricity producers to maximize their profits in circumstances in which the spread (difference between price of power and the cost of fuel) is high. On the other hand, it also signifies that if the spread is negative, that is if the cost of fuel is too high, the plant should not run.

114

REAL ASSET MODELING AND REAL OPTIONS

This introduces the concept of modeling the choice of optimal strategy to real options theory. In this manner, according to the real options approach it is possible to value the plant’s flexibility, and therefore to formulize the optimal strategy whether to commit or decommit a unit (for fuel conversion) in order to take full advantage of the price situation in the market. For instance, to ‘switch on’ the plant only at peak times to capture all the peak prices benefiting from the high returns rather than keeping the plant on continuously (base load) and missing out on the peaks. However, the plant is not only characterized by operational flexibilities, it is also subject to operational constraints classified by Tseng and Barz (2001), Gardner and Zhuang (2000) in six generic categories: ‘commitment/ decommitment lead times’, ‘intertemporal constraints’, ‘minimum and maximum generation capacity constraints’, ‘response rate constraints’, ‘variable heat rate’ and ‘additional costs’. These constraints introduce serial dependence in the commitment decisions sequence and have a significant impact on the plant’s value and optimization policy. If we wish to fully consider and assess the impact of operational constraints, the spark spread approach is not the appropriate way to do so. The most appropriate and complete approach to determine the optimal strategy in this constrained environment is stochastic dynamic programing. Operational constraints establish the state in which the plant is operating in and consequently affecting the future states of the plant since each state is dependent on the others. The constraints along with the expected payoff establish the optimization process maximizing producers’ profits and the plant’s value. In the following sections of this chapter we will fully analyze the potential applications of spark spread and stochastic optimization approaches for power generation modeling. The transition from regulated to liberalized power sectors has not only transformed the concept of optimization moving from a cost-minimization strategy to a more complex option theory analysis, but also marked the separation between marketing and power generation. Due to a previously regulated market, the marketing sector has held the responsibility of establishing the supply of electricity according to realized contracts. This was then followed by power generation, which acted accordingly responding to the decisions from the marketing sector. In a more detailed way, let us assume that in a regulated environment an electricity producer received a fixed price to deliver an uncertain load yielding a profit of:
Reg = l(F − C)

(10.4)

where l = an uncertain load; F = fixed price; and C = cost of production. The birth of a competitive electricity market made feasible the independence between the two sectors. In a liquid and liberalized market the electricity producer is not forced to produce the contracted load (q = l). On

FINANCIAL OPTIMIZATION OF POWER GENERATION

115

the contrary, in every single moment he can decide the optimal production level q(t) in order to maximize his profit given market price conditions, buying or selling the additional load (l − q(t)) in the market. Therefore, the produce would find it profitable to set up the following strategy: if E(t) > C produce q(t) = q(max) In addition, obtaining a profit

(10.5)


unReg (t) = l(t)(F − C) + (q(max) − l(t))(E(t) − C) >
Reg where E = price of electricity. Otherwise, if E(t) < C, he would produce q(t) = 0, with
unReg (t) = l(t)(F − E(t)) >
Reg . The existence of electricity exchanges caused a transformation: power generation no longer had to rely on the choices of marketing based on objectives of cost-minimization in which decisions on ‘how’ and ‘how much’ electricity is produced were established; that is, a backward decision process initiated from the marketing sector to power generation. Now, however, the optimization strategy is concentrated on operational capabilities and strategies of the plant itself, illustrating a forward decision process (Figure 10.1). In simpler terms, the plant according to its production capabilities and constraints devises its own optimal strategy in order to maximize electricity producers’ returns capturing the highest spreads. Not only did the concept of optimization change in respect to the decision production process, but also towards managing its risk. Previously, an electric company would hedge its risk in a static manner; according to the decisions made by the marketing sector, the company’s risk would in turn be managed. By evaluating the power plant as a real option, being able to measure the plant’s flexibility, a more accurate optimization process is made requiring a more dynamic management of risk. Thereby, the management of risk depends on the optimization strategy chosen at that particular time which affects the risk of subsequent periods. The objective of this chapter is to provide a different perspective in the valuation of a generation asset, in particular a thermal power plant. Modeling approaches for other types of generation units such as hydropower or

Forward process Power generation

Marketing Backward process

Figure 10.1 Power generation/marketing decisional process

116

REAL ASSET MODELING AND REAL OPTIONS

nuclear units can be derived in a similar manner, although they will not be explicitly discussed and developed here. Modeling generation assets as a real option allows for a more detailed understanding not only of its valuation, but the risk and optimization strategy the plant is subject to.

10.2 GENERATION ASSET MODELING: THE SPARK SPREAD METHOD Classical option pricing approaches are built around the concept of no arbitrage and contingent claims replicability. We have seen that in electricity markets, due to non-storability, derivatives cannot be replicated. However, this does not mean that known option-pricing formulas cannot be useful as guidelines for what a reasonable derivative price might be. In a very simplistic vision, thermal power generation assets are black boxes useful to transform fuel in electricity. Hence, as we have explained, they can be thought of as a string of spark spread options. This kind of option can be evaluated by means of the classical Margrabe model or by using Monte Carlo simulations. In particular, as we noted in the previous section, according to the spark spread modeling approach a power plant cash flow can be written as follows: Power plant CF =

N


max(Eti − HFti − Cfixed , 0)

(10.6)

i=1

The Margrabe model is usually considered the most direct way of solving the problem of pricing a spark spread option. This model is simply an extension of the classical Black–Scholes and Merton model, which considers the case of an option to exchange one asset for another, as in the spark spread option. In his original paper of 1978, Margrabe presented a formula for valuing European exchange options, developed with the same traditional assumptions of the Black–Scholes theoretical framework. He gives the price of a European exchange option as: European exchange price = S1 exp(−D1 (T − t)) · N(d1 ) − S2 exp(−D2 (T − t)) · N(d2 )   S ln S1 + (D2 − D1 + 0.5σ2 )(T − t) 2 √ ; σ T −t

(10.7)

√ where d1 = d2 = d1 − σ T − t; and  σ = σ12 + σ22 − 2σ1 σ2 ρ. Uppercase Si , Di , σi (i = 1, 2) are the respective parameters representing value, convenience yield and volatility of assets 1 and 2; ρ is the correlation parameter and N(x) is the cumulative normal density function of x.

FINANCIAL OPTIMIZATION OF POWER GENERATION

117

Of course, the closed formulation of the Margrabe model is based on the hypothesis of the joint log-normality of the electricity and fuel price. This assumption is of course not realistic; nevertheless, the Margrabe model has had a lot of success in business practice because of its simplicity and its analytic formulation. The spark-spread methodology can also be applied with a less traditional probabilistic hypothesis, but no closed formulas are, in general, available. Numerical techniques such as Monte Carlo simulations can in general, be applied for spark-spread option value calculations under non-trivial hypotheses concerning electricity and fuel joint dynamics. The main advantage of the spark-spread modeling approach for real asset valuation is its computational simplicity and, eventually, the availability of analytic formulae for price and sensitivities calculation. Moreover, the spark-spread approach is able to produce ‘optimal’ load curves which can be obtained as the delta vector of the string of options. Delta-based optimal load curves react in a coherent way to changes in market conditions and provide useful information for dynamic hedging purposes. The main disadvantage of this modeling approach is that it implies that power plants are infinitely flexible and so they can be turned on or off in continuous time. In reality, we saw that many operational constraints limit the possibility to freely exercise plant flexibilities. Because of this simplistic assumption, the spark-spread evaluation method tends to overestimate the value of the plant and can potentially produce inconsistent load profiles. Some operational constraints can be taken into account by this methodology (peak/off peak capacity swings, minimum/maximum periodic production quantity, quantity dependent heat rate) but, in general, for a full comprehensive valuation of operational constraints we are forced to turn our attention towards more complex but realistic modeling approaches.

10.3 GENERATION ASSET MODELING: THE STOCHASTIC DYNAMIC OPTIMIZATION APPROACH The simplicity of the spark-spread approach is of course appealing, but if we exclude the case of an extremely flexible power generation unit, in the majority of cases power plants are characterized by many operational constraints which prevent a straightforward application of this simple option-pricing valuation approach. In particular, due to operational constraints the unit commitment decision is not always an option. The existence of intertemporal constraints mentioned in the first section of this chapter make the unit-commitment problem a path-dependent (or better a co-dependent) option valuation problem. In this new and much more complex theoretical environment it is not only important to determine the value of the power plant, but also the

118

REAL ASSET MODELING AND REAL OPTIONS

optimal operational strategy. A similar problem is faced in traditional financial applications in the pricing and optimal exercise strategy determination of American-style options. In order to properly consider those physical constraints, which may have a sensible impact on the production value and on the optimal operational strategy, traditional option-pricing methodologies are not sufficient, and stochastic optimization techniques have to be used. As we have mentioned, operational constraints have been classified by Tseng and Barz (2001) and by Gardner and Zhuang (2000) into six generic categories: ‘commitment/ decommitment lead times’, ‘intertemporal constraints’, ‘minimum and maximum generation capacity constraints’, ‘response rate constraints’, ‘variable heat rate’ and ‘additional costs’. Let us now briefly explain their meaning and the economic impact they may have.

10.3.1 Commitment/decommitment lead times These parameters reflect the non-zero response time of the unit from the time a commitment decision is taken. A minimum time is required from the time a certain operational decision is taken to the moment it is physically realized, and this delay may have an economic impact since it forces the plant manager to anticipate operational decisions and to act with incomplete information due to uncertainty. Essentially the start-up/shut-down time belongs to this category.

10.3.2 Intertemporal constraints Let us define intertemporal constraints as all those operational characteristics of the generation plant which someway affect the intertemporal unit-commitment policy. For example, a minimum on/off time, minimum ramp time or cool-down time belong to this category. Minimum on time is the minimum time the unit has to run after it has been switched on, whilst the minimum off time is the minimum time required between a switchoff decision and a new turn-on decision. The unit ramp time is the time necessary for the unit to pass from a minimum capacity production to its maximum regime. Cooling time is the time necessary to cool down the boiler completely after a shut-down decision. Intertemporal constraints drastically limit the exercise of unit operational flexibility and hence have an enormous impact on plant economic value. Variable costs may also be associated with intertemporal constraints. For example, start-up costs usually depend on the cooling time since a cold start is less efficient. These costs may also be considered in the optimization problem.

FINANCIAL OPTIMIZATION OF POWER GENERATION

119

10.3.3 Minimum and maximum generation capacity constraints Obviously, each thermal power generation unit is limited in the maximum power capacity it can generate, but it is also low-bounded by a minimum capacity level below which the plant cannot operate.

10.3.4 Response rate constraints These represent the time required to effect a discrete change in the quantity level.

10.3.5 Variable heat rate The heat rate determines the capacity of a certain unit to transform a unitary quantity of fuel into electricity. In other words, it determines its physical efficiency. The heat rate is not always constant, and varies according to the level of generation. It is usually modeled as a quadratic function of the load (see for example Wood and Wollenberg, 1984).

10.3.6 Additional costs Plants may break down or suffer from operational difficulties and we may thus face maintenance costs or periods of programed or unexpected limited production capacity. Many other operational constraints can also affect a thermal power plant. Those described above are general classes, considered as the most important. Let us see how the power plant optimization problem can be framed within a stochastic optimization theoretical framework and how most of the mentioned operational constraints, and associated costs, can be considered properly within the model. Recent studies have shown how the valuation problem of energy assets, involving the flexibility and physical constraints of their operations, are often best described by recursive treatments such as stochastic dynamic programing. In fact, the unit-commitment problem in a liberalized energy market can be considered as a prototypical example where sequential decisions, each contingent on the last, have to be taken in order to maximize the benefit (utility) deriving from specific contracts or assets (the typical power plant optimization problem). Stochastic dynamic programing is a special type of optimization technique employed for recursive problems under uncertainty. In general, the decision strategy associated with a particular asset (physical but also financial), is represented by a particular choice of a set of ‘control

120

REAL ASSET MODELING AND REAL OPTIONS

variables’ (a(t)) that we can manipulate to achieve our optimization (for example, the number of swings, the switch on/off of the plant, or the exercise of a certain option). On the other hand, another set of variables, ‘state variables’ (x(t)), determine the exact state of our system as a function of time. State variables are important because the physical constraints of the system can be imposed on them. The last group of variables important for our problem definition is the set of ‘stochastic variables’ (S(t)) (for example, the electricity price, fuel price, and so on). The combination of the three mentioned groups of variables generates a certain cash-flow result which can be written as a function of time in the following way: K(t) = K(x(t), a(t), S(t))

(10.8)

Our objective is to determine the set of control variables (the strategy), which maximizes a weighted sum of cash flows over a certain time horizon, subject to some physical constraints. For this reason we define the following value function:  T 
P m−t Vt (xt ) = max E β Km (10.9) at , at+1 ,..., aT

m=t

where β is the discount factor vector, and EP is the expectation operator under the P probability measure. In order to better specify the constraints it is necessary to introduce an equation of motion which describes the evolution of the system state given a particular choice of the control variables: xt+1 = L(xt , at )

(10.10)

Because of the famous Bellman’s optimality principle our expression for the value function V(t) may be simplified as follows:2   Vt (xt , St ) = max K(xt , at , St ) + βEP (Vt+1 (L(xt , at ), St+1 )|St ) (10.11) at

This recursive equation for the value function is known as Bellman’s equation for a stochastic dynamic programing problem.3 The simplest way to solve these kinds of optimization problems is through backward induction. Given the value function of the system in a future instant of time T, VT (x(T), S(T)) = g(x(T), S(T)) (it can also be imposed arbitrarily if T is far enough in the future to not influence the actual value of the 2

See the Appendix for a more detailed discussion of Bellman’s theorem. Here we assume that the power plant manager is risk-neutral. For this reason all the expectation operators are considered to be under the natural probability measure P. According to the derivative valuation approaches discussed in Part III, we may consider the opportunity to substitute the natural probability measure P with the risk-neutral one Q. 3

FINANCIAL OPTIMIZATION OF POWER GENERATION

121

asset), it can be substituted in the right-hand side of the Bellman equation, which can then be solved for VT−1 (x(T − 1), S(T − 1)). Of course this theoretical approach has to be combined with a correct modeling of the thermal power unit cash-flow function and operational constraints. For this purpose we follow the approach suggested by Tseng and Barz (2001). We assume that operating decisions are taken at every single point t of the time horizon equispaced partition p[0, T] = 0, 1, . . ., T. Realistically, the time step of the time partition defined represents a single hour. Moreover, we are going to assume that the thermal plant operating activity is characterized by the following intertemporal constraints: minimum time on line ‘t_on’, minimum time off line ‘t_off’, and cool-down time ‘t_cold’. For simplicity we can assume that for every t the plant condition is fully identified by a simple 0–1 variable a(t) which describes the unit commitment decision a time t,4 while the state variable x(t) indicates the interval of time that the generator has been on or off-line at time t. There is no unique way of defining the equation of motion which relates a(t) and x(t). Here we are going to define x(t) as a variable taking values within the set of unit commitment state space C. The set C is made up of two disjoint sub-sets: Con the on line state space and Con the off line state space, such that: x(t) ∈ C ≡ Con ∪ Coff

(10.12)

where Con = {1, . . ., t_on}; and Coff = {−1, . . ., −t_off, . . ., −t_cold}. The decision variable a(t) can be described as follows, for every t belonging to p[0, T]: a(t) = 1

if 1 ≤ x(t) < t_on

a(t) = 0

if −1 ≥ x(t) > −t_off

a(t) = 0 or 1

if x(t) ∈ O = {t_on ∪ (−t_off ≥ x(t) ≥ −t_cold} (10.13)

while the dynamic equation for the state variable can be defined as follows: x(t + 1) = min(t_on, max(x(t), 0) + 1) > 0

if a(t) = 1

x(t + 1) = max(−t_cold, min(x(t), 0) − 1) < 0

if a(t) = 0

(10.14)

A diagrammatic approach, as in Figure 10.2, can help us to understand the meaning of the transition state equation described above. Once we have described mathematically the intertemporal constraints which affect the power plant operating regime, we have to describe the 4 As explained by Tseng and Barz (2001), this assumption implies that the generator can be synchronized and dispatched in real time. Therefore, the dispatch problem can be solved instantaneously after price realization, hence in a deterministic environment. Given that the unit is committed at time t(a(t) = 1), the optimal quantity q∗t ∈ [qmin , qmax ] can be obtained solving a deterministic problem.

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REAL ASSET MODELING AND REAL OPTIONS

X(t )

X(t ⴙ 1)

⫺t_cold

⫺t_cold

⫺t_off

⫺t_off

⫺1

⫺1

1

1

t_on

t_on

Figure 10.2 Diagrammatic representation of the dynamic equation (10.14)

cost/revenue functions associated to each power plant operating state. The most important cost function is that associated with fuel costs. The fuel cost is a function of the unitary price of the fuel at particular time F(t), which is an external random variable, and of the efficiency of the generator expressed by its heat-rate function H(q(t)). As first proposed by Wood and Wollenberg (1984), the heat-rate function can be described by a quadratic function of the output quantity q: H(q(t)) = α + βq(t) + γq(t)2

(10.15)

and therefore the fuel-cost function can be written as: (q(t), F(t)) = H(q(t)) · F(t)

(10.16)

Fuel costs are not the only costs which affect thermal power plants’ operating regimes. In fact, there exists a series of costs which can be generically be named as ‘operational state transition costs’: ct (xt , xt+1 )

(10.17)

Many operational costs, in fact, depend upon the time spent by the unit on or off-line. Start-up and shut-down costs are essentially the most important costs which can be included in this category. For example, start-up costs vary with the temperature of the boiler and consequently depend on the time

FINANCIAL OPTIMIZATION OF POWER GENERATION

123

spent off-line by the unit. They can be modeled as an increasing function of this time elapsed. In addition, fixed costs can eventually be considered in the model, even if they do not have any impact on the optimal operational strategy. They just affect, as a displacement factor, the production value. We now have the elements to define the power plant operation manager’s objective function. This function should be maximized by the manager to obtain the optimal production value and the optimal operating strategy over the selected time horizon: V(x(t0 ), E(t0 ), F(t0 )) =

max

a(t0 ), a(t1 ), ..., a(T) q∗ (t0 ), q∗ (t1 ), ..., q∗ (T)

⎡  ⎤ N ∗ (t + j) − H(q∗ (t + j)) · F(t + j)
+ j) · q E(t 0 0 0 0 p⎣ βj   ⎦

E · a(t0 + j − 1) − ct0 +j−1 x(t0 + j − 1), x(t0 + j) j=1 (10.18)

It is quite simple to recognize in equation (4.18) a classical example of the constrained Bellman’s problem, which can be solved with backward induction given the following boundary conditions: V(x(T), E(T), F(T)) = E(T) · q∗ (T) − H(q∗ (T)) · F(T)

if x(T) > 0

V(x(T), E(T), F(T)) = 0

if x(T) < 0

(10.19)

In the next chapter we will discuss in detail how some numerical methods can be used to solve Bellman’s problem related to thermal power generator valuation/optimization. However, it is important to emphasize here that the thermal power plant model presented is just one example of how stochastic dynamic programing can be used to model sequential decision problems. The same methodology can be used to model and optimize hydropower generation units, gas or fuel storage assets, pipeline and electric transmission lines. Many other flexibilities and constraints which may characterize power generators can be considered within the model. Emission limits, the possibility to swing between different kind of fuels, warm-start costs, minimum time to swing the fuel, and so on, can be included in the model without a tremendous increase of complexity.

APPENDIX DISCRETE TIME STOCHASTIC DYNAMIC PROGRAMING In this Appendix we give some further details regarding the mathematics of a general stochastic dynamic programing problem. The time axis is discretized and indexed by t = 0, 1, . . ., T < ∞. The environment is stochastic, and uncertainty is introduced through zt , an exogenous random variable which follows a Markov process with a transition

124

REAL ASSET MODELING AND REAL OPTIONS

probability function Q(z0 , z) = Pr (zt+1 ≤ z0 | zt = z) and with z0 given. We assume that zt (but not zt+1 ) is known at time t, so clearly zt is a state of the system. The objective function is defined by the expected sum of instantaneous returns u(xt , ct ) discounted by the factor β < 1. This return function u(·) is a continuous and bounded function of the state variable xt and of the control variable ct . We assume xt ∈ X ⊂ R and ct ∈ C(xt , zt ) ⊂ R, hence we allow the feasible choice set C to depend on the pair of states of the economy (xt , zt ). The law of motion of the state x is: xt+1 = f (xt , ct , zt ), with x0 given. The action ct in each period will depend only on the current states through the (possibly) time varying function gt : X × Z → C, that is, ct = gt (xt , zt ), ∀t. Given the time-separability of the objective function and the assumptions on the Markov process and the law of motion f (·), the pair (xt , zt ) completely describes the state of the system at any time t. The function gt is referred to as decision rule. The sequence of decision rules πT = (g0 , g1 , . . ., gT ) is called policy. A policy is feasible if each generic element gt belongs to C(xt , zt ). It is said to be stationary if it does not depend on time, that is gt (xt , zt ) = g(xt , zt ), ∀t. We now define the expected discounted present value of following a given feasible policy πT from the initial time 0 until the final time T as: WT (x0 , z0 , πT ) = E0

T


βt u(xt , gt (xt , zt ))

(10.A1)

t=0

where it is understood that the expectation is taken with respect to Q(z0 , z), that z0 , x0 are given and that xt+1 follows f (xt , zt , gt (xt , zt )). The dynamic programing (DP) problem is to choose π∗T that maximizes WT by solving: max π(T) WT (x0 , z0 , πT )

(10.A2)

such that x(t + 1) = f (x(t), z(t), gt (x(t), z(t))); gt (x(t), z(t)) ∈ C(x(t), z(t)); x(0), z(0), Q(z , z) known. In the present analysis we are not going to discuss in a detailed way all the properties we should assume on Q to establish the main results. For all practical purposes, it is sufficient to know that Q must satisfy the Feller property, a restriction which guarantees that the expectation function E is bounded and continuous in xt ; we want to know when the DP problem above has a solution. The following theorem covers this: Theorem of the maximum If the constraint set C(xt , zt ) is nonempty, compact and continuous, u(·) is continuous and bounded, f (·) is continuous, and Q has the Feller property, then there exists a solution to the problem above, called the optimal policy π∗T and the value function VT (x0 , z0 ) = WT (x0 , z0 , π∗T ) is also continuous. Notice that the value function is the expected discounted present value of the optimal plan, that is:  T 
t ∗ VT (x0 , z0 ) = E0 β u(xt , gt ) (10.A3) t=1

Corollary If C(xt , zt ) is convex and u(·) and f (·) are strictly concave in ct , then gt (xt , zt ) is also continuous. Given the existence of a solution, we can therefore write:   T
t VT (x0 , z0 ) = max E0 u(x0 , c0 ) + β u(xt , ct ) πT

t=1

(10.A4)

FINANCIAL OPTIMIZATION OF POWER GENERATION

By the law of iterated expectations: E0 (x1 ) = E0 (E1 (x1 )), hence:   T
VT (x0 , z0 ) = max E0 u(x0 , c0 ) + E1 βt u(xt , ct ) πT

125

(10.A5)

t=1

Now let’s cascade the max operator:  VT (x0 , z0 ) = max E0 u(x0 , c0 ) + max E1 πT−1

c0

T


 t

β u(xt , ct )

(10.A6)

t=1

where πT−1 = {c1 , c2 , . . ., cT }. We can perform this step because the recursive structure of the problem guarantees that cs will affect the dynamics of the state for t = s, but not for t < s. With some simple algebra on the discount factor, we obtain: ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T ⎨ ⎬
t−1 (10.A7) VT (x0 , z0 ) = max E0 u(x0 , c0 ) + β max E1 β u(xt , ct ) c0 πT−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t=1 ⎪ ⎪    ⎪ ⎪ ⎩ ⎭ WT−1 (x1 ,z1 )

Hence, if we use the definition of VT−1 (the expected present value of the optimal policy with T − 1 periods left) and pass through the expectation operator, we reach: VT (x0 , z0 ) = maxc(0) {u(x0 , c0 ) + βE0 [VT−1 (x1 , z1 )]}

(10.A8)

Generalizing this result to the case in which we have s periods left to go, inserting the state equation into the next period’s value function, and using the definition of conditional expectation, we arrive at Bellman’s equation of dynamic programing with finite horizon (named after Richard Bellman, 1956):    Vs (x, z) = max u(x, c) + β Vs−1 ( f (x, z, c), z )dQ(z , z) (10.A9) c∈C(x,z)

Z

where x and z denote more precisely xT−s and zT−s respectively, and z0 denotes zT−s+1 . Bellman’s equation is useful because it reduces the choice of a sequence of decision rules to a sequence of choices for the control variable. It is sufficient to solve the problem in (10A.9) sequentially T + 1 times, by backward induction as explained later. Hence, a dynamic problem is reduced to a sequence of static problems. A consequence of this result is the so called Bellman’s principle of optimality which states that if the sequence of functions π∗T = {g0∗ , g1∗ , . . ., gT∗ } is the optimal policy that maximizes WT (x0 , z0 , πT ), then if we consider what remains of the objective function after s periods WT−s (xs , zs , πT−s ), then ∗ , . . ., g∗ ) which were optimal for the original problem, are still the the functions (gs∗ , gs+1 T optimal ones. Thus, as time advances there is no incentive to depart from the original plan. Policies with this property are also said to be time-consistent. Time consistency depends on the recursive structure of the problem, and does not apply to settings that are more general.

Backward induction This is an algorithm to solve finite horizon DP problems. 1 Start from the last period, with 0 periods to go. Then the problem is static and reads: V0 (x(T), z(T)) = max c(T) u(x(T), c(T))

(10.A10)

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REAL ASSET MODELING AND REAL OPTIONS

which yields the optimal choice gt (x(T), c(T)) depending on the final value for x(T) and the final realization of z(T). Hence, given a specification of u(·), we have an explicit functional form for V0 (x(T), z(T)). 2 We can easily go back by one period and use the constraint x(T) = f (x(T − 1), z(T − 1), c(T − 1)) to write:  V1 (x(T − 1), z(T − 1)) = max u(x(T − 1), z(T − 1)) c(t)∈C

 +β

z

V0 ( f (x(T − 1), z(T − 1), c(T − 1)),  z(T))dQ(z(T), z(T − 1))

(10.A11)

which allows us to solve once again for gT−1 (x(T − 1), z(T − 1)) and to obtain V1 (x(T − 1), z(T − 1)) explicitly. 3 We continue until time 0 and collect the sequence of decision rules into the optimal policy vector. 4 Given the initial conditions at time 0, we can reconstruct the completely optimal path for the state and the control, contingent on any realization of {zt }Tt=0 .

C H A P T E R 11

Framing and Solving the Optimization Problem

In the previous chapter, we saw that a thermal power generation asset can be thought of as a highly structured financial derivative whose optimal exercise depends upon market price realization. The asset manager has to ‘exercise’ the asset flexibilities in order to maximize the expected return, respecting operational constraints. Of course, for the solution of this kind of problem the dynamic hypothesis we decide to make for the market price variables is fundamental. We can decide to work within a deterministic environment (optimization of price forecasts) in which case the complexity of the optimization problem defined in the previous chapter is drastically reduced, but obviously also the solution is not then extremely accurate. If we decide to solve the optimization problem within a stochastic environment, as we proposed previously, it is necessary to go through numerical methods which allow us to reach a solution to the Bellman’s problem associated with the generation asset model. In this chapter we will examine these methods starting with the detailed framing of the deterministic problem. This initial step is important to understand the importance of properly considering the stochastic nature of the optimization problem in terms of model output, but also the additional complexity introduced by it.

11.1 OPTIMIZATION PROBLEMS IN A DETERMINISTIC ENVIRONMENT In this section we will solve the optimization problem stated in the previous chapter (with all the assumptions and notations defined before) assuming that the price vector (E(t), F(t)) is completely deterministic and known for 127

128

REAL ASSET MODELING AND REAL OPTIONS

the whole problem’s time horizon [0, T]. Given this strong assumption, the value equation (10.18) defined in the previous chapter can be specified as follows for a generic t belonging to [0, T]: V(x(t)) = max

a(t),q(t)

T


(E(t) · q∗ (t) − H(q∗ (t)) · F(t)) · a(t)

t=0

−ct (x(t), x(t + 1))

⇓

(11.1)

∀t ∈ [0, T] if x(t) = t_on V(x(t)) = E(t) · q∗ (t) − H(q∗ (t)) · F(t) + max[V(x(t + 1) = t_on), V(x(t + 1) = −1) − ct (t_on, −1)] if 0 < x(t) < t_on V(x(t)) = E(t) · q∗ (t) − H(q∗ (t)) · F(t) + V(x(t + 1) = min(t_on, x(t) + 1)) if −t_off < x(t) < 0 V(x(t)) = V(x(t + 1) = x(t) − 1) if −t_cold ≤ x(t) ≤ −t_off V(x(t)) = max(V(x(t + 1) = 1 − ct (x(t), x(t + 1) = max(−t_cold, x(t) − 1)), V(x(t + 1) = max(−t_cold, x(t) − 1)) In order to solve this deterministic dynamic program, backward induction can be employed using the standard terminal conditions: V(x(T), E(T), F(T)) = E(T) · q∗ (T) − H(q∗ (T)) · F(T) V(x(T), E(T), F(T)) = 0

if x(T) > 0 if x(T) < 0

(11.2)

The optimal solution of the problem can be obtained as the initial value V(x(0)), given the initial condition x(0) at the initial time. Obviously, the solution of this deterministic problem it is not only an incorrect representation of the real decision environment, since prices are not known for the whole problem’s time horizon, but it is also a bad representation of the decision process itself. In fact, even if we assume that the deterministic prices realization that we consider is the best possible forecast over the time interval [0,T], this formulation of the problem assumes implicitly that sequential decisions are taken without being influenced by

129

FRAMING AND SOLVING THE OPTIMIZATION PROBLEM

the increasing information set represented by new price realizations. Realistically, the optimal operating strategy of the power plant is updated as new prices are observed. This fundamental feature can only be contemplated if we consider properly the uncertain nature of the stochastic vector [E(t), F(t)].

11.2 NA¨IVE APPLICATION OF MONTE CARLO METHODS The simplest, but also incorrect, way of considering the stochastic nature of electricity and fuel prices is that of repeating the deterministic optimization problem described in the previous section for more than one price trajectory and then calculate the plant value, and all the other interesting outputs, as an average according to the following scheme:
Simulate N paths of the stochastic vector [Ei (t), Fi (t)] for every t belonging to [0, T]
Use the deterministic algorithm to determine V i (x(0))
Determine the plant value as the average value over the N simulated paths N 

V(x(0)) =

V i (x(0))

i=1

N

(11.3)

This straightforward application of a Monte Carlo simulation to solve sequential choice problems in a stochastic environment is of course simple to understand and computationally fast, but unfortunately displays serious problems which are a disincentive to its application. In particular, the MC method presented tends to overestimate the true power plant value. In fact, it implicitly assumes that, for every price path, the decision-maker knows in advance future prices and consequently takes unit commitment decisions ignoring prices uncertainty. In a certain sense, the excess of information assumed by this method implies that the commitment policy is optimal in every price scenario. Moreover, the expected power production, which is a secondary but important output of the model, may be inconsistent with operational constraints since it is calculated as the average of consistent production plans. In conclusion, this method even if it considers the stochastic nature of electricity and fuel prices doesn’t realistically represent the influence that market prices evolution has on sequential decisions taken by the plant manager. It is necessary to solve the plant optimization problem considering the dynamic nature of the decision process, as suggested by the Bellman’s representation of the problem itself.

130

REAL ASSET MODELING AND REAL OPTIONS

11.3 SOLVING BELLMAN’S PROBLEM From the previous chapter we know that the power plant optimization/ valuation problem can be correctly represented as a stochastic dynamic program whose associated Bellman’s equation can be written as follows: V(x(t0 ), E(t0 ), F(t0 )) =

max

a(t0 ), a(t1 ), ..., a(T) q ∗ (t0 ), q∗(t1 ), ..., q∗(T)

⎡

⎤

 N ∗ ∗
E(t0 + j) · q (t0 + j) − H(q (t0 + j)) · F(t0 + j) ⎦ p⎣ βj E · a(t0 + j − 1) − ct0 +j−1 (x(t0 + j − 1), x(t0 + j)) j=1

(11.4) As we did in the deterministic case, we can specify the target equation written above as: Vt (x(t), E(t), F(t)) ⇓ ∀t ∈ [0, T]


if x(t) = t_on and Et [Vt+1 (x(t + 1) = t_on, E(t + 1), F(t + 1))] > Et [Vt+1 (x(t + 1) = −1, E(t + 1), F(t + 1)) − ct (t_on, −1)] ↓ Vt (x(t), E(t), F(t)) = E(t) · q∗ (t) − H(q∗ (t)) · F(t) + Et [Vt+1 (x(t + 1) = t_on, E(t + 1), F(t + 1))] a(t) = 1




if x(t) = t_on and Et [Vt+1 (x(t + 1) = t_on, E(t + 1), F(t + 1))] ≤ Et [Vt+1 (x(t + 1) = −1, E(t + 1), F(t + 1)) − ct (t_on, −1)] ↓

FRAMING AND SOLVING THE OPTIMIZATION PROBLEM

Vt (x(t), E(t), F(t)) = E(t) · q∗(t) − H(q∗ (t)) · F(t) + Et [Vt+1 (x(t + 1) = −1, E(t + 1), F(t + 1)) − ct (t_on, −1)] a(t) = 0


if 0 < x(t) < t_on ↓ V(x(t)) = E(t) · q∗ (t) − H(q∗ (t)) · F(t) +Et [V(x(t + 1) = min(t_on, x(t) + 1), E(t + 1), F(t + 1))] a(t) = 1




if −t_off < x(t) < 0 ↓ Vt (x(t), E(t), F(t)) = Et [V(x(t + 1) = x(t) − 1, E(t + 1), F(t + 1))] a(t) = 0




if −t_cold ≤ x(t) ≤ −t_off and Et [V(x(t + 1) = 1, E(t + 1), F(t + 1)) −ct (x(t), x(t + 1) = max(−t_cold, x(t) − 1))] > Et [V(x(t + 1) = max(−t_cold, x(t) − 1), E(t + 1), F(t + 1))] ↓ Vt (x(t), E(t), F(t)) = Et [V(x(t + 1) = 1), E(t + 1), F(t + 1)) − ct (x(t), x(t + 1) = max(−t_cold, x(t) − 1))] a(t) = 1




if −t_cold ≤ x(t) ≤ −t_off and Et [V(x(t + 1) = 1, E(t + 1), F(t + 1)) −ct (x(t), x(t + 1) = max(−t_cold, x(t) − 1))] ≤ Et [V(x(t + 1) = max(−t_cold, x(t) − 1), E(t + 1), F(t + 1))] ↓ Vt (x(t), E(t), F(t)) = Et [V(x(t + 1) = max( − t_cold, x(t) − 1), E(t + 1), F(t + 1))] a(t) = 0

131

132

REAL ASSET MODELING AND REAL OPTIONS

where Et denotes the conditional expectation operator and, as usual, we impose the following boundary conditions: V(x(T), E(T), F(T)) = E(T) · q∗ (T) − H(q∗ (T)) · F(T) V(x(T), E(T), F(T)) = 0

if x(T) > 0 if x(T) < 0

(11.5)

The explicit introduction of the uncertainty about future price realizations emphasizes the fact that every single choice drastically influences the expected continuation value, which is nested with that. This means that, given the state of the system, a certain state transition is optimal among all the feasible transitions if the associated expected continuation value is the highest. The problem now is how to compute for every time step and for every system state the expected value of continuation of every single choice.

11.3.1 The lattice approach The first and simplest solution to the problem above is represented by the lattice approach. Traditionally, both in financial and decision-science applications, binomial and trinomial trees represent the most appropriate environment to solve backward induction problems, or when optimal exercise policies have to be found. The classical approach of American-option pricing proposed by Cox, Ross and Rubinstein (1979) is the most famous example of such a method, but the evaluation method presented in the previous Part for swing options also belongs to this class. According to this method, the evolution of the stochastic vector [E(t),F(t)] for every t belonging to [0,T] can be fully represented by a price lattice. As we have seen in previous chapters a lattice is determined by various nodes for every time step which represent the values which the stochastic vector can assume at that time step. Every single node is connected to other nodes of the following time step by branches, and transition probabilities are associated to every single feasible price transition. (Figure 11.1 represents the lattice scheme.) The most important feature of a lattice representation is the reduction of the feasible price transition. In fact, for every single node only two (in the binomial scheme) or three (in the trinomial one) nodes of the successive time step can be reached with a positive probability, and consequently conditional expectations of continuation value can be easily computed for every single system state x(t): Et [Vt+1 (x(t + 1), E(t + 1), F(t + 1)]
p(i) · [Vt+1 (x(t + 1), Ei (t + 1), Fi (t + 1))] = i∈I

∀t ∈ [0, T]

(11.6)

FRAMING AND SOLVING THE OPTIMIZATION PROBLEM

133

E (i ⫹1, j ⫹ 1) p (i, i ⫹ 1) p (i, i ) E (i, j ⫹ 1)

E (i, j )

p (i, i ⫺ 1) E (i ⫺1, j ⫹ 1)

Figure 11.1 Trinomial lattice scheme Hence, the lattice works jointly as a discretization of the price and decision spaces. There are many analytical schemes in the literature which relate continuous time stochastic dynamical hypothesis for prices evolution to lattice schemes. The Cox, Ross and Rubinstein (1979) binomial method and the Hull and White (1994) trinomial method are the best-known and used ones. Unfortunately, lattice schemes present some negative features which can be summarized as follows:
Lattice methods become impractical when the number of stochastic factors we want to represent increases.
Only few types of continuous time stochastic processes can be discretized within a lattice scheme, and essentially all of them are inappropriate to represent electricity price dynamics. For these reasons other approximation methods have been explored to consistently solve the problems by simulation.

11.3.2 The least-squares Monte Carlo approach (LSMC) The main intuition behind this method is that conditional continuation values can be estimated from cross-sectional information contained in Monte Carlo simulations of electricity and fuel prices by using the least-squares regression method. In particular, we can estimate conditional expectations regressing ex post plant values on functions of the state variables. By means of such an estimation process, repeated for every decision time t, we are able to determine the optimal exercise strategy and, hence, the power plant

134

REAL ASSET MODELING AND REAL OPTIONS

value. This method was originally developed by Longstaff and Schwartz (2001) for valuing American-style options. The hypothesis on which this approach is based can be theoretically justified in all those situations where the conditional expectation operator is an element of L2 of square-integrable functions relative to some measure. In fact, since L2 is a Hilbert space it is provided with a countable orthonormal basis and the conditional expectation operator can be represented as a linear function of the elements of the basis. For example, if we denote Y = [E, F] as the two-dimensional Markovian process of electricity and fuel prices, then we can write: ∞
Et [Vt+1 (x(t + 1), E(t + 1), F(t + 1)] = γj · fj (Yt ) (11.7) j=0

where the type of basis functions fj (.) could be: Leguerre polynomials, Hermite polynomials, Chebyshev polynomials and many others. Fortunately, empirical tests emphasize how, also, Fourier, trigonometric or even simple power series of the state variables can provide accurate results. Obviously, the infinite series is approximated by the first M-basis functions. Using this method and moving backwards in time from terminal conditions it is possible to estimate continuation values for every time step t and system state x(t), and then by recursive substitutions within the algorithm developed in the previous sections it is possible to obtain the power plant value at the initial time. The LSMC method is extremely interesting for these kinds of problems especially for its flexibility. In fact, since it is based on Monte Carlo simulations of state variables we are allowed to use any class of Markovian stochastic processes to model them. This property is very important for us since we have seen how peculiar are the characteristics of electricity price paths. Unfortunately, this method is extremely intensive from the computational point of view, and consequently the solution of the power plant optimization problem could be difficult to solve for long-term time horizons (one year or more) as emphasised by Tseng and Barz (2001). However, this method represents probably the best possible solution to the optimization problem proposed, and useful hints can be obtained by means of its application, as we will see in the Appendix to this chapter.

11.4 ALTERNATIVE SOLUTION METHODS: ORDINAL OPTIMIZATION In previous sections we have considered various methods that are useful in finding an approximate solution to the power plant optimization problem in a dynamic and stochastic environment. The reason why we focus immediately on approximate solutions of the proposed problem is that the

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135

true optimal solution is effectively unattainable. In fact, the true optimal solution of a unit commitment problem should be associated to that strategy, among the set of admissible ones, which dominates on average all the others over the whole state variables space, represented by a consistent set of price scenarios. In problems with dimensions such as we are examining, it may be impossible to evaluate every admissible policy out of a billion or more over all the price path simulations that we have decided to use in order to span the state variables space. Ordinal optimization is an approach for finding a solution to an optimization problem in which a huge number of possible solutions have to be considered. The main idea behind ordinal optimization is that of ‘goalsoftening’. Quite often, stochastic optimization problems such as those presented here have an admissible policy space with little or even no structure, which increases exponentially or combinatorially as the number of sequential decisions increases. Goal-softening means that we substitute the optimal solution (policy) with a ‘good enough’ one that means one of the best n per cent of all the admissible solutions. As explained by Allen and Ilic (1999), ordinal optimization has three basic steps: 1 First, a sample of N policies, out of the set of admissible ones, is selected by means of random sampling. Usually, the policy space is assumed to have a uniform distribution and consequently the sample size selected should be large enough to improve the possibility of having a sufficient number of good-enough policies (N > 1,000 is recommended). 2 Second, a subset of s elements is selected from the original sample. The sub-sample size is selected in such a way that the alignment probability with the good-enough policies set exceeds a selected threshold for the selected percentile n per cent. The selection rule for the s sub-sample policies can be of various types. Allen and Ilic (1999) suggest two simple methods: blind pick (sampling without replacement) and horse racing.1 3 Finally, all the s selected policies are evaluated in order to determine which one guarantees the best-expected performance. Ordinal optimization can be considered a good method for drastically reducing the dimensions of the search space for all those optimization problems where a complete evaluation of all possible strategies is not feasible. However, there are some problems, defined as ‘needle in haystack’ problems, where the optimal solution may be far ahead of the others. For such problems goal-softening is not helpful. 1

See the original book of Allen and Ilic (1999) for a more detailed discussion of sub-sample selection rules.

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REAL ASSET MODELING AND REAL OPTIONS

APPENDIX GENERATION ASSET MODELING: NUMERICAL RESULTS In this Appendix we present results obtained using the methodology developed above to evaluate the production of a thermal power plant over a seven-day time period (168 hours), and to assess the financial risk involved. The stochastic dynamic problem is solved through backward induction by means of a lattice discretization approach. Some assumptions have been made in order to reduce the numerical complexity of the problem, although the example proposed still remains realistic. In particular, we are going to assume the following:
the fuel price and heat-rate are constant over the operating period (fuel price = 3 and

H = 10);


the discrete dynamics of the electricity price is described by the classical CRR binomial

model (a binomial-shaped case is considered); and
the volatility parameter is assumed constant over the operating period.

In order to determine the importance of different operating constraints on the plant production value, we consider three particular cases summarized in Table 11A.1. The first case proposed represents a very flexible power plant, while the second and the third cases are representative of more constrained power plant operations (the second and third cases differ only in the minimum dispatch level). As outlined, the results presented focused mainly on valuation and risk-management issues, ignoring purely optimization issues. To assess the impact of market risk on the value of the production, a new, lattice-based, risk measure, the ‘worst case production value’, is proposed. This measure is simply attainable by means of the lattice approach based on the economic result the asset manager is going to obtain managing under the optimal policy in the worst possible market conditions.2 We will show also that this measure behaves coherently with standard financial intuitions. Let us start by analyzing how the power production value varies with changes in the spot spark spread values. Of course, the numerical values presented in this example do not refer to any realistic situation. The results are shown in Table 11A.2 and Figures 11A.1–4. The plant production value and the risk measure are also affected by volatility changes, and Tables 11A.3 and 11A.4 and Figures 11A.5 and 11A.6 summarize the volatility impact on both the indicators.

Table 11A.1 Summary of test case characteristics

2

t off

t ramp

t on

t cold

min q

max q

Case 1

1

0

1

0

100

250

Case 2

3

2

4

1

100

250

Case 3

3

2

4

1

200

250

Of course, we refer to the worst possible market conditions available on the lattice.

Table 11A.2 Production value and risk measure versus spark spread Spark value −20

−15

−10

−5

0

5

10

15

20

Plant production value

0

0

3.8641

1,571.6

43,611

213,340

420,170

630,010

840,000

Worst case

0

0

0

0

0

8,772

32,997

69,039

114,390

Plant production value

0

0

3.6

1,530.1

43,087

208,280

410,160

615,010

820,000

Worst case

0

0

0

0

0

4,634

23,982

55,147

95,620

Plant production value

0

0

3.5

1,516.3

42,927

208,260

410,160

615,010

820,000

Worst case

0

0

0

0

0

4,634

23,982

55,147

95,620

Case 1

Case 2

Case 3

137

138

Case 1 Plant production value Worst case 1,000,000

Value

800,000 600,000 400,000 200,000 0 ⫺20

⫺10

0

10

20

Spark spread

Figure 11A.1 Production value versus spark spread (case 1)

Case 2 Plant production value Worst case 900,000 800,000 700,000

Value

600,000 500,000 400,000 300,000 200,000 100,000 0 ⫺20 ⫺15 ⫺10 ⫺5 0 5 Spark spread

10

15

20

Figure 11A.2 Production value versus spark spread (case 2)

139

Case 3 Plant production value Worst case 900,000 800,000 700,000

Value

600,000 500,000 400,000 300,000 200,000 100,000 0 ⫺20 ⫺15 ⫺10 ⫺5

0

5

10

15

20

Spark spread

Figure 11A.3 Production value versus spark spread (case 3)

Relative differences from case 1 Case 2 Case 3

0.1 0.09

Value

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

⫺20 ⫺15

⫺10

⫺5 0 5 Spark spread

10

15

20

Figure 11A.4 Production value versus spark spread (relative differences from case 1)

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REAL ASSET MODELING AND REAL OPTIONS

Table 11A.3 Power production value versus volatility Volatility range 1%

2%

3%

4%

5%

6%

Case 1

43,611

87,110

130,390

173,340

215,860

257,850

Case 2

43,087

86,066

128,830

171,270

213,280

254,760

Case 3

42,927

85,746

128,350

170,630

212,490

253,830

Table 11A.4 Worst case value versus volatility Volatility range 1%

2%

3%

4%

5%

6%

Case 1

8,772

4,086.1

2,520.5

1,744.7

1,271.8

1,001

Case 2

4,634

767.3

0

0

0

0

Case 3

4,634

749.8

0

0

0

0

Plant production value vs volatility Case 1 Case 2 Case 3

300,000 250,000

Value

200,000 150,000 100,000 50,000 0 1%

2%

3% 4% Volatility range

5%

6%

Figure 11A.5 Production value versus volatility From the analysis of the previous figures we can also deduce information about power production value sensitivities, useful for hedging purposes. The main results and conclusions which can be inferred from this simple but instructive example are as follows:
The weekly time horizon is large enough that the impact on production values of

operational constraints is not very significant.

FRAMING AND SOLVING THE OPTIMIZATION PROBLEM

141

Worst case vs volatility Case 1 Case 2 Case 3

10,000

Worst case value

8,000 6,000 4,000 2,000 0 1%

2%

3% 4% Volatility range

5%

6%

Figure 11A.6 Worst case versus volatility
Consequently the classical spark-spread representation can be used with any particular

problem, for analysis with a time horizon bigger than a month.
Classical dynamic hedging strategies for the production value turn out to be extremely

sensitive to the optimal policy. The optimal policy is more stable when the spark spread is deep in or out of the money (gamma effect is almost zero). When the optimal policy is very sensitive to small market movements (at the money spark spread there is a strong gamma concentration) the dynamic hedging portfolio has to be adjusted continuously, incurring high transaction costs.
Market risk, measured by the distance between the expected production value and the

worst case, is extremely high, in absolute terms, for deep in the money spark-spread levels. Financial hedging in this case is advisable. Fortunately, for very positive sparkspread values the optimal policy is stable and robust (‘produce as much as you can’), hence static hedging is a good solution to reduce the risk.
‘Worst case’ seems to be a good risk measure for power production for short time hori-

zons. For longer time horizons, risk measures such as Earning at Risk (EaR) or Profit at Risk (PaR), calculated with spark-spread option methodology (ignoring operational constraints) can be adopted.
The sensitivity of the power production value to volatility changes is almost linear,

while the risk measure displays a non-linear sensitivity to volatility changes (this is a remarkable fact since we often hedge the worst case).
The introduction of a factor shape into the electricity price dynamics has a significant

impact on the power production and worst-case values, but it does not affect the shape of the Greeks and hence does not affect the previous general conclusions. This consideration can be extended to generic model specification problems (errors or strong assumptions on the specification of the stochastic dynamic model).

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PART V

Electricity Risk Management: Risk Control Principles and Risk Measurement Techniques

143

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C H A P T E R 12

Risk Definition and Mapping

12.1 MARKET RISK DEFINITION AND BASIC PRINCIPLES In competitive markets such as the liberalized electricity markets, market risk is an extremely important variable both in day-by-day business activity and in the strategic decision-making process. For this reason its correct definition and assessment is a fundamental issue for companies which operate in those markets. Traditionally, risk is defined as the uncertainty which may affect the economic performance of a certain business or investment. This uncertainty can have a positive or a negative impact on the business, but of course also on people concerned by the negative impacts that unpredictable events may have on their business or investment returns. For purely trading-oriented business activities, such as banking, risk issues are not new concepts. Therefore, even when the trading activity is concentrated on electricity or energy-related products the definition and measurement of the market risk can be based on what is considered a benchmark in traditional financial trading activities. In particular, when the trading activity deals with the dynamic management of a portfolio of liquid financial positions, market risk can be traditionally defined as the impact that market uncertainty may have on the value of the portfolio. Consequently, market risk can be assessed and monitored by means of traditional risk tools such as value-at-risk methodologies. On the other hand, when the business is not, or not only, focused on the trading of financial products related to the power or energy markets a direct translation of standard risk methodologies within a mixed industrial/financial activity is not always possible. This is the case, for example, 145

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ELECTRICITY RISK MANAGEMENT

of an integrated electricity firm which has in its portfolio of investments: real assets, physical contracts and financial positions. In this situation it is clear that a dynamic management of the whole portfolio composition is not possible because a significant part of the investments has a very long time horizon, which would be difficult to liquidate in the short term (real assets and physical contracts). Hence, in this case the economic performance of the portfolio is not determined by the continuous time change of its value,1 but by the economic margin or the financial cash flow which is periodically generated by it. Of course, the risk definition in this more general environment is strongly affected by previous considerations. In particular, a value-based risk strategy is not always applicable, and in any case it may not be sufficient. Another fundamental issue to clarify in order to establish a consistent risk strategy is the relation between the business’s economic goal and the definition of the associated risk. In fact, it may be the case that we are interested in reaching a stable economic margin which is most protected from market uncertainties. In other situations, we may be interested in overperforming a certain benchmark which itself might fluctuate with the market. Hence, the market risk is not necessarily related to absolute market fluctuations, but is essentially related to the relative impact that those fluctuation have on our economic and financial goals. Market factors which may affect the economic performance of a complex portfolio are many, and they may be extremely heterogeneous in nature. The full determination of all market risk factors should be carried out together with their mapping onto the company value-creation chain. Riskfactor mapping is fundamental in understanding the relative importance of a certain market risk factor.

12.2 DIFFERENT RISK FACTORS AND THEIR MAPPING ONTO THE COMPANY VALUE-CREATION CHAIN Many different risk categories can be recovered along the production chain of an electricity firm. Since we are mainly focusing on risk in a financial perspective, let us concentrate on those risks that have a clear financial interpretation, and let us try to allocate them inside the production chain as in Figure 12.1. Looking at the scheme of Figure 12.1 from a financial perspective, it is possible to identify the following macro-categories of risks which can be, more or less, diversified away by means of a financial hedging activity:
commodity price risks due to contract structuring;
FX risks; 1 For portfolios composed by physical, financial and real positions, it is not always possible to give a unique and objective definition and assessment of its ‘fair value’.

RISK DEFINITION AND MAPPING

Bilateral contracts

Storage F U E L M K T

147

ASSETS FIN. / PHYSICAL OPTIMIZATION

Term purchase

Spot exchange transactions

Derivatives exch. / OTC Transportation and shipping

Emission risk

Transmission rights

FINANCIAL HEDGING RISK

Figure 12.1 Company risk map


asset/storage management risks;
intercompany transfer price agreement risks;
financial hedging risks; and
‘hybrid’ risks. We shall briefly describe each category of risk trying to identify its source and correct management.

12.2.1 Commodity price risk An electricity company is a fuel buyer and an electricity seller (in a schematic representation); hence, it is exposed to commodity price fluctuations according to the nature of the physical contracts it holds in its portfolio. We have seen that physical contracts can be represented and priced as derivative products; hence, their price risk exposure can be computed by means of classical ‘Greek measures’ such as delta and gamma. This risk category is mainly located at the extremes of the value-creation chain where fuel agreements and electricity delivery contracts take place. However, commodity-price risk-management decisions cannot be taken independently on the asset/storage management decision process and intercompany transfer price structuring decisions. This is because a firm has to look for internal risk-diversification before using financial derivatives products.

148

ELECTRICITY RISK MANAGEMENT

12.2.2 Foreign exchange (FX) risk For European electricity companies, and all non-US-based ones, foreign exchange (FX) risk is embedded in the fact that fuels are typically traded in US dollars (or US dollars equivalent) and electricity in euros, pounds or other currencies. This foreign currency differential affects, in terms of risk exposure, the whole margin-creation process of the company. The FX riskanalytic measurement can be performed with the same approach adopted for commodity price risk (that is, Greek measures). Of course, commodity price-risk exposure and FX risk exposure are highly dependent risk sources both for statistical and contractual reasons. In fact, quite often in fuel agreement purchase contracts and natural-gas pricing contracts (at least in continental Europe) commodity prices and exchange monthly average rates are combined together in payoff/payout formulas in a multiplicative way, and consequently ‘cross gammas’ are not null. For this reason a power company must accurately consider the impact of FX exposure, structuring its physical contracts in order not to increase and to eventually reduce this kind of financial exposure. For this purpose, recently, structured products which combine commodity and exchange rate risk have been proposed as hedging instruments (quanto derivatives).

12.2.3 Asset/storage management risk A power plant can be considered as a black box transforming fuel into electricity according to a certain efficiency level of the plant itself. Hence, as we have seen in previous chapters, the power plant management and scheduling can be performed with a financial perspective through a correct representation (asset modeling) of its asset structure in terms of a correct financial representation of its operational flexibilities and constraints. Real asset modeling is not only a relevant evaluation issue, but also obviously a relevant risk issue. If real asset flexibilities are real options, then they are risky objects which must be assessed and controlled. Of course, a correct modeling of power plants is not a simple task since the operational constraints are usually very complex to model. Hence, the representation of production assets as a simple string of plain-vanilla options is only a firstorder approximation of the asset-management problem (see Part 4 for a more detailed modeling approach). In addition, the fuel and gas storage management problem can be described using a similar approach based on real option methodology. The financial representation of real assets as complex financial products is useful because it enables us to calculate the physical exposure generated by them, for both fuel and electricity sides, with a methodology which combines market uncertainties effects and optimal management strategies.

RISK DEFINITION AND MAPPING

149

12.2.4 Intercompany transfer price agreement risk Intercompany transfer price agreements (ICAs) are very important within big industrial groups since they determine how risks and returns are distributed along the value-creation chain. In fact, the risk–return target of every business unit or division of the company has to be designed in coherence with the risk–return target of the whole company. Here lies the importance of a good ICA structure. Obviously, the importance of ICAs are more related to organizational issues than to strategic or trading ones, but organizational issues are often prerequisites for a correct application of any risk-management strategy. For example, from the risk side a correct structuring approach for ICA allows concentration of risks in some part of the margin-creation chain and consequently focuses managerial expertise in that particular location for better control of those risks. This risk category can be considered as lying between market and an operational risk.

12.2.5 Hedging risk Once a full identification and mapping of financial risks have been done, the risk management is in a position to check if and where it is possible to hedge the risk internally, by means of internal netting or synthetic deals. The risk management can, moreover, decide to hedge all the risk it was not possible to diversify away internally, by means of financial derivative products. Of course, correct management of a portfolio of financial derivatives is itself a risky activity which has to be monitored by means of methodologies typical of financial trading activity.

12.2.6 ‘Hybrid’ risks Here we list and briefly describe a series of risk sources which cannot actually be considered purely financial, since they cannot be completely hedged away with internal diversification or by means of financial derivative products. The following categories can be considered as hybrid risks:
Volume risk (demand risk on a retail portfolio). We have seen that volumetric flexibilities in trading portfolios or embedded in trading instruments can be priced and hedged as financial options since the holder of those flexibilities will exercise them efficiently on the basis of a financial criterion. Volumetric flexibilities enclosed in retail products may not be exercised on the exclusive basis of a financial rationale, and consequently cannot

150

ELECTRICITY RISK MANAGEMENT

be fully hedged by means of classical electricity derivatives products. In this situation, volumetric fluctuations represent a risk factor independent of market price risk. An efficient way of measuring and managing this risk category is essentially based on portfolio diversification techniques, which are out of the scope of the present discussion.
Emission risk. Limitations on hydrocarbon emissions due to national reduction and allocation plans represent an ulterior constraint to production capability and consequently an economic risk. The recent birth of emission spot and derivatives markets gives utility companies the chance to partially diversify away this risk source or, at least, provide clear information about their market price. However, nowadays the liquidity of emission markets, especially emission derivatives markets, is very low, even if increasing, and consequently a fair valuation of emission risk is not possible. Emission markets effectively represent an alternative to generation reduction, which should always be considered by power plant managers.
Congestion risk. Transmission constraints among different zones of the same country or between different countries of the same continental area may create stress in market price differentials. A geographically spread power generation company may be naturally exposed to this risk source since its production plants may be located in areas where interconnection is not always fluid and cheap. As we have mentioned, from the financial point of view this risk should be measured and managed as a price-spread risk. In many countries in the last few years the number of OTC zonal price-spread derivatives transactions has increased considerably proof of the relevance of this risk category. The above are some of the most common sources of market tradable risks, however it is important to note that many of these risks are generated by local regulations or market conventions, which obviously change rapidly. Hence, new risk types and sources are going to be born continuously as the business changes.

12.3 RISK AND OPPORTUNITY (ENTERPRISE RISK ATTITUDE) So far we have emphasized the negative aspect of risk because we have focused only on the negative impact that market uncertainties have on the economic performance of a power company. However, uncertainty is not necessarily associated to potential losses. While financial institutions like banks or financial trading companies are risk-averse by definition, an industrial company such as an electricity company has a risk attitude which

RISK DEFINITION AND MAPPING

151

may be slightly more complicated. In fact, enterprise risk has always been considered a natural component of every industrial initiative. Obviously, this does not mean that market risk should not be considered as a dangerous event, but of course it is always necessary to correctly balance positive and negative outcomes related to market uncertainties. This continuous assessment of positive and negative economic effects due to market fluctuations turns out to be particularly important when strategic decisions have to be taken. This different dimension of risk analysis is mainly related to business activities not directly related to electricity trading. However, it is nevertheless interesting and important to think of market risk also under alternative perspectives. For example, the impact that the introduction of risk concepts may have on traditional planning and budgeting activities is an interesting issue; we will try to introduce this argument later in Chapter 14. Once we have defined the nature of market risk and identified financial risks that may affect the value-creation chain of a power company, mapping them into a schematic representation of that chain, the next step is the synthetic measurement of the impact that market risks may have on the firm economic performance. Since we have concentrated our attention on financial risks, it is possible to measure the impact of those risks by adapting standard financial techniques. Risk measurement methodologies and all the problems related to limits imposition will be considered in the following chapter.

C H A P T E R 13

Risk Measurement Methods

Market risk is a complex subject with multiple dimensions and implications for electricity business activity. Analytical risk measures such as traditional Greek measures or high-order and cross-sensitivities allow us to control in detail market risk, but sometimes these analytical risk measures are too technical to be understood by non-technical staff or by management. Hence, it is necessary to make a synthesis of the information contained in analytical measures into a more intelligible form. The natural way of creating such a type of risk measure is of course that of assessing the impact of risky events in monetary terms, because non-technical people are also capable of understanding the meaning of a potential monetary (or economic) loss. This is exactly the reason why synthetic risk measures have been introduced and have reached a very high importance in the last ten years. A typical example is represented by value at risk (VaR), but VaR is not the only synthetic risk measure important to consider in the electricity field, especially if the business we are interested to risk-assess is not completely based on financial trading. In this chapter we will go into the details of risk measurement of financial and physical trading activities, examining also some mathematical and technical problems related to their calculation.

13.1 RISK MEASURES FOR FINANCIAL TRADING PORTFOLIOS In this section we concentrate on synthetic risk measures for portfolios made up of liquid financial products. In this situation, the portfolio manager may 152

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153

choose in any moment to liquidate or modify the composition of his portfolio without incurring high financial expenses due to scarce liquidity. Because of that, the economic performance of this kind of business is determined mainly by the fluctuation of the value of the portfolio itself more than by the financial payoff of the deals that compose it. Hence, the market value (or mark-to-market value) potential drop is the true risk driver for a financial trading portfolio. For this reason, the synthetic risk measures we are going to discuss in this section are value-based risk measures, and we will start by talking about the classical risk measure: the VaR.

13.1.1 Value at risk Value at risk has become widely used and popular among bank and mutual fund managers since central bank regulators chose to determine banks’ capital requirements based on it early in the 1990s. For this reason it was also natural in the energy field to evaluate the risks embedded in open trading positions by means of VaR from the very beginning of trading activity. Value at risk effectively measures for electricity the market price risk exposure of an open position, condensing risk factors such as market price volatility and correlation and, potentially in more advanced cases, currency and interest rate risks. It is usually defined as: ‘the minimum potential loss that a trading portfolio may have over a holding period of m days, in the x per cent of worst cases’. In other words this means that we may expect to lose more that the VaR figure in the given holding period of m days, in the x per cent of the cases. Practically speaking, the VaR measure tries to give an answer to the question ‘how bad can things get?’ VaR is a simple and intuitive measure which depends on two main arguments, the size of the holding period (number of days) and the confidence level x per cent. The holding period usually reflects the number of days necessary to completely liquidate the position without incurring additional costs, and hence is a measure of market liquidity. Usually, for portfolios of exchange-traded instruments, this period is something between one and fifteen days. The confidence level x per cent reflects how conservative the measure is. In fact, the more x per cent is close to 1 the more the risk measure is high and consequently the probability to have a worse economic result is low. From the statistical point of view, VaR is a percentile measure. It measures the percentile of the portfolio value variations corresponding to the selected confidence level. Formally, if we define with X the portfolio fair value we will have that: {VaR(x%) = c ∈ R+ : Prob(−
X > VaR(x%)) = (1 − x%)}

(13.1)

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ELECTRICITY RISK MANAGEMENT

Typical confidence levels are 95%, 97.5% or 99% since they correspond to well-known values in the Gaussian distribution tables. As we have mentioned, the VaR calculation is based upon the concept of fair value of the asset portfolio. Hence, VaR calculation methodologies are highly related to portfolio valuation methods, especially when options and non-linear derivatives are present in the portfolio.

13.1.2 VaR calculation methods Traditional VaR calculation methods can be essentially divided into two groups: analytical and numerical (simulation-based) methods. Analytical methods were the first presented and used in traditional financial applications of VaR, and they are essentially based on some theoretical assumptions like ‘normality’ of financial assets returns. The normality assumption is essential to obtain closed formulas for VaR calculations when we manage large and well-diversified portfolios of financial assets, but it is clearly not realistic for electricity markets, due to price spikes. However, the relaxation of this assumption makes the development of an analytical calculation method much more difficult and sometimes impossible. For this reason numerical methods should be preferred in the case of electricity derivatives portfolios even if the calculation time may be much higher. It is important, in any case, to briefly recap traditional analytical models. Analytical models (linear and delta-gamma)

Let us concentrate initially on the case of a portfolio made up only of linear derivative products like swaps, futures or forward contracts. In this simple situation, the value of the portfolio is a linear combination of the values of its building blocks where the weights of the combination are representative of the relative weight in the portfolio of each instrument. Under the assumption of normality for forward price returns, we will have that the percentile of the portfolio value variation distribution (the VaR) can be easily computed as a function of the forward price volatilities and correlations according to the following formula:
⎛ ⎞  X1   √ ⎜ .. ⎟ (X . . . X ) · C · VaR = n · α · (13.2) ⎝ . ⎠ n  1 Xn where ⎛ α = value corresponding to the x% ⎞ percentile; n = time horizon;

(i)

(i) (i) (i) Vk sgnk + Si Vk sgnk Si Xi = σi ⎝     ⎠ is the linear risk position vector; Futures

Swap

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155

Si is the value of the ith underlying; C is the underlying correlation matrix; (i) σi is the volatility of the ith instrument; Vk is the volume of the kth instru(i)

ment on the ith underlying; and sgnk is the sign of the kth instrument on the ith underlying. The linear approach can also be used when non-linear derivatives such as options are included in the portfolio. This may be done by means of the delta-linear approximation of option values. In this case the previous formula becomes:
⎛ ⎞   X1  √ ⎜ .. ⎟ (13.3) VaR(δ) = n · α · (X . . . X ) · C · ⎝ . ⎠ n  1 Xn where ⎛ α = value corresponding to the x% percentile; n = time horizon;

(i)

(i) (i) ⎞ (i) (i) (i) Si Vk sgnk + Si Vk sgnk + Si δk Vk Xi = σi ⎝       ⎠ is the linear risk Futures

Swap

Options

position vector; Si is the value of the ith underlying; C is the underlying (i) correlation matrix; σi is the volatility of the ith instrument; Vk is the vol(i)

ume of the kth instrument on the ith underlying; δk is the delta of the kth (i)

instrument on the ith underlying; and sgnk is the sign of the kth instrument on the ith underlying. Obviously, when options and non-linear derivatives represent the biggest part of a portfolio, their linear-delta representation maybe inappropriate for a correct and realistic VaR calculation. Hence, the information concerning the convex relation between underlying price and option values, expressed by the options gamma, can be used to improve the analytical linear model (delta-gamma approach):

VaR(δ/γ)


⎛ ⎞ ⎛ ⎞   X1 Y1  √ ⎜ .. ⎟ 1 ⎜ .. ⎟  = n · α ·(X1 . . . Xn ) · C · ⎝ . ⎠ + (Y1 . . . Yn ) · C · ⎝ . ⎠ 2 Xn Yn (13.4)

where ⎛ α = value corresponding to the x% percentile; ⎞n = time horizon;

(i)

(i)

(i) (i) (i) (i) Si Vk sgnk + Si Vk sgnk + Si δk Vk Xi = σi ⎝       ⎠ is the linear risk Futures

position vector;

Yi = σi2 S2i 



Swap (i) (i) γk Vk



Options



Options

is the gamma risk position vector; Si is

the value of the ith underlying; C is the underlying correlation matrix; σi is

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(i)

the volatility of the ith instrument; Vk is the volume of the kth instrument on the ith underlying; lying; (i)

(i) γk

(i) δk

is the delta of the kth instrument on the ith under-

is the gamma of the kth instrument on the ith underlying; and

sgnk is the sign of the kth instrument on the ith underlying. Within the classical Black–Scholes (BS) theoretical framework, the deltagamma approach can be extended to consider other relevant risk sources such as volatility changes. This is the idea behind the so-called vega-VaR approach. Vega is typically the measure of the sensitivity of the option value with respect to a change in the underlying volatility. For some derivative products volatility is a risk-driver much more important than the underlying asset price. This is the case for exotic instruments like path-dependent options. Barrier or lookback options are not very common instruments in electricity derivatives markets, but they may become common instruments in the near future. Hence, vega-VaR is an extension of traditional analytical approaches which should be considered. If, finally, we cannot consider the normality assumption even as an approximation for the true underlying asset distribution (in other words we reject the BS theoretical basement) we may approximate the true distribution by its first n analytical moments and use the famous Cornish–Fisher expansion to calculate the distribution percentiles.1 The problem here is that analytical expressions for the moments of portfolio value variations are available only under some assumptions which cannot be considered realistic for electricity products. Hence, in order to improve the power of traditional analytical VaR calculation methods, a better alternative is represented by simulation-based approaches. Simulation-based approaches (Monte Carlo and historical)

As we have seen, the main disadvantage of using analytical VaR methods is related to the unrealistic normality hypothesis which affects the whole calculation. In the third Part of the book, where we discussed pricing models, we noted that the traditional BS-type pricing models are not adequate for the pricing of electricity derivatives since geometric Brownian motions cannot describe electricity price behavior. Now we have seen that the same problem arises when we deal with analytical VaR calculation methods. The best way to circumvent this problem is by means of simulation. Using Monte Carlo simulation we can price a wide range of derivative products and consequently obtain a quite accurate estimate of the portfolio value potential changes. In fact, knowing the actual portfolio value it is possible to simulate a large number of scenarios for the relevant risk-drivers (forward prices 1

See Hull (2000) for a detailed description of VaR calculations with Cornish–Fisher expansion.

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157

and volatilities essentially), and for every scenario we can obtain a potential value change for the portfolio itself. Given the large number of scenarios performed, a probability distribution function can be estimated by means of parametric or non-parametric methods and the selected percentile measure can be extracted. Of course if the portfolio is mainly composed of linear positions, the VaR calculation may be quite quick since the simulation scenarios will reflect only forward curve shocks, while in the case of highly structured portfolios the calculation may slow significantly, especially when the portfolio dimensions are big. In this case we can adopt a partial simulation approach concerned only to simulate principal risk components. Moreover, there is still the problem of selecting and realistically modeling the stochastic behavior of the relevant risk drivers. This task, which is not even simple in the univariate case, as we saw in our discussion of pricing models, may be really hard in the multivariate case when different risk drivers influence the portfolio value. In this case the main difficulty will be that of modeling the dependence structure relating to the risk variables, and unfortunately there is not a benchmark solution for that. In any case, the problem of simulating random draws from a multivariate and complex distribution may be solved substituting Monte Carlo simulation with historical simulation. The historical simulation method consists in estimating VaR by means of historical daily market variables movements over a quite large and significant time horizon. Knowing the actual composition of the portfolio it is possible to compute the VaR, calculating the theoretical portfolio value for every single day of the historical sample. Once the time series of the portfolio value changes has been constructed, its probability distribution can be easily calculated and consequently the VaR will be equal to the appropriate percentile of the distribution itself. The main advantage of the historical VaR method is that it is based on an accurate estimate of the empirical distribution of the major risk factor. Hence, all the problems mentioned above regarding the correct simulation of realistic risk-driver dynamics and dependence structure disappear. However, a number of disadvantages arise. The first that is immediately evident concerns the size of the historical sample. In order to have a robust estimation of the portfolio value changes distribution, a consistent database of historical prices is necessary but not always available. Secondly, it is important to note that this approach can be applied only for portfolios of liquid financial instruments, traded on organized exchanges, since only for those kinds of assets are financial time series available. The third important disadvantage concerns the fact that historical VaR is a backward-looking measure, and we know that the past does not always give us a good indication about the future.

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13.1.3 Expected shortfall and conditional VaR Value at risk has certainly been adopted as the best practice risk measure in the financial industry, but this does not mean that it always represents the best possible synthetic indicator of the economic risk embedded in a financial portfolio. There are some important mathematical and practical reasons that lead us to think that VaR may be inappropriate to measure portfolio risk in some particular situations (or in some specific markets). In this section we focus on some practical disadvantages of VaR, also introducing new synthetic risk measures which may replace it with more success. In the Appendix to this chapter we will briefly discuss the mathematical problem of risk measure ‘coherence’. When the loss-density function of a portfolio (the left tail of a portfolio’s value-changes density function) is particularly thick, non-monotonically decreasing or eventually discontinuous (Figures 13.1–13.3), the percentile value selected for the VaR calculation may be unrepresentative of the real financial risk that the portfolio owner faces. This is mainly because, in this case, a rare and extreme event could potentially occur (with a probability lower that that implicit in the VaR measure) provoking a significant drop in the portfolio’s value. Note that portfolios of electricity derivatives quite often have nonstandard distributions because of the mainly spiky behavior that electricity

Normal

Frequency

Thick

Portfolio value changes

Figure 13.1 Thick loss function vs normal

159

Normal

Frequency

Non-monotonic

Portfolio value changes

Figure 13.2 Non-monotonic loss function vs normal

Normal

Frequency

Discont.

Portfolio value changes

Figure 13.3 Discontinuous loss function vs normal

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spot and forward prices display. The spiky behavior often results in a loss density function (defined p.a) for electricity trading portfolios, which is fatter than that of a normal distribution and often non-monotonic and discontinuous depending on the portfolio composition and on the specific market.2 Expected shortfall was the first risk measure alternative to VaR proposed in the literature and used for practical purposes. Nowadays it is considered the best risk measure both for its mathematical properties and its straightforward interpretation. While VaR is a percentile measure of the distribution of a certain random variable, the portfolio value’s daily changes, the expected shortfall (ES) is the conditional mean of the same random variable conditioned on having realizations worse than the VaR value. In other words, we can say that if VaR is a threshold which is fallen short of in a certain percentage of cases, ES is the expectation of the losses under the condition that the same threshold will be fallen short of certainly. Formally, we have that if, {VaR(x%) = c ∈ R+ : Prob(−
X > VaR(x%)) = (1 − x%)}

(13.5)

then {ES(x%) = d ∈ R+ : E(−
X | −
X ≥ VaR(x%)}

(13.6)

ES is always, by construction, a more conservative risk measure than the VaR, and it has nice mathematical features such as being a ‘coherent’ risk measure with often-explicit expressions for partial derivatives (since it is fundamentally an integral). Conditional value at risk (CVaR) was proposed by Rockafellar and Uryasev (2002) as a measure which combines the good feature of ES with the familiar concept of VaR. CVaR is essentially a linear combination of VaR and ES, with a combination parameter which should reflect the portfolio manager’s risk attitude. Formally, CVaR(x%) = γ · VaR(x%) + (1 − γ) · ES(x%)

with 0 ≤ γ ≤ 1

(13.7)

Rockafellar and Uryasev mainly emphasize the nice mathematical features of CVaR, with a special focus on portfolio optimization with risk–reward constraints. However, the idea of combining the information contained in two different risk measures such as VaR and ES has an intuitive market explanation. In fact, while the VaR measure is particularly suitable to budget a risk-adjusted performance value for our portfolio in ‘normal’ market conditions, ES may be used to assess the survival probability of our trading activity. Figure 13.4 illustrates the VaR, ES and CVaR features. The next 2 We have seen previously that there are some electricity markets where spiky behavior is particularly pronounced (France, Germany, Australia, the USA), and others which display smoother price paths (Nordpool).

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0.14

0.12 Value at risk 0.1 Frequency

Expected shortfall 0.08

0.06

0.04 CVaR area 0.02

0 ⫺10

⫺8

⫺6

⫺4

⫺2 0 2 4 Portfolio value changes

6

8

10

Figure 13.4 Graphical representation of VaR, ES and CVaR chapter will be focused on the use of risk measures for strategic planning and performance budgeting for well-diversified portfolios of real, physical and financial assets.

13.2 RISK MEASURES FOR PHYSICAL TRADING PORTFOLIOS The economic performance of a portfolio made up exclusively of liquid financial products is fully determined by its day-by-day value change, since it is possible to dynamically modify the structure of the portfolio itself, closing some positions or opening new ones without incurring enormous transaction costs. When we manage a portfolio of physical deals, however, it is not often possible to liquidate a position without incurring expensive penalties or high liquidity costs. Hence, a non-profitable physical deal cannot be closed upfront, realizing a negative mark to market, but should be held in the portfolio until its natural maturity. The natural consequence of this fact is that the economic performance of a physical trades portfolio is not related to day-by-day portfolio value changes, but is determined by realized margins

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on trades. As a consequence, the risk of the portfolio cannot be expressed by the potential drop in the portfolio value over a short period of time, but should be measured by some indicator of the uncertainty which characterizes the portfolio’s expected payoff over the whole portfolio tenor (trade by trade), or, if the portfolio is renewed on a rolling basis, over its average duration. The above is the main difference which characterizes synthetic risk measures of financial and physical portfolios. Based on this fact, we present here two examples of synthetic risk measures for physical portfolios which focus on economic performance, profit at risk/earnings at risk (PaR/EaR), and on expected cash, cash flow at risk (CFaR).

13.2.1 Profit at risk (PaR) The characterizing feature of PaR is that it assumes that markets are illiquid, and that in consequence open positions are held to maturity. A formal definition of PaR can be given as follows: PaR represents the minimum potential loss that a portfolio may suffer in the x% worst cases if held to maturity. As we mentioned before, PaR is a risk measure suitable for monitoring and managing portfolios generated by medium–long-term structured contracts, existing or still to be closed. The PaR time horizon should be chosen according to the purpose it is intended to be used. Usually, the economic year is chosen by the management for a better comparison with budget values and balance-sheet results. PaR calculations require the assessment of economic margins coming from business activity that will generate economic results in the future. Typically, analytical methods are not available for the calculation of PaR, and scenario-based simulation approaches are used. Simulating spot price scenarios related to portfolio commodities, and evaluating their path evolution up to the selected time horizon is required. The portfolio margin relative to each of these scenarios is then calculated, and a probability distribution of the portfolio’s margin obtained. The selection of the appropriate distribution percentile, according to the chosen confidence level, gives the PaR measure. Since PaR measures the risk embedded in a non-financial portfolio, market risk variables may not be the only relevant drivers which should be simulated. Often, physical contracts are characterized by volumetric clauses which are not always exercised with a financial rationale (and hence cannot be considered as financial options). Hence, volume scenarios can be created in order to incorporate volume risk in PaR calculations.

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163

According to its calculation, PaR is a suitable risk measure for all continuing activities for which the possibility to quickly unwind the position is difficult. It may also be considered to measure the risk of real asset management activities such as power generation, fuel procurement, storage and origination, but in this case we have to remember that economic performance and risk are not affected exclusively by changes in market risk drivers but also by management strategy. Hence, PaR calculations should be based on simulation models that can also capture the impact of operational and strategic decisions (real option models). PaR is a fundamental control tool for the whole value-creation chain, measuring risk exposure and potential limit overruns, affecting tactical decision for risk reduction and hedging implementation. Moreover, it may be used as an indicator for top management in risk-assessing medium–long-term investment decisions and commercial deals. PaR represents a coherent framework for strategic decisions, and in the next chapter we will return to this kind of application in more depth.

13.2.2 Cash flow at risk (CFaR) The cash flow at risk approach answers the question of how large the deviation between actual cash flow and the planned value (or that used in the budget) is due to changes in the underlying risk factors. Effectively, it is a measure quite similar to PaR regarding calculation methodology and time horizon, but it focuses on cash-flow depreciation instead of book depreciation. Sometimes the time delays which characterize the economic and financial events that reflect a firm’s life, lead to a choice between PaR and CFaR as the most appropriate risk measure. Of course, analytical calculation methodologies are again not available for the case of (CFaR) and simulation approaches (Monte Carlo or historical) prevail.

APPENDIX

ON THE COHERENCE OF RISK MEASURES

Artzner, Delbaen, Eber and Heat, in a famous article published in Risk in 1997, and successively developed, first proposed the question of the ‘coherence’ of risk measures within the mathematical finance area. Acerbi and Tasche (2001) give a clear and not too complex view of the meaning of risk coherence and its importance in practical portfolio management and risk-assessment problems. In this Appendix we present the main points of these papers, focusing on areas which may be central for the specific exigencies of an electricity portfolio. The main scope of this Appendix is that of defining mathematically the properties that a certain statistic of a financial or physical portfolio should have in order to be considered consistent, or better, as a coherent risk measure. Obviously, in order to evaluate if a certain measure may be considered a coherent risk measure or not we need to characterize in an axiomatic manner the properties which define the concept of a coherent risk measure.

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It is surprising to consider that once the coherence axioms have been defined, VaR, the risk measure adopted commonly as best practice, fails to be admitted within the class of coherent risk measures because it does not fulfill the most important of the axioms of coherence: sub-additivity. At first glance one may be tempted to believe that since VaR (the benchmark risk measure) is not a coherent risk measure, coherence is an additional property that a risk measure may or may not display and which is relevant only for mathematical purposes. Unfortunately that is not the case. Exactly like temperature, as a measure of air heat, should respect the axiom: ‘…is higher when air is hotter’, in order to be considered a sensible measure, coherence axioms define the properties that certain portfolio statistics should have in order to base coherent decisions on. In mathematical terms, coherence axioms can be stated as follows: Risk coherence (definition). Let us consider a set V of real valued random variables. A function f : V →  is called a coherent risk measure if: (1) it is monotonous: X ∈ V, X ≥ 0 ⇒ f (X) ≤ 0 (2) it is sub-additive: X, Y, (X + Y) ∈ V ⇒ f (X + Y) ≤ f (X) + f (Y) (3) it is positively homogeneous: X ∈ V, h > 0, hX ∈ V ⇒ f (hX) = h f (X) (4) it is translation invariant: X ∈ V, a ∈  ⇒ f (a + X) = f (X) − a The second axiom above is certainly the most important and intuitive. It implicitly reflects the idea of risk diversification. The basic idea of diversification is that the risk of the portfolio is lower or equal to the sum of the risk of individual portfolio components. The essence of sub-additivity is that a coherent risk measure should reflect this accepted dogma. Sub-additivity is an essential tool for allocation of capital requirements and imposition of a limit structure. In addition, for highly diversified electricity portfolios a sub-additive risk measure helps us to understand and construct a consistent risklimit structure. Sub-additivity is an essential property also in risk/reward-constrained portfolio optimization problems. VaR is not a coherent risk measure because it does not fulfill the axiom of sub-additivity (see Acerbi and Tasche, 2001, for a complete discussion of risk-measure coherence).

C H A P T E R 14

Risk-Adjusted Planning in the Electricity Industry

Deregulated electricity markets are highly dynamic markets and consequently static and backward-looking management tools should be supplemented by new instruments which help management to understand and react promptly to unexpected market changes. Risk is not only a random noise that affects a firm’s economic performance, it is also a strategic opportunity which should be appropriately evaluated. When we talk about risk measures, we are immediately inclined to think about the trading of financial instruments. However, risk-measurement methodologies can also be successfully implemented for business activities, which are not directly related to trading. One of these non-trading activities is certainly planning and project valuation. In the literature, the term ‘riskadjusted planning’ has only recently been introduced to indicate a planning activity which also considers the financial risk of the business as an essential element. In previous chapters we have seen that the electricity industry, especially in those countries where the liberalization process is in a mature phase, is highly exposed to many economic and financial risks which may substantially affect the performance of market participants (producers, traders and, of course, vertically integrated companies). For this reason, it is necessary to consider these risks when we plan a business activity or when we structure business projects. Moreover, due to the complexity which may characterize the business itself, risk measurement and control processes should be the most realistic and accurate possible in order to avoid basing strategic decisions on wrong or inaccurate indicators. 165

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In this chapter we provide a general risk-adjusted planning model for an integrated utility company, specifying relevant risk measures, their appropriate use and the optimal way of imposing and monitoring limits. In order to do this we will make use of the mathematical tools discussed so far, but mathematical detail is deliberately not covered as the main scope of the chapter concerns the correct use of the risk measures; for their detailed quantification the reader can refer to previous Parts and chapters of the book.

14.1 PRODUCTION VALUE AND RISK–RETURN MEASURES When we try to evaluate the performance of a certain business activity in a risk–return perspective, it is of fundamental importance to be able to estimate its value. That is exactly the case of a power generation company. If we want to understand if the risk implicit in the electricity production business, over a certain time, is consistent with our risk attitude, we need to know the expected return coming from that business over the same period. Hence, we need to assess the production value given the market conditions and operational characteristics of the asset portfolio. As a consequence, the expected economic results coming from electricity production and the related uncertainty are the result of the interaction between the dynamics of market conditions, and the strategy employed in managing the asset portfolio (Figure 14.1). For example, let us focus on the simplistic case of a power producer which procures fuel on a spot market or by means of a short-term fuel procurement agreement and dispatches its production on a liquid and competitive power exchange. In this case, the margin is produced by optimizing the asset operations based on expected market spark spreads. In Part 4 we saw that many models have been proposed in the last few years to evaluate and optimize power generation assets within a stochastic environment; models based on stochastic dynamic programming are able to consistently evaluate the impact that physical constraints have on production values.

MARKET

INFORMATION

PORTFOLIO STRATEGY

ECONOMIC RISK–RETURN

OPTIMIZATION

Figure 14.1 Asset optimization scheme

RISK-ADJUSTED PLANNING

167

In particular, Tseng and Barz (2001) have shown how Monte Carlo simulations can be used to value power plant production considering many operational constraints. Without repeating here the details of their model, their basic concept is that of determining the optimal decision rule, estimating step by step with a backward-induction method the level of market prices that should be observed in order to make indifferent the decision to commit or de-commit the unit, whenever there is the option to choose. In particular, if we denote by d(E(t), F(t)) this indifference locus, the following unit commitment decision strategy can be applied whenever there is the option to turn off or turn on the plant: if d(E(t), F(t)) > 0 commit the unit (turn on if it was off-line or stay on if it was on-line) (14.1) if d(E(t), F(t)) < 0 de-commit the unit (turn off if it was on-line or stay off if it was off-line) Given this kind of information, we are in a position to realistically estimate and plan not only the expected profit coming from the generation activity, but also some kind of percentile risk measure to quantify the probability of underperforming the expected target. This ‘performance risk measure’1 will simultaneously evaluate the impact on the economic performance of the business activity of both market risk variables and the optimal portfolio strategy. Here we present a simple example of the least-squares Monte Carlo methodology applied to the case of a thermal power plant with the characteristics shown in Tables 14.1 and 14.2. Figure 14.2 reports the economic performance distribution relative to the optimization exercise proposed previously. By means of this methodology, we are in a position to assess the expected profit (gross of taxes and other non-operational costs) and the risk of realizing an economic performance worse than an established level. The question now is how to optimally select the performance level under which we do not wish to go. In other words, how to establish a limit for our performance risk measure. Since we are talking about a risk-adjusted performance measure, management should be particularly interested in protecting shareholders’ investment values. Shareholders are usually focused on a benchmark which reflects the average economic performance of firms operating in the same industrial and geographic area. Hence, a good way of imposing a limit on the performance risk measure is that of reducing to less than a selected confidence level the probability of realizing an economic performance worse than the average sector performance. This selected confidence level should 1

Note that this measure is similar to the classical PaR measure of long-term physical contracts, but it also evaluates the impact on return of strategic decisions within the portfolio.

168

Table 14.1 Power plant physical characteristics used in the example Power plant characteristics Minimum capacity level

128 MW

Maximum capacity level

300 MW

Heat rate function

H(q) = a + bq + cq 2

Minimum time on

5 hours

Minimum time off

7 hours

Time to cool the boiler

14 hours

Start-up costs

S_up (x) = 3000 + (1 − exp(x/10)) euros (x = hours of inactivity)

Shut-down costs

3,000 euros

Table 14.2 Price model characteristics used in the example Price simulation characteristics E(t) electricity price F (t) fuel price

Mixture of four non-linear OU processes Classical mean reverting process

Intrinsic spark spread with average efficiency approx.1.35 euros/MWh Calibration done on Italian natural gas and electricity prices Simulation period for the experiment 1–7 January 2005 (168 hours)

1.6

⫻10⫺6

1.4

Frequency

1.2 1 0.8 0.6 0.4 0.2 0 ⫺5

0

5 10 Economic performance (€)

15

20 ⫻105

Figure 14.2 Power generation economic performance distribution

RISK-ADJUSTED PLANNING

1.6

169

⫻10⫺6

1.4

Frequency

1.2

Performance risk measure

1 Performance risk limit (average sector performance)

0.8 0.6 0.4 0.2 a 0 ⫺5

0

5

10

Economic performance (€)

15

20 ⫻105

Figure 14.3 Power generation performance risk measure

be reasonably low and dependent on the management risk appetite, but, of course, consistent with the intrinsic market risk level. Referring to the previous example of the generation activity based on a unique asset, a performance risk limit can be imposed in such a way that: Prob (Performance (t, T) < Average sector performance (t, T)) < α (14.2) where t is the evaluation date; T the terminal date of the reference period; and α the confidence level of the risk measure. In Figure 14.3 we can see graphically that this measure is a percentile risk measure exactly like value at risk (here we represent a situation of a limit exceeded). However, the average sector performance is not the only performance benchmark used by the shareholder; there are also financial constraints related to the payment of interest on corporate debt, which must be necessarily respected. We can also identify a second but no less important risk dimension, which concerns the company’s survival. A ‘survival performance level’ should be established as that performance level which guarantees the complete and timely payment of the company’s creditors. What is the right measure for this type of corporate risk?

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14.2 SURVIVAL PERFORMANCE LEVEL AND EXTREME MARKET EVENTS We have seen that a percentile measure could be a good method of assessing what we have referred to as performance risk. In fact, management can accept underperforming a target performance level if the probability to do so is reasonably low (compatible with sector risk and with the management’s risk appetite). On the other hand, management may not be willing to consider the possibility of underperforming the established survival level, even if the associated probability is extremely low. This consideration can drastically affect the appropriate risk measure adopted to monitor this second dimension of risk. Extreme-event risk measures similar to those used in the actuarial field should be considered. This may be particularly important in the electricity sector where, due to the typical spiky behavior of electricity prices, the shape of the left tail of the performance density function (the loss density function) may be particularly thick and not monotonically decreasing. This, of course, was not the case for our previous simple example, but what happens if we consider a more wide-ranging business activity which associates trading (or marketing) activity to generation? This situation characterized by a diversified but perhaps also not fully hedged portfolio strategy, may lead to the shape illustrated in Figure 14.4 for the firm performance density function.2 As already noted, the left tail is much thicker, slowly decaying and sometimes (here is not the case) not monotonically decreasing. Therefore, a percentile measure may be unrepresentative of an extreme and rare event which nevertheless may compromise or strongly affect the firm’s continued existence. In this case it is fundamental to use an appropriate measure to monitor this kind of ‘survival risk’. As proposed by Unger and Luthi (2002), risk measures based on conditional performance expectations like expected shortfall may help in the correct assessment and monitoring of extreme event risks. Formally, our ‘survival risk measure’ can be calculated as the economic performance we can expect given that it will be worse than a certain level, which, for example, could be the performance risk level defined previously as the average sector performance (Figure 14.5). From a statistical point of view, the problem is that of determining a conditional expectation of the economic performance: Surv. risk measure = E[Performance (t, T) | Performance (t, T) < x](14.3) where x is the average sector performance. 2 The example has been obtained by constructing a portfolio made up of the same generation asset considered before and a strip of forward and option contracts with the deliberated aim of stressing the loss function of the portfolio itself.

171

3

⫻10⫺6

2.5

Frequency

2

1.5

1

0.5 0 ⫺6

⫺4

⫺2 0 2 Economic performance (€)

4

6 ⫻105

Figure 14.4 Portfolio’s economic performance distribution

3

⫻10⫺6 Survival risk limit

Conditioning level (average sector performance)

2.5

Frequency

2

1.5

1

Survival risk measure (conditional expected performance)

0.5

0 ⫺6

⫺4

⫺2

0

2

Economic performance (€)

4

6 ⫻105

Figure 14.5 A portfolio’s economic performance survival risk measure

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Economic risk is a multidimensional subject. Here we have seen that two different measures are necessary in order to describe and control two extremely dangerous events for the life of a company, the risk of underperforming the market and the risk of defaulting upon debts. However, management often needs a synthetic risk measure which can be used to optimize the economic performance of the company itself. Hence, it may be necessary to summarize the information contained in the performance risk and survival risk measures, by creating a new measure on the basis of which the management can take decisions. The natural candidate to optimally play this game is the measure obtained as the linear combination of the two by means of the parameter γ (0 ≤ γ ≤ 1). As we saw in the previous chapter, a similar measure has already been used in purely financial risk assessment, the conditional value at risk (CVaR).3 We can call our synthetic risk measure the ‘conditional performance risk measure’, defined formally as follows: Cond. per. risk measure = γ ∗ Performance risk measure (1 − γ) ∗ Survival risk measure

(14.4)

A risk measure and the associated risk-limit structure such as that proposed here, not only possess very nice mathematical properties like coherence, convexity and continuity with respect to the confidence interval of the conditional level, but are also intuitive and simple to apply in strategic decisions. In fact, as much as the coefficient γ assumes values close to one, the more the risk measure is concentrated on performance risk. In the opposite situation, when the parameter of the linear combination is almost zero, the synthetic risk measure is very conservative since the survival risk perception is very high.4 The first case could be that of an extremely aggressive company characterized by a low financial leverage, while the second case could describe a situation where the impact of financial debt is high and it is, therefore, fundamental to guarantee the cash flow.

14.3 A PRACTICAL APPLICATION In order to demonstrate concretely the applicability of the proposed riskadjusted scheme, we give below a simple example concerning the use of the proposed management tools for one of the most common decisionmaking problems in electricity production/trading, that is, how much of the installed capacity is it optimal to hedge against spot-market fluctuations? We will try to answer this question using the planning tools developed so far within a risk/return theoretical environment. 3

See Rockafellar and Uryasev (2002) for details on CVaR. Note that the survival risk measure is constructed to be always more conservative than the performance risk measure. 4

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Table 14.3 Empirical experiments results Investment strategy

Expected value

Lower percentile (20%) (performance risk measure)

Expected shortfall (survival risk measure)

Upper percentile (80%) (opportunity measure)

Base

€382,281

€170,260

€71,154

€575,607

A1

€274,504

€118,950

€44,400

€414,167

B1

€80,942

€35,231

€16,332

€122,226

C1

€53,467

€46,306

€41,957

€60,170

A2

€384,482

€240,413

€171,910

€516,989

B2

€389,300

€350,225

€330,614

€427,270

C2

€391,405

€328,274

€300,111

€447,352

Let us again consider the case of our unique asset-based power company. The question is that of determining the right proportion of physical forward contracts (fixed price contracts) to enter into, with respect to the installed capacity of the generation asset. We will consider four different alternatives: (a) no forward agreements (base case); (b) forward agreements for 25 per cent of the maximum installed capacity (cases A1/A2); (c) forward agreements for 75 per cent of the maximum installed capacity (cases B1/B2); (d) forward agreements for the total installed capacity (cases C1/C2) (Table 14.3). All of the alternatives have been considered in two different background scenarios. In the first set of experiments forward contracts are carried out using a part of the installed capacity (cases A1, B1, C1), while in the second set the same contracts are respected by purchasing the necessary electricity in the spot market and dispatching the plant according to market convenience5 (cases A2, B2, C2). Table 14.3 reports the results obtained in terms of expected profit and risk statistics, while Figures 14.6 and 14.7 visually illustrate the intrinsic risk/return differences between the alternative strategies of the example. The results obtained show that every single strategy has a different risk/return profile. The selection of the most suitable one depends upon our preference structure, and can be performed within a classical utility-based approach. Some other interesting facts emerge from the example. From the first group of experiments we can see that all of the trading strategies based upon the partial dedication of the plant production to forward agreements are statistically dominated by the base case in terms of expected profit and risk measures. The second set of experiments illustrates the risk implied by 5

In this case the unit is dispatched according to the so-called ‘independence principle’ (see Hlouskova, Kossmeier, Obersteiner and Schnabl (2002) for more details).

174

ELECTRICITY RISK MANAGEMENT

Base

2

⫻10⫺6

1 0 ⫺5

A1

4

5

10

15

20

0

5

10

15

20

0

5

10

15

20

5

10

15

⫻10⫺6

2 0 ⫺5 1

B1

0

⫻10⫺5

0.5 0 ⫺5 ⫻10⫺5

C1

5

0 ⫺5

0

Economic performance (€)

20 ⫻105

Figure 14.6 Distributional results from first experiment dynamic portfolio management strategies. In fact, when we dispatch our unit in the market independently of trading decisions, we may have a very small increase in expected profit associated with the portfolio strategy, but with a consistent worsening of the risk situation (both performance and survival risk measures). This can be seen by comparing cases B2 and C2. The reason is mainly due to the physical constraints of the generation unit, which drastically affect the dynamic portfolio decision-making process as well as associated returns. In all the cases proposed it is possible to evaluate the importance of an extensive risk analysis to support the decision-making process. In particular, when the left tail of the economic performance distribution is thin, the absolute difference between the performance risk measure and survival risk measure is very small. Consequently, there is no additional information to be gained from their simultaneous use. On the other hand, when the distribution tail is fatter, the additional information provided by the survival measure is fundamental and should be carefully taken into consideration.

RISK-ADJUSTED PLANNING

Base

2

⫻10⫺6

1 0 ⫺5

A2

4

5

10

15

20

0

5

10

15

20

0

5

10

15

20

5

10

15

2

1 B2

0

⫻10⫺6

0 ⫺5 ⫻10⫺5

0.5 0 ⫺5 1

C2

175

⫻10⫺5

0.5 0 ⫺5

0

Economic performance (€)

20 ⫻105

Figure 14.7 Distributional results from second experiment A rational and coherent combination of the performance and risk measures suggested here can provide management with a dynamic and efficient tool for strategic decisions. Typical risk measurement tools can be used successfully for planning purposes supplementing more standard planning and management instruments which mainly reflect a static picture of the firm. This is particularly important in the modern electricity business. Liberalized electricity markets are highly dynamic and characterized by great uncertainty, which must be assessed and correctly considered in planning activities and strategic decision-making processes. Risk measurement is not the only important aspect of a risk-adjusted planning; a consistent risk-limit structure is also essential. Here we have presented an example of how standard risk-management tools can be modified and used in business areas not directly related to financial trading.

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Index

aggregated demand 5, 8–10, 12, 14 aggregated supply 8, 10, 14 ARFIMA 50 ARIMA 44 ARMA 44–6, 48 Asian options 77 auctions 5–6 Augmented Dickey–Fuller test 33–4 average price option 77 average strike options 77 backward induction 102–3, 120, 123, 125, 128, 132, 136, 167 Barlow model 57–9, 104–6 Barrier options 78 Bellman’s equation 120–1, 130 Black model 74 Black–Scholes model 74 Black–Scholes–Merton theoretical framework 77, 83 block options 74 calibration approach 92–3 Cash Flow at Risk (CFaR) 162–3 characteristic triplet 65–7 co-dependent exercise 80, 106 coherence of risk measures 163–4 commodity price risk 147 commodity-spread options 76 comparative static analysis 10–12 component GARCH 49 compound Poisson process 53–4, 97 Conditional Performance Risk Measure 172 Conditional Value at Risk (CVaR) 160–1, 172 congestion risk 150

contracts for differences 73 control variables 120 Cornish–Fisher expansion 156 correlogram analysis 31–6, 45 CRR binomial tree model 99–102, 136 day-head markets 5–6 derivative pricing 51, 56, 59, 63, 74, 83, 92, 95–6, 100 differential swaps 73 discounted cash flow (DCF) 112–13 discrete Fourier transform 24–5 distributional analysis 31–3 Doleans–Dade exponential 65 dynamic hedging 117, 141 dynamic regression models 44 EGARCH 49 electricity derivatives 71–2, 83–5, 92–5, 150–8 electricity derivatives markets 6–7, 156 electricity spot market models 8–9 electricity swaps 73 electricity swaption 74, 104 emission risk 150 energy swaps 73 Euler’s scheme 96–7 exotic derivatives 75–6 Expected Shortfall (ES) 160 fair price 74, 83–4, 88–9, 95–8 Fast Fourier Transform (FFT) 23–5 FIGARCH 50 forward 72–3 Fourier analysis 23–8 Fourier transform 24 179

180

INDEX

fractionally integrated processes futures 72–3 FX risk 148 GARCH 35, 48–50 German–Roncoroni model 98 Girsanov theorem 65–6

50

57, 60–2,

heat rate 81, 113–14, 117–19, 122 heat rate function 122 heading risk 149 heteroskedasticity 35 historical simulation 157 hourly options 74 ideal model 41–2 IGARCH 49 indifference locus 167 inter-company transfer price agreement risk 149 inter-temporal constraints 114, 117–18, 121 Ito processes 63–4, 96–7 Ito’s lemma 63–4 Kalman’s filtering

55

lattice approach 98–9, 132, 136 Least Squares Monte Carlo Approach (LSMC) 133–34 Lévy processes 97 Lévy–Vorst model 77 liberalization process 3–7, 57, 111, 165 linear dependence 48 long memory 35–6, 49–50 lookback options 78 Margrabe model 82, 116–17 marked point process 61–2 market incompleteness 54–5, 85, 88–91 market models 5–8 Markov processes 42, 54–5, 62 Markov switching regime models 54–5 martingale 52, 65 maximum likelihood (ML) 45, 60 mean reversion 33, 35, 52–4, 62 mean reverting processes 52–3, 97 mean reverting with jumps processes 53

merit order curve 9 minimal distance martingale measure 91 Monte Carlo simulations 78, 95, 103, 116–17, 133–4, 167 multiple regime processes 54 net present value (NPV) 112–13 NLOU process 58–60, 97, 104–6 no-arbitrage 84–93 non-storability 85, 88, 116 operational constraints 114, 117–21, 127–9, 140, 148, 167 operational flexibilities 114, 148 ordinal optimization 134–5 ordinary least squares (OLS) 45 Ornstein–Uhlenbeck process 58, 63, 97 over the counter (OTC) 5–6, 71–6, 82, 92, 150 path-dependent option 75–8, 156 performance risk measure 167–9, 174 performance spread options 76 periodic components 22, 24–5, 31, 36 PGARCH 49 plain vanilla options 55, 72–7, 98, 148 power generation modeling 112–14 probabilistic modeling 51 Profit at Risk (PaR) 141, 162–3 Radon–Nikodym derivative 62, 66 real options 112–14, 148 reduced form models 41, 51 relative pricing theory 83 risk-adjusted planning 166, 175 risk factors mapping 146 risk measures 95, 141, 153, 162–5 risk strategy 146, 152 semimartingales 42, 63–6 serial dependence 33, 43–4, 114 spark spread 73, 76, 81–2, 113, 136, 166 spark spread options 81–2, 113–17 spark spread swaps 73 spikes 10, 28, 41, 54, 58, 78, 88, 154 spiky behavior 53–4, 59–61, 87, 158, 160, 170 spot-forward price relationship 87–9 state price aggregation methods 102–3 state variables 120, 133–5 stationarity 34–6

181

INDEX

stochastic dynamic programming 123–6, 166 stochastic exponentials 64–5 stochastic integration 63 stochastic optimization 112–14, 118–19, 135 stochastic volatility 52, 55–7 subordination 55–7 survival risk 172 survival risk measure 170, 174 swing options 79–80, 99, 132 system marginal price 5, 8 time-spread options tolling 79, 81–2

76

80,

transfer function model 46–8 trinomial tree forest 106–8 trinomial trees 100, 132 utility-based pricing approach

90

Value at Risk (VaR) 152–72 VaR calculation methods 154–6 volume swaps 73 wavelet analysis 28–9