1 MODELLING ELECTRICITY SPOT PRICE TIME SERIES USING COLOURED NOISE FORCES By Adeline Peter Mtunya A Dissertation Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Science (Mathematical Modelling) of the University of Dar es Salaam University Of Dar es Salaam May, 2010
2 i CERTIFICATION The undersigned certify that they have read and hereby recommend for acceptance by the University of Dare es Salaam a dissertation entitled: Modelling Electricity Spot Price Time Series Using Coloured Noise Forces, in partial fulfillment of the requirements for the degree of Master of Science (Mathematical modelling) of the University of Dar es Salaam. Prof. T. Kauranne (First Supervisor) Date:... Dr. W. C. Mahera (Second supervisor) Date:...
3 ii DECLARATION AND COPYRIGHT I, Adeline Peter Mtunya, declare that this dissertation is my own original work and that it has not been presented and will not be presented to any other University for a similar or any other degree award. Signature: This dissertation is copyright material protected under the Berne Convention, the Copyright Act 1999 and other international and national enactments, in that behalf, on intellectual property. It may not be reproduced by any means, in full or in part, except for short extracts in fair dealings, for research or private study, critical scholarly review or discourse with an acknowledgement, without the written permission of the Directorate of Postgraduate Studies, on behalf of both the author and the University of Dar es Salaam.
4 iii ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my supervisors, Prof. Tuomo Kauranne (Lappeenranta University of Technology) and Dr. W. C. Mahera (University of Dar es Salaam) for their constant support, guidance and constructive ideas throughout my research work. I have learned so much from them about stochastic modelling and its application to time series and finance. Special thanks goes to Heads of Mathematics Department of my time of study, Dr. A. R. Mushi and Dr. E. S. Massawe, who made all efforts to provide me with a conducive study environment. I wish to express my sincere appreciation to all staff members in the Department of Mathematics for their support and encouragement. I extend my thanks to Deputy Principal (Academics), Mkwawa University College of Education (MUCE) for the sponsorship that enabled me to undertake this study. Also many thanks to NORAD s programme for Master Studies (NOMA) who sponsored the whole Mathematical modelling program. I would like to thank Lappeenranta University of Technology (LUT) - Finland, for providing me admission under exchange program for the whole period of preparing my dissertation for nine months. It was great opportunity for me to meet different experties in the field of my research and other close related fields. I wish to thank CIMO (Center for International Mobility) for providing scholarship for the whole period of my stay at LUT. Warmest thanks to my fellow master s students in the Department of Mathematics. Their cooperative spirit and contribution during the whole period of my study is appreciated. Last but not least, I would like to express my utmost thanks to my parents, brothers and sisters for their love and encouragement during the whole period of my study.
5 iv DEDICATION To my lovely parents Peter Mtunya and Adela Tarimo
6 v ABSTRACT In this dissertation we develop a mean-reverting stochastic model driven by coloured noise processes for modelling electricity spot price time series. The deregulation of electricity market, which once believed to be natural monopoly, has led to the creation of power exchanges where electricity is traded like other commodities. The physical attributes of electricity and behaviour of electricity prices differ from other commodity market. Electricity spot prices in the emerging power markets experience high volatility, mean-reversion, spikes and seasonal patterns mainly due to non-storability nature of electricity. Uncontrolled exposure to market price risks can lead to devastating consequences for market participants in the restructured electricity industry. A precise statistical (econometric) model of electricity spot price behaviour is necessary for risk management, pricing of electricity-related options and evaluation of production assets. We therefore formulate and discuss the stochastic approach used to model the spot prices of electricity by coloured noise forces. Parameter estimation for the model is carried out by Maximum Likelihood Estimation (MLE) method on mean-reverting stochastic process. Data used for model calibration were collected from Nord Pool for the period starting from January, 1999 to February, With the estimated parameters we simulate the model and found that the simulated and real price series have similar trends and covers the same price ranges. Thus, modelling of electricity spot prices using coloured noise gives a good approximation to real prices and we recommend application of coloured noise when modelling the spot prices of electricity.
7 vi Contents Certification i Declaration and Copyright ii Acknowledgements iii Dedication iv Abstract v Table of Contents vi List of Figures x List of Tables xii List of Abbreviations xiii CHAPTER ONE: INTRODUCTION General Introduction Economic Terminologies in pricing of electricity: Power exchange Demand and Supply Wholesale and Retail markets Energy derivatives Options
8 vii Complete and Incomplete markets Over The Counter (OTC) Markets Electricity trading in Nordic countries Electricity behaviour Special features of electricity Stylized features of Electricity Spot Prices Current state of electricity trade in Tanzania Electricity generation Electricity Transmission and Distribution Electricity selling Mathematical terms in stochastic modelling Statement of the Problem Reseach Objectives General Objectives Specific Objectives Significance of the Study CHAPTER TWO: LITERATURE REVIEW 24 CHAPTER THREE: PRICE MODEL BY COLOURED NOISE 31
9 viii 3.1 Introduction Model development: Mathematical Description of Coloured Noise Process: Parameter estimation Maximum Likelihood Estimation(MLE) of Mean Reverting Process: CHAPTER FOUR: DATA ANALYSIS AND METHODOLOGY Source of Data Statistical Analysis of the Data Data Description Normality test Serial correlation in the return series Calibration of the model Analysis of Coloured Noise used in Simulation Model simulation, results and comparison Application on Pure trading Forward price CHAPTER FIVE: CONCLUSION AND RECOMMENDATIONS Conclusion
10 ix 4.2 Recommendations and Future work
11 x List of Figures 1 Deregulation allows competition in generation and selling leaving transmission and distribution monopolistic An increase in demand (from D1 to D2) resulting in an increase in price (P) and quantity (Q) sold of the product Determination of price from Supply and Demand curves Electricity production in Nordic countries Daily average electricity spot price since 1st January, 1999 until 28th April, 2009 (3712 observations) The logarithm of electricity prices from which the main features of electricity market are observed Normal probability test for electricity prices returns Histogram showing distribution of price returns superimposed with a theoretical normal curve Histogram for logarithm of spot prices showing distribution of logprices for the data, superimposed with the theoretical normal curve Log-returns price series showing the existence of some price spikes ACF for price return series showing some important lags. Where most of the values fall out of the bounds. Seasonality can be observed from the lags with strong 7 - day dependence
12 xi 12 PACF for price return series, where some values are out of the bounds The original log-prices, the trend and the detrended data The logarithm of electricity spot prices with removed spikes The noise processes: (a)white noise ξ(t), (b)coloured noise filtered once ζ 1 (t) and (c)coloured noise filtered twice ζ 2 (t) The white noise ξ(t) and coloured noise filtered twice ζ 2 (t) which is applied in an SDE for modelling the spot log-prices An increase in correlation observed after plotting noise levels against their previous values due to filtering of white noise An increase in correlation observed from the Sample Autocorrelation Function (ACF) due to filtering of white noise. Stationarity of the coloured noise is also clear from the lags Simulation results for logarithm of Prices vs real (original) log-prices Simulated Electricity Spot Prices Time-series versus Real Prices Distribution of the original electricity spot prices (a) and the simulated electricity prices (b) Histogram of the residuals Pure price series since 1st January, 1999 until 28th April, Simulated vs real (original) pure price series
13 xii List of Tables 1 Descriptive statistics for the daily average electricity spot prices Daily electricity log-prices parameter estimates for the model Real (original) spot prices data vs Simulated data Real (original) pure-prices data vs Simulated data
14 xiii ABBREVIATIONS ACF AR ARMA ATM EEX GARCH GBM IPP IPTL MRS OTC PACF SADC SDE Auto-correlation Function AutoRegressive AutoRegressive Moving Average Automated Teller Machine European Power Exchange (Power-exchange in Germany) Generalized AutoRegressive Conditionally Heteroskedastic Geometric Brownian Motion Independent Power Producers/Projects/Plants Independent Power Tanzania Ltd Markov Regime Switching Over The Counter markets Partial Autocorrelation Function Southern African Development Community Stochastic Differential Equation
15 CHAPTER ONE INTRODUCTION 1.1 General Introduction The electricity sector has long been an integral part of the engine of economic growth and a central component of sustainable development. During the 1990s, conventional wisdom about the electricity sector was turned on its head. Previously, electricity had been considered a natural monopoly, and the electricity sector in most countries was either owned or strictly regulated by the government. Particularly in developing countries, government leadership in the development and use of electricity was a part of a broader social compact. Then, with astonishing speed, a revolution in thinking swept the sector. Several countries undertook major reforms, ranging from opening their electricity markets to independent power generators to broad-based reforms remaking the entire sector around the objective of promoting competition. Due in part to these changes, $187 billion was invested in energy and electricity projects in developing countries and the economies in transition in Central and Eastern Europe between 1990 and A 1998 survey of 115 developing countries found that nearly two thirds had taken at least minimal steps toward market-oriented reforms in the electricity sector . In Tanzania for instance, electricity market is not yet deregulated. There is only one public owned company that is in charge of electricity business TANESCO. However the deregulation of electricity seems to be right on the way to its starting point as there are a few companies that produce electricity but at the moment must sell it to TANESCO. Analysis of the electricity industry begins with the recognition that there are four rather distinct activities of it: generation, selling (trading), transmission and distribution. Deregulation has in most cases allowed competition in generation and
16 2 selling activities while leaving transmission and distribution monopolistic (see Figure 1). Once electricity is generated, whether by burning fossil fuels, harnessing wind, solar, or hydro energy, or through nuclear fission, it is sent through high-voltage, high-capacity transmission lines to the local regions in which the electricity will be consumed. Figure 1: Deregulation allows competition in generation and selling leaving transmission and distribution monopolistic. When the electricity arrives in the region in which it is to be consumed, it is transformed to a lower voltage and sent through local distribution wires to enduse consumers. The scope of each electricity market consists of the transmission grid or network that is available to the wholesalers, retailers and the ultimate consumers in any geographic area . Markets may extend beyond national boundaries. Deregulation is one of the key aspects towards a competitive market, where price controls are removed and thus encouraging competition. That is, energy prices are no longer controlled by regulators and now are essentially determined according
17 3 to the economic rule of supply and demand. The earliest introduction of energy market concepts and privatization to electric power systems took place in Chile in the early 1980s. However the oldest electricity market is Nord Pool that started in 1991 for the trading of all hydro electric power generated by Norway. The daily spot market has been operational since May 1999 and in 2001 a total of 8.24 TWh were traded on this market. Nord Pool benefited from the fact that electricity in Scandinavia is in great part hydroelectricity, hence has the very valuable property of being storable. The non storability of the other forms of electricity is an important explanatory factor of the spikes as those which were observed in the United States in the ECAR market in June 1998 . Today, the Nord Pool is a successful exchange, where the electricity players in Europe feel they can place their orders safely. Apart from Nord Pool, some other major European electricity exchanges include: UK Power Exchange (UKPX) England (2001), OMEL Spain (1998), Amsterdam Power Exchange (APX) Netherlands (1999), European Power Exchange (EEX) Germany (2001) and Polish Power Exchange - Poland (2000). These had been governed by EU legislation directives in 1996 and European goal was to have fully competitive electricity markets in all EU Member States by 1st of July 2007, and eventually to have common European electricity market . Provision of reliable and cost-effective electricity sources in the rural communities of developing countries (such as Tanzania) for the achievement of social and economic empowerment and poverty alleviation is imperative within the context of the global millennium development goals (MDGs) . Restructuring of the electricity industry will encourage the availability of reliable and cost-effective power supply in view of the following conditions which will manifest: Removal of monopoly in power generation, transmission and distribution and the encouragement of competition in power delivery, Reliability in power delivery, Lower energy tariffs, Increasing the scope for choice, Incorporation of more energy technologies
18 4 into the energy supply mix. Electricity markets differ from the traditional financial markets and other commodity markets due to the non-storability, uncertain and inelastic demand, restrictive transportation networks and a steep supply curve. And these are the reasons behind high volatility of electricity spot prices. Supply and demand must be in balance at each instance separately. A viable model for the spot price process is of up-most importance in all the areas of deregulated power business, including derivative and sales pricing, risk analysis, portfolio management, investment analysis, and regulatory policy making . The market risk related to trading is considerable due to extreme volatility of electricity prices. This is especially true for spot prices, where the volatility can be as high as 50% on the daily scale, which is over ten times higher than for other energy products (natural gas and crude oil) . In this research we aim at studying the techniques for pricing of electric energy derivatives. In this chapter we explain some terminologies used in pricing, discuss electricity trading in Nordic counties and the behaviour of electricity. We then assess the current state of electricity trade in Tanzania. Also, together with mathematical terms in stochastic modelling, we include the statement of the problem, research objectives and significance of the study. Chapter two is on literature review while chapter three presents the price model in details. Chapter four is on data analysis and methodology and in chapter five we give the conclusion. 1.2 Economic Terminologies in pricing of electricity: Power exchange. The Power Exchange is an entity responsible for receiving bids for sales and purchases of electricity, and to match the bids in such a way that prices and
19 5 quantities are settled . The basic activity of the power exchange is operation of the short term physical electricity market, the spot market. A power exchange is an open, centralized, and neutral market place, where the market price of electricity is determined by demand and supply. A high liquidity ensures that the market price at the power exchange is a correct price. The products sold at the exchange are standard products, and the communication is equitable to all actors on the market. The operation of the power exchange is market-oriented, in other words, the members of the power exchange participate in decision making. Therefore it is possible to make the product structure of the power exchange meet the needs of the market participants Demand and Supply. In economics, demand is the desire to own anything and the ability to pay for it and willingness to pay. The term demand signifies the ability or the willingness to buy a particular commodity at a given point of time. Demand is also defined elsewhere as a measure of preferences that is weighted by income. Economists record demand on a demand schedule and plot it on a graph as an inverse downward sloping curve. The inverse curve reflects the relationship between price and demand: as demand increases, price increases as shown in Figure 2.
20 6 Figure 2: An increase in demand (from D1 to D2) resulting in an increase in price (P) and quantity (Q) sold of the product. Supply on the other hand represents the amount of goods that producers are willing and able to sell at various prices, assuming all determinants of supply other than the price of the good in question, such as technology and the prices of factors of production, remain the same. Under the assumption of perfect competition, supply is determined by marginal cost. Marginal cost is the change in total cost that arises when the quantity produced changes by one unit. Firms will produce additional output as long as the cost of producing an extra unit of output is less than the price they will receive Wholesale and Retail markets. A wholesale electricity market exists when competing generators offer their electricity output to retailers. The retailers then re-price the electricity and take it
21 7 to market, in a classic example of the middle man scenario. While wholesale pricing used to be the exclusive domain of the large retail suppliers, more and more markets like New England are beginning to open up to the end users. Large end users seeking to cut out unnecessary overhead in their energy costs are beginning to recognize the advantages inherent in such a purchasing move. Buying direct is certainly not a novel concept in economics, however it is relatively novel in the electricity context. A retail electricity market exists when end-use customers can choose their supplier from competing electricity retailers. A separate issue for electricity markets is whether or not consumers face real-time pricing (prices based on the variable wholesale price) or a price that is set in some other way, such as average annual costs. In many markets, consumers do not pay based on the real-time price, and hence have no incentive to reduce demand at times of high (wholesale) prices or to shift their demand to other periods. Demand response may use pricing mechanisms or technical solutions to reduce peak demand. Generally, electricity retail reform follows from electricity wholesale reform. However, it is possible to have a single electricity generation company and still have retail competition Energy derivatives. An energy derivative is a financial contract whose value depends on energy price. The emergence of the energy markets has given birth to energy derivative markets. For example, a forward contract is an obligation to buy or sell electricity for a predetermined price at a predetermined future time . By definition, a derivative security is a security whose price depends on or is derived from one or more underlying assets. An option is one example of many derivative securities found in the market. The derivative itself is a contract between two or more parties. Its value is determined by the price fluctuations of the underlying asset. The
22 8 most common underlying assets include: stocks, bonds, commodities, currencies, interest rates and market indexes. Two of the most widely used such derivative securities are the futures contracts and the forward contracts. In futures contract, the settlement of the net value is started immediately after making the contract, and it is carried out daily until the end of the delivery time. A forward is a contract in which delivery of the underlying commodity is referred at a later date than when the contract is written with the price of delivery being set at the time of contracting Options. An option is a contract between a buyer and a seller that gives the buyer the right, but not the obligation, to buy or to sell a particular asset (the underlying asset) on or before the option s expiration time, at an agreed price, the strike price. An option contract binds only the seller (also called writer) of the option. In return for granting the option, the seller collects a payment (the premium) from the buyer as a compensation for the risk taken. Two types of options exist in the market. A call option gives the buyer the right to buy the underlying asset and a put option gives the buyer of the option the right to sell the underlying asset. If the buyer chooses to exercise this right, the seller is obliged to sell or buy the asset at the agreed price. The buyer may choose not to exercise the right and let it expire. The underlying asset can be a piece of property, a security (stock or bond), or a derivative instrument, such as a futures contract. The theoretical value of an option is evaluated according to several models. These models attempt to predict how the value of an option changes in response to changing conditions. Hence, the risks associated with granting, owning, or trading options may be quantified and managed with a greater degree of precision.
23 Complete and Incomplete markets. A market is complete with respect to a trading strategy if there exists a selffinancing trading strategy such that at any time t, the returns of the two strategies are equal. In general, a complete market is a market in which every derivative security can be replicated by trading in the underlying asset or assets. That means a market must be possible to instantaneously enter into any position regarding any future state of the market. An incomplete market is the one missing the above property. At any given time at the stock market, the stock price can increase or decrease slightly or fall a lot. It is not possible to hedge against all these increase or decrease in price simultaneously because there is no opportunity to carry out a continuous changing delta hedge, this leads to impossibility of perfect hedging. The impossibility of perfect hedging means that the market is incomplete, that is not every option can be replicated by a self-financing portfolio. It is not early to mention that a power exchange is an incomplete market Over The Counter (OTC) Markets. In finance, Over-the-counter (OTC) or off-exchange trading is to trade financial instruments such as commodities or derivatives directly between two parties in contrast with exchange trading. Exchange trading occurs via facilities constructed for the purpose of trading (i.e., exchanges), such as futures exchanges or stock exchanges. OTC markets refer to all wholesale trade in electricity outside power exchange. With the services provided by the OTC markets, it is possible for the actors on the market to tailor their portfolios of purchase and sale contracts to accurately meet their needs. Unlike in the trading at the power exchange, there is a risk of a counterparty default. The power exchange and the OTC markets that complement each other together form a well-functioning market mechanism for
24 10 the wholesale of electricity, the objective of which is to control the high volatility of the electricity market prices. 1.3 Electricity trading in Nordic countries. All Nordic countries have liberalised their electricity markets. The electricity markets in the Nordic countries have undergone major changes since the middle of the 1990s. The purpose of the liberalisation was to create better conditions for competition, and thus to improve utilisation of production resources as well as to provide gains from improved efficiency in the operation of networks. Norway was the first Nordic country to launch the liberalisation process of its electricity market with the approval of the Energy Act in 1990, which introduced regulated third-party access. Norway was followed by Sweden and Finland in the middle of the 1990s and by Denmark at the beginning of 1998 when the large electricity customers were given access to the electricity network. The liberalisation process in the middle of the 1990s was followed by an integration of the Nordic markets. The establishment of Nord Pool, the Nordic electricity exchange, was an important part of this integration . The physical market is the basis for all electricity trading in the Nordic market. The spot price set here forms the basis for the financial market. Nord Pool Spot organises the market place which comprises the Elspot and Elbas products. Elspot is the common Nordic day ahead market for trading physical electricity contracts. Elbas is a physical balance adjustment market operating 24 hours time. Elbas is an intraday market which opens two hours after the spot market is done and is open until 1 hour before delivery hour. The Nord pool financial market (Eltermin) provides a market place where the exchange members can trade derivative contracts in the financial market. Financial electricity contracts are used to guarantee prices and manage risk when trading power. Nord Pool offers
25 11 contracts of up to six years duration, with contracts for days, weeks, months, quarters and years. In Elspot, a trading day is divided into 24 hourly markets. Market participants provide separate bids for these 24 hours and the market clears separately for each of these 24 hours. Each participant provides a piece-wise linear bid schedule, where quantity is measured in MW and price in /MWh by 12 noon for delivery the following day. Nord Pool determines the clearing price for each market by 2:00 p.m. at which time the market closes and final clearing prices are determined. All contracts become binding at this point and Nord Pool initiates settlement of these contracts. The bids from each of the participants provide a schedule of how much the bidder is prepared to sell or buy at different prices. The system price is determined by the market equilibrium,i.e, the point where supply and demand curves cross. Supply willingness to generate electricity at a given price depends on the nature of production as shown in Figure 3. Figure 3: Determination of price from Supply and Demand curves.
26 12 Generally, if there are no transmission constraints, the Nord Pool area is a combined market and market participants can buy or sell electricity at the same price anywhere in the area. If the system operator designates zones, Nord Pool arranges separate Elspot markets for each zone. Nord Pool first calculates a theoretical unconstrained price based on all submitted bids, without considering transmission constraints. If transmission constraints are binding, Nord Pool adjusts prices upwards in deficit areas and downwards in surplus areas until transmission constraints are satisfied. It is noted that the quantity and nature of electricity produced varies within Nordic countries, from what is observed in the map shown in Figure 4. Figure 4: Electricity production in Nordic countries Electricity behaviour. The behaviour of electricity can be explained in two ways. On one side is the behaviour of electricity as energy (also called load), that is, the physical attributes of the commodity we are dealing with in the market. On the other side is the
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