MODELLING ELECTRICITY SPOT PRICE TIME SERIES USING COLOURED NOISE FORCES

Transcription

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 [2]. 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 [34]. 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 [15]. 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 [34]. 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) [29]. 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 [33]. 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) [36]. 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 [34]. 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 [12]. 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 [27]. 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

27 13 behaviour of electricity prices when we value it in the market Special features of electricity. Wangensteen [34] asserts that electricity has certain features that make it a rather unique commodity. This must be taken into account in power system economics. The following list captures the essentials: Continuous flow. Electricity is generated and consumed in a continuous manner. Gas transported through a gas grid has basically the same feature. Instant generation and consumption. Electricity is consumed in the same moment of time as it is generated. If we again compare with gas, the transport speed of gas in a pipe is about one meter per second. Electricity travels with the speed of light. Non-storability. Electricity cannot be stored in significant quantities in an economic manner. Only indirect storage can be realized through hydroelectric plants or storage of generator fuel. Non-storability is the most significant element that contribute to the high volatility of electricity prices. Consumption variability. Electricity consumption or demand is variable with a characteristic pattern over day and night, over the week, and over the year. The variability in consumption is one of the root-causes of the seasonality in prices. Non-traceability. There is no physical means by which a unit of electricity (a kwh) delivered to a consumer can be traced back to the producer that actually generated the unit.

28 14 This feature puts special requirements on the metering and billing system for electricity. Essentiality to the community. Electricity is regarded as an absolute necessity in a modern society. Practically, every household and every firm has a connection to the power grid (this refers particularly to Nordic countries). How essential electricity is can be illustrated by the Value of Lost Load (VOLL), which is sometimes estimated to 100 times more than the ordinary price. Breakdown possibility. Due to technical characteristics of a power supply system, not only individual consumers can be affected by a contingency. Large areas can be affected in the case of a complete system breakdown. We have seen some large breakdowns, for instance in New York in 1977 and 2003, with tremendous economic consequences Stylized features of Electricity Spot Prices. Electricity spot markets exhibit a number of typical features that are not found in most financial markets. The most important of those features are: Seasonality:- Electricity spot prices reveal seasonal behavior in annual, weekly and daily cycles. Both the demand side and supply side play part in the seasonality of the spot prices. Business activities and weather conditions are considered to be the major factors that are behind the seasonality of electricity prices. It is well known that electricity demand exhibits seasonal fluctuations. They mostly arise due to changing climatic conditions, like temperature and the number of daylight hours. In some countries also the supply side shows seasonal variations in output. Hydro units, for example, are heavily dependent on precipitation and snow melting,

29 15 which varies from season to season. These seasonal fluctuations in demand and supply translate into the seasonal behavior of spot electricity prices. In Nordic countries for example, the cold winter experiences higher electricity prices and spikes while prices usually settle down during summer. In some places this is different [23], Northern California s primary electricity consumption occurs during the summer when air conditioning is highly needed. Extremely high volatility is experienced in electricity prices. 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, i.e. over ten times higher than for other energy products (natural gas and crude oil). The high volatility pattern is due to the transmission and storage problems and, of course, the requirement of the market to set equilibrium prices in real time. It is not easy to correct provisional imbalances of supply and demand in the short-term. Therefore, the price changes are more extreme in the electricity markets than other financial or commodity ones. With the application of the standard concept of volatility, the standard deviation of the returns on a daily scale, Weron [36] obtains the following volatility estimations: - notes and treasury bills: less than 0.5% - stock indices: 1-1.5% - commodities like natural gas or crude oil: 1.5-4% - very volatile stocks: not more than 4% - and electricity up to 50% Volatility:- Spikes:- A fundamental property of electricity spot prices, already observed by many

30 16 authors, is the presence of spikes, that is, rapid upward price moves followed by a quick return to about the same level. During the peak period, the price process has different properties. In particular, the rate of mean reversion is much higher than during normal evolution. The presence of spikes is a fundamental feature of electricity prices due to the non-storable nature of this commodity and any relevant spot price model must take this feature into account. The presence of long-term stochastic variation in the mean level of electricity prices makes it difficult to establish a range of prices, for which the price process is in peak mode. What is considered as a peak level now may become normal in 3 years. The rate of mean reversion is therefore determined not only by the current price level but also by the previous evolution of the price, which suggests that the behavior of electricity spot prices may be non-markovian [26]. The occurrence of spikes in power prices dynamics can be understood if we consider that electricity is a very special commodity. With the exception of hydroelectric power, it cannot be stored and must be generated at the instant it is consumed; the demand is highly inelastic. The generation process (supply) is characterized by low marginal costs but, when emergency generators are to be put on operation in order to satisfy the demand, marginal costs may be very high. Prices are therefore very sensitive to the demand, to outages and grid congestions. Thus shortages in electricity generation, forced outages, peaks in electricity demand determine spikes. Mean reversion:- Energy spot prices are in general regarded to be mean-reverting or anti-persistent [35]. Mean reversion is the tendency of a stochastic process to return over time to a long-run average value. The speed of mean reversion depends on several factors, including the commodity itself being analyzed and the delivery provisions associated with the commodity. When the price of a commodity is high, its supply

31 17 tends to increase thus putting a downward pressure on the price; when the spot price is low, the supply of the commodity tends to decrease, thus providing an upward lift to the price. Thus, in a long run prices will move towards the level dictated by the cost of production. Moreover Weron[35] mentioned that among all financial time series spot electricity prices are perhaps the best example of anti-persistent data. 1.5 Current state of electricity trade in Tanzania We noted down earlier that electricity business in Tanzania is not yet deregulated, though there are some indications for that to take place in the near future. Tanzania Electricity Supply Company (TANESCO) is the only fully authorized for electricity business in the country. TANESCO is wholly owned by the government of Tanzania and is under the Ministry of Energy and Minerals. However, there are some other companies with partial share in the business; these are the independent power projects (IPPs) Electricity generation TANESCO remains the main power producer in the country. Other sources of generation are from independent power producers (IPPs) which feed the National grid and isolated areas as well. TANESCO s generation system consists mainly of Hydro and Thermal based generation. Hydro contributes the largest share of TANESCO s power generation. The total generation from TANESCO own sources in 2008 was 2,985,275,264kWh out of which 2,648,911,352kWh (90%) was from Hydro Power Plants [31]. Total country demand was 4,425,403,157kWh, of which 33% was supplied by IPPs. The hydro-plants operated by TANESCO are all interconnected with the national grid system. TANESCO has been implementing power generation mix program, whereby a

32 18 substantial amount of generation comes from thermal generation through own generation and independent power plants (IPPs). Own thermal generation comes from the Ubungo 100MW gas-fired plant in Dar es Salaam. Another 45MW gasfired power plant located at Tegeta in Dar es Salaam is expected to enter the grid system soon. By the end of year 2008 IPPs contributed a total installed capacity of 282MW. IPPs powering the national grid include the Independent Power Tanzania Ltd (IPTL) with 100MW (diesel based) installed capacity and SONGAS (Songo Songo gas to electricity project) which by the end of 2007 had 182MW capacity. TANESCO also imports a total of 10MW of electric power from Uganda and about 3MW from Zambia. There are also several diesel generating stations connected to the national grid with installed capacity of 80MW but the only operational grid diesel based station is Dodoma which contributes about 5MW. Some other regions, districts and townships are dependent on isolated diesel-based generators with a total installed capacity of 31MW. Mtwara and Lindi regions are supplied by M/S Artumas Group Ltd, an IPP based in Mtwara. The total capacity of Artumas power plant is 8MW using gas from Mnazi Bay gas wells Electricity Transmission and Distribution Transmission and distribution system is totally owned by TANESCO. Transmission system comprises of 36 grid substations interconnected by transmission lines. The transmission lines comprise of 2,732.36km of system voltages 220kV; 1537km of 132kV; and 534km of 66kV, totaling to km by the end of September, The system is all alternating current (AC) and the system frequency is 50Hz.

33 19 The Distribution System Network Supply Voltage are 33kV and 11kV which serve as the back bone stepped down by distribution transformers to 400/230 volts for residential, light commercial and light industrial supply. There are big commercial and heavy industries supplied directly at 33kV and 11kV. Distribution activities are the most intensive in terms of geographical coverage. There are more than 672, 759 customers linked by these distribution lines [31] Electricity selling This is also done by TANESCO, since it receives electricity from the other companies through the main grid. The metering system for TANESCO customers are of two types. First one is Pre-paid Metering System (LUKU): LUKU is a Swahili abbreviation for LIPA UMEME KADIRI UNAVYOTUMIA which means pay for electricity as you use it and the second type is Credit meters (conventional meters): These meters allow for customers to be billed after consuming electricity for a month. LUKU has been designed mainly for residential and commercial consumers, not industrial type. TANESCO uses different tariff rates to different customers groups depending on the quantity of electricity consumed. These include domestic, small business and industrial consumers. The LUKU system has improved customer services further as now the customers can even buy electricity online through mobile phones and bank ATMs. For example a customer can recharge LUKU via ZAP LUKU recharge if is a ZAIN subscriber or M-PESA LUKU recharge if is a VODACOM subscriber. Recharging LUKU through bank ATMs can be done for customers who own bank account in CRDB or NMB banks. In addition to all those businesses, TANESCO actively cooperates with various Governments and other Power Utility bodies. The major areas of cooperation include Southern African Power Pool (SAPP), Nile Basin Regional Power Trade Project and Nile Equatorial Lakes- Subsidiary Action Program (NELSAP).

34 Mathematical terms in stochastic modelling. In this section we define some mathematical terms that are to be used in modeling in this work. Definition 1 Stochastic process A stochastic process is a family of random variables X(t, ω) of two variables t T, ω Ω on a common probability space (Ω, F, P ) which assumes real values and is P -measurable as a function of ω for a fixed t. The parameter t is interpreted as time, with T being a time interval and X(t, ) represents a random variable on the above probability space Ω, while X(, ω) is called a sample path or trajectory of the stochastic process. The stock prices and electricity prices are good examples of stochastic processes. Definition 2 Stochastic differential equation (SDE) A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. SDEs incorporate white noise which is a derivative of Brownian motion (Wiener Process); however, other types of random fluctuations are possible, such as jump processes or coloured noise. A typical equation is of the form dx t = µ(x t, t)dt + σ(x t, t)db t where µ is the drift function, σ is diffusion function and B t is the standard Brownian motion. Definition 3 Ornstein Uhlenbeck process The Ornstein Uhlenbeck process is a stochastic process r t given by the following stochastic differential equation: dr t = θ(µ r t )dt + σdb t

35 21 It represents the mean-reverting process with the equilibrium or mean-value µ, diffusion constant σ and a mean-reversion rate θ. Ornstein Uhlenbeck process is a Gaussian process that has a bounded variance and admits a stationary probability distribution used to model (with modifications) commodity prices stochastically. Definition 4 Markov process A continuous-time stochastic process X = {X(t), t > 0} is called a Markov process if it satisfies the Markov property, that is, P r(x(t n+1 B X(t 1 ) = x 1,..., X(t n ) = x n ) = P r(x(t n+1 B X(t n ) = x n ) for all Borel subsets B of R, time instances 0 < t 1 < t 2 < < t n < t n+1 and all states x 1, x 2,..., x n R for which the conditional probabilities are defined. A Markov process is a mathematical model for the random evolution of a memoryless system, that is, the likelihood of a given future state, depends only on its present state, and not on any past states. Processes which are not Markov are said to be non-markovian. Definition 5 Stationary process A stochastic process X(t) such that E( X(t) 2 ) <, t T is said to be stationary if its distribution is invariant under time displacements: F x 1, x 2,..., x n (t 1 + h, t 2 + h,..., t n + h) = F x 1, x 2,..., x n (t 1, t 2,..., t n ). That is, all finite dimensional distributions of X are invariant under an arbitrary time shift. If X is stationary, then the finite dimensional distributions of X depend on only the lag between the times {t 1,..., t n } rather than their values. In other words, the distribution of X(t) is the same for all t T. 1.7 Statement of the Problem Electricity pricing techniques are of great importance in insuring marketing efficiency in the case when there are variations of cost and supply of electricity at

36 22 the market. Consumers are interested only in the amount of money that they will spend on their electricity consumption over time. This amount of money is a stochastic variable that depends on the electricity price and the amount of consumption at each moment of time. Sometimes this stochastic variable undergoes rapid and extremely large changes which revert back within a short period of time called spikes. Since the prices are affected by external fluctuations such as weather then it is must be that the stochastic variable is influenced by noise terms which are correlated over time and not just white noise. Now, since different customers have different consumption behaviours, the dynamics of the money amounts are different. Therefore a precise stochastic (econometric) model of electricity spot price behaviour is necessary for energy risk management, fair pricing of electricity-related options and evaluation of the production assets. This work intends to mathematically account for the correlation over time of the noise process that influence the spot prices of electricity by making use of exponentially coloured noise terms in a stochastic differential equation (SDE). 1.8 Reseach Objectives General Objectives The main objective of this research is to develop a model for electricity spot price time series and describe the appropriateness of coloured noise terms in spot pricing of electricity in an environment of deregulated electricity market Specific Objectives The specific objectives of this study are: 1. To capture electricity price spikes and the volatility fluctuations around them by adding an exponentially coloured noise process into the SDE.

37 23 2. To determine the conformity between coloured noise terms in a meanreverting model and the spot price time series in the deregulated electricity market. 3. To determine the fair prices of the various derivative securities entered at deregulated electricity market. 1.9 Significance of the Study In this research we aim at studying the techniques for price modelling and forecasting of electric energy derivatives by employing the coloured noise forces in stochastic models. This is a further move from the price correlation that Ornstein- Uhlenbeck process was able to account for from the random walk process, we now take care of the correlation of the noise term. Tanzania is not yet implementing the deregulation of electricity market; however, it is expected to take place in the near future. Tanzania is among stakeholders in the East African regional power plan which aims to set an Eastern Africa Power Pool (EAPP), whose main objective is to set a framework for power exchanges amongst utilities of the member states. TANESCO for example, participates fully in the East Africa Community Energy Committee whose major objective is to prepare the East Africa Power Master Plan. Also the Southern African Power Pool (SAPP) under SADC has expansive projects; some of the SAPP projects include Zambia-Tanzania-Kenya interconnection. Therefore, this study is worth in its own right as the electricity pricing techniques (or models) will be useful to electricity trading companies in the country to run the exercise of pricing of electricity at the time of deregulated electricity market.

38 CHAPTER TWO LITERATURE REVIEW In addition to the fundamentals on electricity market structure outlined in the introduction, in this chapter we give a review of literature more closely pertaining to the topic of this research. The areas of research reviewed in this chapter are of modeling electricity spot prices time series as well as pricing of derivative securities in the power exchanges. Mavrou (2006) in his Master s thesis tried to make clear explanations on the stylized features of electricity prices such as high volatility and seasonality [25]. He described the non-storability property of electricity as the main reason behind high volatility and seasonality in electricity prices. Moreover, he mentioned out that the high volatility is due to the characteristics of demand and supply in electricity market as well. The intersection of demand and supply is what dictates the spot price in an exchange. So demand being relatively insensitive, together with the possibility of constraints in supply during peak times lead to the fact that short term energy prices are highly volatile. He also explained on the seasonality of electricity prices based on weather conditions and business activities. There are various seasonality patterns found in the data including intra-daily, weekly as well as monthly. He assumed that the factors that generate the seasonality in electricity prices are deterministic. Barlow (2002) and Kanamura (2004) proposed the Structural modes or Equilbrium models [3, 21]. In such kind of models they tried to mimic the price formation in electricity market as a balance of supply and demand. The demand for electricity is described by a stochastic process D t = D t + X t dx t = (µ λx t )dt + σdw t

39 25 where D t describes the seasonal component and X t corresponds to the stationary stochastic part. The price is obtained by matching the demand level with deterministic supply function which must be non-linear to account for the presence of price spikes. The assumption of deterministic supply is too restrictive as it implies that spikes can only be caused by surges in demand. In electricity markets spikes can also be due to sudden changes in supply, such as plant outage. In what referred to as ARMA-type models, Cuaresma et al. (2004) applied variants of AR(1) and general ARMA processes, including ARMA with jumps, to short-term price forecasting in the German market (EEX) [11]. They concluded that specifications where each hour of the day was modeled separately present uniformly better forecasting properties than specifications for the whole timeseries. And also they found that the inclusion of simple probabilistic processes for the arrival of spikes could lead to improvements in the forecasting abilities of univariate models for electricity spot prices. The AutoRegressive Moving Average (ARMA) modelling approach assumes that the series under study is (weakly) stationary [36]. If it is not, then transformation of the series to the stationary form has to be done first. This transformation can be done by differencing. The resulting ARIMA (AutoRegressive Integrated Moving Average) model explicitly includes differencing in the formulation. Carnero et al. (2003) considered general seasonal periodic regression models with ARIMA and ARFIMA (AutoRegressive Fractionally Integrated Moving Average) disturbances for the analysis of daily spot prices of electricity [9]. They made a conclusion that for the Nord Pool market, but not other European markets, a long memory model with periodic coefficients was required to model daily spot prices of electricity. ARIMA-type models relate the signal under study to its own past and do not explicitly use the information in other pertinent time series. Electricity prices are not only related to their own past, but may also be influenced

40 26 by the present and past values of various exogenous factors, mostly load profiles and weather conditions [36]. Karakastani and Bunn (2004) tested several approaches including regression- GARCH (Generalized AutoRegressive Conditionally Heteroskedastic) to explain the stochastic dynamic of spot volatility [22]. The GARCH-model by itself is not attractive for short-term price forecasting, however, coupled with autoregression presents an interesting alternative - the AR-GARCH model. The general experience with GARCH-type components in econometric models is mixed [36]. There are cases when modelling heteroskedasticity is advantageous, but there are at least as many examples of poor performance of such models. Kaminski (1999) applied the Jump diffusion model suggested by Merton, which basically adds a Poisson component to the standard Geometric Brownian Motion (GBM) [20]. The problem with this model was that it could not capture the significant features of the mean-reversion of electricity after a spike to its normal price level. In the occurrence of a price spike the GBM would recognize the new level of the price as a standard event and would not consider the previous price level. Haeussler (2008) in her Masters thesis proposed a model to simulate the spot prices of electricity by combining a mean-reverting process with a jump diffusion [17]. A kind of process often used in financial modelling especially in connection with commodities. To simulate the spot prices of electricity, she modified the standard Brownian Motion (BM) so that the characteristics of the spot prices in the electricity market could be replicated. She used the Stochastic Differential Equation (SDE) with three components ds(t) = α(s S(t))dt + σs(t)dw (t) + S(t )Y (t )dn(t) } {{ } } {{ } } {{ } (1) (2) (3) where S(t) is the spot price of electricity in /MWh at time t. S(t ) is also

41 27 a spot price of electricity, but t = t which means that we take the left-hand limit right before the jump. Furthermore, α > 0 is the mean reversion rate, S is the mean-reversion level, σ > 0 is the volatility and W (t) is a standard Brownian motion. The last part, S(t )Y (t )dn(t), covers the jump component as explained by Shreve [30], where Y (t ) is the multiplier of the spot price S(t ) and N(t) is a Poisson process. The three (3) components (summand) are described as: (1) The mean-reverting part which covers the drift (2) The diffusion part which captures the roughness of the spot price changes (3) The jump part which covers the unusual high price jumps which occur at random In [16], Geman et al. (2006) model the electricity log-price as a one factor Markov jump diffusion dp t = θ(µ t P t )dt + σdw t + h(t)dj t The spikes are introduced by making the jump direction and intensity level dependent, that is, if the price is high, the jump intensity is high and the downward jumps are more likely, whereas if the price is low, jumps are rare and upwarddirected. Their approach produces realistic trajectories and reproduces the seasonal intensity patterns observed in American price series. However, Meyer- Brandis and Takov found that the process reverts to a deterministic mean level rather than the stochastic pre-spike value and hence proposed the Multifactor jump diffusion model [26]. Barz et al. (1999) tested several diffusion models in several markets, including mean-reverting diffusion [7]. Actually, they tested mean-reverting diffusions with and without jumps and they concluded from their findings that the mean-

42 28 reverting jump diffusion model gives the best fitting. A specification of the meanreverting jump diffusion is as follows dp t = κ(ν P t )dt + σ p dw t + ξ t dj t The Wiener process accounts for the fluctuations in the region of the long-term mean, the jump process J t for the large up-jumps and ξ t determines the size of the jump and it is usually specified to follow a normal distribution. The jump process is the compound Poisson process with constant or time dependent intensity. One constraint of the jump diffusion is that the mean-reverting part of the process is assumed to be independent of the Poisson process which doesn t apply on electricity [25]. In the model that was formulated by Geman and Roncoroni [16], the spike regime is distinguished from the base regime by a deterministic threshold on the price process: if the price is higher than a given value, the process is the spike regime otherwise it is the base regime. The threshold value may be difficult to calibrate and it is not very realistic to suppose that it is determined in advance [26]. The regime switching model of Weron [35] removes this problem by introducing a two state unobservable Markov chain which determines the transition from base regime to spike regime with greater volatility and faster mean reversion. The underlying idea behind the regime-switching scheme is to model the observed stochastic behavior of a specific time series by two (or more) separate phases or regimes with different underlying processes. In other words the parameters of the underlying process may change for a certain period of time and then fall back to their original structure. Thus regime-switching models divide the time series into different phases that are called regimes. For each regime one can define separate and independent different underlying price processes. The switching mechanism between the states is assumed to be governed by an unobserved random variable. However, Meyer-Brandis [26] already found that such kind of models are more

43 29 difficult to estimate than a one-factor Markov model. Janczura et al. (2009) being motivated by the findings in [37] focused on Markov Regime Switching (MRS) models for the electricity spot prices themselves; not the log-prices as in the most other studies. Further, they introduced two novel features in the context of MRS modelling of electricity spot prices. One is heteroscedasticity in the base regime and the other one is a shifted spike regime distribution. The rationale for heteroscedasticity comes from the observation that price volatility generally increases with price level. Shifted spike regime on the other hand are required for calibration procedure to correctly separate the spikes from the normal behaviour. They used mean daily data from the German EEX market. In their study, the spot price is assumed to display either normal (base regime R t =1) or high (spike regime R t =2) prices each day. The transition matrix Q contains the probabilities q ij of switching from regime i at time t and regime j at time t + 1, for i, j = {1, 2}: Q = (q ij ) = q 11 q 12 = q 11 q 11 q 21 q 22 1 q 22 1 q 22 There is also a suggestion to model the electricity price as sum of several factors in what referred as Multifactor models. The simplest case in Multifactor models is a two-factor model where the first factor corresponds to the base signal with a slow mean-reversion and the second factor represents the spikes and has a high rate of mean-reversion. Barlow et al. (2002) gives a model of this type by representing the electricity price (or log-price) as sum of Gaussian Ornstein- Uhlenbeck processes [4]. However, while the first factor can in principle be a Gaussian process, the second factor is close to zero most of the time (when there is no spike) and takes very high values during a spike. This behaviour is difficult to be described with a Gaussian process.

44 30 In [36], Weron gives explanations on the concept of liberalization of electricity markets and makes a clear survey in some electricity markets particularly in Europe and North America. He described the stylized facts of electricity loads and prices as well. He also analyzed some important approaches for modelling and forecasting both the electricity loads and electricity prices. He defines quantitative (stochastic or econometric) models as the ones which characterise the statistical properties of electricity prices over time, with the ultimate objective of derivatives evaluation and risk management. Moreover, he asserted that such models are not required to accurately forecast hourly prices but to recover the main characteristics of electricity prices, typically at the daily time scale. Based on the type of market in focus, the stochastic techniques can be divided into two main classes: spot and forward price models. Meyer-Brandis et al. (2007) also pointed out that, most of the variability of electricity prices, as well as all interesting features like mean reversion are contained in the daily price series, and the daily parttern is mostly related to seasonality [26].

45 CHAPTER THREE PRICE MODEL BY COLOURED NOISE 3.1 Introduction In this chapter the mathematical model that describes the electricity spot price time-series is developed. The spot price model is basically a mean-reverting model subjected to exponential coloured noise forces. Both the model and the coloured noise process are analysed. The electricity spot markets are influenced by external fluctuations such as weather. Generally, external noise can be thought of as imposed on subsystem by a larger fluctuating environment in which the subsystem is immersed [18]. While the white noise limit usually leads to a good approximation of internal fluctuations, in the case of external fluctuations, the relevant variables can vary substantially over the correlation time. In this case, it is essential to consider coloured noise [5]. As from Hanggi and Jung [18], any modelling in terms of coloured noise is expected to be more physically realistic since a nonzero correlation time is explicitly accounted for. The whole chapter organization is as follows: the next section explains the model development procedures and theory behind. Section three describes the mathematics of coloured noise processes and section four is on parameter estimation. 3.2 Model development: It is by now obvious that financial models that simply incorporate geometrical Brownian motion(gbm) are not valid for modelling electricity prices as they do not admit of neither the price spikes nor the mean-reversion behaviours. In addition, models that tried to capture spikes by mere jump processes with white noise or Wiener processes, could not mimic the overall price process effectively

46 32 as they assumed zero correlation of the noise increments. In order to capture the mean reversion, price spikes and the volatility fluctuations around the spikes we develop a mean-reversion model driven by an exponentially coloured noise process. Through this approach we can incorporate almost all the features of the electricity spot price time series under the study. We hereby specify the logarithm of the spot price process, lnp t, as lnp t = X t (1) implying that the spot price is given by P t = e Xt from which we define X t as a stochastic mean-reverting process driven by the coloured noise process ζ, such that dx t = µ(x t, t)dt + σζ t dt (2) in which the drift term of the mean reverting model is characterised by the distance between the current price X t and the mean reversion level β as well as mean reversion rate κ, that is: µ(x t, t) = β κx t = κ( β κ X t) (3) This is from the simple market theory that if the spot price is below the meanreversion level, the drift will be positive, resulting to an upward influence on the spot price. This means that, when prices are relatively low, supply will decrease since some of the higher cost producers will exit the market, putting upward pressure on prices. And if it is above the mean reversion level, the drift will be negative, exerting a downward influence on the spot price. That is, supply will increase since higher cost producers of the commodity will enter the market putting a downward pressure on prices. Over time, this results in a price path

47 33 that is determined by the mean-reversion level at a speed determined by the mean reversion rate κ. We then substitute X = β κ and obtain the following SDE dx t = κ( X X t )dt + σζ t dt (4) Where, X is the long-term mean which depends on seasonality function, κ is the mean reversion rate, σ is the volatility term responsible for the magnitude of the randomness of the process that is set to a constant, ζ t is an exponentially coloured noise process generated to mimic the behaviour of both spikes and the usual volatility of the price. 3.3 Mathematical Description of Coloured Noise Process: The coloured noise process ζ produces a sequence of correlated random variables ζ(t 1 ), ζ(t 2 ),..., with the same standard deviation in each. Coloured noise is a Gaussian process and it is well known that these processes can be completely described by their mean and covariance functions [1]. The scalar exponential coloured noise process is given in the form of linear stochastic differential equation (SDE), specifically the Ornstein-Uhlenbeck process as follows: dζ(t) = 1 τ ζ(t)dt + αdw t (5) whose solution is: ζ(t) = ζ(0)e t τ + α t 0 e (t s) τ dw s (6) where τ is the correlation time for coloured noise and α is the diffusion constant. The parameter τ indicates the time over which the process significantly correlated in time. W t is a standard Wiener process with dw t N(0; dt) for an infinitesimal time interval dt. For t > s, the scalar exponential coloured noise process in equation (6) has mean, variance and auto-covariance respectively given by:

48 34 ˆ E[ζ(t)] = ζ(0)e t τ ˆ V ar[ζ(t)] = α2 τ 2t (1 e τ ) 2 ˆ Cov[ζ(t), ζ(s)] = α2 t s τ e τ 2 The general vector form of a linear SDE for coloured noise process is given by: dζ(t) = F ζ(t)dt + GdW t (7) where ζ(t) is a vector of length n, F and G are n n respectively n m matrix functions in time and {W t ; t 0} is an m-vector Wiener process with E[dW t dw T t ] = Q(t)dt. In this work, we extend a special case of the Ornstein- Uhlenbeck process and repeatedly integrate it to obtain the coloured noise forcing along the log-prices X t : dζ 1 (t) dζ 2 (t) dζ 3 (t) = 1 τ ζ 1(t)dt + α 1 dw t = 1 τ ζ 2(t)dt + 1 τ α 2ζ 1 (t)dt = 1 τ ζ 3(t)dt + 1 τ α 3ζ 2 (t)dt (8). =. dζ n (t) = 1 τ ζ n(t)dt + 1 τ α nζ n 1 (t)dt These systems of vector equations are Markovian, usually referred to as a random flight model in modelling dispersion of particles. For the sake of generating the coloured noise forces in this work, we choose the values n = 2 and m = 1 as from above, and obtain the following coloured noise process: dζ 1 (t) dζ 2 (t) = 1 τ ζ 1(t)dt + α 1 dw t = 1 τ ζ 2(t)dt + 1 τ α 2ζ 1 (t)dt (9)

49 35 which can be written in vector form analogous to equation (7) as follows: d ζ 1(t) = 1 0 τ ζ 1(t) dt + α 1 dw t (10) ζ 2 (t) ζ 2 (t) 0 1 τ α 2 1 τ this system of equations, with ζ(0) = 0 (i.e starting with no noise) and s < t, has the following solutions, ζ 1 (t) = α 1 t ζ 2 (t) = 1 τ α 1α 2 t 0 0 e (t s) τ dw s (11) e (t s) τ (t s)dw s (12) The vector equation (10) generates a stationary, zero-mean, correlated Gaussian process ζ 2 (t). The generated coloured noise process ζ 2 (t) is applied in equation (4) to model the price. Therefore, we specifically write the mean-reverting log-price equation as dx t = κ( X X t )dt + σζ 2 (t)dt (13) With the use of coloured noise forces, the correlation of the noise terms that influence the spot price time series is modeled more accurately and becomes possible to take into account the spiking characteristics of the prices. 3.4 Parameter estimation The parameters to be estimated are the ones involved in the generation of the coloured noise process (in equations (9)) and those in the mean-reverting logprice process (in equation (13)). For the coloured noise process in equations (9) we have τ, α 1 and α 2 as parameters to be estimated. Since the data at hand are on daily basis, we take one week period as the correlation time for the coloured noise process, that is, the forces that drive the spot price process are assumed to have significant memory within one week time period. This idea comes from

50 36 Weron [36] that for electricity spot price returns there is a strong, persistent 7-day dependence. In order to estimate the value of α 1 we refer to the solution of ζ 1 (t) in equation (11) from which we find the variance of ζ 1 (t) equal to: V ar[ζ 1 (t)] = α2 1τ 2 2t (1 e τ ) which means that the variance has a maximum value of α2 1 τ 2 as t. We equate the positive square-root of this value to 1 (which means the standard deviation is set-up to 1) and solve for α 1 since our interest is in the behaviour of the noises and not the magnitude of fluctuation. The magnitude of fluctuation should be controlled by σ in the mean-reverting SDE of the log-prices in (13). For the sake of estimating the value of α 2, we refer to the solution of ζ 2 (t) in equation (12) and compute its variance: V ar[ζ 2 (t)] = 1 4τ α2 1α2[τ 2 2 (2t 2 + τ 2 + 2tτ)e 2t the variance has a maximum value of 1 4 τα2 1α 2 2 as t. We similarly equate the square-root of this value to 1 (which means the standard deviation is set-up to 1 as well) and solve for α 2 since the values of τ and α 1 have already been estimated from above. τ ] Having done the estimations for the parameters τ, α 1 and α 2, we now estimate the parameters κ, X and σ of the mean reverting SDE (13). We use the method of Maximum Likelihood estimation of mean-reverting process as is described in the subsection hereunder Maximum Likelihood Estimation(MLE) of Mean Reverting Process: The mean reverting SDE (13) is naturally in the form of Ornstein-Uhlenbeck process. The Ornstein-Uhlenbeck mean-reverting (OUMR) model is a Gaussian

51 37 model well suited for maximum likelihood (ML) method. Therefore, we develop a maximum likelihood (ML) methodology for parameter estimation of Ornstein- Uhlenbeck (OU) mean-reverting process. The methodology ultimately relies on a one dimensional search which greatly facilitates the parameter estimation procedure. Our mean reverting SDE as in equation (13) is dx t = κ( X X t )dt + σζ 2 (t)dt; X(0) = X 0 for constants X, κ and X 0 and where ζ 2 (t) is the coloured noise process. In this model the process X t fluctuates randomly, with some over-time correlation of course, but tends to revert to some fundamental level X. The behaviour of this reversion depends on both the short term standard deviation σ and the speed of reversion κ. It is unlikely to have expert knowledge of all parameters and that is why we are forced to rely on a data driven parameter estimation method. And for this dissertation, the required data are available. We illustrate a maximum likelihood (ML) estimation procedure for finding the parameters of the mean-reverting process. However, in order to do this, we must first determine the distribution of the process X(t). The process X(t) is a Gaussian process which is, therefore, well suited for maximum likelihood estimation (MLE). Therefore, we now derive the distribution of X(t) by solving the SDE (13). Through the method of solving by integrating factor for the SDE (13), we have e κt X t = X 0 + or equivalently, X t = X 0 e κt + t t t κe κs Xds + e κs σζ 2 (s)ds 0 0 t κe κ(t s) Xds + σ e κ(t s) ζ 2 (s)ds (14) 0 0

52 38 the first integral on the right hand side of (14) evaluates to X(1 e κt ) and with reference to equation (12) we substitute for ζ 2 (s) and have X t = X 0 e κt + X(1 e κt ) + σ which simplifies to t 0 e κ(t s) [ 1 τ α 1α 2 s X t = X 0 e κt + X(1 e κt ) + 1 t [ s τ σα 1α e (s k) τ (s k)dw k ] ds e κ(t s) e (s k) τ (s k)dw k ] ds where s < t and since k < s we can interchange the variables of integration and the solution becomes, X t = X 0 e κt + X(1 e κt ) + 1 τ σα 1α 2 t 0 t k 0 e κt e ( 1 τ κ)s+ k τ (s k)dsdwk (15) Manipulation using integration by parts of equation (15) yields X t = X 0 e κt + X(1 e κt ) + ( 1 τ κ) 2 1 τ σα 1α 2 t 0 (1 k τ + κk)e k τ κk (1 + t τ 2k τ κt + 2κk)e t 2k κk+ τ τ dwk (16) from this solution for X t, we find the conditional mean value and variance of X t. The mean of the process is given by E[X t X 0 ] = X + (X 0 X)e κt and its variance is mainly the variance of the integral term since the other terms are constants. We use the following theorem found in [32] to find the variance. Theorem 1 Let g(x) be a continuous function and {B(t), t 0} be the standard Brownian motion process. For each t > 0, there exist a random variable F (g) = t 0 g(x)db(x)

53 39 which is the limiting of approximating sums F (g) = 2 n k=1 ( ) [ ( ) ( )] k k k 1 g 2 t B n 2 t B n 2 t, n as n. The random variable F (g) is normally distributed with mean zero and variance V ar[f (g)] = t 0 g(u)du if f(x) is another continuous function of x then F (f) and F (g) have a joint normal distribution with covariance E[F (f)f (g)] = t 0 f(x)g(x)dx. With the aid of Theorem 1 stated above, we find the variance as V ar[x t X 0 ] = ( 1 τ κ) 4 1 τ 2 σ2 α 2 1α 2 2 t 0 [1 ]dk which means that for the positive parameters σ, α 1, α 2, τ, κ and ( 1 τ κ) > 0 the process behaves like Brownian process with variance ( 1 τ κ) 4 1 τ 2 σ 2 α 2 1α 2 2 as t. Thus the Ornstein-Uhlenbeck mean-reverting model in (13) is normally distributed with E[X t X 0 ] = X +(X 0 X)e κt and V ar[x t X 0 ] = ( 1 τ κ) 4 1 τ 2 σ 2 α 2 1α 2 2 For t i 1 < t i, the X ti 1 conditional density f i of X ti is given by f i (X ti ; X, κ, σ) = (2π) 1 2 exp [ ( ( 1 τ κ) 4 1 ) 1 2 τ 2 σ2 α1α ( Xti X (X X)e ) ] κ(t i t i 1 ) 2 ti 1 2( 1 τ κ) 4 1 τ 2 σ 2 α 2 1α 2 2 (17) The values for the constants τ, α 1 and α 2 are obtained as from the beginning of this section.

54 40 Given n + 1 observations x = {X t0,..., X tn } of the process X, the log-likelihood function corresponding to (17) is given by L(x; X, κ, σ) = n [ 2 log ( 1 τ 1 ] κ) 4 τ 2 σ2 α1α ( n Xti X (X X)e ) κ(t i t i 1 ) 2 ti 1 i=1 ( 1 τ κ) 4 1 τ 2 σ 2 α 2 1α 2 2 (18) The maximum likelihood (ML) estimates ˆ X, ˆκ and ˆσ by maximizing the loglikelihood function. In this work we shall rely on the first order conditions as a method to maximize the log-likelihood function. This method requires the solution of a non-linear system of equations. We, therefore, attempt to obtain an analytic alternative for ML-estimation, based on the first conditions. This approach is based on the approach found in Barz [6]. The first order conditions for maximum likelihood estimation (MLE) requre the gradient of the log-likelihood function to be equal to zero. In other words, the maximum likelihood estimators ˆ X, ˆκ and ˆσ satisfy the first order conditions: L(x; X, κ, σ) X = 0 ˆ X L(x; X, κ, σ) = 0 κ ˆκ L(x; X, κ, σ) = 0 σ ˆσ Finding solution to this non-linear system of equations can be done through using a variety of numerical methods. However, in this work we illustrate an approach that simplifies the numerical search by exploiting some convenient analytical manipulations of the first order conditions. We first turn our attention to the first element of the gradient. We have that L(x; X, ( κ, σ) n ) ( 1 e κ(t i t i 1 ) X ti X = X (X X)e ) κ(t i t i 1 ) ti 1 ( 1 τ κ) 4 1 σ τ 2 α1α i=1

55 41 Under the assumption that κ and σ are non-zero, the first order conditions imply n ( ) ( i=1 1 e ˆκ(t i t i 1 ) X ti X ) ˆ X e ˆκ(t i t i 1 ) = f(ˆκ) = ti 1 n i=1 (1 e ˆκ(t i t i 1) ) 2 (19) The derivative of the log-likelihood function with respect to σ is L(x; X, κ, σ) = n σ σ + 2 ( n Xti X (X X)e ) κ(t i t i 1 ) 2 ti 1 σ 3 ( 1 τ κ) 4 1 α τ 1α which together with the first order conditions implies ˆσ = g( ˆ X, ˆκ) = 2 n n i=1 i=1 ( X ti ˆ X (Xti 1 ˆ X)e ) 2 ˆκ(t i t i 1 ) ( 1 τ ˆκ) 4 1 τ 2 α 2 1α 2 2 (20) The expressions (19) and (20) define functions that relate the maximum likelihood estimates. Specifically we have ˆ X as a function f of ˆκ and ˆσ as a function g of ˆκ and ˆ X. In order to solve for the maximum likelihood estimates, we could solve the system of non-linear equations given ˆ X = f(ˆκ), ˆσ = g( ˆ X, ˆκ) and the first order condition L(x; X, κ, σ)/ σ ˆσ = 0. However, the expression for L(x; X, κ, σ)/ σ is algebraically complex and would not lead to a closed form solution, requiring a numerical solution. In this dissertation we apply a rather simpler approach whereby we substitute the function ˆ X = f(ˆκ) and ˆσ = g( ˆ X, ˆκ) directly into the log-likelihood function and maximize with respect to κ. So our problem becomes maxv (κ) κ where V (κ) = n [ 2 log ( 1 τ 1 ] κ) 4 τ g(f(κ), 2 κ)2 α1α ( n Xti f(κ) (X ) ti 1 f(κ))e κ(t i t i 1 ) 2 i=1 ( 1 τ κ) 4 1 τ 2 g(f(κ), κ) 2 α 2 1α 2 2 (21) Once we have obtained ˆκ we can then find ˆ X = f(ˆκ) and ˆσ = g( ˆ X, ˆκ). The advantage of this approach is that the problem in (21) requires a one dimensional

56 42 search and requires the evaluation of a less complex expressions than solving for all three first order conditions. Moreover, this method trivially accommodates fundamental knowledge of any of the process parameters by simply substituting the known parameter(s) into the corresponding equations.

57 CHAPTER FOUR DATA ANALYSIS AND METHODOLOGY 4.1 Source of Data The data set used in this study has been collected from Nord Pool, it consists of 3712 daily averages of the Elspot electricity prices (7 days a week) from 1st January 1999 to 28th February This is around 10 years data from the Elspot system prices data recorded in terms of Euros per Mega-Watt-hour ( /MWh). The system price in the day-ahead market such as Elspot is, in principle, determined by matching offers from generators to bids from consumers to develop a classic supply and demand equilibrium price and then calculated separately for subregions in which constraints will bind transmission imports. We have, therefore, focused on the system daily prices irrespective of the transmission constraints but paying attention to supply and demand equilibrium prices. 4.2 Statistical Analysis of the Data Data Description The values of the most important distribution parameters for the daily average electricity price series from 1 st January, 1999 to 28 th February, 2009 are in summary collected in Table 1. The reported statistics are for electricity prices (P t ), the change in electricity prices (dp t ), the logarithm of electricity prices (ln(p t )) and the log returns of electricity prices (d(ln(p t ))) from one day to the next. The price series P t of electricity, as from the data collected, reflects the stylized features of electricity prices explained in section The prices are quite volatile mainly due to the shocks in demand and supply. The price series which is the basic data in this work, has a constant general mean of 29.41, the standard de-

58 44 viation of 14.71, positive skewness and excess kurtosis of respectively 1.22 and The minimum value in this data is 3.89 and the maximum value is , giving a data range of Mean Std. Dev Skewness Kurtosis Minimum Maximum P t dp t ln(p t ) d(ln(p t )) Table 1: Descriptive statistics for the daily average electricity spot prices. Note: Here and the remainder of this dissertation, the column labeled Std. Dev reports the standard deviation. Skewness and Kurtosis are respectively the third and the fourth moments around the mean, namely Skewness = E[X E[X]]3 (V ar[x]) 1.5 and Kurtosis = E[X E[X]]4 (V ar[x]) 2 for a random variable X. Skewness measures the asymmetry of the distribution of a random variable while Kurtosis measures the peakedness of the distribution. Higher Kurtosis means more of the variance is the result of infrequent extreme deviations. For a normal distribution, Skewness is equal to 0 and Kurtosis is equal to 3. Thus, ExcessKurtosis = Kurtosis 3. Due to variations in the balancing point of demand and supply, the spot prices of electricity are not uniform. The prices experience intra-day and intra-week periodical fluctuations. Our task in this work is not to address the issue of intraday and intra-week variations, but we analyze only the daily average prices. The daily price time series of the data used in this dissertation is displayed in Figure 5. The log-price values are also displayed in Figure 6.

59 Daily Average price[euro/mwh] Time in days[1st Jan, 1999 to 28th Feb, 2009] Figure 5: Daily average electricity spot price since 1st January, 1999 until 28th April, 2009 (3712 observations) Log(Prices) Time in days[1st Jan, 1999 to 28th Feb, 2009] Figure 6: The logarithm of electricity prices from which the main features of electricity market are observed. All the stylized features of electricity prices as explained in section are realized just from first visual inspection of Figures 5 and 6. Seasonality is

60 46 clear from the two figures where the general path of the prices shows a wave-like pattern. High volatility and occasional jumps dominate the whole time frame. Mean-reversion is also quite clear as the price oscillates around the mean level and whenever there is a spike the price is pulled back to the mean level rapidly after the spike. We additionally observe that the prices follow an increasing linear trend mainly being a result of inflation which is a property found in all commodity markets Normality test In this subsection we analyse whether the returns of electricity spot price are normally distributed and the prices are log-normally distributed. In the Black- Schole model the stock prices are assumed to be log-normally distributed. In order to get the values of the log-price we simply find the logarithm of the spot price values. To find the returns (log-return) we use its mathematical definition. That is where r t = ln P t P t 1 * r t is the return at the time t, * P t is the price value at time t, * P t 1 is the price at time t 1 There is more departure from Normality for the electricity prices. Figure 7 shows a Normality test carried out for returns of the electricity spot prices from the data. A solid line connects the 25 th and 75 th percentile in the data and a dashed line extends it to the ends of the data. If the returns were indeed normally distributed the graph would be a straight line as all the points would fall along the dashed

61 47 line. The fat tails we observe reveal out that this is not the case for electricity prices. With a probability of 0.006, for example, we find return values which are higher than 0.2, but the dashed line suggests the probability of such returns to be zero for perfectly normally distributed data. This behaviour of electricity spot price returns is in contrast to most financial theories and models which usually assume the price returns to be normally distributed. That the distribution of electricity spot price returns deviate from normal is also noticed in Figure 8 where the histogram showing distribution of price returns is superimposed with a theoretical normal curve. Normal Probability Plot Probability Electricity Price returns Figure 7: Normal probability test for electricity prices returns.

62 Frequencies Log Returns Figure 8: Histogram showing distribution of price returns superimposed with a theoretical normal curve. Figure 9 is the histogram of the log-prices also compared to a pure normal distribution curve. We clearly observe that the log-prices are slightly skewed to the left which imply the existence of spikes in Nord Pool spot price time series. The histogram seems to be closer to normal distribution simply because the prices experience both positive and negative spikes as how it appears in Figure 5. Spiking is a fundamental property of electricity spot prices. Since our data is on daily basis, spikes do not last more than one time point (a day in this case). According to Weron (2006) and as shown in Figure 10, a positive jump is followed by a negative one of approximately the same magnitude.

63 Frequencies Log prices Figure 9: Histogram for logarithm of spot prices showing distribution of log-prices for the data, superimposed with the theoretical normal curve Returns Time in days Figure 10: Log-returns price series showing the existence of some price spikes.

64 Serial correlation in the return series The dependence in the data {x 1,..., x n } are ascertained by computing correlations for data values at varying time lags. This is done by plotting the sample autocorrelation function (ACF): ACF (h) = ρ(h) = γ(h) γ(0), against the time lags h = 0, 1,..., n 1 and where γ(h) is the sample autocovariance function (ACVF) given by: ACV F (h) = γ(h) = 1 n h (x t+h x)(x t x), n t=1 and x is the sample mean. If the time series is an outcome of a completely random phenomenon, the autocorrelation should be near zero for all time-lag separations. Otherwise, one or more of the autocorrelations will be significantly non-zero. Another useful method to examine serial dependencies is to examine the partial autocorrelation function (PACF) an extension of autocorrelation, where the dependence on the intermediate elements (those within the lag) is removed. The partial autocorrelation is similar to autocorrelation, except that when calculating it, the autocorrelation with all the elements within the lag are eliminated. Figure 11 shows the Sample Autocorrelation Function (ACF) for the spot price returns and Figure 12 is for Sample Patial Autocorrelation Function (PACF) for the returns. High values at fixed intervals in Figure 11 indicate that the return series is subjected to seasonality. Also many values are out of the bounds which means the return series is not essentially random. There is a strong seven-day dependence in both ACF and PACF for electricity spot price returns. Similar results were found in [36].

65 51 Sample Autocorrelation Function (ACF) 0.8 Sample Autocorrelation Lag Figure 11: 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. 1 Sample Partial Autocorrelation Function 0.8 Sample Partial Autocorrelations Lag Figure 12: PACF for price return series, where some values are out of the bounds.

66 Calibration of the model. We present estimates for the parameters used to generate the coloured noise process in (11) and (12) and those for the mean-reverting Stochastic Differential Equation driven by this coloured noise process in (13). For the parameters in the coloured noise process we rely on the assumption that the length of the correlation time τ is equal to 7 as explained in section 3.4 where other parameters α 1 and α 2 were then estimated from this value of τ. For the Stochastic Differential Equation we first removed the trend as shown in Figure 13 and then we cleared out the spikes as shown in Figure 14. The spikes were removed out since they were considered as outliers in the data set and retaining them could distort the parameter estimation procedure. The process of removing the spike considered one and a half (1.5) standard deviation of the moving window of seven days as the threshold value for determining the existence of spikes in the window. The positions handling values that are higher or lower from mean with a difference greater than this threshold value were considered as spikes. The position with spike was then substituted with a mean value of the window. With this approach both positive and negative spikes were considered. Then the estimates for the mean-reversion level ( X) and volatility (σ) were taken as the averages of respectively the mean values and standard deviation of an analysis window of 90 days which is approximate to 3 months period. The mean reversion rate (κ) was continuously estimated depending on previous 90 days data by the Maximum Likelihood method described in section and its mean value is which can be used in further computations. The results for the estimated parameters are in summary presented in Table 2.

67 Original The trend Detrended Log(Prices) Time in days Figure 13: The original log-prices, the trend and the detrended data Log(Prices) Time in days Figure 14: The logarithm of electricity spot prices with removed spikes. Parameters: τ α 1 α 2 X σ The estimates: Table 2: Daily electricity log-prices parameter estimates for the model.

68 Analysis of Coloured Noise used in Simulation As described in section 3.3 the coloured noise used in this work is a stationary, zero-mean and correlated Gaussian markov process which by [5] it is essentially in the form of Ornstein-Uhlenbeck process as the one in equation (5) and/or (7). This fluctuation process is called coloured noise in analogy with the effects of filtering on white light [18]. We adopt the term filtering in our study and apply it twice on white noise to get more correlated noise process as in [10]. In order to generate the coloured noise in two-step filtering for ζ 1 (t) and ζ 2 (t) from the white noise ξ(t) the following algorithm found in [5, 13] was used: At the sample points t n (t 0 < t 1 <... < t N 1 ), ζ 1 (0) = sξ(0) ζ 1 (n) = ρ n ζ 1 (n 1) + 1 ρ 2 nsξ(n) where ξ are independent Gaussian random numbers with zero mean and unit variance, s is the standard deviation for the coloured noise and ρ n are the correlation coefficients given by ρ n = e tn tn 1 /τ. The algorithm is similarly used to generate ζ 2 (n) with ξ(n) being replaced by ζ 1 (n). Figure 15 shows the plots for white noise ξ(t) and coloured noise ζ 1 (t), ζ 2 (t) separately and combined ξ(t) and ζ 2 (t) in Figure 16. An increase in the autocorrelation due to filtering of white noise is seen in Figure 17 and Figure 18.

69 55 4 (a) White noise (b) Coloured noise filtered once (c) Coloured noise filtered twice Time Figure 15: 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). 5 4 White noise Coloured noise 3 2 Noise level Time Figure 16: 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.

70 56 4 (a) White noise 4 (b) Coloured noise filtered once 4 (c) Coloured noise filtered twice Figure 17: An increase in correlation observed after plotting noise levels against their previous values due to filtering of white noise. 1 (a) ACF for White noise Sample Autocorrelation Sample Autocorrelation Sample Autocorrelation Lag (b) ACF for Coloured noise filtered once Lag (c) ACF for Coloured noise filtered twice Lag Figure 18: 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.

71 Model simulation, results and comparison The parameters obtained in Table 2 were used for simulating the electricity spot price time series using Matlab software. The white noise was taken in its discrete form, that is, random numbers generated by the function randn found in Matlab. Simulation of the coloured noise process followed the algorithm described in the previous section. The log-prices were then simulated following the mean-reverting Stochastic Differential Equation (13) in which the coloured noise process was attached and the trend was then restored back. The required prices were then simulated out of these log-prices by taking out the logarithm. Figure 19 shows the real (original) log-prices (in blue colour) and the simulated log-prices with the trend restored (in green colour) on the same plane. The spot prices from real data (in blue colour) and the simulated prices (in red colour) were together plotted as in Figure 20. Figure 21 shows the histograms of both real and simulated spot prices in a comparable manner. The histograms of the real and simulated spot prices appear to be of the same nature in distribution. For more comparison between the real and simulated spot prices we have presented the histogram of the residuals which are the differences of real and simulated prices as in Figure 22. We see that much of the residuals are concentrated at zero with a standard deviation of Table 3 gives the statistical comparison between real and simulated spot prices focusing on the mean, standard deviation, kurtosis and skewness.

72 Real Simulated Log(Prices) Time in days Figure 19: Simulation results for logarithm of Prices vs real (original) log-prices. 120 Real price Simulated price 100 Daily Average price[euro/mwh] Time in days Figure 20: Simulated Electricity Spot Prices Time-series versus Real Prices.

73 (a)histogram of real prices 900 (b)histogram of simulated prices Figure 21: Distribution of the original electricity spot prices (a) and the simulated electricity prices (b) Histogram of the residuals 2000 Frequencies residuals Figure 22: Histogram of the residuals.

74 60 Real prices Simulated prices Mean Std. Dev Skewness Kurtosis Minimum Maximum Table 3: Real (original) spot prices data vs Simulated data. 4.6 Application on Pure trading By pure-trading we mean the price series (behaviour) independent of the influence of the trend and seasonality. In one of the recent studies, [?] was able to get such a series by doing detrending and two steps of deseasonalizing. First he removed the weekly and monthly seasonality by an additive model and next by building regression model with the use of background variables (temperatures and water reservoirs). Thou he analysed the prices differently in each country we adopt the system prices with background variables taken from Norway. The series he obtained is as given in Figure 23. We used the approaches of section 4.5 on this series and obtained the results as ploted in Fugure 24 and corresponding statistics in Table 4. Since the series has negative values which are not suitable for our model, converting them into some positive values by some additions before and then subtraction after simulation was necessary. Though there appears frequent short spikes for simulated data, the simulation was able to produce spikes which are as high as the original spikes. Table 4 shows close results for the simulated and original series in the statistics as well.

75 Pure price series[euro/mwh] Time in days Figure 23: Pure price series since 1st January, 1999 until 28th April, Real price Simulated price 30 Pure price series[euro/mwh] Time in days Figure 24: Simulated vs real (original) pure price series.

76 62 Real pure-prices Simulated pure-prices Mean Std. Dev Skewness Kurtosis Minimum Maximum Table 4: Real (original) pure-prices data vs Simulated data. 4.7 Forward price The only optimal hedging strategy in electricity markets is the use of forwards. Other usual hedging strategies adopted in other financial assets, such as holding certain quantities of the underlying, which is electricity in this case, is not a feasible solution. The reason behind is, as earlier mentioned, that electricity can not be physically and economically stored. It must be consumed almost immediately, once purchased. In this subsection we derive a formula for estimating the forward prices based on our model. The price at time t of the forward expiring at time T is obtained as the expected value of the spot price at expiry under an equivalent Q-martingale measure, conditional on the information set available up to time t; namely F (t, T ) = E Q t [S T F t ] (22) Thus, we need to replace X t = lns t and integrate the resulting SDE in order to extract S T and later calculate its expectation. Regarding the expectation, we must calculate it under an equivalent Q-martingale measure. In a complete market this measure is unique, ensuring only one arbitragefree price of the forward. However, in incomplete markets (such as the electricity

77 63 markets) this measure is not unique, thus we are left with the difficult task of choosing an appropriate measure for the particular market in question. Another approach, common in the literature, is simply to assume that we are already under an equivalent measure, and thus proceed to perform the pricing directly. This latter approach would rely however in calibrating the model through implied parameters from a liquid market. This is certainly difficult to do in young markets and those which are about to be initiated, as there will be no liquidity of instruments that would enable us to do this. But for the case of Nord Pool where our reference data have been taken, this is possible as there is sufficient liquidity of the market. We follow instead Lucia and Schwartz approach in [24], which consists of incorporating a market price of risk in the drift, such that ˆ X X λ and λ λ σ κ where λ denotes the market price of risk per unit risk linked to the state variable X t. This market price of risk, to be calibrated from market information, pins down the choice of one particular martingale measure. Under this measure now we may then rewrite the stochastic process in (13) for X t as dx t = κ( ˆ X Xt )dt + σˆζ 2 (t)dt (23) where the long term mean is assumed to have some seasonality factor g(t), that is ˆ X = 1 κ dg dt + g(t) λσ κ and ˆζ 2 (t) is the coloured noise process in which the incorporated dŵ (t) is the increment of Brownian motion in the Q-measure specified by choice of λ. Integrating the process (23) from t to T we get X T = X t e κ(t t) + T t κe κ(t s) ˆ Xds T + σ e κ(t s) ˆζ2 (s)ds (24) t

78 64 We now introduce the market price of risk and we get T T X T = g(t )+(X t g(t))e κ(t t) λ σe κ(t s) ds+σ e κ(t s) ˆζ2 (s)ds (25) t t Further manipulation with the expansion of ˆζ 2 (s) gives T X T = g(t ) + (X t g(t))e κ(t t) λ σe κ(t s) ds + ( 1 t τ 1 κ) 2 τ σα 1α 2 T (1 k τ + κk)e k τ κk (1 + T t 2k (T t) κk+ κ(t t) + 2k 2κk)e τ τ d Ŵ k (26) τ τ t Since S T = e X T, we can replace (26) and then substitute into (22) to get the forward price F (t, T ) = E Q t [S T F t ] ( ) = e λ T t σe S(t) κ(t s) ds κ(t t) G(T ) e G(t) [ ] E t e aσ T t (1 k τ +κk)e τ k κk (1+ T t τ 2k (T t) κ(t t)+2κk)e τ κk+ 2k τ τ dŵk F(27) t where a = ( 1 τ κ) 2 1 τ α 1α 2 In order to evaluate the expectation above we make use of Ito s Isometry theory and probability theory, which is stated in Theorem 2 below. Theorem 2 If f belongs to H 2 [0, T ], the space of random functions defined for all t in [0, T ], and T 0 E[f(t)]2 dt <, then E[ T 0 f(t)dw t] = 0 and E[( T 0 f(t)dw t) 2 ] = T 0 E[f(t)]2 dt Thus the expectation is obtained as [ E t e aσ T t = e a2 σ 2 2 T t (1 k τ +κk)e k τ κk (1+ T t τ (1 k τ +κk)e k τ κk (1+ T t τ 2k τ 2k τ ] (T t) κ(t t)+2κk)e τ κk+ 2k τ dŵk F t (T t) κ(t t)+2κk)e τ κk+ 2k τ dk e ( 1 τ κ) 4 1 τ 2 σ2 2 α2 1 α2 2 (28)

79 65 We now substitute equation (28) into (27) to get an approximation for the forward price, ( ) F (t, T ) e λ T t σe S(t) κ(t s) ds G(T ) e κ(t t) e ( 1 τ κ) 4 1 τ G(t) 2 σ2 2 α2 1 α2 2 (29) Since our analysis of the data series was not concerned with seasonality therefore g(t) = 0. Also from the definition, G(t) = e g(t) and G(T ) = e g(t ) Thus we have G(T ) = G(t) = 1 With these values, together with estimated parameters in Table 2, the Forward prices for different expiries can be computed. However, unfortunately we have no data for forward prices which could be used for comparison in this part.

80 CHAPTER FIVE CONCLUSION AND RECOMMENDATIONS 4.1 Conclusion. In this dissertation, we have developed a stochastic mean-reverting model driven by coloured noise process in modelling of electricity spot price time series data. The data used were collected from Nord Pool market, the Elspot. Analysis of these data was carried out and some important statistical behaviours such as mean-reversion, spikes, seasonality and the trend were found as in Figures 5 and 6. Also we performed the normality test and ploted the autocorrelation functions as shown in Figures 7, 11 and 12. And thus the data was found to be appropriate for stochastic modelling. The coloured noise process were also mathematically described as in section 3.3 and analysed in section 4.4. The analysis showed significant autocorrelation within the coloured noise process as compared to white noise where there is no any significant autocorrelation as shown in Figures 17 and 18. The model was then used to simulate the spot prices. Matlab software were used to do both data simulation and ploting of the corresponding figures. Results from the model show that the simulated price series is similar to the real evolution of electricity spot price time series observed in the market, since it covers the price series interval both above and below and was able to reflect some higher values (spikes) of the price series as it appears in Figures 19 and 20. As to what is diplayed in Table 3, the decriptive statistics for both emperical and those from the simulated data are also close to each other, indicating that the model has been a good representation of the real spot price process. Concerning the forward price, the prices depend much on the market price of risk which is inevitable for incomplete markets. These forward prices are very useful since

81 67 power markets are one hour/day ahead markets for spot markets. Though the data used in this work were from Scandinavian countries, the relevance of this study to developing countries like those found in Sub-Saharan Africa and Tanzania in particular was insisted as found in the first chapter of this dissertation. 4.2 Recommendations and Future work. The application of coloured noise process in market price models such as those of electricity seems to be a novel idea. From the results of this work it has been shown that the coloured noise process is appropriate in modelling electricity prices. We hereby recommend that; 1. Power generating companies, electricity traders and all the participants in electricity markets should consider the results of this work for price forecasting and proper pricing of electricity. 2. Tanzania and other developing countries in general should speed up the grid interconnection among different regions and liberalize the power sector for efficient and reliable power supply to consumers. Apart from all the efforts which have been taken by various scholars to model the spot prices of electricity, including this work, there are always holes left uncovered. We recommend in future, therefore, first to take more steps in filtering of the coloured noise process in order to reduce the frequency of spikes observed in the simulation. Secondly to combine the coloured noise approach together with other modelling methods such as Multifactor models and the Structural models so that we may come up with a more efficient model.

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