A STOCHASTIC MEAN-REVERTING JUMP-DIFFUSION MODEL WITH MULTIPLE MEAN REVERSION RATES HASIFA NAMPALA

Transcription

1 A STOCHASTIC MEAN-REVERTING JUMP-DIFFUSION MODEL WITH MULTIPLE MEAN REVERSION RATES By HASIFA NAMPALA A dissertation submitted in partial fulfillment of the requirements for the degree of Master of Science, mathematical modeling of the University of Dar es Salaam University of Dar es Salaam October, 29

2 ABSTRACT The nature of electricity and the behavior of electricity prices differ from that of other commodity markets. One reason for this difference is that electricity is a non-storable good so it must be consumed almost at the instant it is generated. In most countries where electricity is deregulated, the demand for electricity is highly inelastic and varies rapidly yet supply can not increase at the same rate as demand. This results into rapidly rising prices, know as price spikes which stand out from the base price. The aim of this work is to identify and analyze the most important features of electricity spot prices, and propose a simulation algorithm that synthetically reconstructs the behavior of the real spot prices. In order to achieve this, we are suggesting a stochastic mean reverting jump-diffusion model with multiple mean reversion rates to cater for low and high spot prices. The results show that with the use of theoretical and empirical distributions it is possible to simulate a jumping mean reverting process with parameters close to real data values.

3 CERTIFICATION

4 DECLARATION AND COPYRIGHT I, HASIFA NAMPALA, declare that this dissertation is my own original work and 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 of 1999 and other international and national enactment, 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 dealing; for research or private study, critical scholarly review or discourse with an acknowledgment, without written permission of the Director of Postgraduate Studies, on behalf of both the author and the University of Dar es Salaam.

5 DEDICATION To the three girls who have brightened each day of my work, Mum Mrs.M. Namakula, friend Matylda Jabłońska and daughter Fahima. It is because of you three that I have come this far.

6 ACKNOWLEDGMENTS I thank you GOD for making it possible. My sincere thanks to NOMA for the scholarship in Tanzania. In the same vain I wish to thank CIMO for the full scholarship while in Finland. I thank the departments of mathematics at LUT and UDSM for all the assistance rendered. I specially thank Dr. Juma Kasozi the head of mathematics department of Makerere University for the opportunity and all the support. My sincere thanks go to Dr. Wilson Mahera the coordinator of NOMA and CIMO, you have been a mentor throughout my graduate studies. Prof. Matti Heiliö and Prof. Tuomo Kauranne, thank you for making it possible. You have inspired me and guided me throughout my work, you had working holidays in summer so as to provide supervision, I sincerely thank you! My classmates both at UDSM and LUT thank you for providing the synergy, it was lovely working with you. Isambi Sailon (LUT) and Fred Mayambala (UDSM), you provided me a right hand, I thank you. Finally my best fried Ibrahim Mukisa, you were virtually present and that kept me going.

7 Contents ABSTRACT ii CERTIFICATION iii DECLARATION AND COPYRIGHT iv DEDICATION v ACKNOWLEDGMENTS vi 1 Introduction The Nordic electricity market Modeling electricity prices Definition of terms used in stochastic modeling Statement of the problem Research objective Specific objectives Justification of the study Literature review 1 3 The salient features of electricity spot prices Volatility Spikes Seasonality Mean reversion Data analysis and methodology The data Statistical analysis

8 4.2.1 Normality test Log-returns Analysis of the jumps Analysis of mean reversion Mean-reversion in Nord Pool market A Stochastic mean reverting jump-diffusion model with multiple mean reversion rates The stochastic differential equation Model fitting Parameter estimation Model simulation Comparing the original and simulated data Conclusion and recommendations 5

9 List of Tables 1 Electricity generation (TWh) in the Nord Pool area in 27. Source: Nordel Percentages of spikes on different days of the week in different areas in the Nord Pool electricity market Percentage of spikes at different hours of the day in all the system and areas of the Nord Pool electricity market General statistics of Nord Pool system and area electricity spot prices General statistics of prices after the spikes were replaced by log-normally distributed numbers Normality tests on Nord Pool system and area log-returns Normality tests on Nord Pool system and area log-returns after replacement of spikes Parameter estimates Comparing the general statistics of the spikes extracted from the original and simulated prices in the System, Finland and Denmark East Comparing the general statistics in the original and simulated prices in System, Finland and Denmark East

10 List of Figures 1 System and area hourly electricity spot prices in Nord Pool electricity market Independent system and area hourly electricity spot prices in Nord Pool electricity market Spikes extracted from system and area hourly electricity spot prices in Nord Pool electricity market Sweden intra day spot prices for spiky days Sweden intra day spikes rescaled to the maximum = Nord Pool electricity hourly spot prices with the spikes replaced by lognormally distributed random numbers System and area prices after the spikes replaced by mean value of the analysis window Histogram for log-returns of Nord Pool system and area electricity spot prices Histogram for log-returns of Nord Pool system and area electricity spot prices after replacement of the spikes Log prices of Nord Pool system and area spot prices Log returns of Nord Pool system and area spot prices QQ-plots verifying that log-returns are not normal, since the tails are deviating from the horizontal line Histograms of the number of jumps against mean value of the electricity price of the analysis window one hour before the spike Histograms of the number of jumps against value of the electricity price one hour before the occurrence of the spike

11 15 Normalised histograms of jump sizes of the system and area prices, against the theoretical exponential probability distribution Dependence of mean reversion rate on the price levels Probability of occurrence of a spike from a given price level Behavior of the spike prices at a particular spike time but on consecutive days Scatter plots for dependence of the price level and log-price level on the magnitude of the difference between the current price and the price at the same time of the following day Behavior of Finland spot prices on four consecutive days at the spike time with different spike magnitudes Behavior of Sweden spot prices on four consecutive days at the spike time with different spike magnitudes Behavior of Denmark West spot prices on four consecutive days at the spike time with different spike magnitudes Behavior of Norway spot prices on four consecutive days at the spike time with different spike magnitudes System simulated prices plotted against system historical electricity spot price Finland simulated prices plotted against Finland historical electricity spot price DenmarkEast simulated prices plotted against Denmark East historical electricity spot price Distribution of the original Denmark East electricity prices (a) and the simulated Denmark East electricity prices(b) Price differences in original Finland electricity prices (top panel) and in the simulated Finland electricity prices(bottom panel)

12 29 Distribution of returns in original system electricity prices (a) and in the simulated system electricity prices(b) Spikes extracted from the original Finland electricity prices (a) and from the simulated Finland electricity prices(b) Distribution of spikes extracted from the original Finland electricity prices (a) and from the simulated Finland electricity prices(b)

13 General introduction Prior to deregulation, electricity supply in all countries was organized in either of the two ways; the first and most common was (and still is in most developing countries), as a state owned enterprise, and second as a privately owned but regulated enterprise. A state owned enterprise is a legal entity created by the government to undertake commercial activities on behalf of an owner government, and is usually considered to be an element or part of the state. State owned enterprises are usually very big enterprises with little accountability of day to day operations within the enterprise, with less sensitivity to customer needs. In other words in state owned enterprise, there is no independent regulator who ensures fair trading. In some countries however, the management of electricity has shifted from state owned to private owned enterprises with an independent regulatory body to bridge the gap between state interests and customer service. In countries where there is no significant political interference or micro management from the state, privatization has yielded more positive results as compared to state owned enterprises. Take a case of Uganda, my home country in Africa, before the privatization of Uganda Electricity board into, three separate private companies, for the generation, transmission and distribution of electricity, in 1999, it would take between 6 months to 2 years to approve a request for electricity supply in a private home. With the help of electricity regulatory authority, an independent body formed in 2, there has been some noticeable improvement in customer care, though most people still feel the reform has not been worth while. For example as reported in The New Vision newspaper on 12 th July 29, "The minister needs to tell Ugandans that the power crisis in Uganda lies in the privatization process of the power sector that has been messed up by the incompetence, dishonesty and corruption of some government electricity regulatory bodies and other leaders". "It is corruption and incompetence that has allowed corrupt private companies

14 to continue profiting from the sweat and blood of Ugandans without value for money". Due to a number of problems in state owned enterprises and privatized-regulated businesses, electricity markets around the world are currently transforming from regulated to competitive environments. The motive of deregulation has been to provide electricity more efficiently, more reliably and with higher quality at a cheaper price. Electricity has transformed from a primarily technical business, to one in which the product is treated in much the same way as any other commodity, with trading and risk management as key tools to run a successful business, [12], [3], [37]. Different countries use different kinds of models during the deregulation of their electricity. For example in Argentina the deregulation of electricity was part of a wider process in which the country opened up to capital markets, privatization, and deregulation of public services. "Law 2465 of Argentina s Congress established the pillars for the Argentinean electricity market, and created ENRE (Ente Nacional Regulador de la Electricidad, National Regulator of the electricity)" [2]. The benefits as well as the short falls from deregulation vary from one country or market to another. For example the Nord Pool is the world s first international power exchange market and it had been ranked one of the most successful power markets in the world [46]. Its success according to [1] has been attributed to four factors namely, "a simple but sound market design, which to a large extent was made possible by the large share of hydropower", "successful dilution of market power, attained by the integration of the four national markets into a single Nordic market", strong political support for a market-based electricity supply system" and " Voluntary, informal commitment to public service by the power industry". The nature of electricity trading varies from one market to the other but in all markets the spot prices are very volatile, and the volatility is both from the demand and supply sides. In markets where purchasing is through bidding, the generators will be happier to sell bigger amounts when the consumers bid high and the demand side will prefer the

15 opposite. This creates nervousness among both sides in an effort to ensure neither under nor over bidding. Unlike other commodities, electricity can not be physically and economically stored, thus it should be consumed at the same rate it is produced. At the same time, electricity consumption in most markets is weather dependent, with the highest consumption in the winter time because of excessive heating needs. To ensure availability of electricity, sometimes the buyers bid extremely high causing very high prices up to or even more than ten times as high, these high prices are commonly called price spikes. However, one other key characteristic of electricity prices is the tendency of the prices to always revert to a "reasonable value" which in most cases is some long-run mean value. However high the prices spike, in a short while they always revert to a mean value and this tendency is called mean reversion. One of the consequences that have resulted from deregulation of electricity markets is the desire to model the dynamics of electricity prices. Compared to other commodities, electricity is extremely volatile, up to two magnitudes higher than other assets, and this has triggered all market participants to hedge not only against the volume risk but also against price movements. And hence, this has led to research in electricity price modeling and forecasting, [46]. Adequate models are the ones that are able to capture most aspects of electricity prices, that is, seasonality, formation of price spikes, mean reversion and volatility. In our study, we have propose a stochastic mean-reversion jump diffusion model, with multiple mean reversion rates, to cater for the spiky and non-spiky regimes. From the model we generate a formula for the spot price of electricity, which we will use to simulate electricity spot prices using Matlab software and compare how the model replicates the dynamics of electricity prices in Nord Pool electricity market. The structure of this thesis is as follows. Section 2 introduces the Nordic electricity market and provides a brief summary of the theory of stochastic processes used in this

16 thesis. Section 3 outlines literature of the related work, Section 4 discusses the main characteristics of electricity spot prices. Section 5 covers the statistical analysis of the Nord Pool spot prices and the methodology used, in Section 6 we introduce the stochastic model used to simulate the spot prices and present the simulation results. Finally section 7 concludes and gives proposals for future work. 1 Introduction 1.1 The Nordic electricity market Power market liberalization was pioneered by Chile in That was followed by the British electricity sector in 199; the wholesale included only England and Wales until 25. Starting with Norway in 1991, followed by Sweden in 1996, Finland in 1998, Denmark West in 1999 and finally Denmark East in 2, the Nordic countries (except Iceland) gradually deregulated national generation and marketing of electricity. They merged their wholesale markets into a common Nordic power market called Nord Pool, [46]. Nord Pool has had a number of achievements including its ability to sustain shocks in which [1] quoted as "Unlike the California electricity market that collapsed following from severe demand and supply shocks in 2-21, the lights have stayed on in the Nordic market in spite of similar adverse supply and demand shocks in 22-23". On the other hand, despite all its achievements, there are still some complaints from the consumers about the high tariffs which are explained by the new European system of carbon emission permits, that has led to rising wholesale prices [1]. The nature of trading at Nord Pool is through bidding. For hourly bidding the market participants (generators and buyers) present their bids specifying the price and volume of electricity they require for every single hour of the next 24 hours of the following day starting at 1: a.m of the next day. These bids must be submitted to the administrator by 12 noon.

17 The point of intersection between the demand and supply determine the system price as well as volume and if there is no such a point then there will be no transactions on that hour. After determining system price Nord Pool verifies whether it is cost effective to transmit electricity in the whole Nord Pool area at the system price. If it realizes that there are some constraints, then extra cost will be incurred by areas that are likely to cause congestion in the transmission grids. The logic behind area prices is to create advantages to local geographical areas so as not to exceed the capacity of the transmission grid, [46]. 1.2 Modeling electricity prices Several models of electricity price dynamics have been proposed in literature by different researchers. Models proposed by [34] were single-factor and multi-factor affine diffusion to account for the mean-reversion property. To include jumps and spikes, jump-diffusion models of [15], [19] and [21] have been proposed. To distinguish the normal stable motion from the turbulent and spike dynamics, regime-switching models have been introduced in [27], [35] and [14]. On the other hand [44] used ARMA and ARMAX to asses its forecasting performance. Recently, [24] used a simple spot price model that is the exponential of the sum of an Ornstein-Uhlenbeck and an independent mean reverting pure jump process. In all these models, the key motive is to develop a model that can as closely as possible depict all salient characteristics of electricity, and it is for this reason that standard formulas in financial practice can not be used to model electricity prices. 1.3 Definition of terms used in stochastic modeling 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

18 as a function of ω for a fixed t. Parameter t is interpreted as time with T being the time interval and X(t, ) represents a random variable on the above probability space Ω, while X(, ω) is called the sample path or trajectory of the stochastic process. A nondecreasing family F t of sub- σ-fields of F defines a filtration for which F s F t F for s < t < T. A stochastic process X(t, ω) is adopted to F t if X(, ω) : [, T ] Ω S is F t - measurable for each t. Definition 2 Stationary stochastic 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 X1,X 2,...,X n (t 1 + h, t 2 + h,..., t n + h) = F X1,X 2,...,X n (t 1, t 2,... t n ). (1) That is, all finite dimensional distributions of X are invariant under an arbitrary time shift. If X is stationary, then the finite t-dimensional distributions of X depend on only the lag between the times t 1, t 2,..., t n rather than their values. In other words, the distribution of X(t) is the same for all t T. Definition 3 Markov process A continuous time stochastic process X = X(t), t 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 ) (2) For all Borel subsets B of R, time instances < 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. Definition 4 Stochastic differential equation 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

19 process. SDEs incorporate white noise which is a derivative of Brownian motion (Wiener process). A one factor SDE is of the form: ds t = µ(s t, t)dt + σ(s t, t)dw t, (3) where µ is the drift of the stochastic process, σ is the diffusion of the process, and dw t is standard Brownian motion with E(dW t ) = and E((dW t ) 2 ) = dt. Definition 5 Poisson distribution Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The distribution is associated with a parameter (λ) which is the mean number of occurrence of events in a given time period. Considering a time interval t, the probability distribution of a number of events to occur in this interval is; P r(n(t + t) N(t) = k) = e λ λ k, k =, 1, 2, 3,... (4) k! Definition 6 Poisson process The Poisson process is a continuous-time counting process (N(t) : t ) of random variables, where N(t) is the number of events that have occurred up to time t (starting from time ). The increments in a Poisson process are independent and stationary, and N() =. The number of events between time x and time y is given as N(y) N(x) and has a Poisson distribution. Definition 7 Exponential Distribution

20 Given a Poisson distribution with rate of change λ, for a random variable X, the distribution of waiting times between successive changes with k = ) is D(x) = P r(x x) (5) = 1 P r(x x) (6) = 1 e λx (7) and hence the probability distribution P r(x) = D (x) = λe λx is called an exponential distribution. 1.4 Statement of the problem America, Australia, New Zealand and most countries in Europe have deregulated their electricity markets, [15], and various scholars have developed different models to replicate the dynamics of prices in those electricity markets. The key factor in most models in the inclusion of price spikes because [46] asserts that firms that are not prepared to manage the risk that arise from the price spikes can see their earnings for the whole year evaporate in a few hours. And [24] suggested that the use of two mean reversions rates to capture the spikes is the best way to model electricity spot prices in highly spiky electricity markets like Nord Pool. Much as [15] and [24] incorporated two mean reversion rates to capture the spikes, in the Nord Pool market, neither considered the fact that Nord Pool is a day ahead market thus the logical mean reversion of the prices will be in at least 24 hours because of the nature of electricity trading. 1.5 Research objective To develop a mathematical model that captures all the major characteristics of electricity trading which will replicate the trends of electricity spot prices in a deregulated electricity markets.

21 1.5.1 Specific objectives To analyse the occurrence of price spikes in electricity prices. To determine the dependence of mean reversion rate on the price level. To determine the accuracy of a stochastic mean-reversion jump diffusion model with two mean reversion rates, with the time lag of 24 hours, in replicating the trend of electricity spot prices. 1.6 Justification of the study Stochastic modeling of electricity prices has been done by various scientists employing various models. Considering the recent models, the most commonly used mean reversion model is by [34], though, much as it captures the mean reversion character of electricity, it does not mimic the spikes. [4] improved [34] and included the jumps, the loophole in their study though, was the fact that they never considered the fact that prices revert to mean level at a rate determined by the price level at a given time point, thus they used a mean reversion rate which was not price dependent. The most recent is by [24] who improved on the previous models and considered the fact that the spikes require a higher reversion rate than the ordinary price fluctuations. They used the Ornstein-Uhlenbeck process with jumps and mean reversion in the spike process as well. Much as this seemed a perfect model to replicate electricity spot prices, they considered a constant volatility which is not the case because deviation from the mean in electricity prices is never constant during spiky and non-spiky regimes. As emphasized by [15] that risk management and asset valuation needs require indepth understanding and sophisticated modeling of commodity spot prices. Thus it is crucial to have a model that captures the most significant characteristics of electricity trading. In this study therefore we develop a stochastic model that mimics the jumps in

22 electricity market with time dependant volatility, mean reversion that is dependent on price levels as well as depicting the trading technicalities in the electricity markets. 2 Literature review The deregulation of electricity made forecasting a necessity for all market players. The cost of under-or-over-contracting and then selling or buying of electricity on a balancing market has increased so much that they can lead to financial distress of the utility, [46]. Many scholars have come up with various models to forecast electricity prices: These include classic models, regime switching models as well as stochastic mean reverting models. One of the earliest examples is in [29], where the spiky characteristic is addressed through a random walk jump-diffusion model, adopted from Merton (1976). However, the model ignores another fundamental feature of electricity prices, the mean-reversion in the baseline regime. This property is established in [4]. The comparison of alternative models (Geometric Brownian motion and mean-reversion with/without jumps) across several deregulated markets suggests a mean-reverting model with a jump component as more adequate specification. Although this type of financial asset modeling addresses crucial characteristics of price dynamics, namely meanreversion and spikes, it still assumes deterministic price volatility, which clearly contradicts empirical evidence, [17]. Despite their intuitive interpretation and non-linear behavior, jump-diffusion models present some limitations. Firstly, it is assumed that all shocks affecting the price series die out at the same rate. In reality however, two types of shocks exist implying different reversion rates; large disturbances, which diminish rapidly due to economic forces, and moderate ones, which might persist for a while. Estimation bias is inevitable, as Maximum Likelihood methods tend to capture the smallest and most frequent jumps in the data. As emphasized by [26], stochastic jump-models do not disentangle mean-reversion

23 from the reversal of spikes to normal levels. Secondly, the model assumptions for jump intensity (constant or seasonal) are convenient for simulating the distribution of prices over several periods of time, but restrictive for actual short-term predictions for a particular time. The Poisson assumption for jumps may be valid empirically, but only provides the average probability of jumps for particular transitions. Tight demand-supply conditions and hence spikes, could emerge irrespectively of season, for instance due to anomalous fuel prices or a technical failure in supply. For this reason, a cause, structural representation, instead of probabilistic seasonal formulations could be more appealing for the parameters of the jump distribution for specific day ahead forecasting. In a different approach [11] applied a variant of AR(1) and general ARMA process including ARMA with jumps to STPF in the German Market. They concluded that specifications where each hour of the day was modeled separately present uniformly better forecasting properties than specifications for the whole time series and that the inclusion of simple probability processes for the arrival of extreme price events (jumps) could lead to improvements in the forecasting abilities of univariate models for electricity spot prices. To improve on [11], [44] used various autoregressive schemes for modeling and forecasting prices in California. They observed that AR models where each hour of the day is modeled separately performed better than a single for all hours. In an effort to improve on prior studies, [9] considered general seasonal periodic regression models with ARIMA and ARFIMA disturbances for the analysis of daily spot prices of electricity. They concluded that for Nord Pool market (but not for the other European markets) a long memory model with periodic coefficients was required to model daily spot prices effectively; however the model s forecasting performance was not evaluated. [23] concluded on almost the same study that the long memory in Nord Pool market is due to the fact a significant amount of supply is from hydro plant and it is an empirical finding that rivers flow and water reservoir levels exhibit long memory.

24 Other scholars used Regime switching models, with the conception that the spiky character of electricity prices suggest that there exists a non-linear mechanism between normal and high price states known as regimes. A model with two regimes, a stable mean reverting AR(1) regime and a spike regime for the deseasonalised log prices was proposed by [27]. They assumed that the dynamics of the spikes regime can be modeled with a simple normal distribution whose mean and variance are higher than those of the mean reverting base regime process and it was extended by [4] and [5] by allowing the log-normal and Pareto distributed spike regime. On the other hand, [31] compared Markov regime switching specifications with regimes driven by AR(1) processes to an AR(1) model using average daily prices from the German EEX market. He concluded that for forecasts of 3-8 days ahead, regime switching models were more accurate. To integrate previous models, [23] calibrated seasonal ARFIMA and regime switching seasonal ARFMA models to Nord Pool area prices. They reported that both specifications seem to perform similarly in terms of out of sample predictions for the individual series but the non linear specifications out performed the linear model in forecasting of relative prices of neighboring regions with in the Nordic area. In a different approach, [36] compared Markov regime switching models with various time series specifications and evaluated their predictive capabilities in the California power market. The model had two latent states governed by mean reverting AR(1) price processes and the seasonality component composed of dummy variables for daily effects and sinusoidal function with linear trend to capture long-term seasonal effects. The model gave better results in the spiky periods as compared to calm periods. They concluded, that linear and threshold non-linear models provide better forecasting results than the Markov regime switching models. Much as the stochastic models of finance found their way to the power market, the most prominent model of geometric Brownian motion can not be used independently in

25 modeling electricity prices since it does not capture spikes and mean reversion. So it was modified to capture the characteristics of electricity prices and this resulted in to jump diffusion models. In a comparative study [4] evaluated the effectiveness of various diffusion models in describing the evolution of electricity prices in several markets. They tested the mean reverting diffusion model, known as the Ornstein-Uhlenbeck process: dp t = (α βp t )dt + σdw t (8) which was originally proposed by Vasicek in 1977 for modeling interest rate dynamics and geometric mean-reverting diffusions. They concluded that geometrical meanreverting jump diffusion model gave the best performance and that all models without jumps were inappropriate for modeling electricity prices. A general specification of jump diffusion model that comprises of all the characteristics of electricity prices was proposed by [45]. He considered a stochastic differential equation below, as the model which governs the dynamics of electricity price processes. dp t = µ(p t, t)dt + σ(p t, t)dw t + dq(p t, t) (9) where q(p t, t) is a pure jump process which produces infrequent but large upward jumps. Likewise [22] suggested the use of mean reversion coupled with upward and downward jumps, with the direction of the jump being dependant on the current price level. Though [43] suggested that positive jumps should be followed by negative jumps of approximately the same size so as to capture the rapid decline of electricity prices after the jumps. Also [8] proposed the drift to be given by a potential function which forces the price to return to its seasonal level after the upward jump and should allow the rate of the mean reversion to be a continuous function of the distance from this level. However, the mean-reverting diffusion models assume the diffusion process to be independent of the Poisson component, which is not the case in electricity, that is, prices

26 are less likely to spike when the demand is low, for example in the night. Empirical data also suggest that the homogeneous Poisson process may not be the best choice for the jump component since spikes are seasonal; they typically show up in high price seasons, like in winter in Scandinavia and summer in central USA. And for this reason [4] used non-homogeneous Poisson process with deterministic periodic intensity function instead of homogeneous Poisson process with a constant jump intensity. Despite their intuitive interpretation and non-linear behavior, jump-diffusion models present some limitations. Firstly, it is assumed that all shocks affecting the price series die out at the same rate, [17]. In reality however, two types of shocks exist implying different reversion rates; large disturbances, which diminish rapidly due to economic forces, and moderate ones, which might persist for a while. Estimation bias is inevitable, as maximum likelihood methods tend to capture the smallest and most frequent jumps in the data. As emphasized by [27], stochastic jump-models do not disentangle mean-reversion from the reversal of spikes to normal levels. Secondly, the model assumptions for jump intensity (constant or seasonal) are convenient for simulating the distribution of prices over several periods of time, but restrictive for actual short-term predictions for a particular time. The Poisson assumption for jumps may be valid empirically, but only provides the average probability of jumps for particular transitions. Tight demand-supply conditions and hence spikes, could emerge irrespectively of season, for instance due to anomalous fuel prices or a technical failure in supply. For this reason, a cause, structural representation, instead of probabilistic seasonal formulations could be more appealing for the parameters of the jump distribution for specific day a head forecasting. On the other hand, [33] discussed mean reversion process for energy prices that includes one to three factor models for energy pricing. They pointed out major problem in the practical use of these multifactor models, is that, direct market data for analyzing these models only exists for spot prices of the commodity, whereas the models have

27 additional hidden variables such as long run mean or the stochastic volatility. Finally, [37] analyzed the problem of pricing electricity tariffs in energy open markets with the assumption that both the customers power consumption and the market prices are stochastic processes. They showed the more the uncertainty about the customers consumption, the higher the fixed charge of the price tariff contract, should be. In his pricing model, every customer is only interested in the amount of money that he will spend on his electricity consumption. This amount is a stochastic variable that depends on the electricity price and the amount of consumption at each moment of time. Because different customers have different consumption behaviors the dynamics of the money amounts are different. They assume that each customer will consume electricity in future according to a given stochastic consumption model. 3 The salient features of electricity spot prices 3.1 Volatility Electricity is the most volatile of all tradable commodities. Its volatility is caused by both the sellers and the buyers. According to [46], statistics reveal the following percentages of how the daily volatility of electricity compares with other commodities: treasury bills and notes have volatility less than.5%, stock indices have a moderate volatility between 1% and 1.5%, commodities like crude oil or Natural gas have volatility between 1.5% and 4%, very volatile stocks have volatility not exceeding 4%, electricity has a volatility of up to 5%!!. The volatility of electricity prices can be explained by many factors but the most important reason is the fact that electricity can not be stored. The physical properties of electricity production and transmission coupled with its non-storability makes it hard

28 to match supply and demand, since there are constraints on the amount of electricity that can be delivered at a given moment. At a given grid, the mismatch between one generator and buyer can disrupt delivery of electricity to every buyer on the same grid, [48]. Once electricity is generated, it travels along a network of distribution and transmission lines designed to take the electricity from the generation source to the demand source. Each line within this network has a capacity or a maximum amount of electricity that it can carry at a given moment. Once constrained, the marginal cost of transmission becomes infinite. This implies, and is often the case, that sections of an electricity market can become isolated from the rest of the market. Once this occurs, the generators in the isolated market enjoy a greater level of market power, or influence over prices, [13]. Volatility is also due to capital intensive electricity generation, that is, the biggest percentage of marginal production costs is fixed costs. Thus in case of low demand, a plant would rather operate below capacity than not, for as long as the market price is above the marginal operating costs. This implies that the excess capacity of electricity will cause the prices to go down causing losses to the suppliers (generators) to the extent that some times they (generators) pay the buyers (for example big factories) to consume the generated electricity. On the other hand, if the levels of demand are high then more generators will be put to function thus increasing the production costs and hence higher electricity prices. 3.2 Spikes As mentioned earlier, in a deregulated electricity market, prices are determined by the intersection points of the aggregated demand and supply functions. Any unexpected eventuality on either the supply side or demand side would either shift the supply curve to the left or lift up the demand curve, therefore causing a price jump. When the contingency making the spot price to jump high has disappeared, the high price will

29 quickly fall back down to the normal range, [17]. By definition, a spike is defined as a price that surpasses a specified threshold, [46]. If we consider the price series of system and area prices in the Nord Pool electricity market, as shown in Figures 1 and 2, it is visible that there are times when the prices are extremely high. For example in Figure 2, on Monday at 17: the spot price was 26.4 Eur/MWh and at 18: hours it shoot to Eur/MWh (about seven times as high) and at 2: hours, it had settled to 24.3 Eur/MWh. So Eur/MWh is a price spike. Identification of spikes is a very important issue as it bears on the estimation of the deterministic and stochastic components for models of electricity spot price dynamics. However, the definition of a spike so far has been a rather subjective matter since it is difficult to gain any consensus on what the threshold or time interval should be, [42]. Some authors use fixed price thresholds to identify the spikes [32]. Other references suggest the use of fixed log-price change thresholds, for example log-price increments or returns exceeding 3%, [5], or variable log-price change thresholds, example, logprice increments or returns exceeding three standard deviations of all price changes, [1] and [43]. What [8] considered as spikes were those price moves that were outside 9% prediction intervals, implied by the normal distribution with the mean and variance given by the 6-days moving average and 6-days moving variance of the price moves. Yet another approach was used by [22], who filtered raw price data using different thresholds and selected the one leading to the best calibrated model in view of its ability to match the kurtosis of observed daily price variations. It is important to note that different definitions and identification techniques of price spikes may lead to quite different results, [3]. In this study we use definition proposed by [28], i.e, a spike the difference between the price exceeding a moving average µ t in a specified window w by more than s times a moving standard deviation σ t. Considering high frequency of data (hourly), we set

30 w = 96 meaning 4 days and s = 3. Figure 3 shows the spikes extracted from the price series. Figure 4 shows the zoomed extract of 24 spikes in Sweden on spiky days and Figure 5 when the spikes are rescaled to maximum of 1. The significant feature is that when the prices go up, they do it suddenly but when they are descending back to the base value they do it haphazardly. Spot price in Eur\MWh NordPool System and area hourly electricity spot prices DenmarkW System Finland DenmarkE Norway Sweden Time in Hours x 1 4 Figure 1: System and area hourly electricity spot prices in Nord Pool electricity market. System Finland Sweden x DenmarkE x DenmarkW x Norway x x x 1 4 Figure 2: Independent system and area hourly electricity spot prices in Nord Pool electricity market.

31 Spikes of electricity prices in Nord Pool areas Spot price differences in Eur\MWh Finland Sweden DenmarkE DenmarkW Norway System Time in Hours x 1 4 Figure 3: Spikes extracted from system and area hourly electricity spot prices in Nord Pool electricity market Figure 4: Sweden intra day spot prices for spiky days.

32 1 Intraday distribution of Sweden spikes scaled to max= Figure 5: Sweden intra day spikes rescaled to the maximum = 1. After identifying the spikes it is important to filter them out so as they do not bias the findings of the analysis. There are various techniques that can be used to replace the spikes and parameter estimates will be dependant on the chosen technique for replacement. Some authors suggest to dampen prices exceeding a certain threshold with a logarithmic function or to replace the observed outliers by the thresholds themselves, [46]. An alternative may be to replace the extreme observations by the mean of the two neighboring prices [47], or by one of the neighboring prices [22]. It is discovered that this can lead to complications when there are two or more consecutive spikes. Also seasonal behavior of electricity prices may alter the prices too much. Due to seasonality, for example, weekend prices are generally significantly lower than during the week, [6] suggested that the outliers can be replaced by the median of all prices having the same weekday and month as the spike. [45] proposed the use of wavelet decomposition to filter out the spikes. The choice of the method is at the discretion of the scholar since no research has revealed which method is best. In our study, spikes are replaced by log normally distributed numbers with mean and standard deviation equal to those of the current analyzing win-

33 dow (since the log-returns are expected to be normally distributed and, hence, the prices to be log-normal). Figure 6 shows the Nord Pool area and system prices with the spikes removed and replaced by log-normally distributed numbers. Figure 7 shows the prices when the spikes are replaced by the mean value of the analysis window. Going by the highest value, there is no significant difference between using either method of replacing spikes. NordPool spot system and area Hourly prices with spikes removed Spot price differences in Eur\MWh DenmarkW System Finland DenmarkE Norway Sweden Time in Hours x 1 4 Figure 6: Nord Pool electricity hourly spot prices with the spikes replaced by lognormally distributed random numbers. System Finland x 1 4 Sweden x 1 4 DenmarkE x 1 4 DenmarkW x 1 4 Norway x x 1 4

34 Figure 7: System and area prices after the spikes replaced by mean value of the analysis window. 3.3 Seasonality Seasonality of electricity prices is influenced by both the demand and supply sides. The most common factors that explain the seasonality in the demand side are business activities as well as the weather conditions. In central and Northern Europe as well as Canada the demand peak normally occurs during winter season due to excessive heating, and low during the long day light hours in summer. In other geographical areas such as mid-western USA, demand peak is in summer due to humidity and heat leading to extensive use of air conditioning, [46]. The supply side also shows seasonal variations in output, for example, hydro units are heavily dependent on weather conditions, that is, precipitation and snow melting, which varies from season to season. These seasonal fluctuations in demand and supply are directly translated into the seasonal behavior of electricity spot prices. The seasonality patterns can also be in form of intra-day, weekly or monthly. Tables 2 and 3 for example show the percentage of spikes that occur at a given hour of the day as well as the day of the week in system and area prices in the Nord Pool market. The time intervals are 6: to 1: hours, 11: to 15: hours, 16: to 2: hours and 21: to 5: hours, for morning, afternoon, evening and night respectively. It is observed that morning hours exhibit high price spikes, after which it gets "quiet" for the afternoon, it gets spiky again in the evening and finally "silent" in the night. The week days are more spiky than weekends, though it is also important to note that Sunday morning is very spiky in all areas and exceptionally spiky in Norway. This could be probably because every body bids high to ensure availability of electricity at the beginning of the week. On the other hand, using Table 1 extracted from [2], we can see that about 5% of electricity is generated from hydro, and according to statistic, [1], "electric heating is a major electricity consumption sector", at the same time, the production of hydro power

35 is dependant on the water level, so when the weather changes the production level will depend on precipitation as well as snow melting abilities. This implies that variations in climatic conditions may, and indeed do, cause supply or demand shocks. Table 1: Electricity generation (TWh) in the Nord Pool area in 27. Source: Nordel. Denmark Finland Norway Sweden Total Hydro power Nuclear power Other thermal Wind power Total Table 2: Percentages of spikes on different days of the week in different areas in the Nord Pool electricity market. Sys Fin Swe DenE DenW Nor Monday Tuesday Wednesday Thursday Friday Saturday 2 Sunday Table 3: Percentage of spikes at different hours of the day in all the system and areas of the Nord Pool electricity market. 3.4 Mean reversion sys Fin Swe DenE DenW Nor Morning Afternoon Evening Night To explain the concept of mean reversion, it is best appreciated when compared to a random walk which is used to model prices under geometric Brownian motion (GBM) is based on the assumption that price changes are independent of one another. In other words, the historical path of the price followed to achieve its current price is irrelevant for predicting the future price path (prices follow a Markov process). Mean reversion can be thought of as a modification of the random walk, where price changes are not