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A Joint State-Space Model for Electricity Spot and Futures Prices

Report no. 965

Kjetil F. Kåresen Egil Husby

December 2000

© Copyright Norsk Regnesentral

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Rapport/Report

Norsk Regnesentral / Norwegian Computing Center Gaustadalléen 23, Postboks 114 Blindern, 0314 Oslo, Norway Telefon 22 85 25 00, telefax 22 69 76 60

© Copyright Norsk Regnesentral

Tittel/Title: A Joint State-Space Model for Electricity Spot and Futures Prices

Dato/Date: December År/Year: 2000 ISBN: 82-539-0470-3 Publikasjonsnr.: Publication no.: 965

Forfatter/Author: Kjetil F. Kåresen and Egil Husby

Sammendrag/Abstract: We develop a joint model for electricity spot and futures prices, taking the special features of power markets into account. We use a stochastic multi-factor spot price model, and derive the futures price as the expected value of the future spot price, adjusted for risk preferences and market imperfections. The model is fit to observed spot and futures prices using the extended Kalman filter. We obtain good predictions of future prices and quantitative bounds on the uncertainty. The estimated model provides the necessary input for derivative pricing, production planning, and risk management, although these applications are not detailed in the paper.

Emneord/Keywords:

Tilgjengelighet/Availability:

Prosjektnr./Project no.: Satsningsfelt/Research field:

Antall sider/No. of pages:

A Joint State-Space Model forElectricity Spot and Futures Prices

Kjetil F. Karesen and Egil Husby�

May 13, 2002

Abstract

We develop a joint model for electricity spot and futures prices, taking thespecial features of power markets into account. We use a stochastic multi-factorspot price model, and derive the futures price as the expected value of the futurespot price, adjusted for risk preferences and market imperfections. The model isfit to observed spot and futures prices using the extended Kalman filter. We obtaingood predictions of future prices and quantitative bounds on the uncertainty. Theestimated model provides the necessary input for derivative pricing, productionplanning, and risk management, although these applications are not detailed inthe paper.

�Karesen is from the Norwegian Computing Center, Box 114 Blindern, N-0314 OSLO, Norway, e-

mail: [email protected]. Husby is from Norsk Hydro ASA, N-0240 Oslo, Norway. We are gratefulto Fred Espen Benth, Norwegian Computing Center and University of Oslo, for useful comments anddiscussion on the paper. This work was supported by the Norwegian Research Council, project SIP1211441420.

1 Introduction

International energy markets are rapidly being deregulated. In particular, previouselectricity monopolies are increasingly being replaced by power exchanges whereprices are freely determined by market demand and supply. Participants in the powermarkets typically include power producers, large consumers and end-user distributors.The players in the deregulated power markets need to make decisions in the presence oflarge uncertainties about future electricity prices. This has spurred a demand for realis-tic price models, needed for tasks such as production planning, portfolio optimization,derivatives pricing and risk management. The solutions to these problems do not de-pend on expectations of future prices alone. More importantly, they also requires athorough understanding of the uncertainty, i.e. a representation of the underlying priceprocess.

In this paper we develop a stochastic model for the joint evolution of spot andfutures prices for electricity, taking into account many of the special features inherentin the power markets. We fit the model to data from the Nordic power exchange,NordPool, which was opened in 1991 as the first power exchange in the world. Today,NordPool provides a highly liquid spot and futures market for electricity. We lookspecifically at the model’s ability to predict the future spot price and to replicate theobserved term structure of the futures price volatility. The scope of the paper is limitedto the derivation and validation of the model. We shall not detail its application to thespecific applications mentioned above.

Standard stock market theory inspired by the seminal work of Black and Scholes(1973), Merton (1973), and others relies on probabilistic models for the underlyingstock and uses arbitrage arguments to price derivatives such as futures and options.The most famous example is the Black-Scholes formula, which prices European op-tions assuming the underlying stock price is a geometric Brownian motion. In thisframework the futures price is simply the current price increased at the risk free in-terest rate. Commodity prices have some distinctive features that are not found inordinary stock prices, however. Most notable are seasonal patterns and mean rever-sion (Bessembinder, Coughenour, Seguin, and Smoller 1995). Such systematic varia-tions should of course not be present in ordinary stocks, since they would quickly bepicked up by speculators and counteracted by their actions. In order to use the arbi-trage theory for commodity markets, it is common to include storage costs and a conve-nience yield in the model, see e.g. (Gibson and Schwartz 1990, Schwartz 1997, Hillardand Reis 1998, Filipovic 1998). (The convenience yield accounts for benefits accruingto the holder of the actual commodity, but not to the holder of the futures contract.)

Electricity may be seen as an extreme case of a commodity since there is no wayof directly storing it. It is true that hydro power can be indirectly stored as waterin reservoirs and thermal power can be stored as coal or oil. However, the capacityof water reservoirs and the possible variations in production output of thermal powerplants are limited. Furthermore, these forms of storage are not easily available tospeculators. We thus find it unnatural to model storage costs and convenience yields.

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Instead we assume that the futures price is the expected value of the future spot price,contaminated with some additional noise accounting for market imperfections, andadjusted by a factor which accounts for the risk preferences of the market. This risk-preference adjustment is in some sense similar to the usual change from the data drivenspot price measure to the risk neutral measure, which is fundamental to derivativepricing in the standard arbitrage theory. However, since electricity cannot be stored,our market is highly incomplete and the adjustment cannot be derived from arbitragearguments. Instead, we estimate the risk-preference factor directly from the observedspot and futures data along with the rest of the model.

We start with a simple one-factor model for the spot price. This is a geometric AR-1 series, which is the discrete analog of the geometric Ornstein-Uhlenbeck processused by Schwartz (1997) for oil and copper prices. We then obtain a flexible modelclass by forming the product of many geometric AR-1 factors and a seasonal trend.This multi-factor model is different from the two- and three-factor models of Schwartz(1997) who explicitly models a stochastic convenience yield and interest rate. From thepostulated spot price model we derive the futures prices and show that these essentiallybecome products of geometric random walks.

In a series of papers on energy derivatives Clewlow and Strickland (1999a, 1999b)follow the opposite approach. They postulate a model for the futures prices and derivethe spot price from this. Our model is strongly related to theirs. However, their focus ison pricing of derivatives, while the issue of model fitting is not discussed. They workexclusively in a risk neutral futures price measure and do not relate their model to theobserved spot price data. For some commodities the spot price is hard to observe andit is natural to focus on the futures price alone. In our case, the electricity spot price isreadily available and it is natural to use this information for model fitting. We fit ourmodel to historical spot and futures data using similar techniques to Schwartz (1997).The model is first manipulated into a dynamical linear model framework. This allowsus to use the Kalman filter for fitting the model as well as predicting future pricesand uncertainties. Some moderate non-linearities in the relationship between spot andfutures prices are linearized by extended Kalman filter techniques. Fixed parametersthat enter linearly into the model are directly estimated by the Kalman filter, while theparameters that enter non-linearly are estimated by multi-variate non-linear maximumlikelihood optimization. The Kalman filter requires a model that is discrete in time.This may be achieved by discretizing a continuous model, but we have chosen to workentirely in a discrete-time setting.

Using a simulated dataset we show that the parameters in our model can be identi-fied. We then apply the model to real electricity data from the Nordic market and showthat the model gives good predictions of the future spot price and plausible estimatesof the uncertainty.

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2 Data

Every day, NordPool determines a spot price for electricity with an hourly resolution.The spot price is determined one day in advance by matching demand and supplycurves from all the participants in the market. In addition to the spot price there isa liquid market for futures contracts1. A typical futures contract specifies a constantdelivery of electricity over a specified period in the future. The period may be a givenweek, season or year.

Our data is collected in the period from late 1995 until early 2000. We use data witha weekly resolution because these are less affected by outliers and generally closer tothe log-normal distribution than hourly or daily data. For the spot price (Fig. 1) we usethe weekly average prices while the futures prices (Fig. 2) are Friday closing prices.

Later on we shall examine the ability of the futures prices to predict the futurespot price. To do this we have smoothed the interval futures prices to obtain a smoothfutures curve as illustrated in Fig. 3. Fig. 4 compares these smooth futures curveswith the spot price data at some selected dates.

3 Single-factor model

We now consider a simple single-factor model. In the next section we consider howto combine several factors like this and a seasonal trend into a realistic multi-factormodel.

3.1 Spot price

As with other commodity prices, we expect the electricity price to be mean reverting,i.e. to have a tendency to return to some “normal” level. In a period with high priceconsumers will tend to switch to alternative energy sources such as oil. Power intensiveindustry may move abroad or close down. Thus, the demand will be reduced and theprice will fall. When the price is low, the opposite happens. In a somewhat longer timespan, new power plants may be started or closed down according to price. In statisticalterms we want a stationary model. This is clearly not satisfied by the non-stationarygeometric Brownian motion that is commonly used to model ordinary stock prices.The simplest possible stationary model is probably an Auto-Regressive (AR) series oforder one (Priestley 1981). The AR-1 can not be used directly as a price model sinceit may become negative. To avoid this we exponentiate it and consider the Geometric

1Both futures and forward products are traded at NordPool. These differ only in the way paymentsare settled. Under the assumption that the interest rates are deterministic, the prices may be assumedto be the same (Cox, Ingersoll, and Ross 1981, Miltersen and Schwartz 1998). We shall not make anydistinction here. This is reasonable since power prices are typically very volatile and other sources ofuncertainty are much more important than the interest rate.

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Figure 1: NordPool spot price of electricity for the Nordic market (weekly averages) for the period1996-2000.

AR-1 series �� On the log-scale we have the ordinary AR-1 series� ��

(1)

where lowercase letters denote the logarithm of corresponding uppercase letters. Theparameter "! �$# �

determines the mean reversion rate of the series. Small�

gives a strongly mean reverting series while�

close to one gives a series that tends toreturn very slowly to the mean level. We assume that the noise term

��is temporally

white and has a normal distribution with mean 0 and standard deviation % . (In the nextsection we shall add a deterministic trend function that can absorb any non-zero meanin the AR-1 term. Consequently, no generality is lost by fixing & �'��( .)

Iterating the AR-1 model (1) gives� ��)*��+-,/. � ��0� +-,/.�,21346587 � 4 �'��)9�;:<��5

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Figure 2: The upper panel shows NordPool futures prices as a function of current time (horizontalaxes) and delivery time (vertical axes). The lower panel shows the product structure with arbitrarycoloring. Note that futures products with short delivery periods (weeks and blocks of four weeks) areavailable in the near future, while only seasons and whole years are available further ahead.

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Weeks until delivery

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Figure 3: An example of a smooth futures curve obtained from the interval futures prices. Eachsegment in the step function gives the price and delivery interval of a futures product that is traded inthe market. The smooth line is a constructed futures curve with weekly time resolution. This curve wascomputed as the “smoothest possible” curve with the following property: For each interval, the mean ofthe smooth curve is the same as the observed futures price.

It follows that the conditional distribution for the series at future time)

, given thevalue at the current time

�, is� ��)*�6= � ��?>A@CBD��+E,/. � ��FHGJI2��FK)L�NM (2)

where I2��FO)L�( �P��RQ�S +-,/.UT�V�� Q W % QG �(We let

@;��XF % Q � denote the normal distribution with meanX

and variance % Q andthe tilde character means “is distributed as”.) Letting

��)��Y��[Z \in (2) gives the

stationary distribution � ��>A@^] F ��P�_� Q % QJ` �On the original scale the stationary distribution of

��is log-normal and therefore

skewed. We have that& ��a�?�bNcedgf ��P�_� Q W % QGihkj median��i�?l�D�

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Spot priceFuture prices

Figure 4: Spot price and smoothed futures curves from some selected dates. Each futures curve (solidlines) represents the market products available at the particular date corresponding to the starting pointof the curve. Note that the market futures prices predict future spot price patterns fairly well, exceptin 1996-97. In 1996 the the spot price rose sharply due to a very dry year with unusually low waterreservoir filling.

The volatility is

vol��nm G�o�_� % � (3)

(We define the volatility of a series by

vol� . m

var prq � .� .�,21 (4)

throughout.)

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3.2 Futures price

We now define the futures price of a series as the expected value of the series at afuture time

)given the value at the current time

�s*tu��FO)L�( &�v ��)L�w= ��xy� (5)

This expectation may be interpreted as the fair price. It does not account for risk prefer-ences or market imperfections. These elements will be introduced when we turn to themulti-factor model of the next section. An immediate consequence of definition (5) isthat the futures price become a martingale; If we fix the delivery time

), the expected

value of tomorrow’s futures price is equal to the futures price today.Using (2) and the well known expectation of log-normal variables gives thats t ��FK)L�(�bNced{z|��+-,/. � ��I2��FO)*�~}J� (6)

This expression is different from the futures price commonly assumed for ordinarystocks which is s t ��FO)*�?��N� S +-,/.UT��F

(7)

where � is the risk free interest rate. Even if � is adjusted by storage costs and conve-nience yield (6) and (7) are fundamentally different.

Fig. 5 shows some futures price curves obtained by plotting (6) as a function of)��for various current prices

��. We note that the futures curves start at the

current price and approach the unconditional mean as the time to delivery goes toinfinity. This tendency to “forget the initial price” happens quickly for small

�and

slowly for�

close to one. When the current price is far away from the unconditionalmean, the futures curves are monotonic, but when the current price is close to theunconditional mean the curves are actually slightly bumped. The bumped behavioris mainly notable for large % . Large % corresponds to the situation where

�_��has a

highly skewed distribution with the mean far away from the median. An interpretationof this bumped behavior is as follows: If the current price is at the unconditional mean,this is actually a quite high price since we are far above the median. In a situation withhigh price and high volatility there is a considerable chance that the price will rise evenhigher in the near future. For the parameter values estimated in Section 5 the bumpedbehavior is not very noticeable, though.

Some algebra gives the following recursive representation of the futures prices t ��FK)L�?�s t �� DFO)*� �� D�6�& �� w� � (8)

Thus, for fixed maturity time)

,s t ��FK)L�

is a martingale. The expected value oftomorrow’s price is equal to the price today. Furthermore,

s t ��FO)*�is also easily

seen to be a geometric random walk, having normal log-increments with time-varyingvolatility. (The derived futures price model is thus the discrete analog of a geomet-ric Brownian motion with time-dependent volatility.) We find this noteworthy: Even

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Figure 5: Futures price curves in the single factor model for various current prices; �� ,�� (the median), �g��?��O�6��6�� ¢¡��~£¥¤y¦�¡�§©¨6ª (the unconditional mean), and ��©�«� .The futures curves are shown with dashed lines and the unconditional mean is indicated by a solid line.Upper left: ¬�� , ¦u�� N¨ . Upper right: ¬��w�®��¯ , ¦u�� . Lower left: ¬��®� , ¦u��°� . Lowerright: ��®�©¯ , ¦±��² . Note that larger gives slower mean reversion, and larger ¦ gives moreasymmetrical curves

though the spot price is mean reverting and stationary, the futures price follows therandom walk model commonly used for ordinary stocks.

Using (4) and (8) it follows immediately that the volatility of the futures price is

vols t ��FO)L�(³��+E,/. % � (9)

The futures price volatility thus decreases with increasing time to maturity. This isconsistent with the observed properties of el-futures in the Nordic market. Mathemat-ically this behavior is a result of the mean-reverting properties of the spot price. Froma market point of view it can be explained by the fact that a lot of the informationreceived today will affect the spot price in the near future, but not in the distant future.For example, the Nordic market has a large percentage of hydro power. The amount ofprecipitation therefore influences the price strongly. A week with more precipitationthan normal for the season will increase the filling of the reservoirs. This reduces theprice in the near future. However, some extra water today has very little to say aboutthe reservoir filling in two years time. This information should thus have a minimaleffect on futures contracts with a long time to maturity.

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4 Multi-Factor Model

4.1 Spot price

We now model the spot price by a multi-factor model of the form´ ��?�µ*�� W¬¶·¸ 5 1 � ¸ ��©� (10)

We assume that the ¹ factors,� ¸ �� , are stochastically independent of each other.

Each is a geometric AR-1 series, as defined in the previous section, with AR parameter� ¸ and noise standard deviation % ¸ . The trendµ*��

is a deterministic function thataccounts for seasonal variations and absorbs the mean level of the process. Since thetrend models yearly cycles, a convenient parameterization is a truncated Fourier serieswith period equal to one year. On the log scale we assume that the trend, º ��»prq µ*�� , has the form

º ��?�¼ 7 � ½3¾ 5 1 ¼ 1�¿ ¾�ÀHÁ q�Â GÄÃ¢Å/�Æ G�Ç �È¼ Q ¿ ¾'É�Ê/À Â GJÃ2ÅD�Æ G³Ç � (11)

The order of the series is a compromise. A high-order series can potentially model thetrue trend more accurately, but is more susceptible to estimation error. Based on somepreliminary experiments we used Ë ÌG

for all results reported here. The model (10)does not include inflation. This could easily be accounted for by adding a linear factorto the log-trend (11). However, in our present data there is no evidence of any risingtrend in the spot price. A possible explanation is that energy prices have been decliningat a rate exactly compensating the general monetary inflation. For the results given herewe have therefore not included inflation in the model.

The volatility of the spot price in the multi factor model is easily calculated. Sincethe trend is deterministic, it will not affect the volatility. Furthermore, since the factorsare independent, the squared volatility will simply be the sum of the squared volatilitiesof the individual factors. Using the single-factor volatility (3), we thus find that

vol´ ��? ÍÎÎÏ ¶3¸ 5 1 G�o�_� % Q¸ � (12)

4.2 Futures price

We now consider the futures prices in the multi-factor model. We first derive a “theo-retical” futures price that is similar to the futures price defined in the previous section.This is based on the assumption that the participants in the market are perfectly ratio-nal and the market is perfectly liquid. We then model an “observed” futures price that

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allows for imperfections such as bid-ask spread, deviances from the theoretical modeletc. We define the theoretical futures price ass[ÐE��FO)*�(³�wÑ S +-,/.rTrÒÔÓ Q W &�v ´ ��)*�6= � 1 ��F6�6�6�wFO� ¸ ��~x~� (13)

The expectation is similar to the definition for the single-factor model. Note, however,that we condition on all the

� ¸ �� , assuming that these represent the knowledge avail-able to the market at time

�. The factor

� Ñ S +-,/.UTrÒÔÓ Qmodels the risk preferences of the

market, with Õ×Ö indicating deviations from risk neutrality. For example, it is possi-ble that the sellers of futures contracts are mainly producers who want to hedge theirprice risk, while the buyers are mainly speculators. Since the speculators will want apremium for taking the risk, Õ should be negative in this scenario.

By independence of the components, the futures price (13) can be split into theproduct of the futures prices of each component,s Ð ��FO)*�?�� Ñ S +E,/.rTrÒÔÓ Q W µ*��)L� W ¶·¸ 5 1 s t�Ø ��FO)*�©� (14)

Using (6) and taking logarithms givesÙ Ð ��FO)*�? Õ ��)9�;��Ú Æ G�� º ��)*�� ¶3¸ 5 1 ��+E,/.¸ � ¸ ��ÈI ¸ ��FO)L�� (15)

Note that the logarithm of the futures price is linear in the underlying � ¸ factors.We now consider the observed futures prices. At each time point

�there are Û ��

futures contracts with different maturity times,)�ÜK��

, that are traded in the market. Weassume that these are equal to the theoretical futures prices with some additional noiseresulting from imperfections in the market,Ù ÐÄÝÜ ��? Ù ÐRÞO��FO)EÜO��Ôß*��àáÜO��F â�l�DFHG8F6�w�6�wF Û ��©� (16)

We take the noise terms,àPÜO��

, to be normal with covariance structure

covz«àãÜO��FOà ¾ ��y}�³äeÞK)-ÜO��©FK) ¾ ��Ôß¥å Q �

The correlationäÄ��)EÜÔFO) ¾ � jÌ says that if one of the observed futures prices is some-

what above the theoretical value then it is likely that this is true for the others as well.We expect this correlation to be stronger for futures contracts with closely spaced ma-turity times than for those further apart. We model this by an exponentially decayingcorrelation function, äÄ��)-ÜÔFO) ¾ �?�bNced f �Læ = )-Ü��;) ¾ =çÚ Æ Gè h �The exponential functional form ensures that the covariance matrix of all the

à�Üis posi-

tive definite, see e.g. (Cressie 1993). Apart from that, we have no particular motivation

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for this functional form, and many other choices are possible. The parameterè

can beinterpreted as the effective range of the correlation in years. When

= )<Ü��) ¾ =çÚ Æ G� èthe correlation has been reduced to approximately 5%. The above correlation onlyconcern futures contracts observed at the same

�. It is possible that the errors will also

be correlated from one observation date to the next. For simplicity we have not mod-eled such temporal correlations. We stress that the futures prices themselves do indeedhave temporal correlation, which is induced by the correlation in the underlying � ¸ ��factors. It is only the difference between the theoretical and observed futures pricesthat is modeled as temporally white.

Finally, we derive the volatility of the observed futures prices in the multi-factormodel. Using (9), (14), and the independence of the components, it follows that

vols ÐÄÝ ��FO)L�( ÍÎÎÏ é ¶3¸ 5 1 � QKS +-,/.UT¸ % Q¸Hê ��G/å Q � (17)

4.3 State-space form

The model from the previous section links the observed (log-scale) spot and futuresprices linearly to the unobserved state factors, � ¸ �� . This is very convenient, since themodel can then be formulated into a general linear state space form. Once this is done,Kalman filter methodology can be used for computing minimum variance estimatesof the unobserved states and predicting futures prices. See Harvey (Harvey 1989) fora thorough treatment. The Kalman filter can also be used to estimate all the fixedparameters that enter linearly into the model by adding constant equations of the form¼ 7 ��o³¼ 7 ��

. This allows us to estimate all the¼

coefficients from the trend andthe risk preference parameter Õ . Furthermore, the Kalman filter can be used to evaluatethe likelihood of the observed data through the prediction error decomposition. Weuse this together with a general non-linear optimization routine to obtain maximumlikelihood estimates of all the unknown parameters that enter non-linearly into themodel. These are the AR coefficients and the variance and covariance parameters. Anexact statement of the state-space form of the model is given in Appendix A.

4.4 Interval Futures

So far we have only defined the futures price for delivery at one time point in the future.Traded futures contracts specify constant delivery over an interval such as a month ora year. The futures price for the interval v )Lë'FO)�ì(x is naturally defined as the averageof the point-wise prices, s Ð ��FO) ë FO) ì �? �í +8î3+ 5 +8ï s Ð ��FO)*�©F (18)

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where í ") ì �_) ë �A�. Using this expression for the theoretical futures prices we

define the observed futures prices analogously to (16). On the log-scale (18) becomesÙ Ð ��FO) ë FO) ì �? prq�ðñ ò �í +8î3+ 5 +8ï b6c¥d»z Ù Ð ��FK)L�~}eó ôõ F (19)

where the point-wise futures are given by (15). Unfortunately, this expression is nolonger linear in the � ¸ �� . If all the observed futures have short delivery periods, asimple solution is to approximate the interval futures price with the point-wise futuresprice for the midpoint of the period. For longer delivery periods, this is not satisfac-tory and a better approximation is needed. We have used the extended Kalman filtertechnique as detailed in Appendix B.

We now check the validity of this approximation on a simulated data set. (Thisdata set will be considered in more detail in the next section.) For illustration, weconsider the

¼ 1~¿®1 coefficient which is responsible for the main sinusoidal variation inthe seasonal trend. We estimate this parameter both by the Extended Kalman filter andby the ordinary Kalman filter. In the latter case we approximate the interval futures bypoint-wise futures corresponding to the mid-point of the interval. As seen in Fig. 6 themidpoint approximation does not work satisfactorily for season and year futures whilethe Extended Kalman filter works very well in all cases.

5 Results

To test whether the parameters in the model can be successfully identified, we first fitthe model to a simulated data set. The data were generated according to the modelin the previous section with three underlying state-factors,

� 1 , � Q , ��ö . The dataset had the same length and the same futures product structure as the real data (seeFig 2). The parameters used for the simulations were the same as those estimated forthe three-factor model on the real data (see below.) Fitting the model to these datagave maximum-likelihood parameter estimates that agreed well with the true values(Table 1). Furthermore, the Kalman filter estimates of the underlying states follow thetrue states closely (Fig 7). We conclude that all the parameters are well estimated fromthe simulated data. Note that the actual non-linear expressions were used to simulatethe interval futures in the test data. The good parameter estimates thus provide furtherevidence that the Extended Kalman filter linearization is adequate.

In general, the non-linear maximum likelihood optimization is not guaranteed tolocate the global optimum of the likelihood. For the current model, we have observedthat the optimization converges to the same estimates for a wide range of reasonablestarting values, however. To initialize the Kalman filter we also had to provide initial (apriori) estimates and standard deviations for the linear parameters. For all the resultspresented here, we used 0 as the initial value for the � ¸ , Õ , and the

¼coefficients,

except¼ 7 where we used the overall mean of the spot price data, and

¼ 1Ô1 where we

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Figure 6: State estimates of the ÷ ¤�ø�¤ coefficient on simulated data. Note that the Extended Kalmanfilter estimate converges to the true value while the ordinary Kalman filter estimate degrades seriouslyin late 1997. This corresponds exactly to the first time when season futures are present in the data (seeFig. 2). A further degradation is seen in early 1999 which corresponds to the first time yearly futuresare introduced into the data.

used 0.5. The non-zero value for¼ 1�1 was used to reflect the a priori knowledge that

prices should be lower in the summer than in the winter. The initial standard deviationswere

�for

¼ 7 , �� for Õ , and � � for all the others.We fit four models to the real data. These differed only in the number of underlying

state factors, which we varied from one to four. The estimated parameters are given inTable 2. Fig 8 shows the state estimates and the estimated seasonal trend for the three-factor model. (The plotted trend is based on the final Kalman filter estimate of the

¼coefficients, i.e. using all the data.) Unlike the simulated data there is no underlyingtruth to compare these estimates to. Note that the slowest mean reverting factor (

�closest to 1) accounts for the major deviations of the spot price from the seasonal trend.The factors with faster mean reversion account for small-scale variation. Together withthe seasonal trend these factors explain all the variation in the spot price data.

Prediction in the general state-space model is easy. We simply run the Kalmanfilter forwards into the future, but omit the observation update steps. Fig 9 illustratesthe predictions from the three-factor model. The predictions are quite close to thefutures curve. This is not surprising since these futures data have been used in themodel fitting. Note, however, that the model prediction is smoother than the futurescurve. More importantly, the model prediction is available at any desired time-point

15

Parameter Estimate Actual value� 1 0.997 0.997� Q 0.946 0.942�ùö0.630 0.650% 1 0.030 0.033% Q 0.060 0.057% ö 0.119 0.120å0.044 0.045è0.327 0.319Õ 0.002 0.005¼ 7 5.168 5.179¼ 1�¿�1 0.171 0.171¼ 1�¿ Q 0.046 0.044¼ Q ¿�1 -0.040 -0.040¼ Q ¿ Q 0.007 0.008

Table 1: Parameter estimates for simulated data.

and can be used to predict further ahead than the available futures data. Furthermore,the model gives the uncertainty of the predictions. As an example we have plotted ú Gstandard deviations in the ‘predictive distribution’ of the spot price, which is easilyavailable from the Kalman filter. We expect the future spot price to be within theselimits with approximately 95% probability. Note that this is a point-wise statement. Atany given point the probability is 95% that the realized price falls between the limits.The probability that the entire curve falls between the limit is lower. It is thereforenot surprising that the realized spot price curve is slightly outside the limits at a fewpoints. (Unfortunately, all of the years 1997-1999 had more than average precipitationand low prices. The realized spot price therefore tends to be lower than the predictionfor the whole period.) We should note that the given limits include the uncertaintyin all linear parameters of the model, but do not include the estimation uncertaintyof the AR parameters and the variance parameters. For simplicity we also used theglobal estimates of these non-linear parameters instead of refitting them to the dataused for prediction only. (We find this reasonable since in a practical application wewould probably use at least this much data to estimate the parameters.) In conclusion,we think the estimated uncertainties from the three-factor model are reasonable. Theprediction results from the one-factor model are quite different (Fig. 10). First, theprediction does not follow the futures curve as closely as the three-factor model. Withonly one underlying factor, this model appears to be to rigid for an adequate modelingof the futures curve. The resulting model errors have inflated the futures price errorstandard deviation

å(see Table 2), which is much larger than the estimate for the three-

16

X1

0.7

0.9

1.1

1.3

1996 1997 1998 1999 2000

X2

0.6

0.8

1.0

1.2

1996 1997 1998 1999 2000

True stateEstimate

X3

0.6

1.0

1.4

1996 1997 1998 1999 2000

Figure 7: Estimates from simulated data of the underlying states � ¤ , � ¡ , �Pû in the three-factor model.Note that the estimated states follow the true states closely.

parameter model. This leads to unreasonably wide uncertainty limits for the one-factormodel. Still, the mean prediction from the one-factor model is very good. We do,however, think that this is partly coincidental, particularly because the period 1997-1999 does not include any dry years. The problems with inflated standard deviationsappear to reduce quickly with higher model order. The prediction results from the two-and four-factor models (not shown) were quite similar to the three-factor model.

We now compare root mean square (RMS) prediction errors between the predictedspot price and the actually realized values. We compare the model predictions with theresults from using the smoothed futures curves alone. To obtain a fair comparison, onlypredictions corresponding to available futures prices were used. We see from Fig. 11that the prediction errors tend to increase with the distance ahead. However, whenincluding the predictions for 1996, there are some very large short-term predictionerrors. These are due to the extreme behavior of the spot price in 1996, which was anunusually dry year. All the models have performed better than the futures alone. Thebest predictions were in fact obtained by the one-factor model. However, we think thedata set includes too few years to be entirely conclusive on this issue. It is also notable

17

ParameterOne-factormodel

Two-factormodel

Three-factormodel

Four-factormodel� 1 0.984 0.994 0.997 0.999� Q - 0.812 0.942 0.972�ùö

- - 0.650 0.893�0ü- - - 0.632% 1 0.137 0.044 0.033 0.024% Q - 0.120 0.057 0.046% ö - - 0.120 0.074% ü - - - 0.124å

0.117 0.055 0.045 0.041è2.190 0.469 0.319 0.259Õ -0.004 -0.031 0.005 0.015¼ 7 4.775 5.270 5.179 4.968¼ 1�¿�1 0.179 0.175 0.171 0.170¼ 1�¿ Q 0.046 0.045 0.044 0.042¼ Q ¿�1 -0.039 -0.040 -0.040 -0.040¼ Q ¿ Q 0.007 0.010 0.008 0.008

Table 2: Parameter estimates for the electricity price data from NordPool.

that the simple strategy of predicting the future spot price to be constantly equal totoday’s spot price works quite well for this data set.

As a further validation, we compare the volatility estimated directly from the ob-served futures price series with the volatility predicted by the model using equa-tion (17). We find that the volatility of the observed futures price decreases withincreasing time to maturity. This fact is well known in the market. All the modelsreplicate this qualitative behavior well (Fig. 12), but the model predicted volatilitiesare somewhat higher than the directly estimated volatilities. In particular, this is truefor the one-factor model. Again, we think this is due to model error inflating the fu-tures price error parameter

å. One should bear in mind that we are trying to capture the

essential structure of a high dimensional futures curve problem with a low-dimensionalparametric model. If we assume that most of the error reflected by

åis due to model

error, it is of interest to compute the model volatilities withå� . It turns out that

except for the one-parameter model this gives an excellent fit to the directly estimatedvolatility curve. We note that the model with

åý will have the same long-termproperties as the model with

å Ö since kappa only affects the additional white-in-time noise that is added to the futures prices at each time point.

Realizing that the model is never a perfect fit to reality, it is not obvious that it isbest to use the Maximum Likelihood estimates for all the non-linear parameters. In

18

NO

K /

MW

h

0.5

1.0

1.5

1996 1997 1998 1999 2000

X1X2X3

State estimates

NO

K /

MW

h

5010

020

030

0

1996 1997 1998 1999 2000

Spot pricesEstimated trend

Trend estimate

Figure 8: Top: State estimates for the three-factor model fitted to the NordPool el-price data. Bottom:Estimated seasonal trend and spot price data.

particular, one may want to adjust the standard deviation parameters of the model inorder to fit the observed futures price volatility as well as possible. As an example wefit the model volatility (17) to the observed futures price volatility by a least squaresfit (Fig. 13). We adjusted % 1 F6�6�6��F % ü to obtain the best possible fit while keepingåa . The one-factor model is not able to fit the observed volatility very well. Theexponential shape of the one-factor model gives too low volatility for products closeto maturity and very far from maturity. The higher order models, on the other hand,are able to reproduce the observed volatility curve very well by combining severalexponential curves.

6 Model Discussion

The parametric models fitted here are interpretable and mathematically tractable. Theyprovide predictions of the future spot price that may deviate from the futures pricesof the market. This is natural since the models assume that market futures pricesare affected by risk preferences and market imperfections. In some cases it may bedesirable that the initial prediction from the model coincides exactly with the currentfutures curve observed in the market. This may easily be achieved, for example byadjusting the trend factor

µ*��in (10). This would be similar to the approach favored

by Clewlow and Strickland (1999a).

19

NO

K /

MW

h

020

040

060

0

1996 1997 1998 1999 2000


NO

K /

MW

h

020

040

060

0

1996 1997 1998 1999 2000


Figure 9: Predictions of the future spot price from the three-factor model compared to the marketfutures curves. Top: Prediction using the data up to week 13 in 1996. Bottom: Prediction using the dataup to week 26 in 1997.

20

NO

K /

MW

h

020

040

060

0

1996 1997 1998 1999 2000


NO

K /

MW

h

020

040

060

0

1996 1997 1998 1999 2000


Figure 10: Predictions of the future spot price from the one-factor model compared to the marketfutures curves. Top: Prediction using the data up to week 13 in 1996. Bottom: Prediction using the dataup to week 26 in 1997.

21

Prediction distance (weeks ahead)

Roo

t mea

n sq

uare

err

or (

NO

K /

MW

h)

0 20 40 60 80 100 120

2040

6080

100

Futures aloneTodays spot priceOne factor modelTwo factor modelThree factor modelFour factor model

Prediction distance (weeks ahead)

Roo

t mea

n sq

uare

err

or (

NO

K /

MW

h)

0 20 40 60 80 100 120

2040

6080

100

Futures aloneTodays spot priceOne factor modelTwo factor modelThree factor modelFour factor model

Figure 11: Root mean square prediction errors as a function of prediction distance. Top: Errorsaveraged over the whole data set. Bottom: Errors averaged over the period from 1997 and onwards.

22


Vol

atili

ty (

perc

ent)

0 20 40 60 80 100

050

100

150

Empirical volatilityOne-factor modelTwo-factor modelThree-factor modelFour-factor model


Vol

atili

ty (

perc

ent)

0 20 40 60 80 100

020

4060

8010

0

Empirical volatilityOne-factor modelTwo-factor modelThree-factor modelFour-factor model

Figure 12: Observed (empirical) futures price volatility compared to model predicted volatility. Top:Model volatility calculated from (17) with parameters taken from Table 2. Bottom: As above, but withþ �� .

23

weeks until delivery

Vol

atili

ty (

perc

ent)

0 20 40 60 80 100

2040

6080

100

One-factor model


Vol

atili

ty (

perc

ent)

0 20 40 60 80 100

020

4060

Two-factor model

Empirical volatilityVolatility of single factorsBest fit of multi-factor model


Vol

atili

ty (

perc

ent)

0 20 40 60 80 100

020

4060

Three-factor model


Vol

atili

ty (

perc

ent)

0 20 40 60 80 100

020

4060

Four-factor model

Figure 13: An example of adjusting the model volatility to the volatility of the observed futures pricesby a least-squares fit.

There is a price to pay for imposing a low-dimensional model on a high dimen-sional problem. The high estimates for the standard deviation of the futures priceerror term,

å, is probably partly due to model error. This may possibly be improved

by introducing correlations between the futures price errors corresponding to differ-ent trading dates. A suitable diagnostic for checking whether the model has capturedthe temporal correlations in the data, is to compute the auto-correlation function ofthe one-step-ahead prediction errors. For the three-factor model, there is clearly someresidual temporal correlation of the innovations (Fig. 14). For the lower-order models(not shown) the correlation is more pronounced and for the higher-order model it isless pronounced. Introducing temporal correlation in the futures price error terms willincrease the order of the state-space model, however, and lead to longer computingtimes. It will also destroy the attractive model feature that all temporal dependenciesgo through the underlying

��state factors. An alternative way to try to model the

temporal correlations in the futures prices better, is to increase the number of statefactors. Using too few factors can clearly lead to an inflated estimate of the futuresprice error,

å. This is clearly illustrated with simulated data. The excellent parame-

ter estimates in Table 2 are obtained by first simulating from the three-factor modeland then fitting the three-factor model. When we tried to fit a two-factor model to thethree-factor data, we got a too high estimate of

å, similar to that observed with the

real data. This may seem to indicate that we should use a very high order for the real

24

0 5 10 15 20

Lag

0.0

0.4

0.8

Spot prices

0 5 10 15 20

Lag

0.0

0.4

0.8

First future price

0 5 10 15 20

Lag

-0.2

0.2

0.6

1.0

10’th future price

Figure 14: Auto-correlation functions for the one-step-ahead prediction errors in the three-factormodel

data. Alternatively, we might try to choose the order with Akaike’s model selectioncriterion or similar criteria based on penalized likelihood (see e.g. (Harvey 1989)). Forthis highly structured dataset we find that the likelihood based criteria almost invari-ably chooses the higher-order model, however, and is of little interest. Furthermore, ifthe order is too high, some of the AR coefficients (the

�’s) will be very close to each

other. It will then be difficult to estimate these reliably from the data. It may also beunreasonable to assume that the

�_��factors are independent any longer.

A high number of underlying state factors may also lead to other unidentifiabilityproblems. For example, we have observed that the highest

�tend to be very close to

one with a very small standard deviation (cf. the estimates for the four-factor modelin Table 2). This factor then becomes almost constant and is confused with the meanlevel of the trend. This may partly be the case for the three- and four-factor modelsfitted here. Note that the trend estimate in Fig. 8 appear to be somewhat above themean level of the spot price data. We should keep in mind, however, that the trendestimate depends on the futures prices as well as the spot price, and the futures pricesare above the spot price for most of the period (cf. Fig. 4). The upwards sloping trendof the futures prices as compared to the spot price may be interpreted in two ways:

1. The spot price in 1997-1999 is consistent with the long-term seasonal trend andthe upwards slope of the futures prices is due to the risk preferences of the par-ticipants of the market. This can be represented in the model by a positive risk

25

preference parameter, Õ .

2. The spot price in 1997-1999 is below the long-term seasonal trend. The upwardsslope of the futures prices reflects the expectation that prices will return to nor-mal. This can be represented in the model by a zero risk preference parameter, ahigh trend, and small state-factors for the period 1997-1999.

The model can not easily discriminate between these explanations. This is not neces-sarily serious because short and medium range predictions from the model will typ-ically be quite similar in any case. However, the interpretation of the model will bedifferent. Also, very long term predictions may be different. This should not be sur-prising since we generally need a series of data that is long compared to our predictionhorizon in order to make good predictions.

A solution to the above problems is to provide more a priori input. When all theparameters were determined completely freely, we found that the estimates of the riskpreference parameter, Õ , were quite unstable for the real data. To avoid this we useda small a priori standard deviation (0.01) for Õ . This led to estimates close to zero.(We thus think one should not place too much confidence in the estimates of Õ .) Analternative solution is to estimate the overall mean of the spot-price data from a longerstretch of data than used here and fix the

¼ 7 coefficient to this value. This shouldallow an accurate estimate of Õ . Note, however, that there is also the possibility ofconfusing the risk preferences with the expectation of monetary inflation. The upwardssloping futures curves in Fig. 4 may be due to the investors expecting inflation, ratherthan requiring a risk premium. If an independent estimate of power price inflation isavailable, this can easily be included in the trend as noted in Section 4.

The models considered here allow efficient computations with the Kalman filterdue to the state-space formulation. The underlying factors are Markov processes andthe state-space model is a vector Markov model. However, for the multi-factor models,the marginal distribution of the observed spot price is highly non-Markov. This is dueto the fact that the sum of Markov processes is not necessarily Markov. Modeling thelog-spot price as a sum of AR-1 processes leads to very simple expressions for thefutures prices. These depend on the past only through the current state. If we had usedan AR-2 or a higher order AR process for the spot price instead, this would not be true.

The current framework is easily extensible as long as the observed log-prices de-pend linearly, or approximately linearly, on the underlying state factors. One may forexample want the model standard deviations to vary with the season. (In the Nordicmarket electricity prices are typically more volatile in the summer than in the winter.)This modification only requires notational changes.

7 Conclusions

We have studied an analytically tractable model that describes the temporal develop-ment of the spot price and futures curve with a moderate number of parameters. The

26

spot price model incorporate the crucial features of seasonality and mean reversion.The derived futures prices, on the other hand, are neither mean reverting nor seasonal.If the risk preference parameter Õ the theoretical futures price (13) is a martin-gale by construction. Furthermore, it is a product of geometric random walks. Thisis theoretically pleasing. Unlike the spot asset, the futures contract is a financial pa-per that is easily traded and stored. It is therefore a desirable property of the modelthat the derived futures prices are consistent with the usual stock market assumptions.Derivatives written on futures contracts may thus be priced with the standard arbi-trage theory. For example, the Black–Scholes equation can be used to price Europeanoptions on futures contracts. Such options are heavily traded at NordPool and are typ-ically priced by Black–Scholes by market practitioners. The volatility required in theBlack–Scholes equation can easily be calculated from the estimated parameters in ourmodel. Note, however, that the log-normal distribution required by Black–Scholes isexactly true only for the futures price (13) which specifies delivery at a given timeinstant. Fortunately, it may be shown that this assumption is approximately true alsofor the interval futures price (18).

We think the presented two- or three-factor models provide a reasonable compro-mise between realism on the one hand and simplicity and interpretability on the other.We have shown that these models give good predictions of the future spot price anda realistic description of the uncertainty. Furthermore, they can reproduce qualitativeproperties of observed futures prices such as the shape of the downwards sloping termstructure of the futures price volatility. There is evidence that the description of thehigh-dimensional futures curve by a moderate number of parameters leads to someresidual model error. Most notably, the model tends to overestimate the volatility ofthe futures prices if the maximum likelihood estimates are used for all parameters. Asolution is to adjust the model standard deviation parameters by directly matching his-torical futures price volatilities to the corresponding model quantities as in Fig 13. Thiswould force the model to mimic the term structure of historical futures prices. Alterna-tively, the model volatility may be matched to implicit market volatilities. There is alsoevidence that some of the model parameters are not very well estimated from the data.In particular, the estimates obtained for the risk preference parameter Õ varied a lot be-tween the different models. The model fitting of this paper has been quite ambitious intrying to fit all unknown parameters to the data alone. In a commercial application onemight want to incorporate additional a priori information into the model, possibly in aBayesian fashion. This could easily be incorporated into the current framework, andwould lead to more stable parameter estimates. With judicious parameter estimationwe believe models of this kind should have important applications for price prediction,pricing of derivatives, and risk management. Our exposition has been focused on elec-tricity prices, but the model should be equally relevant for other commodities that areexpensive or impossible to store.

27

A State-Space Form of the Model

The general linear state space model can be written in matrix form asÿ ��?�� ÿ �� á��F(state eq.) (20)

� ��L�� ÿ ��0�� k�� (observation eq.) (21)

The noise terms�á��

andk��

are white and independent with variances

var�o��?��»��

and vark��?� ��©�

We translate our model into this framework as follows:State equation

ÿ ��?��

� 1 �� Q ��...� ¶ ��Õ¼ 7¼ 1 7¼ 1Ô1¼ Q 7¼ Q 1

��

F � ��?��

� 1 � Q. . . � ¶ �

. . . �

� ��F

�{��?��

% Q1. . . % Q¶

. . .

��

Observation equation

� ��? ��

� ��Ù ÐÄÝ1 ��...Ù ÐÄÝ� ��

� �� F � ��? ��

� � ¶ ¸ 5 1 I ¸ ÞK��FK) 1 ��ß...� ¶ ¸ 5 1 I ¸ ÞK��FO) � ��ß

� ��F

�L��

� �6�w� � � É�Ê/À .Ó Q ÀHÁ q .Ó Q É�Ê/À Q .Ó Q À�Á q Q .Ó Q� +�� S .UT�,/.1 �6�w�� +�� S .UT�,/.¶ +�� S .UT�,/.Ó Q � É�Ê/À +�� S .rTÓ Q ÀHÁ q +�� S .rTÓ Q É�Ê/À Q +�� S .UTÓ Q À�Á q Q +�� S .UTÓ Q...

. . ....

......

......

......� +�� S .rT�,/.1 �6�w�� +�� S .UT�,/.¶ +�� S .rT�,/.Ó Q � ÉNÊDÀ +�� S .rTÓ Q À�Á q +�� S .rTÓ Q É�Ê/À Q +�� S .rTÓ Q À�Á q Q +�� S .rTÓ Q

� ��F

28

��? �� 6�6� ä��) 1 FO) 1 � �6�6�"ä��) 1 FO) � �...

.... . .

... ä��) � FO) 1 � �6�6�"äÄ��) � FO) � �� å Q �

Note that the number of futures, Û Û �� will depend on�

although this is not madeexplicit in the notation. The matrices in the observation equation are thus of variabledimension. Apart from computational issues, this does not create any complications,however.

B Application of the Extended Kalman Filter to Inter-val Futures

To handle the non-linearities introduced by the interval futures in Section 4.4 we usethe extended Kalman filter technique. In the prediction step of the Kalman filter weuse the actual non-linear relationship (19), while for the covariance structure we use anapproximate

�matrix obtained by linearizing (19) around the current state-estimate,

see (Harvey 1989). To do this, we need the partial derivative ofÙ Ð ��FK) ë FO) ì �

withrespect to � ¸ �� . An easy calculation shows that

�� ¸ �� Ù Ð ��FK) ë FO) ì � � + î+ 5 + ï s Ð ��FO)*�'��+-,/.¸� + î+ 5 +8ï s Ð ��FO)*� �(22)

Note that for)�ì )Vë

this specializes to� +-,/.¸ which is the constant linear slope

of (15). For) ì j ) ë

the partial derivative of the interval futures price is a weightedaverage of the slopes of the individual futures, with the weights given by the point-wise futures prices. We also need the partial derivative of

Ù Ð ��FO) ë FO) ì �with respect

to the Õ and¼

parameters. It turns out that these are also given by weighted averagessimilar to (22). This allows us to implement the extended Kalman filter by a simplemodification of the

�matrix given in the previous section: For each line of the matrix

corresponding to an interval future, we simply average the expressions over all therelevant point-wise futures.

C The Kalman Filter

For easy reference we now state the Kalman filter equations for model (20)-(21). Thebest predictor for ÿ �� given � �O�! �� 2 is the conditional expectation

"ÿ ��w= ��( & � ÿ ��w= � �K�! Ä�� (23)2The notation #��%$�� denote #��H�'&(#��U¨N�'&O�K�O�)&(#��

29

This predictor is optimal under mean squared error. The mean square errors are givenby the conditional variance-covariance matrix:

* ��w= ��( & z�� ÿ ��R� "ÿ ��w= �� ÿ �� "ÿ ��w= ��,+�= � �K�! Ä��~}V var� ÿ ��w= � �O�! ��©� (24)

The recursions in the Kalman filter are:

1. Predict the next state

"ÿ ��w= � �� "ÿ �� = � ��.-×��(25)

2. Compute expected prediction error of the state estimate

* ��w= � ��?��*�� ¥= � ��/��'+��0�»��(26)

3. Predict the next observation

"� ��w= ��?��L�� "ÿ ��w= � ��(27)

4. Compute the expected prediction error of the observation estimate1 ��w= � ��L��* ��w= � ��L�� + �� k��(28)

5. Compute the “Kalman gain”:

2;��?�* ��w= ��L��,+ 1 ��w= � �� ,21(29)

6. Update the state prediction based on the new observation at time�:

"ÿ ��w= ��? "ÿ ��w= ��2;��N� � ��R� "� ��K� (30)

7. Update the prediction error for the state

*��w= ��?�* ��w= � ��R�2;��)�L��)* ��w= � ��(31)

Using the quantities computed in the Kalman filter, the logarithm of the likelihood canbe computed by:�*G prq43 � � 1 F � Q Fw�6�6�wF �65 � 53. 5 1 prq = 1 ��FK� ��6=w�

53. 5 1 Þ � ��R� "� �� ß + 1 ��FO� ��H,21 Þ � ��R� "� �87Ä� ß �const. (32)

30

References

Bessembinder, H., J. F. Coughenour, P.J. Seguin, and M.M. Smoller, 1995, Mean rever-sion in equilibrium asset prices: Evidence from the futures term structure, Journalof Finance 50, 361–375.

Black, F., and M. Scholes, 1973, The Pricing of Options and Corporate Liabilities,Journal of Political Economy 81, 637–654.

Clewlow, L., and C. Strickland, 1999a, A Multi-Factor Model for Energy Derivatives,Working paper, http://www.lacima.co.uk.

Clewlow, L., and C. Strickland, 1999b, Valuing Energy Options in a One Factor ModelFitted to Forward Prices, Working paper, http://www.lacima.co.uk.

Cox, J. C., J. E. Ingersoll, and S. A. Ross, 1981, The Relation between Forward Pricesand Futures Prices, Journal of Financial Economics 9, 321–346.

Cressie, N., 1993, Statistics for Spatial Data. (Wiley, NY).

Filipovic, D., 1998, Energy Risk. (McGraw-Hill).

Gibson, R., and E. S. Schwartz, 1990, Stochastic Convenience Yield and the Pricingof Oil Contingent Claims, Journal of Finance 45, 959–976.

Harvey, A. C., 1989, Forecasting, Structural Time Series Models and the Kalman Fil-ter. (Cambridge University Press).

Hillard, J. E., and J. Reis, 1998, Valuation of Commodity Futures and Options underStochastic Convenience Yields, Interest Rates, and Jump Diffusions in the Spot,Journal of Financial and Quantitative Analysis 33, 61–86.

Merton, R. C., 1973, Theory of Rational Option Pricing, Bell Journal of Economicsand Management Science 4, 141–183.

Miltersen, K. R., and E. S. Schwartz, 1998, Pricing of Options on Commodity Futureswith Stochastic Term Structures of Convenience Yields and Interest Rates, Journalof Financial and Quantitative Analysis 33, 33–59.

Priestley, M. B., 1981, Spectral Analysis and Time Series. (Academic Press).

Schwartz, E. S., 1997, The Stochastic Behavior of Commodity Prices: Implicationsfor Valuation and Hedging, Journal of Finance LII, 923–973.

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a joint state-space model for electricity spot and … · mail: [email protected]. husby is...

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