Weather Derivatives and Electricity Demand Simulation

SCOR PrizeAbstract
The Master’s Thesis underlying this paper was written at the Technical University ofMunich in cooperation with Munich Re. It was sponsored by Munich Re and supervised
by Prof. Dr. Matthias Scherer (TU Munich) and Dipl.‐Math. Ralf Hungerbühler (MunichRe).
The intention of this paper is to build an appropriate model for the electricity demand
and temperature in New South Wales (NSW), Australia. Because a strong correlation
exists between temperature and electricity demand, the proposed model accounts forthe interaction between both of them and uses temperature information to improve themodeling accuracy of the electricity demand.
The following figure shows the dependency structure between temperature andelectricity demand, which is mainly shaped in this way because of the use of air‐condition during hot days and intensive heating during cold days. One can clearly noticethat a temperature around 18 °C leads to the lowest electricity demand for thoseobservations. This is caused by the fact that human people feel comfortable at around 18°C without having a need for cooling or heating. One can observe such a dependency
structure at many places over the world. Therefore, the topics this paper deals with caneasily be transferred to other locations in the world.

First, an overview of weather derivatives is given and the basic underlyings fortemperature derivatives, namely Cooling Degree Days and Heating Degree Days, areintroduced. Then different modeling and valuation approaches are mentioned.
The following section deals with a model for the daily average temperature of NSW. The
model considers seasonality in the temperature during each year as well as variability inthe variation. Finally, the autocorrelation between different days is also captured within
the model.
Afterwards, a structurally related model for the daily average electricity demand isestablished. For this, an autoregressive model including temperature as an exogenous
variable to account for the interaction between temperature and electricity demand isintroduced.

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Subsequently, the model is extended to generate half‐hourly electricity demand values.
Therefore, multiple linear regressions are used and further explanatory variables likethe day of the week are taken into account.
Finally, a stylized financial contract whose pay‐off depends on the temperature as wellas on the half‐hourly electricity demand is evaluated using Monte‐Carlo simulations
generated by the developed model.

1 Weather Derivatives
In the temperature market, especially with focus on the energy industry, traded
temperature derivatives are mostly written on Degree Day Indices. Those are useful
because they are designed to correlate well with the domestic demand for heating andcooling. The Cooling Degree Day Index is defined in the following way:Definition 1: Cooling Degree Days (CDDs) (Brix, 2005)
The value of CDDs on a particular day i is defined as≔ max;0

where is the average temperature on day i and, ,∈ ,over a period fromall days during that period, i.e.

,,
is a baseline temperature. A CDD index
is defined as the sum of the CDDs over.

Analogously, the Heating Degree Day Index is defined the other way around. For thebaseline temperature normally a temperature of 18 °C is used, justified by the reasonsexplained in the abstract. Most of the temperature derivatives, like options or futures,are written based on those underlying indices, and therefore, one can think of different
evaluation methods.
A common method is Burn Analysis, where the fair value of a contract is estimated fromthe distribution of historical payouts. Another method is to use Simulation Technique.
Here, one builds a model for a specified index like CDD (Index Modeling), or one
simulates the underlying time series, for example the daily mean temperature (DailyModeling). In both cases Monte Carlo simulation is used for pricing. In this paper, thedaily modeling method is used, as the stylized financial contract depends on thetemperature as well as on the electricity demand. Therefore, in the following sections, amodel for temperature and a model for electricity demand including temperature
information are presented.

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2 Modeling temperature of New South Wales
The main focus of this paper is to build a model to price a contract contingent on theaverage temperature and the electricity demand. Therefore, the model is kept as general
as possible to stay in the same structural model class for both time series. The intention
is to use a discrete ARMA/GARCH setting with some modifications. In a very generalform, a temperature model has the following structure:′,∈.

Here,captures a trend over a long period, i.e. many decades, and describes theseasonal trend, which exists during every year. deals with the variability in thetemperature variation over the season and ′ describes the remaining temperature
anomalies.
The daily mean temperature data used for the temperature analysis is delivered by theAustralian Bureau of Meteorology for the station located at Bankstown Airport in New
South Wales.
To account for the long‐term trend
in the temperature observations, a linear
trend function is fitted to the yearly mean temperature over 42 available years of data.
This trend can be explained by the global warming phenomena. For the underlying 42years of data, the yearly mean temperature differs by 0.64 °C.
To capture the seasonal trend, following the Ansatz of Benth (F.E. Benth, 2008), atruncated Fourier series is fitted to the trend‐removed 15330 points of data
representing 42 years without leap year days. The function has the following structure
sin

2πt
365

2πt
365

and the parameters are estimated by using the standard least square method.
The following figure shows a sequence of detrended temperature data together with thefitted function
in red.

To account for the variability in the temperature variation over the season, again atruncated
Fourierseries,namely,isused.
Thefunction
sin

2πkt
365

2365

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Weather Derivatives and Electricity Demand Simulation

SCOR Prizeis fitted to the calculated empirical variance values for each day of the year, i.e. to 365
values for the variance generated from 42 observations for each day. Therefore, the data
is filled into a matrix to generate one column for every day of the year. According to theavailable data, every column has 42 values. Then the empirical variance for everycolumn, i.e. for every day of the year, is calculated.
As a result, one observes a clear seasonal volatility effect in the temperatureobservations. During summer months the variation is higher than during winter months.
Oneends

upwiththemodelfortemperature′anomaliestobe.

Then an AR(1) process is used to capture the dependence of the remaining temperature
anomalies ′ to the values of the previous day, i.e.
′′,2, … , 15330.

The histogram and the qq‐plot of the remaining innovations in the following figuresuggest a good adequacy of the proposed model for the New South Wales data.

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3 Modeling Electricity Demand for New South Wales
In this section, a structurally similar model to the temperature model is proposed for thedaily mean electricity demand simulation of New South Wales. The main idea of thismodel is to include the knowledge of the temperature to enhance the electricity demandsimulation. The daily mean electricity demand data comes from the Australian Energy
Market Operator (AEMO) website, where also the half‐hourly electricity demand for
New South Wales, used in the following section, is publishedAt first, similar to the temperature model, a long‐term trend is removed from the data.
As the macroeconomic environment influences the electricity demand in a long‐termview, a polynomial function of grade two is fitted to the data, i.e.
.
The following figure shows that one has to account for a weekly pattern additionally to ayearly cyclic seasonality for modeling electricity demand.

The weekly pattern is removed from the data by using the Moving Average Technique(Weron, 2006) to fit a deterministic weekly function
to the data. As a result,explicit values are estimated for each day of the week. The results are given in thefollowing table.
Sun‐782.70

Mon
111.92

Tue311.62

Wed291.93

Thu313.36

Fri221.25

Sat‐467.38

One can clearly observe that the electricity demand is estimated to be the lowest every
Sunday and lower than during weekdays on Saturday. This is consistent to theobservations from the historical electricity demand data and those observations makesense because on weekends less electricity is needed by the industry.

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The seasonality is again modeled by a truncated Fourier series but with moresummands
thantheone
usedforthetemperaturemodel,i.e.
sin

2πkt
365

2πkt
365.

To capture the variation in the seasonality, the same function foris used as in thetemperature modeling case. The only difference now is that one has only 11 years ofobservations to estimate the daily variance compared to 42 years of temperatureobservations. In the electricity demand case one obtains a seasonal cyclic structure forthe variation in the seasonality which has two local maxima within the year. Thus,
variation in electricity demand is highest during summer months but also high during
winter months. It is low during spring and autumn.
Summarizing the electricity demand model up to this point, the remaining demand
anomalies ′ are given by′.

To include the temperature information in the next step, an extension of an AR‐model,namely an ARX‐model, is used for modeling the remaining dependency between thedemand anomalies.
Definition 2: ARMAX Model (Weron, 2006)
For ∈ Ζ the autoregressive moving average model with exogenous variables , … ,, , ,…,canbecompactlywritten′

where theoperator,with, oras,

are the orders of the exogenous factors. Withisashorthand⋯denoting the backstep
notationfor

being the correspondent coefficients.

For the exogenous variable influencing the ARX(1)‐process, statistically significant
variables are used. The remaining temperature anomalies ′ are taken into the model aswell as their interaction with a summer and winter dummy variable. Therefore, theinteraction terms are highly significant and one accounts for the smile shaped
correlation structure, seen in the first figure of this paper, between temperature andelectricity demand. Including the interaction between each season or the squared

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Taking a look at the modeling results in the following figure, one can clearly see theadditional model accuracy when using the temperature information as exogenous
variable in the electricity demand modeling procedure. The figure shows the out‐ofsample results for 2011 while the model was fitted to data from 1969 until 2011 for thetemperature and from 2000 until 2010 for the electricity demand. The blue series
represents the historically observed electricity demand and the red series shows one
single simulation path with temperature influence (upper graph) and without
temperature influence (lower graph).

4 Modeling Half‐Hourly Electricity Demand
For being able to evaluate different contracts which depend in their pay‐off on the half‐hourly electricity demand, the proposed model is extended to model half‐hourly values
in addition to daily average electricity demand values. The basic idea is to use linear
regression technique to predict 48 model points for each day. Therefore, 48 linear
regressions are calculated simultaneously using the daily average electricity demand
coming from the previous model as explanatory variable. Additionally, dummy variables
for the day of the week as well as for summer and winter are included into theregression. This ensures a very comfortable r‐squared of 0.92 in mean over the 48
regressions.
In summary, the regression model has the following setting: The historically half‐hourly
electricity demand observations are grouped so that one gets 4018 observations foreach half‐hourly point in time to fit the regression. As basic configuration, a Saturday inautumn or spring is used. So all regression betas have to be viewed in comparison to a
Saturday in autumn or spring.

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As a result, the model captures the daily shape of electricity demand, i.e. low during thenight and peaks during the day dependent on the day of the week and the season. The
out‐of‐sample results for a few days in December 2011 are given in the following figure(in the upper graph without temperature influence and in the lower graph withtemperature influence).

5 Monte Carlo Pricing and Further Applications
Finally, one is able to use the model to price various financial contracts written on thedaily mean temperature, the daily mean electricity demand and/or the half‐hourly
electricity demand. Such contracts can help companies to insure against higher or lower
electricity demand caused by abnormal high or low temperature evolutions. Due to thefact that the underlying series are modeled instead of pre‐specified indices like theintroduced Cooling Degree Days for example, the proposed model is very flexible for
pricing all kinds of pay‐off functions.
For further research, one can think of including the electricity price into the model.
There should also exist a high correlation between the electricity price and theelectricity demand as well as the temperature which could be modeled in a moresophisticated setting. Another approach to extend the model could be to incorporate
weather forecasts to increase the accuracy of the forecasting simulations.

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Major Reference List
Brix, S. J. (2005). Weather Derivative Valuation. Cambridge University Press.
Benth, F. E. et all (2008). Stochastic modelling of electricity and related markets. World Scientific.
Weron, R. (2006). Modelling and Forecasting Electricity Loads and Prices. Wiley.

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