Juliusz Pres

Piotr Fiszeder

English

Research Papers in Economics

© Copyright by The Nicolaus Copernicus University Scientific Publishing House
DYNAMIC ECONOMETRIC MODELS 
Vol. 8 – Nicolaus Copernicus University – Toruń – 2008 
Piotr Fiszeder 
Nicolaus Copernicus University in Toruń 
Juliusz Preś 
Szczecin University of Technology 
Pricing of Weather Options for Berlin Quoted  
on the Chicago Mercantile Exchange  
1. Introduction  
  At present the global weather derivatives market is developing very fast. 
Only in the recent period (April 2007 - March 2008) the notional value of all 
weather contracts reached over 32 billion USD. As a result of such a dynamic 
increase on this market the problem of appropriate weather options pricing ap-
pears more often. Usually in these situations, the Black-Scholes formula is used. 
Unfortunately, many observers and weather market participants have noticed 
that this approach cannot be applied because of the different nature of weather 
underlying
1. It must be added, that the unique and complex features of weather 
indices made it impossible until now to create any complete and universal pro-
cedure for pricing this class of instruments. For this reason many different ap-
proaches of pricing weather derivatives have been proposed. The most popular 
are: historical burn analysis, index modelling and daily modelling. Among 
these, the last one seems to have the greatest potential in creating one precise 
approach of pricing all weather contracts
2. Therefore in this paper we concen-
trate on daily modelling as an approach which makes use of models of stochas-
tic processes. Moreover this work we analyse not only daily time series, but also 
monthly values of weather indices.  
                                                 
1 See Dischel (1998).  
2 Brix, Jewson and Ziehmann (2005). © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Piotr Fiszeder, Juliusz Preś  164 
  The main goal of this work is to present the approach of pricing weather op-
tions using the ARFIMA-FIGARCH model
3. It has been shown that this model 
neglects seasonally changing autocorrelation, which leads to significant devia-
tions in option pricing. We propose in the paper two pricing models which is the 
authors' contribution. First the model ARFIMA-FIGARCH used for daily data 
(see Moreno, 2003 for analysis of temperature) has been extended by including 
seasonality in variance what turned out to be statistically significant. The sec-
ond model bases on monthly values of temperature indices, which is a kind of 
compromise between simple pricing approaches like historical burn analysis 
and more advanced methods, like econometric daily modelling.  
  In section 2 we describe all the stochastic models that we used in our re-
search. In this part we propose a small improvement that allows the inclusion of  
seasonality in variance. Section 3 contains the description of all statistical fea-
tures of selected time series and the results of estimation for analysed models. 
In the next section the analysed models are used in an empirical example for 
pricing Chicago Mercantile Exchange’s weather options for Berlin. Section 5 
contains conclusions.  
2. Models of Stochastic Processes  
  In the process of air temperature we can usually set apart such distinctive 
features like seasonality, trend as a result of global warming and urbanization, 
long memory, which is typical for climate processes (see Hurst, 1951, Kwiat-
kowski and Osiewalski, 2002), and random fluctuations. Additionally, we can 
find seasonality in variance. Volatility of air temperature is higher in winter 
months and smaller in summer months. Hence all the above features ought to be 
included in modelling of air temperature.  
  The models proposed in this work are an extended version of the ARFIMA-
FIGARCH model and allow the inclusion of seasonality in mean and variance. 
The ARFIMA model, as a generalization of the ARIMA model, allows the ef-
fective description of short and long memory in mean. The FIGARCH model, 
as a generalization of the GARCH model, allows similar features to be de-
scribed in variance. The extended model ARFIMA ) , , ( 1 Q d P  - FIGARCH
) , , ( 2 q d p  can be presented in the following form:  
t t t
d L y L L ε ϑ μ ϕ ) ( ) ( ) 1 )( (
1 = − − ,      ) , 0 ( ~ 1 t t t h D − ψ ε , (1) 
) sin cos (
0
2 /
1
t t t i i
r
j
i
m
i
i
j
j t ω λ ω θ γ μ + + =∑∑
==
, (2) 
                                                 
3 This model is used by practitioners on the financial market and it exists in popular 
application “SWS 6.0” made by Speedwell Weather Derivatives Ltd. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Pricing of Weather Options for Berlin Quoted on the Chicago Mercantile Exchange  165
t
m
i
i i i i t
d L t t L L ν β ω τ ω κ α ε φ )] ( 1 [ ) sin cos ( ) 1 )( (
2 /
1
0
2 2 − + + + = − ∑
=
, (3) 
where  L  is the backshift operator ( s t t
s L − =ε ε ),  ∑
=
− =
P
j
j
jL L
1
1 ) ( ϕ ϕ , 
∑
=
+ =
Q
j
j
jL L
1
1 ) ( ϑ ϑ ,  ∑
=
− =
q
j
j
jL L
1
1 ) ( φ φ ,  ∑
=
=
p
j
j
jL L
1
) ( β β ,  t t t h − =
2 ε ν , all roots 
of polynomials  0 ) ( = L ϕ  and  0 ) ( = L φ  are beyond unit circle,  5 . 0 1 1 < < − d , 
1 0 2 < < d , 
m
i
i
π
ω
2
= ,  365 = m . 
In order to obtain positive variance  t h  without imposing additional restrictions 
on the parameters in the equation for variance, a logarithmic formula of vari-
ance can be applied ( t t t h ln
2 − =ε ν ). The seasonal component in equation (2) 
can be presented in a different than harmonic way: 
∑ ∑
= =
+ =
12
1 0 k
kt k
r
j
j
j t m c t γ μ , (4) 
where mkt  are dummy variables that mean successive months (eg.  1 1 = t m  for 
January and  0 1 = t m  for other months). In a similar way we can describe a sea-
sonal component for variance: 
t
k
kt k t
d L m L L ν β δ α ε φ )] ( 1 [ ) 1 )( (
12
1
0
2 2 − + + = − ∑
=
. (5) 
3. Modelling Time Series of Air Temperature and Weather Indices  
  Historical data for Berlin were obtained from Deutscher Wetterdienst and 
they were collected over the period from January 1, 1948 to December 31, 2004 
(20808 daily observations). Distributions of daily values of temperatures are 
different in specific months (periods) of a year. Research in this area has shown 
that in winter months there is left asymmetry while in summer months asymme-
try is positive. The hypothesis about normality of distribution, using a variety of 
tests, has been rejected for specific months and also for the entire year
4. The re-
sults of statistical tests indicate the existence of the following properties of daily 
average temperature: increasing linear trend, seasonality both in mean and vari-
ance, autocorrelation (higher in winter months), long memory in mean and the 
ARCH effect.  
                                                 
4 Owing to the limited size of this paper some results of performed tests and estima-
tions had to be omitted. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Piotr Fiszeder, Juliusz Preś  166 
  Apart from the daily average of air temperature, also monthly temperature 
indices (HDD and CAT) were analysed. These indices can be calculated using 
the following formulae: 
∑
=
− ° =
m
t
t y C ,
1
) 18 0 max( HDD , (6) 
∑
=
=
m
t
t y
1
CAT , (7) 
where  t y - average daily temperature, m- number of days in any given period. 
The hypothesis about normality of distribution was rejected only for the HDD 
index for the months of March and December. In the current research on pricing 
weather contracts, only the distribution of the cumulated monthly indices (HDD 
and others) is analysed. In this paper we treat the monthly time series of these 
indices as a realization of stochastic processes and explore their features. The 
performed tests indicate the existence of the following features in monthly indi-
ces HDD and CAT: linear trend, seasonality in mean and in variance, long 
memory in mean (weaker than for daily time series).  
  The model described in equations (1-3) was selected to explain the average 
daily temperature. The long memory has been described by ARFIMA model  
(d1 = 0.1269 (0.0283)). The short-term dependences in the time series of tem-
perature have been explained by autoregressive and moving average parts with 
lags equal to two ( 2 = P  and  2 = Q ). In the equation for conditional variance 
the parameter  2 d  turned out to be insignificant, hence the process of volatility 
has no long memory. The GARCH(1,1) model with annual periodical fluctua-
tions and normal conditional distribution turned out to be sufficient to describe 
variance of temperature. 
  For monthly values of HDD and CAT two models with different parame-
terisation of seasonality were considered. If the model consisted of a harmonic 
seasonal component and linear trend, the error term for the HDD index was best 
described by the AR(1) model. For the seasonal model with dummy variables 
and linear trend (equation 4), the error term was best described by 
ARFIMA(0,d1,0). 
  For the CAT index, for any parameterisation of seasonality, linear trend and 
long memory were observed. In both cases, the ARFIMA model (0,d1,0) turned 
out to be the best. For all indices unconditional variance of random term in all 
models was variable. The ARCH effect was not present. 
  The final parameterisation (values P, Q, p, q, r and the number of parts of 
trigonometric polynomials) were selected by the Schwarz criterion, taking into 
account results of proper diagnostic tests. The analysed models were verified in 
terms of the average mean and variance using Monte Carlo simulations. The re-
sults are presented in Table 1. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Pricing of Weather Options for Berlin Quoted on the Chicago Mercantile Exchange  167
Table 1. Results of models’ verification for mean and variance based on Monte Carlo 
simulations
5  
Parameters 
 
Type of model 
Model 1 (daily)  Model 2 (monthly)  Model 3 (monthly) 
January 
HDD 
June 
CAT 
January 
HDD 
June 
CAT 
January 
HDD 
June  
CAT 
Mean  546.01 515.33 529.87 548.16 546.29  526.73 
Variance 6746.95  4640.91  7243.87 1655.75 7483.06  1688.18 
Δ for mean   2.14  11.00 14.00  21.82* 2.42  0.39 
Δ for variance   818.45 3045.30* 321.53  60.14  82.34  92.57 
Model 2 includes harmonic structure of seasonality, whereas in model 3 this component consists of dummy 
variables. The asterisk indicates that the null hypothesis about the equality of the expected value (variance) of 
the index assuming that the analysed model is true with expected value (variance) for population was rejected 
at the 5% level. Test statistics Δ for mean (variance) were calculated as absolute values of the differences be-
tween means (variances) calculated for generated data and sample.  
Mean and variance values for the January HDD index were 555.39 and 7727.93 
respectively, while for the June CAT index these parameters were 515.29 and 
1536.38. Comparisons of the examined models have also been performed using 
other criteria, like adjusted determination coefficient or root mean squared error 
(Table 2).  
Table 2. Evaluation of quality of models  
Type of model 
January - HDD  June  - CAT 
RMSE  Adj. R²  RMSE  Adj. R² 
Model 1(daily)  11.07  0.997  11.48  0.997 
Model 2 (monthly with harmonic  
seasonality)  55.04 0.922 57.85 0.929 
Model 3 (monthly with dummy variables)  52.46  0.928  63.42  0.914 
  The obtained results show that despite the overestimation of variance during 
summer months, the model constructed for daily observations of temperature 
has the best ability to describe monthly values of HDD and CAT indices. The 
effect of the variance overestimation is caused by omitting seasonal variability 
of autocorrelation of temperature (see Figure 1).  
                                                 
5 For the selected model 100 000 time series of temperature with length 30 or 31 ob-
servations were generated and afterwards indices were calculated. For models of indic-
es, the values of indices were generated directly. The complete procedure of verification 
is given in Caballero, Jewson and Brix (2002). © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Piotr Fiszeder, Juliusz Preś  168 
  
Figure 1. Autocorrelation function for average daily temperature in different periods of 
the year. 
4. Pricing Weather Options  
  The models presented above have been applied in pricing two distinct 
monthly option contracts (call) with cap value
6. Underlying for these options 
were indices: HDD for January 2004 and CAT for June 2004. Specification of 
these contracts is shown in Table 3. 
Table 3. Specification of examined CME weather options for Berlin  
Name  Contract 1  Contract 2 
Type of option  Call  Call 
Index  HDD Berlin  CAT Berlin 
Period of life  January 2004  June 2004 
Strike value   600 HDD  550 HDD 
Tick value  10 000 GBP  10 000 GBP 
Maximum payout (cap)  500 000 GBP  500 000 GBP 
  In order to compare pricing approaches, three methods mentioned above: 
historical burn analysis (HBA), index modelling (IM) and daily modelling 
(DM) were used. The last method includes all analysed models. The results are 
given in Table 4. 
  In the case of January’s contracts a clearly lower evaluation was obtained 
using the model with harmonic seasonality (monthly model 2), for which the 
mean from the simulation turned out to be lower by 14 points than the mean 
from the sample.  
                                                 
6 An option contract with cap value can be obtained by buying a call option and sell-
ing the call option with the same expiration date but with different strike prices.  © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Pricing of Weather Options for Berlin Quoted on the Chicago Mercantile Exchange  169
Table 4. Valuations of weather options using different approaches  
Valuation 
methods 
Price of  
 contract 1  
[GBP] 
Price of  
 contract 2 
[GBP] 
Relative price of 
contract 1  
as a percentage of 
maximum payout
[% limit] 
Relative price of 
contract 2  
as a percentage of 
maximum payout 
[% limit] 
HBA  86 550  45 150  17.31  9.03 
IM  99 550  39 450  19.91  7.89 
DM – model 1  84 900  96 650  16.98  19.33 
MM – model 2  67 250  133 600  13.45  26.72 
MM – model 3  92 900  67 150  18.58  13.43 
  The different results were obtained in pricing June’s contract for the CAT 
index. For the first two methods the obtained valuations were below 10% of the 
maximum payout. The methods based on time series modelling estimated the 
price of this contract between 13.43% and 26.72% of the maximum payout. The 
valuation received on the basis of daily modelling is strongly overestimated be-
cause of the clearly overestimated level of variance for June (Table 1). The sec-
ond model clearly overestimated the mean value and that is why the price of the 
option was overvalued. The most reliable pricing seems to be the one obtained 
from the last model (13.43%). Unfortunately, in this case the estimation’s error 
will be higher, because the analysed model provided an inferior fit to the data. 
Besides, the model for monthly values does not take into account the variable 
number of days in a month and it cannot be used directly to estimate the price of 
weather contracts for all lengths of time. 
  Daily modelling of temperature allows the better usage of historical data. If 
one wants to price a monthly weather contract using historical burn analysis or 
index modelling approaches one can use the data only from the expiration pe-
riod. For example for 10 years of data the estimation will be based on 10 his-
torical values. Application of daily modelling allows the use of 3652 observa-
tions in the pricing process. Therefore this approach is potentially better. Only 
potentially, because it assumes that the model is correct and able to describe all 
properties of weather time series. Hence, the risk of using any model may be an 
important factor in derivative pricing
7. The bayesian model pooling gives possi-
bility to formally include specification uncertainty in statistical inference (see 
for example Osiewalski, 2001). 
5. Conclusions  
  The models constructed for monthly observations of indices do not explain 
volatility of temperature indices in a sufficient way. The model constructed for 
daily observations of temperature is the best in describing temperature indices. 
This model describes most of the important features. Despite the application of 
                                                 
7 See, for example, Jajuga (2007). © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Piotr Fiszeder, Juliusz Preś  170 
such an advanced model, it is not able to describe the real temperature process 
completely, because it neglects seasonal variability of autocorrelation, which 
causes over- or underestimations of variance in some periods.  
  Caballero and Jewson (2003) suggest as an alternative using the SAROMA 
model. This model considers seasonality in the autocorrelation function, but the 
number of parameters that need to be used in this model, makes the estimation 
process difficult. Besides, the SAROMA model requires many observations in 
order to avoid a spurious explanation of data. In addition, this model omits long 
memory. The other solution could be to include in the ARFIMA-FIGARCH 
model a set of time-varying parameters of long memory in mean (and perhaps 
also in variance). Unfortunately, the specification of this model will be more 
complex than the one presented in this paper. 
References  
Brix, A., Jewson, S., Ziehmann, Ch. (2005), Weather Derivative Valuation, University 
Press, Cambridge. 
Caballero, R., Jewson, S. Brix, A. (2002), Long Memory in Surface Air Temperature: 
Detection, Modelling and Application to Weather Derivative Valuation, Climate 
Research, 21, 127–140. 
Caballero, R., Jewson, S. (2003), Seasonality in the Statistics of Surface Air Tempera-
ture and the Pricing of Weather Derivatives, Meteorological Applications, 10, (4), 
367–376. 
Dischel, R. (1998), Black-Scholes Won’t Do, Energy Power and Risk Management, 10, 
27–33. 
Hurst, H. E. (1951), Long Term Storage Capacity of Reservoirs, Transactions of Ameri-
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Jajuga, K. (2007), (red.) Zarządzanie ryzykiem (Risk Management), PWN, Warszawa. 
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Moreno, M. (2003), Weather Derivatives Hedging and Swap Iliquidity, Weather Risk 
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(red.) Zarządzanie ryzykiem (Risk Management),

Ekonometria bayesowska w zastosowaniach (Bayesian Econometrics in Applications),

Long Memory in Surface Air Temperature: Detection, Modelling and Application to Weather Derivative Valuation,

Long Term Storage Capacity of Reservoirs,

Modele ARFIMA: podstawowe własności i analiza bayesowska,

Seasonality in the Statistics of Surface Air Temperature and the Pricing of Weather Derivatives,

Weather Derivatives Hedging and Swap Iliquidity, Weather Risk Management Association.

Pricing of Weather Options for Berlin Quoted on the Chicago Mercantile Exchange

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Pricing of Weather Options for Berlin Quoted on the Chicago Mercantile Exchange

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