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Model Comparison for Temperature-based WeatherDerivatives in Mainland China
Lu Zong, Manuela EnderDepartment of Mathematical SciencesXi’an Jiaotong-Liverpool University
April 14, 2013
Abstract
In this paper, we provide a comprehensive comparison of two models for the simu-lation and pricing of temperature-based weather derivatives. The model of Alaton et al(2002) and the CAR model of Benth et al (2007) are applied to temperature data fromtwelve cities in Mainland China. The objective of this paper is to analyse whether theCAR model, as a more advanced model has a better performance in fitting the dailyaverage temperature (DAT). A higher accuracy of the CAR model can be found indeedin terms of normality of residuals and in terms of smaller relative errors. However, theshortcomings of both models are revealed in this study as well. The Chinese citiesinvolved cover all seven climatic zones in the standard of climatic regionalization thatis used as a partition of China to get representative clusters.
Keywords: weather derivatives, temperature modelling, China, pricing, simulation
JEL classification: C51, G13, Q54
1 Introduction
Worldwide, the economy is influenced by the weather. The industries with the highestweather sensitivity are in the sector of energy, agriculture, retail, construction and trans-portation (Weather Risk Management Association, 2011). Although the great influence ofweather fluctuations has been known for a long time, financial instruments to hedge weatherrisk, namely weather derivatives were firstly launched by the Chicago Mercantile Exchangein the year of 1999. The main advantages of weather derivatives are:
1. Compared to weather insurance, there is no industrial limitation on weather derivativecontracts. These contracts are designed based on the region with different strike pricesand cold/warm seasons, but are valid for all industries within the same region.
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2. Weather derivatives compensate the buyers for all abnormal weather events and notonly for weather catastrophes with high severity and low possibility like insurancecontracts.
3. As the underlying is a weather index, it cannot be controlled by the buyers. Henceproblems like moral hazard or adverse selection do not exist.
4. Transaction costs related to weather derivatives are much lower than for other hedginginstruments like insurance. So, more investors of a certain industry which is exposedto weather risk are encouraged to hedge their investments.
As in China, agriculture is one of the most important parts in the GDP and 70% ofthe population live on farms, managing weather risk is extremely important, especially forsmall scale farm households. Weather derivatives can provide an alternative to subsidizations(Heimfarth and Musshof, 2011). Considering the great potential demand of China, more re-search is needed for this market. Currently, very few studies exist on Chinese weather data.Goncu (2012) applied a seasonal volatility model to capture the fluctuations of DATs of Bei-jing, Shanghai and Shenzhen and to price temperature-based weather derivatives for thesethree cities. This paper is the first that apply two models to twelve cities in China from allseven climatic zones in order to find the most suitable model for pricing temperature-basedcontracts nationally wide.
As a non-tradable asset, temperature is priced with indexed underlyings. The cooling-degree day (CDD) and heating-degree day (HDD) are the most common underlyings whilepricing temperature-based weather derivatives. They are respectively given by:
HDD(t1, t2) =
∫ t2
t1
max(18− Tt, 0) dt, (1)
CDD(t1, t2) =
∫ t2
t1
min(Tt − 18, 0) dt, (2)
where Tt denotes the daily observed temperature.
Since the underlyings are non-tradable, the Black-Scholes framework cannot longer beapplied to price weather derivatives. Different methods for an incomplete market, approxi-mation formulas and assumption on the market price of risk are used instead.
In the remaining part of the paper, we introduce two temperature and pricing models,namely the Alaton et al’s model (2002) and the continuous-time autoregressive CAR modelof Benth et al (2007). In the next section, both models are applied to thirty years of dailyaverage temperature (DAT) data of twelve cities in Mainland China. Finally, we comparethe fitting performance of both models in terms of simulation errors and pricing results.
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2 Modelling temperature
There are different ways to price temperature-based weather derivatives. Figure 1 gives anoverview of the most important models for the modelling of temperature dynamics.
Figure 1: An overview of models for temperature modelling
The simplest method is a so called Burn Analysis. The basic idea is to use the observa-tions of the index from the past to model the future movements. Empirical studies revealedthat the Burn Analysis has comparatively large errors as events that were not observed inthe past cannot be modelled as possible future events (e.g. Schiller et al (2012), Cao andWei (2004)).
Dornier and Queruel (2000) proposed to use a continuous-time Ornstein-Uhlenbeck (OU)process to model the temperature evolution. The volatility is assumed to be constant in thisfirst model of this type. As temperature data shows heteroskedasticity of the volatility, animprovement of the model was made by Alaton et al (2002) who introduced a monthly con-stant volatility.
The OU process itself cannot model auto-correlation. To capture the correlation Brodyet al (2002) introduced a fractional Brownian motion. Benth and Saltyte-Benth (2005) useda hyperbolic Levy-process to model the residuals instead of a Brownian motion.
Beside advanced stochastic models, time-series models used in econometrics were applied
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to temperature data as well. Caballero et al (2002) suggested to use an ARMA and ARFIMAmodel. Jewson and Caballero (2003) proposed a model called AROMA to deal with the slowdecay of the autocorrelation function. An ARCH model was suggested by Campbell andDiebold (2005). In Benth et al (2007), the OU process was combined with an econometricsapproach to get a higher-order continuous-time autoregressive process, the CAR model.
To sum up, an advanced stochastic model usually measures the average daily temper-ature with a seasonal function. Different stochastic methods are applied to the residualsobtained by subtracting the seasonal function from the daily temperature. Usually, a Brow-nian motion, a mean reverting process, an Ornstein-Uhlenbeck process, and optionally anautoregressive process are included in the temperature modelling.
For this model comparison we choose the Alaton et al (2002) model, because as the mostclassic model for the evaluation of weather derivatives it can be used as a benchmark modelsimilar to the Black-Scholes model for options on shares’ prices. However, the most restrictivepart of the Alaton et al’ model (2002) is the monthly constant volatility. As the Benth etal’s model (2007) overcomes this restriction and allows the modelling of daily volatility, it isstraight forward to compare these two models to analyse how much the modelling could beimproved in terms of lower errors. This is the first model comparison based on data fromChina. The few existing studies comparing different temperature models like Schiller et al(2012) and Goncu (2013) focus on data from the more mature market in the USA. Schilleret al (2012) included the Burn Analysis, Alaton et al’s model (2002), Benth et al’s model(2007) and a new proposed spline model. Goncu (2013) compared the models of Alaton et al(2002), Benth and Benth (2005), Brody et al (2002), and Campbell and Diepold (2005). Moreadvanced models have usually more parameter which makes the estimation more complexand less stable. In order to find the best model for temperature modelling in China, thetrade-off between a better fit and a more complex modelling needs to be elaborated whichis done in the following sections.
2.1 Data of the empirical study
In this paper, thirty years of DATs of twelve cities in Mainland China are used. The citiesare selected according to the standard of climatic regionalization used in architecture whichdivides Mainland China into seven climatic zones (Ender and Zong, 2012). Figure 2 gives adetailed map of partition. Table 1 gives an overview of the temperature data of the twelvecities.
2.2 Alaton et al’s model (2002)
Alaton et al’s model (2002) fits the DATs with two parts, i.e. the seasonality part and therandom process. The first part is measured with a sine function showing both, the seasonalityand the global warming trend of temperature. It is expressed as:
Tms = A+Bt+ C sin(ωt+ ϕ), (3)
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Figure 2: Map of climatic regionalization in China
where t denotes the time measured daily and ω = 2π/365.The random process is modelled with a Brownian motion with a mean reverting process
which could be solved by the Ornstein-Uhlenbeck process. Finally, the model of DATs isgiven by:
Tt = (Ts − Tms )e−α(t−s) + Tms +
∫ t
s
e−α(t−τ)στ dWτ , (4)
where Tms is given by (3) and Wt refers to a Brownian motion.Table 2 gives the results of the four parameters A, B, C and ϕ of equation (3) and of
parameter α in equation (4) of twelve different Chinese cities. The estimation is based on aregression method.
According to Alaton et al (2002), the volatility parameter σ is measured on the assump-tion to be monthly constant. The estimated values are obtained by using both, the regressionmethod and the discretizing method. Table 13, shown in the Appendix gives the estimatorsof the volatility using sequentially the quadratic variation, regression method and the meanvalue of these two results.
2.3 CAR model (Benth et al, 2007)
A model for the underlying without the restriction of a piecewise constant volatility is thecontinuous-time autoregressive (CAR) model (Benth et al, 2007). Similar with Alaton et
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Table 1: Overview of DAT samples (January 1981 - December 2010)Climatic Mean Standard Max Min
zone deviationHaerbing I 4.95 14.91 30.9 -30.9Changchun I 6.20 14.07 30.4 -30.1Beijing II 12.96 11.03 34.5 -12.5Tianjin II 12.96 11.15 32.9 -14.1Shanghai III 16.38 8.75 34.2 -4.8Hangzhou III 17.01 8.93 35.0 -4.7Nanjing III 15.95 9.38 34.5 -7.8Guangzhou IV 22.43 6.17 34.2 3.3Hainan IV 25.28 4.35 32.6 9.8Kunming V 15.53 4.85 24.6 -3.0Lhasa VI 8.54 6.70 22.6 -10.5Urumchi VII 7.41 13.71 33.1 -27.2
al’s model (2002), the CAR model keeps the deterministic seasonal function:
Λ(t) = a0 + a1t+ a2 cos(2π(t− a3)/365). (5)
Finally, the temperature under a CAR(p) model is expressed as
T (t) = Λ(t) +X1(t), (6)
where Xq is the qth coordinate of the vector X, q = 1, . . . , p.The explicit form of the stochastic process X(t) in Rp for p ≥ 1 is given by
X(s) = exp(A(s− t))x+
∫ s
t
exp(A(s− u))epσ(u) dWu, (7)
for s ≥ t ≥ 0 and X(t) = x ∈ Rp, where eq is the qth unit vector in Rp, q = 1, . . . , p andWt denotes the Brownian motion. The volatility function σ(t) > 0 is a square integrable andreal-valued function.
The parameter A represents the mean-reverting p× p matrix given by:
A =
0 1 0 . . . 00 0 1 . . . 0...
.... . .
...−αp −αp−1 −αp−2 . . . −α1
, (8)
where αq, q = 1, . . . p, are assumed to be constants.To estimate the parameters involved, we start the estimation with an AR(p) process to
use the link between the CAR(p) and the AR(p) model (Benth et al, 2007). The temperatureTi on day i = 0, 1, 2, . . . is expressed as:
Ti = Λi + yi, (9)
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Table 2: Estimated values of A, B, C, ϕ and α of twelve cities in Mainland China
A B C ϕ αHaerbing 3.986 1.776 ∗ 10−4 20.165 -1.851 0.2821Changchun 5.527 1.25 ∗ 10−4 18.876 -1.863 0.3008Beijing 12.406 1.022 ∗ 10−4 15.017 -1.852 0.3121Tianjin 12.814 0.278 ∗ 10−4 15.205 -1.874 0.2800Shanghai 16.29 2.212 ∗ 10−4 11.703 -2.071 0.3008Hangzhou 15.951 1.937 ∗ 10−4 11.82 -2.015 0.2636Nanjing 14.999 1.75 ∗ 10−4 12.561 -1.974 0.2515Guangzhou 21.819 1.123 ∗ 10−4 7.649 -2.021 0.2279Hainan 24.916 0.673 ∗ 10−4 5.237 -1.86 0.1954Kunming 14.505 1.872 ∗ 10−4 5.898 -1.739 0.2911Lahsa 8.060 1.441 ∗ 10−4 8.496 -1.84 0.2616Urumchi 5.942 3.241 ∗ 10−4 17.81 -1.839 0.1977
where Λi = Λ(i) as defined in equation (5). The process yi is an AR(p) process thatfollows:
yi+p =
p∑j=i
bjyi+p−j + σiεi, (10)
where εi are independent, standard normally distributed random variables.As it was shown in Benth at al (2007) that the optimal choice is p = 3, we follow this
approach. Hence, we have:
yi+3 = b1yi+2 + b2yi+1 + b3yi + σiεi. (11)
Finally, we transfer the AR(3) process into a CAR(3) process by using Benth et al’ssolution (2007):
3− a1 = b1,
2a1 − a2 − 3 = b2,
a2 + 1− (a1 + a3) = b3.
Table 3 lists the estimated values of b1, b2 and b3, α1, α2 and α3.Different from Alaton et al’s model (2002), Benth et al (2007) proposed a truncated
Fourier series as a functional volatility σi = σ(i) to model the observed seasonal heteroskedas-ticity of residuals after removing the linear trend, seasonal component and AR(3) process:
σ2(t) = c1 +4∑
k=1
(c2kcos(2kπt/365) + c2k+1sin(2kπt/365)). (12)
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Table 3: Estimated parameters of AR(3) and CAR(3) process
b1 b2 b3 α1 α2 α3
Haerbing 0.8444 -0.2094 0.1204 2.1556 1.5206 0.4854Changchun 0.8645 -0.2714 0.1322 2.1355 1.5424 0.5391Beijing 0.7726 -0.1165 0.0900 2.2274 1.5713 0.4339Tianjin 0.8788 -0.2568 0.1337 2.1212 1.4992 0.5117Hangzhou 0.9492 -0.3187 0.1003 2.0508 1.4203 0.4698Nanjing 0.9488 -0.3174 0.1025 2.0512 1.4198 0.4711Shanghai 0.8781 -0.28 0.1065 2.1219 1.5328 0.2954Guangzhou 1.0782 -0.3993 0.0956 1.9218 1.2429 0.4167Hainan 1.0393 -0.3105 0.0780 1.9607 1.2319 0.3492Kunming 0.8117 -0.1538 0.0901 2.1883 1.5304 0.4322Lahsa 0.7896 -0.0926 0.0811 2.2104 1.5134 0.3841Urumchi 0.9953 -0.2622 0.0627 2.0047 1.2716 0.3296
Table 4 gives the result of the parameters c1 to c9 estimated using the least squaresmethod.
Table 4: Estimated parameters of seasonal volatility function of the CAR model
c1 c2 c3 c4 c5 c6 c7 c8 c9Haerbing 7.8542 4.0041 1.2152 0.5701 -1.9496 0.4113 0.1414 0.0758 0.3161Changchun 8.7863 4.9923 1.2327 0.0051 -2.1259 0.1087 -0.2900 0.0609 0.2408Beijing 3.9743 0.3922 0.9666 0.0708 -0.7373 -0.0986 -0.3272 0.0492 -0.0163Tianjin 3.5554 -0.1899 0.9953 -0.3571 -0.8115 -0.0290 -0.1305 0.0380 0.0365Hangzhou 3.7754 0.6239 1.7720 -0.4665 -0.2793 -0.0153 -0.2585 0.0455 -0.2055Nanjing 3.4677 0.6274 1.1431 -0.1427 -0.5266 -0.0220 -0.2949 0.0069 -0.0939Shanghai 3.6615 1.2009 1.5928 -0.0453 0.2622 -0.2355 -0.5166 0.0348 -0.2897Guangzhou 2.7028 1.1581 1.1987 -0.1467 0.2805 -0.1051 -0.2850 0.0896 -0.1150Hainan 1.5156 0.7415 0.7582 -0.0767 0.1460 0.0474 -0.1332 0.0339 -0.0839Kunming 2.5353 0.7192 1.0103 -0.2222 -0.2884 -0.0394 0.2453 -0.0713 -0.0399Lahsa 2.7588 0.4149 1.0084 0.6218 0.2384 0.1368 0.2000 0.0408 -0.1436Urumchi 6.6512 0.1391 0.8918 0.1284 -0.9269 1.6059 0.5340 -0.1125 0.2341
Figure 3 shows an example of the fitted Fourier process after removing the linear trend,the seasonal effect and the AR(3) process.
2.4 Temperature simulation
Figure 4 and Figure 5 give an example for a simulated path of DAT in 2010 applying Alatonet al’s model (2002) and the CAR model (Benth et al, 2007). Instead of a smooth curveof DATs as in Figure 4, the CAR model (Benth et al, 2007) is able to produce fluctuationsfrom day to day which stresses the more realistic assumptions of this model.
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Figure 3: Fitted Fourier volatility process together with empirical daily squared volatility ofHaerbing, Beijing, Hangzhou and Hainan
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2.5 Normality of residuals
As the stochastic process used for example for the simulation above is modelled by a Brownianmotion, the residuals after removing all other components of the model should follow a normaldistribution hypothetically. However, when dealing with DATs over many years, there is alarge amount of data available. When testing many data points, normality tests like theKolmogorov-Smirnov test reject the hypothesis even when the departure from normalityis very small. So it is important to check histograms and Q-Q plots as well to judge thedistribution of residuals eventually. Benth et al (2008) pointed out that errors of using thenormal distribution only have little effects on modelling and pricing. However, accordingto the assumptions of regression, the non-normality of residuals could be interpreted as theresult of non-constant variance and interdependence, this issue will be carefully checked inthe next chapter of model comparison and model performance.
3 Model comparison
Compared with Alaton et al’s model (2002), the CAR model (Benth et al, 2007) is appar-ently a more advanced model. In the first instance, an autoregressive process was taken intoaccount. Secondly, the functional volatility of temperature is added instead of the assump-tion of a monthly constant volatility. Therefore, we test whether the CAR model can reducethe estimation errors and increase the normality of the residual distribution in comparison to
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Figure 4: DAT simulation using Alaton et al’s model (2002) (Year: 2010)
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the performance of Alaton et al’s model (2002). As there are no historical prices for weatherderivatives in China available yet, the comparison of accuracy is mainly based on tempera-ture modelling. As the seasonal functions of the two models are essentially the same, it issufficient to compare the two models in terms of fitting the deseasonalized temperature data.
In details, the model comparison covers three main steps. Firstly, in terms of the nor-mality of residuals, three tests, i.e. the Kolmogorov-Smirnov test, the Jarque-Bera test andthe Lilliefors test are applied to the residuals. This analysis is supported by histograms andQ-Q plots of the residuals. In the second step, the errors of temperature simulation of thetwo models are compared to find out which model can reduce simulation errors. Finally,weather derivative contracts are priced to get an impression of the impact on prices fromthe model choice and an indicator of the model risk involved.
3.1 Test of residuals
In this section, the residuals that are left after calibrating the model of Alaton et al (2002)and the CAR model (Benth et al, 2007) to the temperature data are compared. The resid-uals are shown in Figure 6 and Figure 7, the squared residuals are plotted in Figure 8 and9. It cannot be neglected that a seasonal pattern exists for both models. Even though theCAR model (Benth et al, 2007) includes a CAR(3) process and measures the daily volatil-ity with a seasonal function, the seasonality in the residuals could not be entirely eliminated.
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Figure 5: DAT simulation using CAR model (Benth et al, 2007) (Year: 2010)
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In Figure 10 and in Figure 11, the histograms of the residuals’ distributions and the cor-responding normal distributions are shown. Although a bias between the distributions of theresiduals and the normal distribution is visible, the similarity between the two distributionsis close.
As histograms can barely show the tail behaviour and correlation, additionally the Q-Qplots and the autocorrelation function ACF of the residuals in plotted in Figure 12 to Figure15. The residuals’ distributions from both models tend to show a bias at the end of thequantile line against the standard normal quantiles. The bias is caused by a steeper trend inthe Q-Q plot which indicates that the residuals’ distributions are more dispersed comparedto the normal distribution.
The ACFs show a difference between the Alaton et al’s model (2002) and the CAR model(Benth et al, 2007). As the ACFs of the CAR model are flatter than the ACFs of Alatonet al’s model (2002), it can be concluded that the CAR model manage to eliminate more ofthe existing trends than the Alaton et al’s model (2002).
Finally, three different tests for normality are applied to the residuals of both models,namely the Kolmogorov-Smirnov test, the Jarque-Bera test and the Lilliefors test. As longas the null hypothesis is accepted at least in one of the tests, we accept it for this city under
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Figure 6: Residuals of DATs after removing all components of Alaton et al’s model (2002)
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the given model. Table 5 and Table 6 give the results for all cities in both models. Theproportion of acceptance for each city is presented as well.
From Table 5 and Table 6, it is plain to see that there are more cites with normaldistributed residuals from the calibration of the CAR model (Benth et al, 2007) (five outof twelve cities) than from the calibration of the Alaton et al’s model (2002) (three out oftwelve cities). Further, the proportions of acceptance among the cities that are ultimatelyaccepted are slightly higher for the CAR model (Benth et al, 2007) than those for Alaton etal’s model (2002). We can conclude that the CAR model (Benth et al, 2007) when appliedto Data from China is the better model in terms of normality of residuals. However, as thenormal distribution is still rejected for roughly half of the cities, the assumptions of the CARmodel (Benth et al, 2007) seem to fit Chinese Data not completely perfectly which impliesthat more (advanced) models should be tested in future research.
3.2 Monthly relative errors
Beside the normality of residuals, error measures should be studied in order to compare theperformance of the model of Alaton et al (2002) and of the CAR model (Benth et al, 2007).We compute the monthly relative error measure that is defined as (Mraoua and Bari, 2005):
ERrelative =Testimated − Tobserved
Tobserved. (13)
Monthly means that all errors of one month are added up to form one aggregated per-
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Figure 7: Residuals of DATs after removing all components of CAR model (Benth et al,2007)
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formance measure. For the model comparison, the difference of the errors of the two modelsis of particular importance. In Table 7 and Table 8, the differences of errors are respectivelyobtained from the monthly errors based on the fitted DATs and on simulated DATs. Thereported values are calculated by subtracting the error of the CAR model (Benth et al, 2007)from the corresponding error of Alaton et al’s model (2002). This method is chosen to easilyidentify the months in which the relative errors of the CAR model (Benth et al, 2007) aresmaller, or in other words, superior to Alaton et al’s model (2002). The last columns ofTable 7 and Table 8 summarizes the total number of months in which the performance ofthe CAR model (Benth et al, 2007) is better.
In the case of fitted temperature data when there is no random process involved, the dif-ferences of errors between the two models tend to be very close to zero. However, the numberof positive differences that indicates that the CAR model (Benth et al, 2007) is a better fitin this specific month is larger than 6 in Table 7 and in Table 8 where the only exception isHainan. Especially for more northern cities like Haerbing, Changchun and Urumchi wherethe mean temperature is comparatively lower, but where standard deviation is higher (seeTable 1). This tells us that the CAR model (Benth et al, 2007) can indeed capture fluctu-ation in temperatures better than the model of Alaton et al (2002). From these result, weconclude that the CAR model (Benth et al, 2007) has a better performance.
In Figure 16 and 17, the bar plots of the monthly relative errors of the twelve cities based
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Figure 8: Squared residuals of DATs after removing all components of Alaton et al’s model(2002)
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on simulated temperatures are shown. As we have already noticed, the monthly relativeerrors are in general smaller of the CAR model (Benth et al, 2007). The bar plots help tounderstand how the errors are distributed over the year. Here, the result is very similar forall cities and all models. There are large differences between the relative errors in cold seasonand those in warm season. During the summer where the volatility is typically lower, theerror measure is very close to zero. However, for winter months with higher volatility theerrors tend to be much higher. The more advanced CAR model (Benth et al, 2007) cannotcapture the seasonality of DATs’ fluctuations. Even though the errors are on average smallerof the CAR model (Benth et al, 2007), this analysis reveals shortcomings of both models.
3.3 Temperature-based weather derivatives pricing
In the third section of the model comparison, we look at prices for futures and options de-termined by the Alaton et al’s model (2002) and the CAR model (Benth et al, 2007).
A weather future is a compulsory contract between the buyer and the seller to tradean asset at a negotiated price on a fixed date afterwards. In this case, the asset is thecurrency value of a specified weather index. The payment is done by cash settlement. Tohedge weather risk, the traders should buy or sell future contracts that are in contrary tothe weather condition that is positive for them. For example, a farmer who wants to reducethe loss due to lower temperature than usual should buy a CDD contract (see equation (2)for definition) before the cold season starts. Similar to common options, weather options
14
Figure 9: Residuals of DATs after removing all components of CAR model (Benth et al,2007)
0 2000 4000 6000 8000 100000
200
400
600
800Haerbing
0 2000 4000 6000 8000 100000
100
200
300
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500Changchun
0 2000 4000 6000 8000 100000
100
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500Beijing
0 2000 4000 6000 8000 100000
100
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500Tianjin
0 2000 4000 6000 8000 100000
100
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500Nanjing
0 2000 4000 6000 8000 100000
100
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500Hangzhou
0 2000 4000 6000 8000 100000
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500Shanghai
0 2000 4000 6000 8000 100000
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500Hainan
0 2000 4000 6000 8000 100000
100
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500Guangzhou
0 2000 4000 6000 8000 100000
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500kunming
0 2000 4000 6000 8000 100000
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500Lahsa
0 2000 4000 6000 8000 100000
500
1000Urumchi
as calls and puts give the right to buy or sell respectively the weather future at a specifiedstrike price on (European options) or before (American options) the exercise day.
Since there are no traded contracts in China that could be used as benchmarks or tocalibrate the market price of risk, we compare the price differences in order to find somepatterns that may exist. In the absence of market prices and further knowledge of the riskaversion of potential buyers, we let the market price of risk (MPR) be equal to a constant λthat is set to be 0 in this study. The converting ratio (a.k.a. principal nominal) is one unitof currency paid for one degree Celsius.
According to Alaton et al (2002), the pricing method is based on the assumption that theprobability of the daily temperature in the HDD contract (see equation (1) for definition)period being larger than 18 ◦C is zero. Hence, the future price of such a degree day basedcontract is normally distributed under measure Q with the mean µn and standard deviationσn. The approximate formula of an CDD future contract for c = 18 at time t ≤ t0 follows:
FAlatonCDD (t, t0, tn) = EQ
[∫ tn
t0
max(T (s)− c, 0)ds|Ft], (14)
where t0 stands for the first day and tn for the last day of the contract.Practically, the payoff of a CDD contract is the accumulation of daily CDDs during the
period of interest:
15
Figure 10: Histograms of residuals from Alaton et al’s model (2002)
−30 −20 −10 0 10 20 300
100
200
300
400Haerbing
−30 −20 −10 0 10 200
200
400
600Changchun
−20 −10 0 10 200
100
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400Beijing
−30 −20 −10 0 10 20 300
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600Tianjin
−20 −10 0 10 20 300
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600Nanjing
−20 −10 0 10 200
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400Hangzhou
−20 −10 0 10 200
100
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400Shanghai
−20 −10 0 10 200
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600Hainan
−20 −10 0 10 200
100
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400Guangzhou
−30 −20 −10 0 10 20 300
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600kunming
−20 −10 0 10 200
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600Lahsa
−30 −20 −10 0 10 20 300
200
400
600Urumchi
Xn =n∑t=1
max (Tt − 18, 0) =n∑t=1
Tt − 18n. (15)
Hence, the CDD call option price is given by
CAlatonCDD (t0, tn) = e−r(tn−t0)EQ[max(Xn −K, 0)|Ft0 ], (16)
where K stands for the strike price and r for the risk free rate. From the assumptionthat Xn ∼ N(µn, σ
2n) and Φ representing the cdf of the standard normal distribution, we get
CAlatonCDD (t0, tn) = e−r(tn−t0)
[(µn −K)Φ
(K − µnσn
)+
σn√2πe
−(K−µn)2
2σ2n
]. (17)
Similar to the model of Alaton et al (2002), the pricing for the CAR model (Benth etal, 2007) is also based on a martingale approach. Benth et al (2007) give the approximationformula for a CDD future with c = 18 at time t ≤ t0 as follows:
FCARCDD(t, t0, tn) = EQ
[∫ tn
t0
max(T (s)− c, 0)ds|Ft]
=
∫ tn
t0
v(t, s)Ψ
(m(t, s, e′1exp(A(s− t)X(t))
v(t, s)
)ds,
(18)where
m(t, s, x) = Λ(s)− c+
∫ s
t
σ(u)θ(u)e′1exp(A(s− u))ep du+ x, (19)
16
Figure 11: Histograms of residuals from CAR model (Benth et al, 2007)
−60 −40 −20 0 20 400
200
400
600Haerbing
−30 −20 −10 0 10 20 300
200
400
600Changchun
−20 −10 0 10 20 300
100
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400Beijing
−40 −20 0 20 400
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600Tianjin
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600Guangzhou
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600kunming
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600Lahsa
−40 −20 0 20 400
100
200
300
400Urumchi
x = e′1exp(A(s− t))X(t), (20)
v2(t, s) =
∫ s
t
σ2(u)(e′1exp(A(s− u))ep)2 du, (21)
Ψ(x) = xΦ(x) + Φ′(x). (22)
The definitions of Λ(t), T (t), X(t) and the matrix A can be found in section 2.3. Analogueto the derivation for Alaton et al’s model (2002), the option price for a CDD contract forthe CAR model (Benth et al, 2007) is defined by
CCARCDD(t, τ, t0, tn) = e−r(τ−t)EQ
[max(FCAR
CDD (τ, t0, tn)−K, 0) |Ft]. (23)
Besides using the approximation formulas for future and option prices, a Monte Carlosimulation is possible in each case. Usually a Monte Carlo simulation needs less assumptionsas closed form solutions are not necessary in this case. However, simulation is time consumingas a large number of paths are required to ensure a certain stability of the solution. Further,discretisation errors are made. As market prices are not available to compare with, we use atleast both pricing methods, the approximation formulas and Monte Carlo simulation. Theresults for HDD and CDD futures are shown in Table 9 and Table 10. Call option prices aregiven in Table 11 and 12. The contract period is for HDD contracts January 2010 and forCDD contracts July 2010. These kind of contracts could be used to hedge against lower tem-peratures than usual in January and against higher temperatures than usual in July. Other
17
Figure 12: Q-Q plot of residuals from Alaton et al’s model (2002)
−4 −2 0 2 4−40
−20
0
20
40
Standard Normal Quantiles
Qua
ntile
s of
Inpu
t Sam
ple Haerbing
−4 −2 0 2 4−40
−20
0
20
Standard Normal Quantiles
Qua
ntile
s of
Inpu
t Sam
ple Changchun
−4 −2 0 2 4−20
−10
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s of
Inpu
t Sam
ple Beijing
−4 −2 0 2 4−40
−20
0
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Standard Normal Quantiles
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ntile
s of
Inpu
t Sam
ple Tianjin
−4 −2 0 2 4−20
0
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Standard Normal Quantiles
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ntile
s of
Inpu
t Sam
ple Nanjing
−4 −2 0 2 4−20
−10
0
10
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Standard Normal Quantiles
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ntile
s of
Inpu
t Sam
ple Hangzhou
−4 −2 0 2 4−20
−10
0
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Standard Normal Quantiles
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ntile
s of
Inpu
t Sam
ple Shanghai
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−10
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Standard Normal Quantiles
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ntile
s of
Inpu
t Sam
ple Hainan
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−10
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ntile
s of
Inpu
t Sam
ple Guangzhou
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−20
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40
Standard Normal Quantiles
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ntile
s of
Inpu
t Sam
ple Kunming
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−10
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Standard Normal Quantiles
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ntile
s of
Inpu
t Sam
ple Lahsa
−4 −2 0 2 4−40
−20
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Standard Normal Quantiles
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Inpu
t Sam
ple Urumchi
contracts that might fit better to a given risk management situation could be priced similarly.
The differences between the prices based on the approximation formulas and on MonteCarlo simulation are for all contracts and for all cities very small. In most cases the differenceis less than 0.5 per cent. We can conclude that the assumptions for the approximation for-mulas for the models are acceptable and that the simulation is sufficiently stable. Comparingthe HDD contracts, the model of Alaton et al (2002) has always higher prices. Respectivelyfor CDD contracts, Alaton et al’s (2002) prices are always lower. The differences can be justa few percentage points, but the prices can vary up to 20 per cent. This shows that modelrisk exist. Given the analysis before, that the CAR model (Benth et al, 2007) has in morecases normal distributed residuals and on average smaller relative errors, the prices of thismodel should be closer to real prices and therefore preferable.
Finally, that the prices for futures and options within one climatic zone are very sim-ilar, but different between climatic zones supports the use of the standard of climatic re-gionalization as a partition of Mainland China in order to reduce dimensions when pricingtemperature-based derivatives (Ender and Zong, 2012).
4 Conclusion
With a non-tradable underlying, the modelling and pricing of assets based on temperatureface many challenges. The publication of some temperature models in the last decade in-
18
Figure 13: Q-Q plot of residuals from CAR model (Benth et al, 2007)
−4 −2 0 2 4−50
0
50
Standard Normal Quantiles
Qua
ntile
s of
Inpu
t Sam
ple
Haerbing
−4 −2 0 2 4−40
−20
0
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40
Standard Normal Quantiles
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ntile
s of
Inpu
t Sam
ple
Changchun
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−20
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Inpu
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Beijing
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Inpu
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Tianjin
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−20
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Inpu
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Nanjing
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−20
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Standard Normal Quantiles
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Inpu
t Sam
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Hangzhou
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Shanghai
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Hainan
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Guangzhou
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kunming
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Lahsa
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Urumchi
creases the possibilities for overcoming the obstacles, but increases also the uncertainty whichmodel to choose especially in non-mature markets like in China. The purpose of the pre-sented model comparison was to find a more suitable model for the Chinese market amongtwo promising candidates, the Alaton et al’s model (2002) and the CAR model (Benth et al,2007).
The results of the study have shown that the calibration to Chinese data of twelve main-land cities from all climatic zones of China gives a feasible modelling of temperature withacceptable error terms. The CAR model (Benth et al, 2007) as a more advanced model hasa little bit higher accuracy in terms of normality of residuals and in terms of relative errors.However, both models failed to capture the temperature fluctuations perfectly. For a largeproportion of cities the residuals do not follow a normal distribution. Especially to deal withthe seasonality of the volatility is problematic for both models. Consequently, the prices forfutures and options still lack reliability.
For future research in order to find the most suitable model for Chinese data, stochasticvolatility models (like Benth and Benth, 2011) need to be included in the empirical study.This means that the volatility is modelled by a stochastic process itself and is no longerdeterministic. Then, capturing the seasonal behaviour of the volatility should be possibleand the error terms for the cold season should be reduced. Further, a spline model approachlike proposed by Schiller et al (2012) should be applied to Chinese data to check whetherthe normality of residuals can be assured by this approach.
19
Figure 14: ACF of residuals from Alaton et al’s model (2002)
0 200 400 600 800−0.5
0
0.5
1Haerbing
0 200 400 600 800−0.5
0
0.5
1Changchun
0 200 400 600 800−0.5
0
0.5
1Beijing
0 200 400 600 800−0.5
0
0.5
1Tianjin
0 200 400 600 800−0.5
0
0.5
1Nanjing
0 200 400 600 800−0.5
0
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1Hangzhou
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0.5
1Shanghai
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1Hainan
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0
0.5
1Guangzhou
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1Kunming
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0.5
1Lahsa
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0
0.5
1Urumchi
References
Alaton, P., Djehiche, B., and Stillberger, D. (2002). On Modeling and Pricing WeatherDerivatives. Applied Mathematical Finance, Vol 9.
Basawa, I.V., Rao, P., and B, L.S. (2008). Statistical Inference for Stochastic Processes.New York: Academic Press.
Benth, F.E. and Saltyte Benth, J. (2005). Stochastic Modelling of Temperature Vari-ations with a View towards Weather Derivatives. Applied Mathematical Finance, Vol12.
Benth, F.E. and Saltyte Benth, J. (2007). The Volatility of Temperature and Pricingof Weather Derivatives. Quantitative Finance, Vol 2.
Benth, F.E. and Saltyte Benth, J. (2011). Weather Derivatives and Stochastic Mod-elling of Temperature. International Journal of Stochastic Analysis, Vol 2011.
Benth, F.E., Saltyte Benth, J. and Koekebakker, S. (2007). Putting a Price on Tem-perature. Scandinavian Journal of Statistics, Vol 34.
Benth, F.E., Saltyte Benth, J. and Koekebakker, S. (2008). Stochastic Modelling ofElectricity and related Markets. Singapore: World Scientific Publishing.
Brody, D.C., Syroka, J. and Zervos, M. (2002). Dynamical Pricing of Weather Deriva-tives. Quantitative Finance, Vol 2.
20
Figure 15: ACF of residuals from CAR model (Benth et al, 2007)
0 200 400 600 800−0.5
0
0.5
1Haerbing
0 200 400 600 800−0.5
0
0.5
1Changchun
0 200 400 600 800−0.5
0
0.5
1Beijing
0 200 400 600 800−0.5
0
0.5
1Tianjin
0 200 400 600 800−0.5
0
0.5
1Nanjing
0 200 400 600 800−0.5
0
0.5
1Hangzhou
0 200 400 600 800−0.5
0
0.5
1Shanghai
0 200 400 600 800−0.5
0
0.5
1Hainan
0 200 400 600 800−0.5
0
0.5
1Guangzhou
0 200 400 600 800−0.5
0
0.5
1kunming
0 200 400 600 800−0.5
0
0.5
1Lahsa
0 200 400 600 800−0.5
0
0.5
1Urumchi
Caballero, R., Jewson, S. and Brix, A. (2002). Long Memory in Surface Air Tempera-ture: Detection, Modelling and Application to Weather Derivative Valuation. ClimateResearch, Vol 21.
Caballero, R. and Jewson, S. (2003). Seasonality in the Statistics of Surface Air Tem-perature and Pricing of Weather Derivatives. Meteorological Applications, Vol 10.
Campbell, S.D. and Diebold, F.X. (2005). Weather Forecasting for Weather Deriva-tives. Journal of the American Statistical Association, Vol 100.
Cao, M. and Wei, J. (2004). Weather Derivatives Valuation and Market Price of WeatherRisk. Journal of Future Markets, Vol 24.
Dornier, F. and Querel, M. (2000). Caution to the Wind. Energy and Power Risk Man-agement, Weather Risk Special Report.
Ender, M. and Zong, L. (2012). Analysis of Temperature-based Weather Derivatives inMainland China: Temperature Joint Modelling Proceedings of The 9th InternationalConference on Applied Financial Economics (AEF2012), Greece.
Goncu, A. (2012). Pricing Temperature-based Weather Derivatives in China. Journal ofRisk Finance, Vol 13.
Goncu, A. (2013). Comparison of Temperature Models using Heating and Cooling DegreeDays Futures. Journal of Risk Finance, Vol 14.
21
Table 5: Residual normality tests for Alaton et al’s model (2002)
Kolmogorov-Smirnov test Jaerque-Bera test Liliefors test Proportion Final resultHaerbing Rejected Rejected Rejected 0 RejectedChangchun Rejected Rejected Rejected 0 RejectedBeijing Rejected Rejected Rejected 0 RejectedTianjin Rejected Rejected Accepted 1/3 AcceptedHangzhou Rejected Rejected Rejected 0 RejectedNanjing Rejected Accepted Accepted 2/3 AcceptedShanghai Rejected Rejected Rejected 0 RejectedGuangzhou Rejected Rejected Rejected 0 RejectedHainan Rejected Rejected Rejected 0 RejectedKunming Rejected Rejected Rejected 0 RejectedLahsa Rejected Rejected Accepted 1/3 AcceptedUrumchi Rejected Rejected Rejected 0 Rejected
Heimfarth, L. and Musshof, O. (2011). Weather index-based Insurances for Farmersin the North China Plain: An Analysis of Risk Reduction Potential and basis Risk.Agricultural Finance Review, Vol 71.
Jewson, S. and Brix, A. (2005). Weather Derivative Valuation: The Meterological, Sta-tistical, Financial and Mathematical Foundations. Cambridge: Cambridge UniversityPress.
Mraoua, M. and Bari, D. (2005). Temperature Stochastic Modeling and Weather Deriva-tives Pricing: Empirical Study With Moroccan Data. Afrika Statistika, Vol 2.
Schiller, F., Seidler, G. and Wimmer, M. (2012). Temperature Models for PricingWeather Derivatives. Quantitative Finance, Vol 12.
Weather Risk Management Association (2011). Weather Risk Derivative Survey May2011.
Appendix
22
Table 6: Residual normality tests for CAR model (Benth et al, 2007)
Kolmogorov-Smirnov test Jarque-Bera test Liliefors test Proportion Final resultHaerbing Rejected Rejected Rejected 0 RejectedChangchun Rejected Rejected Rejected 0 RejectedBeijing Rejected Accepted Accepted 2/3 AcceptedTianjin Rejected Accepted Accepted 2/3 AcceptedHangzhou Rejected Rejected Accepted 1/3 AcceptedNanjing Rejected Accepted Accepted 2/3 AcceptedShanghai Rejected Rejected Accepted 1/3 AcceptedGuangzhou Rejected Rejected Rejected 0 RejectedHainan Rejected Rejected Rejected 0 RejectedKunming Rejected Rejected Rejected 0 RejectedLahsa Rejected Rejected Rejected 0 RejectedUrumchi Rejected Rejected Rejected 0 Rejected
Table 7: Montly relative error difference between Alaton et al’s model (2002) and CARmodel (Benth et al, 2007) (∗103)
Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Total monthHaerbing -3.4318 0.6807 2.0349 -0.0989 0.3052 0.4277 0.4621 0.4446 0.4137 0.1085 0.8328 0.6064 10Changchun -5.3224 0.7591 2.8751 -0.0846 0.2461 0.3574 0.3910 0.3723 0.3326 0.0795 1.0180 0.6440 10Beijing -17.2830 7.5535 -0.6266 -0.0321 0.1498 0.2368 0.2679 0.2514 0.1917 0.0272 -0.7272 4.1954 8Tianjin -11.4836 13.1579 -0.7351 -0.0514 0.1621 0.2621 0.2963 0.2775 0.2144 0.0439 -0.6635 6.4654 8Nanjing 19.6234 -1.8231 -0.5516 -0.0979 0.1114 0.2248 0.2586 0.2533 0.2122 0.0804 -0.2411 -1.4187 7Hangzhou 8.3239 -1.2039 -0.4727 -0.0997 0.0877 0.1944 0.2238 0.2270 0.1951 0.0807 -0.1593 -0.7753 7Shanghai -0.2573 -0.8841 -0.4395 -0.1061 0.0648 0.1631 0.1943 0.1994 0.1691 0.0812 -0.1082 -0.5278 6Guangzhou 0.7267 -0.2678 -0.1567 -0.0439 0.0437 0.1041 0.1354 0.1328 0.1003 0.0367 -0.0608 -0.1996 7Hainan -0.4135 -0.1794 -0.1044 -0.0164 0.0555 0.1075 0.1324 0.1230 0.0813 0.0138 -0.0715 -0.1631 6Kunming 0.8958 -0.2327 -0.1008 0.0076 0.0790 0.1241 0.1415 0.1197 0.0711 -0.0112 -0.1340 -0.3241 7Lahsa -14.3621 -2.7152 -0.5110 -0.0382 0.1692 0.2377 0.2999 0.2832 0.1984 0.0279 -0.9836 2.7988 7Urumchi 1.4891 0.7352 5.6827 -0.0506 0.2340 0.3442 0.3656 0.3416 0.2865 0.0501 1.2419 0.7783 11
Table 8: Montly relative error difference between Alaton et al’s model (2002) and CARmodel (Benth et al, 2007) - simulated temperature
Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Total monthHaerbing 0.1246 0.1586 0.1314 0.2341 0.0097 -0.0081 -0.0198 -0.0017 0.0266 -0.2911 -0.3266 0.1050 7Changchun 0.3518 -0.4918 0.0982 0.0126 -0.1404 -0.0416 0.0015 0.0152 0.0224 -0.0148 0.0927 1.0051 8Beijing 0.2194 0.1296 0.1365 0.0006 0.0162 0.0129 0.0144 0.0134 -0.0240 -0.0347 0.1094 -0.0121 9Tianjin 0.1609 0.0372 0.2080 0.1473 0.0044 0.0204 -0.0326 0.0133 0.0633 0.1129 -0.4214 0.1162 10Nanjing 0.1711 0.2233 -0.2179 0.0136 -0.0002 0.0575 0.0400 0.0073 -0.0062 0.1093 -0.0200 -0.1604 7Hangzhou -0.1215 0.0929 0.1187 0.0722 0.0551 -0.0145 0.0100 -0.0092 -0.0171 -0.0760 0.0559 0.0299 7Shanghai 0.1576 0.2502 -0.0575 -0.0589 -0.0474 -0.0037 0.0112 0.0244 0.0220 0.0510 -0.1464 0.0116 7Guangzhou 0.0755 0.0150 0.0209 -0.0897 -0.0331 -0.0049 -0.0046 0.0016 -0.0064 0.0121 0.0243 0.0408 7Hainan -0.1159 -0.0213 0.0656 0.0229 0.0206 0.0047 -0.0045 -0.0229 -0.0139 -0.0173 -0.0657 -0.0454 4Kunming 0.0742 0.0046 0.1429 -0.1031 0.0582 0.0051 0.0154 -0.0010 -0.0159 0.0646 -0.0484 -0.0093 7Lahsa 0.3925 0.4983 -0.0768 0.2567 -0.0504 0.0514 0.0292 0.0428 0.0107 0.0889 0.3087 -0.7051 9Urumchi 0.0137 0.0654 3.1617 -0.0358 0.0078 0.0107 -0.0082 0.0129 0.0461 0.0734 -0.1593 0.0005 9
23
Figure 16: Monthly error of temperature using Alaton et al’s model (2002)
1 2 3 4 5 6 7 8 9 10 11 12−1
−0.5
0
0.5
1
1.5Haerbing
1 2 3 4 5 6 7 8 9 10 11 12−2
−1
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1Changchun
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Urumchi
Figure 17: Monthly error of temperature using CAR model (Benth et al, 2007)
1 2 3 4 5 6 7 8 9 10 11 12−1
0
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2Haerbing
1 2 3 4 5 6 7 8 9 10 11 12−1
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1Beijing
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2Tianjin
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1Nanjing
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Table 9: HDD future pricing using Monte Carlo simulation (MC) and approximation formu-las (Contract Period: Jan. 2010)Climatic City Future price Future price Future price Future price
zone MC - Alaton et al Alaton et al MC - CAR model CAR modelI Haerbing 149.02 148.51 144.50 143.21
Changchun 141.79 141.46 133.92 131.87II Beijing 95.61 95.62 86.59 87.02
Tianjin 98.59 98.66 98.44 98.47III Shanghai 53.71 53.63 48.08 48.01
Hangzhou 51.96 52.22 50.99 50.89Nanjing 61.21 61.19 59.72 59.07
IV Guangzhou 14.85 14.66 12.12 11.99Hainan 0 0 0 0
V Kunming 30.01 29.99 25.23 24.59VI Lahsa 78.52 78.44 74.41 73.98VII Urumchi 120.64 121.01 116.03 116.16
Table 10: CDD future pricing using Monte Carlo simulation (MC) and approximation for-mulas (Contract Period: Jul. 2010)Climatic City Future price Future price Future price Future price
zone MC - Alaton et al Alaton et al MC - CAR model CAR modelI Haerbing 26.21 26.02 32.94 32.35
Changchun 24.68 25.08 32.55 32.87II Beijing 44.42 44.42 43.42 42.89
Tianjin 42.23 42.19 42.20 42.31III Shanghai 46.96 46.87 50.73 50.91
Hangzhou 47.03 47.00 49.02 49.76Nanjing 45.05 45.16 46.72 47.11
IV Guangzhou 50.12 50.07 53.89 53.78Hainan 53.07 53.05 56.93 56.98
V Kunming 15.31 15.25 17.31 16.97VI Lahsa 3.86 3.69 4.02 4.23VII Urumchi 27.58 27.96 29.60 29.34
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Table 11: HDD call options pricing using Monte Carlo simulation (MC) and approximationformulas (Contract Period: Jan. 2010Climatic City Strike price Option price Option price Option price Option price
zone /RMB MC - Alaton et al MC CAR modelAlaton et al CAR model
I Haerbing 750 43.01 42.85 36.8 37.70Changchun 35.08 34.75 27.02 27.25
II Beijing 500 24.42 24.42 16.11 15.44Tianjin 27.45 27.44 16.98 17.26
III Shanghai 200 25.25 25.15 19.42 19.57Hangzhou 23.59 23.70 22.61 22.40Nanjing 32.75 32.74 31.36 31.24
IV Guangzhou 50 7.73 7.53 5.25 5.00Hainan 0 0 0 0
V Kunming 100 15.75 15.74 10.72 10.98VI Lahsa 400 21.66 21.52 17.45 17.62VII Urumchi 600 35.12 35.50 30.24 30.21
Table 12: CDD call options pricing using Monte Carlo simulation (MC) and approximationformulas (Contract Period: Jul. 2010)Climatic City Strike price Option price Option price Option price Option price
zone /RMB MC - Alaton et al MC CAR modelAlaton et al CAR model
I Haerbing 100 11.92 11.79 18.61 18.70Changchun 10.40 10.65 18.59 18.34
II Beijing 150 23.07 23.06 21.87 21.99Tianjin 20.95 20.81 21.54 20.86
III Shanghai 200 18.48 18.39 22.68 22.25Hangzhou 18.60 18.54 21.32 20.55Nanjing 16.61 16.7 18.64 18.24
IV Guangzhou 250 15.48 15.37 18.44 18.31Hainan 17.46 17.44 21.65 21.14
V Kunming 50 8.12 8.08 11.93 10.32VI Lahsa 20 0.99 0.770 1.43 1.20VII Urumchi 100 13.36 13.73 14.72 15.30
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Table 13: Estimated values of volatility for Alaton et al’s model (2002) of twelve cities inmainland China
Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.Beijing 2.03 2.11 2.31 2.33 2.25 2.08 1.89 1.55 1.61 1.86 2.25 2.20
2.19 2.41 2.85 2.89 2.63 2.31 2.09 1.73 2.07 2.49 2.78 2.50mean 2.11 2.26 2.58 2.61 2.44 2.19 1.99 1.64 1.84 2.17 2.52 2.35Tianjin 1.81 1.92 2.26 2.43 2.28 2.03 1.75 1.50 1.65 1.88 2.04 1.81
1.92 2.23 2.81 3.01 2.75 2.27 1.94 1.65 2.07 2.55 2.62 2.06mean 1.86 2.07 2.54 2.72 2.51 2.15 1.85 1.58 1.86 2.21 2.33 1.94Haerbing 3.51 3.15 3.03 3.17 2.91 2.39 1.73 1.69 2.23 3.02 3.59 3.61
3.83 3.67 3.77 3.85 3.32 2.61 1.86 1.92 2.72 3.76 4.35 4.13mean 3.67 3.41 3.40 3.51 3.11 2.50 1.79 1.80 2.48 3.39 3.97 3.87Changchun 3.66 3.47 3.30 3.37 2.87 2.27 1.69 1.75 2.40 3.37 3.89 3.89
3.97 3.97 3.92 4.08 3.32 2.49 1.81 1.99 2.88 4.15 4.59 4.42mean 3.81 3.72 3.61 3.73 3.09 2.38 1.75 1.87 2.64 3.76 4.24 4.16Shanghai 2.41 2.73 2.96 2.89 2.23 2.14 1.74 1.50 1.65 1.86 2.62 2.71
2.35 3.18 2.67 2.84 2.11 2.31 1.58 1.28 1.93 1.50 2.70 2.72mean 2.38 2.96 2.82 2.86 2.17 2.23 1.66 1.39 1.79 1.68 2.66 2.71Hangzhou 2.16 2.45 2.67 2.46 2.25 1.91 1.65 1.39 1.61 1.64 2.08 2.23
2.34 2.69 3.08 2.97 2.46 2.15 1.77 1.58 1.90 2.01 2.49 2.48mean 2.25 2.57 2.88 2.71 2.36 2.03 1.71 1.48 1.75 1.83 2.29 2.36Nanjing 2.18 2.47 2.93 2.85 2.36 1.98 1.80 1.74 1.83 2.16 2.51 2.54
2.15 2.21 2.47 2.39 2.15 1.89 1.63 1.43 1.61 1.73 2.24 2.22mean 2.19 2.41 2.76 2.65 2.29 1.97 1.72 1.55 1.73 1.90 2.34 2.37Guangzhou 2.23 2.50 1.99 2.54 1.58 1.26 1.25 1.16 1.24 1.46 1.88 2.14
2.35 2.62 2.27 2.80 1.73 1.35 1.35 1.25 1.41 1.71 2.20 2.33mean 2.29 2.56 2.13 2.67 1.65 1.31 1.30 1.21 1.32 1.58 2.04 2.24Hainan 1.68 1.77 1.90 1.55 1.19 0.95 0.82 0.88 0.85 0.97 1.37 1.64
1.76 1.85 2.13 1.73 1.29 1.00 0.87 0.94 0.96 1.11 1.56 1.77mean 1.72 1.81 2.02 1.64 1.24 0.98 0.84 0.91 0.91 1.04 1.46 1.70Kunming 1.86 1.92 2.05 1.86 1.97 1.44 1.08 1.26 1.36 1.60 1.60 1.78
2.08 2.13 2.33 2.04 2.13 1.56 1.18 1.41 1.60 1.82 1.84 1.96mean 1.97 2.02 2.19 1.95 2.05 1.50 1.13 1.33 1.48 1.71 1.72 1.87Lahsa 2.16 2.08 1.90 1.76 1.80 1.82 1.55 1.41 1.22 1.23 1.43 1.84
2.37 2.29 2.14 1.89 2.13 1.98 1.68 1.52 1.41 1.69 1.74 2.03mean 2.27 2.19 2.02 1.83 1.97 1.90 1.62 1.47 1.32 1.46 1.58 1.94Urumchi 2.77 2.58 2.54 3.28 3.25 2.64 2.44 2.61 2.65 2.63 2.71 3.05
2.91 2.82 3.39 3.73 3.67 2.87 2.65 2.98 3.08 3.12 3.45 3.27mean 2.84 2.70 2.97 3.50 3.46 2.76 2.55 2.80 2.87 2.87 3.08 3.16
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