Model Comparison for Temperature-based Weather Derivatives in Mainland China

Transcription

1 Model Comparison for Temperature-based Weather Derivatives in Mainland China Lu Zong, Manuela Ender Department of Mathematical Sciences Xi an Jiaotong-Liverpool University April 4, 23 Abstract In this paper, we provide a comprehensive comparison of two models for the simulation and pricing of temperature-based weather derivatives. The model of Alaton et al (22) and the CAR model of Benth et al (27) are applied to temperature data from twelve cities in Mainland China. The objective of this paper is to analyse whether the CAR model, as a more advanced model has a better performance in fitting the daily average temperature (DAT). A higher accuracy of the CAR model can be found indeed in terms of normality of residuals and in terms of smaller relative errors. However, the shortcomings of both models are revealed in this study as well. The Chinese cities involved cover all seven climatic zones in the standard of climatic regionalization that is used as a partition of China to get representative clusters. Keywords: weather derivatives, temperature modelling, China, pricing, simulation JEL classification: C5, G3, Q54 Introduction Worldwide, the economy is influenced by the weather. The industries with the highest weather sensitivity are in the sector of energy, agriculture, retail, construction and transportation (Weather Risk Management Association, 2). Although the great influence of weather fluctuations has been known for a long time, financial instruments to hedge weather risk, namely weather derivatives were firstly launched by the Chicago Mercantile Exchange in the year of 999. The main advantages of weather derivatives are:. Compared to weather insurance, there is no industrial limitation on weather derivative contracts. These contracts are designed based on the region with different strike prices and cold/warm seasons, but are valid for all industries within the same region.

2 2. Weather derivatives compensate the buyers for all abnormal weather events and not only for weather catastrophes with high severity and low possibility like insurance contracts. 3. As the underlying is a weather index, it cannot be controlled by the buyers. Hence problems like moral hazard or adverse selection do not exist. 4. Transaction costs related to weather derivatives are much lower than for other hedging instruments like insurance. So, more investors of a certain industry which is exposed to weather risk are encouraged to hedge their investments. As in China, agriculture is one of the most important parts in the GDP and 7% of the population live on farms, managing weather risk is extremely important, especially for small scale farm households. Weather derivatives can provide an alternative to subsidizations (Heimfarth and Musshof, 2). Considering the great potential demand of China, more research is needed for this market. Currently, very few studies exist on Chinese weather data. Goncu (22) applied a seasonal volatility model to capture the fluctuations of DATs of, and Shenzhen and to price temperature-based weather derivatives for these three cities. This paper is the first that apply two models to twelve cities in China from all seven climatic zones in order to find the most suitable model for pricing temperature-based contracts nationally wide. As a non-tradable asset, temperature is priced with indexed underlyings. The coolingdegree day (CDD) and heating-degree day (HDD) are the most common underlyings while pricing temperature-based weather derivatives. They are respectively given by: HDD(t, t 2 ) = CDD(t, t 2 ) = t2 t max(8 T t, ) dt, () t2 where T t denotes the daily observed temperature. t min(t t 8, ) dt, (2) Since the underlyings are non-tradable, the Black-Scholes framework cannot longer be applied to price weather derivatives. Different methods for an incomplete market, approximation 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 (22) and the continuous-time autoregressive CAR model of Benth et al (27). In the next section, both models are applied to thirty years of daily average temperature (DAT) data of twelve cities in Mainland China. Finally, we compare the fitting performance of both models in terms of simulation errors and pricing results. 2

3 2 Modelling temperature There are different ways to price temperature-based weather derivatives. Figure gives an overview of the most important models for the modelling of temperature dynamics. Figure : An overview of models for temperature modelling The simplest method is a so called Burn Analysis. The basic idea is to use the observations of the index from the past to model the future movements. Empirical studies revealed that the Burn Analysis has comparatively large errors as events that were not observed in the past cannot be modelled as possible future events (e.g. Schiller et al (22), Cao and Wei (24)). Dornier and Queruel (2) proposed to use a continuous-time Ornstein-Uhlenbeck (OU) process to model the temperature evolution. The volatility is assumed to be constant in this first model of this type. As temperature data shows heteroskedasticity of the volatility, an improvement of the model was made by Alaton et al (22) who introduced a monthly constant volatility. The OU process itself cannot model auto-correlation. To capture the correlation Brody et al (22) introduced a fractional Brownian motion. Benth and Saltyte-Benth (25) used a hyperbolic Levy-process to model the residuals instead of a Brownian motion. Beside advanced stochastic models, time-series models used in econometrics were applied 3

4 to temperature data as well. Caballero et al (22) suggested to use an ARMA and ARFIMA model. Jewson and Caballero (23) proposed a model called AROMA to deal with the slow decay of the autocorrelation function. An ARCH model was suggested by Campbell and Diebold (25). In Benth et al (27), the OU process was combined with an econometrics approach to get a higher-order continuous-time autoregressive process, the CAR model. To sum up, an advanced stochastic model usually measures the average daily temperature with a seasonal function. Different stochastic methods are applied to the residuals obtained by subtracting the seasonal function from the daily temperature. Usually, a Brownian motion, a mean reverting process, an Ornstein-Uhlenbeck process, and optionally an autoregressive process are included in the temperature modelling. For this model comparison we choose the Alaton et al (22) model, because as the most classic model for the evaluation of weather derivatives it can be used as a benchmark model similar to the Black-Scholes model for options on shares prices. However, the most restrictive part of the Alaton et al model (22) is the monthly constant volatility. As the Benth et al s model (27) overcomes this restriction and allows the modelling of daily volatility, it is straight forward to compare these two models to analyse how much the modelling could be improved in terms of lower errors. This is the first model comparison based on data from China. The few existing studies comparing different temperature models like Schiller et al (22) and Goncu (23) focus on data from the more mature market in the USA. Schiller et al (22) included the Burn Analysis, Alaton et al s model (22), Benth et al s model (27) and a new proposed spline model. Goncu (23) compared the models of Alaton et al (22), Benth and Benth (25), Brody et al (22), and Campbell and Diepold (25). More advanced models have usually more parameter which makes the estimation more complex and less stable. In order to find the best model for temperature modelling in China, the trade-off between a better fit and a more complex modelling needs to be elaborated which is done in the following sections. 2. Data of the empirical study In this paper, thirty years of DATs of twelve cities in Mainland China are used. The cities are selected according to the standard of climatic regionalization used in architecture which divides Mainland China into seven climatic zones (Ender and Zong, 22). Figure 2 gives a detailed map of partition. Table gives an overview of the temperature data of the twelve cities. 2.2 Alaton et al s model (22) Alaton et al s model (22) fits the DATs with two parts, i.e. the seasonality part and the random process. The first part is measured with a sine function showing both, the seasonality and the global warming trend of temperature. It is expressed as: T m s = A + Bt + C sin(ωt + ϕ), (3) 4

5 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 is given by: T t = (T s T m s )e α(t s) + T m s + t s e α(t τ) σ τ dw τ, (4) where Ts m is given by (3) and W t 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 a regression method. According to Alaton et al (22), the volatility parameter σ is measured on the assumption to be monthly constant. The estimated values are obtained by using both, the regression method and the discretizing method. Table 3, shown in the Appendix gives the estimators of the volatility using sequentially the quadratic variation, regression method and the mean value of these two results. 2.3 CAR model (Benth et al, 27) A model for the underlying without the restriction of a piecewise constant volatility is the continuous-time autoregressive (CAR) model (Benth et al, 27). Similar with Alaton et 5

6 Table : Overview of DAT samples (January 98 - December 2) Climatic Mean Standard Max Min zone deviation I I II II III III III IV IV Kunming V Lhasa VI VII al s model (22), the CAR model keeps the deterministic seasonal function: Λ(t) = a + a t + a 2 cos(2π(t a 3 )/365). (5) Finally, the temperature under a CAR(p) model is expressed as T (t) = Λ(t) + X (t), (6) where X q is the q th coordinate of the vector X, q =,..., p. The explicit form of the stochastic process X(t) in R p for p is given by X(s) = exp(a(s t))x + s t exp(a(s u))e p σ(u) dw u, (7) for s t and X(t) = x R p, where e q is the q th unit vector in R p, q =,..., p and W t denotes the Brownian motion. The volatility function σ(t) > is a square integrable and real-valued function. The parameter A represents the mean-reverting p p matrix given by: A =......, (8) α p α p α p 2... α where α q, q =,... 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, 27). The temperature T i on day i =,, 2,... is expressed as: T i = Λ i + y i, (9) 6

7 Table 2: Estimated values of A, B, C, ϕ and α of twelve cities in Mainland China A B C ϕ α Kunming where Λ i = Λ(i) as defined in equation (5). The process y i is an AR(p) process that follows: y i+p = p b j y i+p j + σ i ɛ i, () j=i where ɛ i are independent, standard normally distributed random variables. As it was shown in Benth at al (27) that the optimal choice is p = 3, we follow this approach. Hence, we have: y i+3 = b y i+2 + b 2 y i+ + b 3 y i + σ i ɛ i. () Finally, we transfer the AR(3) process into a CAR(3) process by using Benth et al s solution (27): 3 a = b, 2a a 2 3 = b 2, a 2 + (a + a 3 ) = b 3. Table 3 lists the estimated values of b, b 2 and b 3, α, α 2 and α 3. Different from Alaton et al s model (22), Benth et al (27) proposed a truncated Fourier series as a functional volatility σ i = σ(i) to model the observed seasonal heteroskedasticity of residuals after removing the linear trend, seasonal component and AR(3) process: σ 2 (t) = c + 4 (c 2k cos(2kπt/365) + c 2k+ sin(2kπt/365)). (2) k= 7

8 Table 3: Estimated parameters of AR(3) and CAR(3) process b b 2 b 3 α α 2 α Kunming Table 4 gives the result of the parameters c to c 9 estimated using the least squares method. Table 4: Estimated parameters of seasonal volatility function of the CAR model c c 2 c 3 c 4 c 5 c 6 c 7 c 8 c Kunming 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 2 applying Alaton et al s model (22) and the CAR model (Benth et al, 27). Instead of a smooth curve of DATs as in Figure 4, the CAR model (Benth et al, 27) is able to produce fluctuations from day to day which stresses the more realistic assumptions of this model. 8

9 Figure 3: Fitted Fourier volatility process together with empirical daily squared volatility of,, and Normality of residuals As the stochastic process used for example for the simulation above is modelled by a Brownian motion, the residuals after removing all other components of the model should follow a normal distribution hypothetically. However, when dealing with DATs over many years, there is a large amount of data available. When testing many data points, normality tests like the Kolmogorov-Smirnov test reject the hypothesis even when the departure from normality is very small. So it is important to check histograms and Q-Q plots as well to judge the distribution of residuals eventually. Benth et al (28) pointed out that errors of using the normal distribution only have little effects on modelling and pricing. However, according to the assumptions of regression, the non-normality of residuals could be interpreted as the result of non-constant variance and interdependence, this issue will be carefully checked in the next chapter of model comparison and model performance. 3 Model comparison Compared with Alaton et al s model (22), the CAR model (Benth et al, 27) is apparently a more advanced model. In the first instance, an autoregressive process was taken into account. Secondly, the functional volatility of temperature is added instead of the assumption of a monthly constant volatility. Therefore, we test whether the CAR model can reduce the estimation errors and increase the normality of the residual distribution in comparison to 9

10 Figure 4: DAT simulation using Alaton et al s model (22) (Year: 2) kunming the performance of Alaton et al s model (22). As there are no historical prices for weather derivatives in China available yet, the comparison of accuracy is mainly based on temperature modelling. As the seasonal functions of the two models are essentially the same, it is sufficient 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 normality of residuals, three tests, i.e. the Kolmogorov-Smirnov test, the Jarque-Bera test and the Lilliefors test are applied to the residuals. This analysis is supported by histograms and Q-Q plots of the residuals. In the second step, the errors of temperature simulation of the two 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 from the model choice and an indicator of the model risk involved. 3. Test of residuals In this section, the residuals that are left after calibrating the model of Alaton et al (22) and the CAR model (Benth et al, 27) to the temperature data are compared. The residuals are shown in Figure 6 and Figure 7, the squared residuals are plotted in Figure 8 and 9. It cannot be neglected that a seasonal pattern exists for both models. Even though the CAR model (Benth et al, 27) includes a CAR(3) process and measures the daily volatility with a seasonal function, the seasonality in the residuals could not be entirely eliminated.

11 Figure 5: DAT simulation using CAR model (Benth et al, 27) (Year: 2) kunming In Figure and in Figure, the histograms of the residuals distributions and the corresponding normal distributions are shown. Although a bias between the distributions of the residuals and the normal distribution is visible, the similarity between the two distributions is close. As histograms can barely show the tail behaviour and correlation, additionally the Q-Q plots and the autocorrelation function ACF of the residuals in plotted in Figure 2 to Figure 5. The residuals distributions from both models tend to show a bias at the end of the quantile line against the standard normal quantiles. The bias is caused by a steeper trend in the Q-Q plot which indicates that the residuals distributions are more dispersed compared to the normal distribution. The ACFs show a difference between the Alaton et al s model (22) and the CAR model (Benth et al, 27). As the ACFs of the CAR model are flatter than the ACFs of Alaton et al s model (22), it can be concluded that the CAR model manage to eliminate more of the existing trends than the Alaton et al s model (22). 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 long as the null hypothesis is accepted at least in one of the tests, we accept it for this city under

12 Figure 6: Residuals of DATs after removing all components of Alaton et al s model (22) kunming the given model. Table 5 and Table 6 give the results for all cities in both models. The proportion 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 normal distributed residuals from the calibration of the CAR model (Benth et al, 27) (five out of twelve cities) than from the calibration of the Alaton et al s model (22) (three out of twelve cities). Further, the proportions of acceptance among the cities that are ultimately accepted are slightly higher for the CAR model (Benth et al, 27) than those for Alaton et al s model (22). We can conclude that the CAR model (Benth et al, 27) when applied to Data from China is the better model in terms of normality of residuals. However, as the normal distribution is still rejected for roughly half of the cities, the assumptions of the CAR model (Benth et al, 27) seem to fit Chinese Data not completely perfectly which implies that 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 the performance of the model of Alaton et al (22) and of the CAR model (Benth et al, 27). We compute the monthly relative error measure that is defined as (Mraoua and Bari, 25): ER relative = T estimated T observed T observed. (3) Monthly means that all errors of one month are added up to form one aggregated per- 2

13 Figure 7: Residuals of DATs after removing all components of CAR model (Benth et al, 27) kunming formance measure. For the model comparison, the difference of the errors of the two models is of particular importance. In Table 7 and Table 8, the differences of errors are respectively obtained from the monthly errors based on the fitted DATs and on simulated DATs. The reported values are calculated by subtracting the error of the CAR model (Benth et al, 27) from the corresponding error of Alaton et al s model (22). This method is chosen to easily identify the months in which the relative errors of the CAR model (Benth et al, 27) are smaller, or in other words, superior to Alaton et al s model (22). The last columns of Table 7 and Table 8 summarizes the total number of months in which the performance of the CAR model (Benth et al, 27) is better. In the case of fitted temperature data when there is no random process involved, the differences of errors between the two models tend to be very close to zero. However, the number of positive differences that indicates that the CAR model (Benth et al, 27) is a better fit in this specific month is larger than 6 in Table 7 and in Table 8 where the only exception is. Especially for more northern cities like, and where the mean temperature is comparatively lower, but where standard deviation is higher (see Table ). This tells us that the CAR model (Benth et al, 27) can indeed capture fluctuation in temperatures better than the model of Alaton et al (22). From these result, we conclude that the CAR model (Benth et al, 27) has a better performance. In Figure 6 and 7, the bar plots of the monthly relative errors of the twelve cities based 3

14 Figure 8: Squared residuals of DATs after removing all components of Alaton et al s model (22) kunming on simulated temperatures are shown. As we have already noticed, the monthly relative errors are in general smaller of the CAR model (Benth et al, 27). The bar plots help to understand how the errors are distributed over the year. Here, the result is very similar for all cities and all models. There are large differences between the relative errors in cold season and those in warm season. During the summer where the volatility is typically lower, the error measure is very close to zero. However, for winter months with higher volatility the errors tend to be much higher. The more advanced CAR model (Benth et al, 27) cannot capture the seasonality of DATs fluctuations. Even though the errors are on average smaller of the CAR model (Benth et al, 27), 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 determined by the Alaton et al s model (22) and the CAR model (Benth et al, 27). A weather future is a compulsory contract between the buyer and the seller to trade an asset at a negotiated price on a fixed date afterwards. In this case, the asset is the currency value of a specified weather index. The payment is done by cash settlement. To hedge weather risk, the traders should buy or sell future contracts that are in contrary to the weather condition that is positive for them. For example, a farmer who wants to reduce the 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 4

15 Figure 9: Residuals of DATs after removing all components of CAR model (Benth et al, 27) kunming as calls and puts give the right to buy or sell respectively the weather future at a specified strike 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 to calibrate the market price of risk, we compare the price differences in order to find some patterns that may exist. In the absence of market prices and further knowledge of the risk aversion of potential buyers, we let the market price of risk (MPR) be equal to a constant λ that is set to be in this study. The converting ratio (a.k.a. principal nominal) is one unit of currency paid for one degree Celsius. According to Alaton et al (22), the pricing method is based on the assumption that the probability of the daily temperature in the HDD contract (see equation () for definition) period being larger than 8 C is zero. Hence, the future price of such a degree day based contract is normally distributed under measure Q with the mean µ n and standard deviation σ n. The approximate formula of an CDD future contract for c = 8 at time t t follows: [ tn ] FCDD Alaton (t, t, t n ) = E Q max(t (s) c, )ds F t, (4) t where t stands for the first day and t n 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: 5

16 Figure : Histograms of residuals from Alaton et al s model (22) kunming n X n = max (T t 8, ) = t= Hence, the CDD call option price is given by n T t 8n. (5) t= C Alaton CDD (t, t n ) = e r(tn t ) E Q [max(x n K, ) F t ], (6) where K stands for the strike price and r for the risk free rate. From the assumption that X n N(µ n, σn) 2 and Φ representing the cdf of the standard normal distribution, we get [ ( ) CCDD Alaton (t, t n ) = e r(tn t ) K µn (µ n K)Φ + σ n e (K µn) 2 ] 2σn 2. (7) 2π Similar to the model of Alaton et al (22), the pricing for the CAR model (Benth et al, 27) is also based on a martingale approach. Benth et al (27) give the approximation formula for a CDD future with c = 8 at time t t as follows: F CAR CDD(t, t, t n ) = E Q [ tn where t ] max(t (s) c, )ds F t = m(t, s, x) = Λ(s) c + s t σ n tn t ( ) m(t, s, e v(t, s)ψ exp(a(s t)x(t)) ds, v(t, s) (8) σ(u)θ(u)e exp(a(s u))e p du + x, (9) 6

17 Figure : Histograms of residuals from CAR model (Benth et al, 27) kunming v 2 (t, s) = x = e exp(a(s t))x(t), (2) s t σ 2 (u)(e exp(a(s u))e p ) 2 du, (2) 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. Analogue to the derivation for Alaton et al s model (22), the option price for a CDD contract for the CAR model (Benth et al, 27) is defined by C CAR CDD(t, τ, t, t n ) = e r(τ t) E Q [ max(f CAR CDD (τ, t, t n ) K, ) F t ]. (23) Besides using the approximation formulas for future and option prices, a Monte Carlo simulation is possible in each case. Usually a Monte Carlo simulation needs less assumptions as closed form solutions are not necessary in this case. However, simulation is time consuming as 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 at least both pricing methods, the approximation formulas and Monte Carlo simulation. The results for HDD and CDD futures are shown in Table 9 and Table. Call option prices are given in Table and 2. The contract period is for HDD contracts January 2 and for CDD contracts July 2. These kind of contracts could be used to hedge against lower temperatures than usual in January and against higher temperatures than usual in July. Other 7

18 Figure 2: Q-Q plot of residuals from Alaton et al s model (22) Kunming 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 Monte Carlo simulation are for all contracts and for all cities very small. In most cases the difference is less than.5 per cent. We can conclude that the assumptions for the approximation formulas for the models are acceptable and that the simulation is sufficiently stable. Comparing the HDD contracts, the model of Alaton et al (22) has always higher prices. Respectively for CDD contracts, Alaton et al s (22) prices are always lower. The differences can be just a few percentage points, but the prices can vary up to 2 per cent. This shows that model risk exist. Given the analysis before, that the CAR model (Benth et al, 27) has in more cases normal distributed residuals and on average smaller relative errors, the prices of this model should be closer to real prices and therefore preferable. Finally, that the prices for futures and options within one climatic zone are very similar, but different between climatic zones supports the use of the standard of climatic regionalization as a partition of Mainland China in order to reduce dimensions when pricing temperature-based derivatives (Ender and Zong, 22). 4 Conclusion With a non-tradable underlying, the modelling and pricing of assets based on temperature face many challenges. The publication of some temperature models in the last decade in- 8

19 Figure 3: Q-Q plot of residuals from CAR model (Benth et al, 27) kunming creases the possibilities for overcoming the obstacles, but increases also the uncertainty which model to choose especially in non-mature markets like in China. The purpose of the presented model comparison was to find a more suitable model for the Chinese market among two promising candidates, the Alaton et al s model (22) and the CAR model (Benth et al, 27). The results of the study have shown that the calibration to Chinese data of twelve mainland cities from all climatic zones of China gives a feasible modelling of temperature with acceptable error terms. The CAR model (Benth et al, 27) as a more advanced model has a 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 large proportion of cities the residuals do not follow a normal distribution. Especially to deal with the seasonality of the volatility is problematic for both models. Consequently, the prices for futures and options still lack reliability. For future research in order to find the most suitable model for Chinese data, stochastic volatility models (like Benth and Benth, 2) need to be included in the empirical study. This means that the volatility is modelled by a stochastic process itself and is no longer deterministic. Then, capturing the seasonal behaviour of the volatility should be possible and the error terms for the cold season should be reduced. Further, a spline model approach like proposed by Schiller et al (22) should be applied to Chinese data to check whether the normality of residuals can be assured by this approach. 9

20 Figure 4: ACF of residuals from Alaton et al s model (22) Kunming References Alaton, P., Djehiche, B., and Stillberger, D. (22). On Modeling and Pricing Weather Derivatives. Applied Mathematical Finance, Vol 9. Basawa, I.V., Rao, P., and B, L.S. (28). Statistical Inference for Stochastic Processes. New York: Academic Press. Benth, F.E. and Saltyte Benth, J. (25). Stochastic Modelling of Temperature Variations with a View towards Weather Derivatives. Applied Mathematical Finance, Vol 2. Benth, F.E. and Saltyte Benth, J. (27). The Volatility of Temperature and Pricing of Weather Derivatives. Quantitative Finance, Vol 2. Benth, F.E. and Saltyte Benth, J. (2). Weather Derivatives and Stochastic Modelling of Temperature. International Journal of Stochastic Analysis, Vol 2. Benth, F.E., Saltyte Benth, J. and Koekebakker, S. (27). Putting a Price on Temperature. Scandinavian Journal of Statistics, Vol 34. Benth, F.E., Saltyte Benth, J. and Koekebakker, S. (28). Stochastic Modelling of Electricity and related Markets. Singapore: World Scientific Publishing. Brody, D.C., Syroka, J. and Zervos, M. (22). Dynamical Pricing of Weather Derivatives. Quantitative Finance, Vol 2. 2

21 Figure 5: ACF of residuals from CAR model (Benth et al, 27) kunming Caballero, R., Jewson, S. and Brix, A. (22). Long Memory in Surface Air Temperature: Detection, Modelling and Application to Weather Derivative Valuation. Climate Research, Vol 2. Caballero, R. and Jewson, S. (23). Seasonality in the Statistics of Surface Air Temperature and Pricing of Weather Derivatives. Meteorological Applications, Vol. Campbell, S.D. and Diebold, F.X. (25). Weather Forecasting for Weather Derivatives. Journal of the American Statistical Association, Vol. Cao, M. and Wei, J. (24). Weather Derivatives Valuation and Market Price of Weather Risk. Journal of Future Markets, Vol 24. Dornier, F. and Querel, M. (2). Caution to the Wind. Energy and Power Risk Management, Weather Risk Special Report. Ender, M. and Zong, L. (22). Analysis of Temperature-based Weather Derivatives in Mainland China: Temperature Joint Modelling Proceedings of The 9th International Conference on Applied Financial Economics (AEF22), Greece. Goncu, A. (22). Pricing Temperature-based Weather Derivatives in China. Journal of Risk Finance, Vol 3. Goncu, A. (23). Comparison of Temperature Models using Heating and Cooling Degree Days Futures. Journal of Risk Finance, Vol 4. 2

22 Table 5: Residual normality tests for Alaton et al s model (22) Kolmogorov-Smirnov test Jaerque-Bera test Liliefors test Proportion Final result Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Accepted /3 Accepted Rejected Rejected Rejected Rejected Rejected Accepted Accepted 2/3 Accepted Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Kunming Rejected Rejected Rejected Rejected Rejected Rejected Accepted /3 Accepted Rejected Rejected Rejected Rejected Heimfarth, L. and Musshof, O. (2). Weather index-based Insurances for Farmers in the North China Plain: An Analysis of Risk Reduction Potential and basis Risk. Agricultural Finance Review, Vol 7. Jewson, S. and Brix, A. (25). Weather Derivative Valuation: The Meterological, Statistical, Financial and Mathematical Foundations. Cambridge: Cambridge University Press. Mraoua, M. and Bari, D. (25). Temperature Stochastic Modeling and Weather Derivatives Pricing: Empirical Study With Moroccan Data. Afrika Statistika, Vol 2. Schiller, F., Seidler, G. and Wimmer, M. (22). Temperature Models for Pricing Weather Derivatives. Quantitative Finance, Vol 2. Weather Risk Management Association (2). Weather Risk Derivative Survey May 2. Appendix 22

23 Table 6: Residual normality tests for CAR model (Benth et al, 27) Kolmogorov-Smirnov test Jarque-Bera test Liliefors test Proportion Final result Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Accepted Accepted 2/3 Accepted Rejected Accepted Accepted 2/3 Accepted Rejected Rejected Accepted /3 Accepted Rejected Accepted Accepted 2/3 Accepted Rejected Rejected Accepted /3 Accepted Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Kunming Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected Table 7: Montly relative error difference between Alaton et al s model (22) and CAR model (Benth et al, 27) ( 3 ) Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Total month Kunming Table 8: Montly relative error difference between Alaton et al s model (22) and CAR model (Benth et al, 27) - simulated temperature Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Total month Kunming

24 Figure 6: Monthly error of temperature using Alaton et al s model (22) kunming Figure 7: Monthly error of temperature using CAR model (Benth et al, 27) kunming

25 Table 9: HDD future pricing using Monte Carlo simulation (MC) and approximation formulas (Contract Period: Jan. 2) Climatic City Future price Future price Future price Future price zone MC - Alaton et al Alaton et al MC - CAR model CAR model I II III IV V Kunming VI VII Table : CDD future pricing using Monte Carlo simulation (MC) and approximation formulas (Contract Period: Jul. 2) Climatic City Future price Future price Future price Future price zone MC - Alaton et al Alaton et al MC - CAR model CAR model I II III IV V Kunming VI VII

26 Table : HDD call options pricing using Monte Carlo simulation (MC) and approximation formulas (Contract Period: Jan. 2 Climatic City Strike price Option price Option price Option price Option price zone /RMB MC - Alaton et al MC CAR model Alaton et al CAR model I II III IV V Kunming VI VII Table 2: CDD call options pricing using Monte Carlo simulation (MC) and approximation formulas (Contract Period: Jul. 2) Climatic City Strike price Option price Option price Option price Option price zone /RMB MC - Alaton et al MC CAR model Alaton et al CAR model I II III IV V Kunming VI VII

27 Table 3: Estimated values of volatility for Alaton et al s model (22) of twelve cities in mainland China Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec mean mean mean mean mean mean mean mean mean Kunming mean mean mean