Regime Switching models on Temperature Dynamics

Transcription

1 Department of Mathematics Regime Switching models on Temperature Dynamics Emanuel Evarest, Fredrik Berntsson, Martin Singull and Wilson Charles LiTH-MAT-R--216/12--SE

2 Department of Mathematics Linköping University S Linköping

3 Regime Switching models on Temperature Dynamics Emanuel Evarest 1,2, Fredrik Berntsson 1, Martin Singull 1 and Wilson Charles 2 1 Department of Mathematics, Linköping University 2 Department of Mathematics, University of Dar es Salaam Abstract Two regime switching models for predicting temperature dynamics are presented in this study for the purpose to be used for weather derivatives pricing. One is an existing model in the literature (Elias model) and the other is presented in this paper. The new model we propose in this study has a mean reverting heteroskedastic process in the base regime and a Brownian motion in the shifted regime. The parameter estimation of the two models is done by the use expectation-maximization (EM) method using historical temperature data. The performance of the two models on prediction of temperature dynamics is compared using historical daily average temperature data from five weather stations across Sweden. The comparison is based on the heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The expected HDDs, CDDs and CAT of the models are compared to the true indices from the real data. Results from the expected HDDs, CDDs and CAT together with their corresponding daily average plots demonstrate that, our model captures temperature dynamics relatively better than Elias model. Key words. Weather derivatives, Regime switching, temperature dynamics, expectation-maximization, temperature indices 1

4 1 Introduction Weather derivatives business came into existence in the 199s, where one of its strongest cause being the El Niño winter of During this event many companies faced the problem of significant decline of their income flow because of the unusual mild winter. This forced them to find the means of protecting themselves against unpredictable weather risk. The inception of weather derivatives, made weather a tradable commodity, so individuals and business organizations no longer live at the mercy of weather [28, 7]. It is estimated that the impact of weather accounts for $5.3 billion of the $16 trillion U.S. gross domestic product. Generalizing these figures to the entire world, you can have an astonishing results of its impact [23, 6]. Indeed, one-third of businesses worldwide are directly affected by weather conditions, thus having relatively better models for weather dynamics will be of great importance in pricing weather derivatives for protection against these weather related risks. Weather derivatives cover low-risk, highprobability events while weather insurance, on the other hand, typically covers high-risk, low-probability events, as defined in a highly tailored, or customized policy, see [1, 18]. In general, weather derivatives are financial instruments used for risk management purposes to hedge against losses due to adverse or unpredictable weather conditions. The payoff of these financial instruments is derived from weather variables such as temperature, rainfall, snowfall, wind and humidity. It is estimated that about 98% of the weather derivatives now traded in the market are based on temperature [1, 25]. Therefore in this study we focus on modelling temperature dynamics for the use in weather derivatives pricing. Currently temperature indices used for constructing weather derivatives contracts, includes Heating degree days (HDDs), cooling degree days (CDDs), cumulative average temperature (CAT) mainly used in European cities for summer months and Pacific Rim which are used in Japan and other Asian countries, see [4, 2, 23]. HDDs and CDDs accumulated over the contract period is the sum of positive differences between daily average temperature and the reference temperature for winter and summer respectively. Pacific Rim is the average of cumulative average temperature over the contract period. Most of the available literature on weather derivatives modelling and pricing, have modelled the dynamics of weather variables using single regime stochastic models where it is assumed that there is no changes in state of the underlying weather 2

5 variable, see [5, 1, 22, 27, 26]. Many financial time series data occasionally depict abrupt changes in their behavior, due to changes in economic environments, see [3, 14, 12]. This led many researchers in financial and commodity modelling to focus on regime switching models in order to capture such abrupt and discrete shifts in the behaviour of financial variables [11, 15, 16]. This class of models offers the possibility to represent the dynamics of the underlying variable(s) by more than one different stochastic models or using the same model with different parameters varying with states of the dynamical system [13]. In the context of weather derivatives market, dramatic changes in weather induces the regime shift behavior in weather of a particular location or region. For instance, weather regimes can be reflected by different abrupt changes in temperature behaviour of a specified location due to urbanization and changes of measurement station and anthropogenic climate change which is a continuous process [24]. The regime switching behavior in temperature can also be associated with the switching behaviour in energy prices, where weather variations influences the volume of sales in energy markets. During the mild winter the demand for energy for heating decreases and during the cold summer demand for electricity for cooling decreases. Elias [9], presented a two state regime switching model where the volatility of either states are considered as constant. This consideration might not always be true due to the fact that when temperature dynamics shifts from one state to another it will not always maintain the same volatility for each time it switches to that particular state. Therefore, in this study we present a regime switching model for temperature dynamics whose volatility varies with its state and its underlying process. It is a two state regime switching model with base regime governed by a mean-reverting heteroskedastic process and shifted regime governed by a Brownian motion. The model is capable of capturing the discrete shifts in temperature dynamics. The switching mechanism from one state to another is a Markov process and is governed by the latent random variable. The individual regimes of the temperature process are assumed to be independent of each other but joined together by membership probabilities. Parameters for the model we propose together with that of Elias model are estimated by the expectation-maximization (EM) algorithm introduced by Dempster [8]. Based on this EM algorithm, we develop explicit expressions for each of these model parameters. The EM algorithm is quite robust with respect to poorly 3

6 chosen initial values and moves quickly to a likelihood surface region [17, 21]. Since temperature time series are used for generating indices for weather derivatives pricing, we use accumulated HDDs, CDDs and CAT indices to verify how our model performs relatively better than Elias model in comparison to the actual indices from historical temperature time series data. The remaining part of this study is organized as follows: In Section 2, temperature dynamics regime switching models are presented and the solution of each stochastic regime switching model are presented as well. In Section 3, the method for parameter estimation is presented and closed form expression for each of the model parameters are produced. Also using the deseasonalized data obtained from historical temperature data we estimate model parameters using the EM algorithm. In Section 4, results and discussions about the results are presented. Also, the two models are compared based on the daily average temperature times series, HDDs, CDDs and CAT indices for three chosen months for winter and summer seasons respectively. Finally, in Section 5 concluding remarks is provided. 2 Temperature Dynamics Models In this section, we formulate the stochastic model for weather derivatives based on temperature as its underlying variable. The two state Markov regime switching model we propose here assumes that the daily average temperature switches between the two regimes. The base regime is governed by a mean-reversion heteroskedastic process and shifted regime is governed by a Brownian motion. The heteroskedasticity in the base regime is introduced due to the fact that temperature volatility varies with changes in the temperature process as it undergoes discrete shifts between the states S t of the process. The transition of the process from one regime to another is stochastic and driven by a transition probability matrix. For an N states regime switching model, the transition matrix for the switching process from one regime to another is given by p 11 p 1N P =..... p N1 p NN 4, (2.1)

7 where one step transition probabilities p ij of the process X n are given by p ij = P {X t+1 = j X t = i}, i, j = 1, 2,, N, (2.2) N p ij 1 and p ij = 1. j=1 In almost all available literature on weather dynamics modelling and weather derivatives pricing, the presence of predictable annual seasonal component in temperature data has been shown to exist. Thus, we can define the daily average temperature as the sum of deseasonalized temperature and seasonal component. Letting time scale be in days, the temperature at time t is denoted by T d (t) and defined as T d (t) = T d (t) + s d (t), (2.3) where T d (t) is the deseasonalized temperature and s d (t) is the seasonal component. The seasonality in this case can be defined as the sum of a linear function and trigonometric function as ( ) 2π s d (t) = a sin 365 (t a 1) + a 2 t + a 3, (2.4) where the constants a and a 1 defines the amplitude of temperature and phase angle, while the constants a 2 and a 3 represent the coefficient and constant of the linearity of the seasonal trend respectively. The deseasonalized temperature T d (t) is obtained by removing the seasonal component from (2.3), i.e. T d (t) = T d (t) s d (t). (2.5) Therefore, the model we propose is based on (2.5) while the seasonal cycle is given by (2.4). An existing regime switching model for temperature dynamics referred in this study is that presented by Elias [9]. They modelled temperature time series dynamics by a two state regime switching model, where the temperature process switches between a mean-reverting process and a Brownian motion. The daily 5

8 deseasonalized temperature T d (t) at time t is given by { Tt,1 : d T d (t) = T t,1 = (L K T t,1 )dt + σ 1 dw t, if Td (t) is in regime 1, T t,2 : d T t,2 = µ 2 dt + σ 2 dw t, if Td (t) is in regime 2, (2.6) where K is the mean reversion speed, L K is the long term mean, µ 2 is the mean of the Brownian motion while σ 1 and σ 2 are the volatilities of the mean-reversion process and Brownian motion respectively, and W t is a Wiener process. The mean reversion speed, long term mean and volatility σ 1 in the base regime are taken as constants. Also the drift µ 2 and volatility σ 2 in the shifted regime are constants. The probability that the temperature will be in the base regime is p 1 and the probability that it will be in the shifted regime is p 2 where Recall that for an Ito process p 1 + p 2 = 1. dy t = µdt + σdw t, (2.7) and for any function G(t, y) that is twice differentiable, then the process ( G dg(t, Y t ) = t + µ G ) + σ2 2 G dt + σ G dw Y t 2 Yt 2 t (2.8) Y t is also an Ito process. Then, using Ito s Lemma to each of the independent processes in (2.6) we get the explicit solution for the Markov regime switching temperature dynamics. The base regime process becomes T t,1 = L ( K + T,1 L ) t e K(t s) + σ 1 e K(t s) dw s, (2.9) K while for the shifted regime we get T t,2 = T,2 + µ 2 t + t σ 2 dw s. (2.1) The volatility ˆσ 2 e = t σ2 1e 2K(t s) ds in (2.9) is fixed since K is a fixed value. 6

9 2.1 New Regime-switching Temperature Model The Markov regime switching model for evolution of temperature dynamics we propose, is a two state regime switching model consisting of a base regime governed by a mean-reverting process with heteroskedasticity and a shifted regime governed by a Brownian motion. The heteroskedasticity in the base regime, is introduced due to the fact that the volatility of temperature process varies with changes in the temperature at the given time. This kind of mean reverting process is derived from a broad class of mean-reverting constant elasticity variance processes developed by Jones [19] which are used for describing the dynamics of stochastic volatility given by dx t = κ(µ X t )dt + σx γ t dw t, (2.11) where γ > 1 when volatility increases with increase in prices and γ < 1 when volatility increases with a decrease in prices. The base regime for the two state Markov regime switching temperature model, assumes the process given in (2.11) with parameter γ = 1. Therefore the new regime-switching model for deseasonalized temperature dynamics model is given by { Tt,1 : d T t,1 = (µ 1 β T t,1 )dt + σ 1 Tt,1 dw t, if Td (t) is in regime 1, T d (t) = T t,2 : d T t,2 = µ 2 dt + σ 2 dw t, if T d (t) is in regime 2, (2.12) where β is the mean-reversion rate for the base regime mean-reverting stochastic process, µ 1 β is the long term mean on which the process reverts to, µ 2 is the mean of the shifted regime, while σ 1 Tt,1 and σ 2 are the volatilities of the base and shifted regimes respectively. The probabilities for the process to be in regime 1 and 2 are p 1 and p 2, respectively. Also here, {W t } t> is the standard Wiener process. Applying Ito s Lemma to (2.12) the integral form of the base regime is obtained by setting U t = T t,1 µ 1 β, where du t = βu t dt + σ 1 (U t + µ 1 )dw t. (2.13) Using the term e βt U t for developing the differentiation we get ( d(e βt U t ) = e βt σ 1 U t + µ ) 1 dw t. (2.14) β 7

10 Upon integration on (2.14) and substituting back U t = T t,1 µ 1, we get the β solution for the base regime as T t,1 = µ ( 1 β + T,1 µ ) t 1 e βt + σ 1 Ts,1 e β(t s) dw s. (2.15) β Similarly for the shifted regime its solution is given by T t,2 = T,2 + µ 2 t + t σ 2 dw s. (2.16) Thus, the general solution of the temperature regime switching process is now given by { Tt,1 : T d (t) = T ( ) t,1 = µ 1 + T,1 µ 1 e βt + t σ T β β 1 s,1 e β(t s) dw s, T t,2 : T t,2 = T,2 + µ 2 t + t σ 2dW s, (2.17) with probabilities p 1 and p 2 for the process to be in the base and shifted regimes, respectively. 3 Parameter estimation The problem of estimating parameters in a Markov regime-switching model is not a straight forward task because the regimes are not directly observable rather the switching process is controlled by the transition probabilities. Among the various methods used in the literature for regime switching parameter estimation is an optimization based approach called the expectation-maximization (EM) algorithm. The EM algorithm is an iterative statistical estimation method for model parameter estimation in incomplete data problems developed by Dempster [8] and its application in Markov regime switching parameters estimation was introduced by Hamilton [11] and Kim [2]. It is a two step algorithm consisting of an expectation step and a maximization step. The two-step iterative procedure alternates between the conditional expectation computation and solving the unconstrained optimization problem with respect to the set of unknown model parameters. In the expectation step, the expectation of likelihood function is computed by considering the missing variables as observable ones. In the maximization step, 8

11 the maximum likelihood estimation of the unknown parameters are computed by maximizing its expected likelihood function obtained in the expectation step. Thus, the EM algorithm is used for estimating the parameters of the temperature dynamic processes (2.6) and (2.12) using historical daily average temperature data. For the model specified by (2.6), its discretized version is given by T d (t) = { Tt,1, if S t = 1, T t,2, if S t = 2, (3.18) where the individual regimes are given by T t,1 = L + (1 K) T t 1,1 + σ 1 ɛ t,1 (3.19) and T t,2 = µ 2 + T t 1,2 + σ 2 ɛ t,2, (3.2) with random noise ɛ t,1 and ɛ t,2 for base and shifted regimes, respectively. For the model defined by (2.12), the discrete version is given by T d (t) = { Tt,1, if S t = 1, T t,2, if S t = 2, (3.21) where individual regimes corresponding to T t,1 and T t,2 are given by T t,1 = µ 1 + (1 β) T t 1,1 + σ 1 Tt 1,1 ɛ t,1 (3.22) and T t,2 = µ 2 + T t 1,2 + σ 2 ɛ t,2. (3.23) From (2.12) with a vector of unknown parameters λ = (µ 1, β, σ 1, p 1, p 2, µ 2, σ 2 ), the EM algorithm iteratively computes the expected value of the log-likelihood function Q(λ λ (n) ), and finding the new maximum likelihood estimate λ n+1 that maximizes the log likelihood function. For individual regime processes, the vector of unknown parameters are λ 1 = (µ 1, β, σ 1, p 1 ) for the base regime and λ 2 = (µ 2, σ 2, p 2 ) for the shifted regime. For convenience, the notation y t is used instead of T t,j, for t = 1, 2,, T. Using the initial guess for parameters λ (), the expectation step makes inference about the state process followed by computation of the conditional distribution of the states P (S t = j y 1, y 2,, y T ; λ), 9

12 for the process to be in regime j at time t. The complete data likelihood function for the vector of unknown parameters λ is given by L(λ; y t, S t ) = P (y t, S t λ) = T 2 1 (St=j)f (y t ; U j, V j ) p j, t=1 j=1 where U j and V j are the means and variances of the j th regime, p j is the probability of being in j th regime and 1 St is the indicator function. Based on these conditional probabilities obtained from the expectation step, the maximization step computes new maximum likelihood estimates of the vector of unknown parameters λ. 3.1 The Expectation Step Suppose Y t = (y 1, y 2,, y t ), and assume that λ (n) is the calculated vector of parameters in the maximization step in the n th iteration. The conditional probability distribution of the states S t computed are considered to be proportional to the height of conditional normal density function, weighted by its probability of being in the particular regime. Using the Bayes rule the probability distribution for a partition X i of the event space given Y is P (X i Y ) = P (Y X i)p (X i ) i P (Y A i)p (X i ). (3.24) The conditional distribution of S t is determined by j,t = P (S t = j y t ; λ (n) ) = p(n) j f ( y t S t = j; y t 1 ; λ (n)), (3.25) 2 p (n) j f (y t S t = j; y t 1 ; λ (n) ) j=1 for the time update values t = 1, 2,, T used to reference the observations or measurement update step. The function f ( y t S t = j; y t 1 ; λ (n)) represent the conditional density function of the underlying regime process j at time t. For the base regime of the Markov regime switching model given by (2.12) and its discrete version (3.22), has a drift coefficient µ 1 + (1 β) T t 1,1 and diffusion coefficient σ 1 Tt 1,1. The process 1

13 T d (t) given T d (t 1) corresponding to Y t given Y t 1 has a conditional normal distribution with mean µ 1 + (1 β)y t 1 and standard deviation σ 1 y t 1. Thus, the conditional probability density function for the base regime is given by f ( y t S t = 1; y t 1 ; λ (n)) = 1 2πσ (n) exp 1 y t 1 ( y t ( 1 β (n)) y t 1 µ (n) 1 ( ) 2 y 2 (3.26) Likewise, for the shifted regime with drift coefficient µ 2 and diffusion coefficient σ 2, the process Y t given Y t 1 will have a conditional normal distribution with mean µ 2 and standard deviation σ 2 whose conditional probability density function is given by f ( y t S t = 2; y t 1 ; λ (n)) = 1 2πσ (n) 2 exp ( 2 y t µ (n) 2 ( 2 σ (n) 2 σ (n) 1 ) 2 ) 2 t 1 ) 2. (3.27). The probabilities given by (3.25) are considered as the output of the expectation step called membership probabilities. The corresponding Q-function is given by Q(λ λ (n) ) = E [log L(λ; y t, S t )] 2 = j=1 t=1 H n j,t [ log(p (n) j 1 2 log(2πv (n) j ) 1 ] 2 (y t U (n) j ) 2 (V (n) j ) 1, (3.28) for the given set of unknown parameters λ (n), mean U j and variance V j in the j th regime. 3.2 The Maximization Step In the maximization step, the maximum likelihood estimate λ (n+1) for the vector of unknown parameters is computed by maximizing the expected log-likelihood function λ (n+1) = arg max Q(λ λ (n) ). (3.29) λ 11

14 The maximum likelihood estimates for the unknown parameters are computed as follows: To compute the estimates of p j with the constraint p 1 + p 2 = 1, we have which gives p (n+1) = arg max Q(λ λ (n) ) p {[ = arg max p t=1 1,t ] p (n+1) j = log p 1 + t=1 t=1 j,t 2 1 [ j,t t=1 2,t ] log p 2 }, (3.3). (3.31) For the base regime with probability density function given by (3.26), the loglikelihood function is log[l(λ (n) 1 ; y t, S t )] = β (n+1) = [ 1,t log p 1 log( 2πσ 1 y t 1 ) 1 ] 2 (y t (1 β)y t 1 µ 1 ) 2 σ1 2 yt 1 2. (3.32) Differentiating (3.32) with respect to each of the unknown parameters and equating to zero, we obtain 1,t y 1 1,t y t y y 2 t 1 (yt y t 1) t 1 µ (n+1) 1 = t 1 1,t yt 1 2 y 2 t 1 1,t y 2 t 1 1,t y2 t 1 1,t y 2 t 1, (3.33) 1,t yt 1(y 2 t (1 β (n+1) )y t 1 ), (3.34) 1,t yt

15 and ( σ (n+1) 1 ) 2 = 1,t y 2 t 1(y t (1 β (n+1) )y t 1 µ (n+1) 1 ) 2 1,t y 2 t 1. (3.35) Similarly, for the shifted regime with probability distribution given by (3.27) the log-likelihood function is log[l(λ (n) 2 ; y t, S t )] = [ 2,t log p 2 log( 2πσ 2 ) 1 ] 2 (y t µ 2 ) 2 σ2 2. (3.36) Differentiation of (3.36) with respect to each of the unknown parameters and equating to zero gives and ( σ (n+1) 2 µ (n+1) 2 = ) 2 = 2,t y t, (3.37) 2,t 2,t (y t µ (n+1) 2 ) 2 2,t. (3.38) Estimation of parameters for the seasonality process defined by Equation (2.4) is done by using the Gauss-Newton method. First (2.4) is rewritten as ( ) ( ) 2π 2π S d (t) = a 11 sin 365 t + a 22 cos 365 t + a 33 t + a 44, (3.39) and then fitted to daily average temperature data, where a = (a a 2 22), a 1 = 365 2π tan 1 ( a22 a 11 ), a 2 = a 33 and a 3 = a

16 3.3 Temperature Data sets The historical temperature data sets we use were made available by the Swedish Meteorology and Hydrology Institute (SMHI) 1 open data initiative. We have five hourly temperature data sets selected from different locations from south, central and north of Sweden. These include temperature data from Rynge, Linköping- Malmslätt, Stockholm-Bromma Airport, Mattmar and Tarfala for the period ranging from January 1998 to December 215. Rynge is located in the south of Sweden in Malmö town, Stockholm-Bromma and Mattmar are located in the central of Sweden but Mattmar is in the forest. Linköping-Malmslätt is between South and central of Sweden, while Tarfala is located to the far north of Sweden in Kiruna Municipality. The parameter estimation for both models are based on daily average temperature data for the period of January 1998 to December 21 for each location. For Malmslätt data set the conversion from hourly to daily average temperature is presented in Figure 1. The daily average temperature data for the period of January 22 to December 21 is used for comparing the performance of the two models Temperature [ C] Temperature [ C] Time(hours) Time(days) Figure 1: Daily average temperature data for Malmslätt (right) obtained from hourly data (left) from the same measurement station for the period of 1 st January, 1998 to 31 st December, 212. The conversion of data from hourly to daily has been done using the average of all 24 hour measurements and not just average of daily minimum and maximum. The red colour (right) shows the section of data used for parameter estimation from 1 st January, 1998 to 31 st December, Visit for access to a large range of weather related data. 14

17 The results of the estimation for unknown parameters for seasonality given by (2.4) are given in Table 1. The daily average temperature with seasonal cycle and deseasonalized daily temperature based on the seasonality estimates given in Table 1 are shown in Figure 2. Using the deseasonalized temperature data presented in Figure 2, the estimates for unknown parameters from (2.12) and (2.6) are given in Table 2. Simulated deseasonalized daily temperature for the new model and Elias model are given in Figure 3. Parameter a a 1 a 2 a 3 Estimates Table 1: Parameter estimates for seasonality, estimated by using Gauss-Newton method for daily average temperature time series from Malmslätt for the period January 1998 to December Temperature[ C] Deseasonalized Temperature Time(days) Time(days) Figure 2: Daily average temperature data plot for Malmslätt with seasonal cycle (left) and deseasonalized temperature data (right) after removing the seasonal cycle from daily average temperature for the period of 1 st January, 1998 to 31 st December, Results and Discussions Using the results of parameter estimates in Table 2, simulated daily average temperature values were generated using the two models. The simulated daily tem- 15

18 Parameter p 1 p 2 µ 1 β σ 1 µ 2 σ 2 Estimates Parameter p 1 p 2 L K σ 1 µ 2 σ 2 Estimates Table 2: Parameter estimates for the regime switching model we propose (top) and the Elias model (bottom). The parameters has been estimated using EM algorithm based on daily average temperature from Malmslätt for the period January 1998 to December Deseasonalized Temperature Deseasonalized Temperature Time Time Figure 3: Simulations of new model (left) and Elias model (right) for deseasonalized daily temperature values for the first 12 days. perature values were used to compare the performance of the two models based on sum of HDDs, CDDs and CAT for chosen months in winter and summer. The winter months for most of Swedish cities where HDDs index is used are October, November, December, January, February, March and April. In these months temperature is less than the reference temperature T ref which is 18 C in our case. We choose the months of December, January and February for comparing the expected HDDs of two models with actual HDDs from historical temperature data for the period of January 22 to December 21, see Figure 5 for temperature data from Malmsätt and Table 3 for temperature data sets from Stockholm- Bromma, Tarfala and Rynge. It is clear that sum of expected HDDs for the new model in all three months from 5 are close to the true HDDs from real data compared to the sum of expected HDDs for Elias model. Similarly, Table 3 shows that expected HDDs from new model are 16

19 the closest (bold values) for all chosen locations, except for month of January and February in Tarfala measurement station, where expected HDDs from Elias model are closer to the true HDDs compared to our model. The accumulated expected HDDs from Elias model are smaller compared to both accumulated HDDs from our model and true HDDs from real data. Likewise, the daily average temperature for the two models are compared to real data shown in Figure 4. The simulated daily average temperature for the new model are closer to the real data compared to simulated daily average temperature for Elias model, that are higher than real data. Thus, it can deduced that our model is a relatively better representation of temperature dynamics compared to Elias model. 4 3 Real data New Model 5 4 Real data Elias Model Daily Average Temperature Daily Average Temperature Days Days Figure 4: Simulated daily average temperature of new model with real data (left) and Elias model with real data (right) for the period of January 22 to December 21. The summer months where CDDs index can be used are May, June, July, August and September. In this months daily average temperature is expected to be above the reference temperature T ref. We select months of June, July and August for comparing the sum of expected CDDs from the two models to the actual CDDs from real data for the period of 9 years, see Figure 6. It can be observed that sum of expected CDDs for the new model are closer to the real data compared to sum of CDDs from Elias model for almost all years. Therefore, it can also be deduced that, the new model captures temperature dynamics relatively better than Elias model. On the other hand it can be observed that the sum of CDDs shown in Figure 6 17

20 8 December 8 January 8 February Sum of HDDs Sum of HDDs Sum of HDDs Years Years Years Figure 5: Comparison of sum of Heating Degree days for the months (December, January and February) for the period January 22 to December 21. The colours blue, green and yellow represent real data, new model and Elias model accumulated HDDs. for the true data were very small or even zero. This imply that real daily average temperature data from Malmslätt were above the reference temperature T ref with very small margin for some years or completely less than T ref for other years. This implies that there were cool summer for areas around Malmslätt. This results justifies the use cumulative average temperature (CAT) instead of CDDs for European cities in summer for pricing weather derivatives contracts based on temperature. The expected CAT for the months of June, July and August from the two models are also compared to the true CAT from historical data, see Figure 7 for temperature data from Malmslätt and Table 3 for historical temperature data from Stockholm-Bromma, Tarfala and Rynge. It can be observed that the CAT from the new model is relatively closer to the true CAT compared to CAT from Elias model. Hence, our model is a relatively good representation of temperature dynamics compared to Elias model. The results indicates that our model is a good representation of the temperature dynamics since it produces temperature indices that are relatively close to the true temperature indices. Furthermore, weather is local and a particular model can sometimes be favourable to a particular area. The Elias model was developed for Canadian data, using it for Swedish data could cause it to be less accurate compared to our model. Also lattice construction method (pentanomial lattice in particular) was used for parameter estimation in Elias model, we instead used the EM algorithm for parameter estimation for both models. Therefore, both models 18

21 Bromma Tarfala Rynge August July February January August July February January August July February January Year Real New Elias Real New Elias Real New Elias Real New Elias Real New Elias Real New Elias Real New Elias Real New Elias Real New Elias Real New Elias Real New Elias Real New Elias Table 3: Comparison accumulated HDDs (for month of January and February) and CAT (for months of July and August) for real data from Stockholm-Bromma, Tarfala and Rynge and the two models, for the period of January 25 to December 21. The bold values from the models are closest values to the real data. 19

22 4 June 45 July 35 August Sum of CDDs Sum of CDDs 25 2 Sum of CDDs Years Years Years Figure 6: Comparison of sum of CDDs for Malmslätt temperature data and the two models for the months of June, July and August for the period of January 21 to December 21. The colours blue, green and yellow represent real data, new model and Elias model accumulated CDDs. 1 June 1 July 9 August CAT 5 CAT 5 CAT Years Years Years Figure 7: Comparison of CAT for Malmslätt temperature data and the two models for the months of June, July and August for the period of January 21 to December 21. The colours blue, green and yellow represent real data, new model and Elias model CAT. can be considered to be good for modelling temperature dynamics because of its locality nature. But for purpose of comparison, we have used the same data sets and the same method for parameter estimation for both models. Thus, in this paper we can still conclude that our model is relatively good for modelling temperature dynamics. 2

23 5 Conclusion This study investigates the problem of modelling weather dynamics, particularly temperature dynamics for the purpose of weather derivatives pricing. Though the study has not gone as far as pricing weather derivatives, the main goal at hand was to find the appropriate model for temperature dynamics that could be used for pricing weather derivatives contracts based on temperature indices such as HDDs, CDDs and CAT. In this study we have proposed a regime-switching model for temperature dynamics that is compared with a regime switching model proposed by Elias [9]. Based on the results presented in Section 4, on accumulated HDDs, CDDs, CAT and the corresponding daily average temperature, the proposed model for temperature dynamics gives relatively better results compared to an existing regime-switching model given by Elias. Though our proposed model gives fairly good results, but like other models it is based on some assumption like Gaussian assumption for Gaussian mixture during parameter estimation. This might reduce the accuracy of prediction results as observed in Table 3 for Tarfala measurement station. The true CAT from real data for July and August for all years in Tarfala were negative and our model produced positive CAT values. This means that, the areas around Tarfala has cold summer and both models have failed to capture this extreme temperature dynamics. This can be solved possibly by introducing timedependent parameters in the model to capture the extreme temperature variations. References [1] Peter Alaton, Boualem Djehiche, and David Stillberger. On modelling and pricing weather derivatives. Applied Mathematical Finance, 9(1):1 2, 22. [2] Fred Espen Benth and Jurate Saltyte Benth. The volatility of temperature and pricing of weather derivatives. Quantitative Finance, 7(5): , 27. [3] Nicolas P. B. Bollen. Valuing options in regime-switching models. Derivatives, 6:38 49,

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