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Department of Mathematics

Regime Switching models on

Temperature Dynamics

Emanuel Evarest, Fredrik Berntsson, Martin Singull and

Wilson Charles

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Department of Mathematics

Link¨

oping University

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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 tem-perature dynamics is compared using historical daily average temtem-perature data from five weather stations across Sweden. The comparison is based on the heating degree days (HDDs), cooling degree days (CDDs) and cu-mulative 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 corre-sponding daily average plots demonstrate that, our model captures tempera-ture dynamics relatively better than Elias model.

Key words. Weather derivatives, Regime switching, temperature dynamics, expectation-maximization, temperature indices

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1

Introduction

Weather derivatives business came into existence in the 1990s, where one of its

strongest cause being the El Ni˜no winter of 1997-98. 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 them-selves 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 re-sults 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 protec-tion against these weather related risks. Weather derivatives cover low-risk, high-probability 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 unpre-dictable 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 [10, 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

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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 in-duces the regime shift behavior in weather of a particular location or region. For instance, weather regimes can be reflected by different abrupt changes in temper-ature behaviour of a specified location due to urbanization and changes of mea-surement 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 en-ergy 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 dy-namics 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 gov-erned by the latent random variable. The individual regimes of the temperature process are assumed to be independent of each other but joined together by mem-bership probabilities.

Parameters for the model we propose together with that of Elias model are esti-mated 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

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chosen initial values and moves quickly to a likelihood surface region [17, 21]. Since temperature time series are used for generating indices for weather deriva-tives 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, temper-ature 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 switch-ing model we propose here assumes that the daily average temperature switches between the two regimes. The base regime is governed by a mean-reversion het-eroskedastic 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 Stof 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 =    p11 · · · p1N .. . . .. ... pN 1 · · · pN N   , (2.1)

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where one step transition probabilities pij of the process Xnare given by pij = P {Xt+1 = j|Xt= i}, i, j = 1, 2, · · · , N, (2.2) 0 ≤ pij ≤ 1 and N X j=1 pij = 1.

In almost all available literature on weather dynamics modelling and weather derivatives pricing, the presence of predictable annual seasonal component in tem-perature 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 Td(t) and

defined as

Td(t) = ˜Td(t) + sd(t), (2.3)

where ˜Td(t) is the deseasonalized temperature and sd(t) is the seasonal

compo-nent. The seasonality in this case can be defined as the sum of a linear function and trigonometric function as

sd(t) = a0sin

 2π

365(t − a1)



+ a2t + a3, (2.4)

where the constants a0 and a1 defines the amplitude of temperature and phase

angle, while the constants a2 and a3 represent the coefficient and constant of the

linearity of the seasonal trend respectively.

The deseasonalized temperature ˜Td(t) is obtained by removing the seasonal

com-ponent from (2.3), i.e.

˜

Td(t) = Td(t) − sd(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 dy-namics by a two state regime switching model, where the temperature process switches between a mean-reverting process and a Brownian motion. The daily

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deseasonalized temperature ˜Td(t) at time t is given by ˜ Td(t) = ( ˜Tt,1 : d ˜Tt,1 = (L − K ˜Tt,1)dt + σ1dWt, if T˜d(t) is in regime 1, ˜ Tt,2 : d ˜Tt,2 = µ2dt + σ2dWt, if T˜d(t) is in regime 2, (2.6)

where K is the mean reversion speed, KL is the long term mean, µ2 is the mean

of the Brownian motion while σ1 and σ2 are the volatilities of the mean-reversion

process and Brownian motion respectively, and Wtis a Wiener process.

The mean reversion speed, long term mean and volatility σ1 in the base regime

are taken as constants. Also the drift µ2and volatility σ2 in the shifted regime are

constants. The probability that the temperature will be in the base regime is p1

and the probability that it will be in the shifted regime is p2 where

p1+ p2 = 1.

Recall that for an Ito process

dYt = µdt + σdWt, (2.7)

and for any function G(t, y) that is twice differentiable, then the process

dG(t, Yt) =  ∂G ∂t + µ ∂G ∂Yt +σ 2 2 ∂2G ∂Y2 t  dt + σ∂G ∂Yt dWt (2.8)

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

˜ Tt,1 = L K +  ˜ T0,1− L K  e−K(t−s)+ Z t 0 σ1e−K(t−s)dWs, (2.9)

while for the shifted regime we get ˜ Tt,2 = ˜T0,2+ µ2t + Z t 0 σ2dWs. (2.10) The volatility ˆσ2 e = Rt 0 σ 2 1e

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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 gov-erned 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 pro-cess 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

dXt= κ(µ − Xt)dt + σXtγdWt, (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 deseason-alized temperature dynamics model is given by

˜ Td(t) = ( ˜Tt,1 : d ˜Tt,1 = (µ1− β ˜Tt,1)dt + σ1T˜t,1dWt, if ˜Td(t) is in regime 1, ˜ Tt,2 : d ˜Tt,2 = µ2dt + σ2dWt, if ˜Td(t) is in regime 2, (2.12) where β is the mean-reversion rate for the base regime mean-reverting stochastic

process, µ1

β is the long term mean on which the process reverts to, µ2 is the mean

of the shifted regime, while σ1T˜t,1and σ2are the volatilities of the base and shifted

regimes respectively. The probabilities for the process to be in regime 1 and 2 are

p1 and p2, respectively. Also here, {Wt}t>0is the standard Wiener process.

Applying Ito’s Lemma to (2.12) the integral form of the base regime is obtained

by setting Ut= Tt,1− µβ1, where

dUt= −βUtdt + σ1(Ut+ µ1)dWt. (2.13)

Using the term eβtUtfor developing the differentiation we get

d(eβtUt) = eβtσ1  Ut+ µ1 β  dWt. (2.14)

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Upon integration on (2.14) and substituting back Ut = Tt,1 − µβ1, we get the

solution for the base regime as ˜ Tt,1 = µ1 β +  ˜ T0,1− µ1 β  e−βt+ Z t 0 σ1T˜s,1e−β(t−s)dWs. (2.15)

Similarly for the shifted regime its solution is given by ˜

Tt,2 = ˜T0,2+ µ2t +

Z t

0

σ2dWs. (2.16)

Thus, the general solution of the temperature regime switching process is now given by ˜ Td(t) = ( ˜T t,1 : ˜Tt,1 = µβ1 + ˜T0,1− µβ1  e−βt+R0tσ1T˜s,1e−β(t−s)dWs, ˜ Tt,2 : ˜Tt,2 = ˜T0,2+ µ2t + Rt 0 σ2dWs, (2.17)

with probabilities p1 and p2 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 op-timization based approach called the expectation-maximization (EM) algorithm. The EM algorithm is an iterative statistical estimation method for model param-eter 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 [20]. It is a two step algorithm consisting of an expec-tation 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,

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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

˜ Td(t) = ( ˜Tt,1, if St= 1, ˜ Tt,2, if St= 2, (3.18) where the individual regimes are given by

˜

Tt,1 = L + (1 − K) ˜Tt−1,1+ σ1t,1 (3.19)

and

˜

Tt,2 = µ2+ ˜Tt−1,2+ σ2t,2, (3.20)

with random noise t,1 and t,2 for base and shifted regimes, respectively. For the

model defined by (2.12), the discrete version is given by ˜ Td(t) = ( ˜Tt,1, if St= 1, ˜ Tt,2, if St= 2, (3.21)

where individual regimes corresponding to ˜Tt,1 and ˜Tt,2are given by

˜

Tt,1 = µ1+ (1 − β) ˜Tt−1,1+ σ1T˜t−1,1t,1 (3.22)

and

˜

Tt,2 = µ2+ ˜Tt−1,2+ σ2t,2. (3.23)

From (2.12) with a vector of unknown parameters λ = (µ1, β, σ1, p1, p2, µ2, σ2),

the EM algorithm iteratively computes the expected value of the log-likelihood

function Q(λ|λ(n)), and finding the new maximum likelihood estimate λn+1 that

maximizes the log likelihood function. For individual regime processes, the vector

of unknown parameters are λ1 = (µ1, β, σ1, p1) for the base regime and λ2 =

(µ2, σ2, p2) for the shifted regime. For convenience, the notation ytis used instead

of ˜Tt,j, for t = 1, 2, · · · , T .

Using the initial guess for parameters λ(0), the expectation step makes inference

about the state process followed by computation of the conditional distribution of the states

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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; yt, St) = P (yt, St|λ) = T Y t=1 2 X j=1 1(St=j)f (yt; Uj, Vj) pj,

where Ujand Vjare the means and variances of the jthregime, pjis the probability

of being in jthregime and 1St 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 Yt = (y1, y2, · · · , yt), and assume that λ(n) is the calculated vector of

parameters in the maximization step in the nth iteration. The conditional

proba-bility distribution of the states St 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 Xi of the event space given Y is

P (Xi|Y ) =

P (Y |Xi)P (Xi)

P

iP (Y |Ai)P (Xi)

. (3.24)

The conditional distribution of Stis determined by

Hj,t(n)= P (St = j|yt; λ(n)) = p(n)j f yt|St= j; yt−1; λ(n)  2 P j=1 p(n)j f (yt|St= j; yt−1; λ(n)) , (3.25)

for the time update values t = 1, 2, · · · , T used to reference the observations or measurement update step.

The function f yt|St = j; yt−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

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˜

Td(t) given ˜Td(t − 1) corresponding to Yt given Yt−1 has a conditional normal

distribution with mean µ1+ (1 − β)yt−1and standard deviation σ1yt−1. Thus, the

conditional probability density function for the base regime is given by

f yt|St= 1; yt−1; λ(n) = 1 √ 2πσ1(n)yt−1 exp      −  yt− 1 − β(n) yt−1− µ (n) 1 2 2σ1(n)2y2 t−1      . (3.26)

Likewise, for the shifted regime with drift coefficient µ2and diffusion coefficient

σ2, the process Yt given Yt−1 will have a conditional normal distribution with

mean µ2and standard deviation σ2 whose conditional probability density function

is given by f yt|St= 2; yt−1; λ(n) = 1 √ 2πσ(n)2 exp      −  yt− µ (n) 2 2 2σ(n)2 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; yt, St)] = 2 X j=1 T X t=1 Hj,tn  log(p(n)j − 1 2log(2πV (n) j ) − 1 2(yt− U (n) j ) 2(V(n) j ) −1  , (3.28)

for the given set of unknown parameters λ(n), mean Uj and variance Vj in the jth

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

λ

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The maximum likelihood estimates for the unknown parameters are computed as

follows: To compute the estimates of pj with the constraint p1+ p2 = 1, we have

p(n+1)= arg max p Q(λ|λ(n)) = arg max p (" T X t=1 H1,t(n) # log p1+ " T X t=1 H2,t(n) # log p2 ) , (3.30) which gives p(n+1)j = T P t=1 Hj,t(n) T P t=1 2 P 1 Hj,t(n) . (3.31)

For the base regime with probability density function given by (3.26), the log-likelihood function is log[L(λ(n)1 ; yt, St)] = T X t=2 H1,t(n) h log p1− log( √ 2πσ1yt−1) − 1 2(yt− (1 − β)yt−1− µ1) 2 σ1−2yt−1−2i. (3.32) Differentiating (3.32) with respect to each of the unknown parameters and equat-ing to zero, we obtain

β(n+1)= T P t=2 H1,t(n)yt−1−1  yt− yt−1 T P t=2 H1,t(n)y−2t−1(yt−yt−1) T P t=2 H1,t(n)y−2t−1   T P t=2 H1,t(n)yt−1−2  y2 t−1− T P t=2 H1,t(n)y2t−1 T P t=2 H1,t(n)y−2t−1   , (3.33) µ(n+1)1 = T P t=2 H1,t(n)yt−1−2(yt− (1 − β(n+1))yt−1) T P t=2 H1,t(n)yt−1−2 , (3.34)

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and  σ1(n+1) 2 = T P t=2 H1,t(n)yt−1−2(yt− (1 − β(n+1))yt−1− µ (n+1) 1 )2 T P t=2 H1,t(n)y−2t−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 ; yt, St)] = T X t=2 H2,t(n)  log p2− log( √ 2πσ2) − 1 2(yt− µ2) 2σ−2 2  . (3.36) Differentiation of (3.36) with respect to each of the unknown parameters and equating to zero gives

µ(n+1)2 = T P t=2 H2,t(n)yt T P t=2 H2,t(n) , (3.37) and  σ(n+1)2  2 = T P t=2 H2,t(n)(yt− µ (n+1) 2 )2 T P t=2 H2,t(n) . (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

˜ Sd(t) = a11sin  2π 365t  + a22cos  2π 365t  + a33t + a44, (3.39)

and then fitted to daily average temperature data, where

a0 = q (a2 11+ a222), a1 = 365 2π tan −1 a22 a11  , a2 = a33 and a3 = a44.

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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 rang-ing from January 1998 to December 2015. Rynge is located in the south of Swe-den 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 aver-age temperature data for the period of January 1998 to December 2001 for each location. For Malmslätt data set the conversion from hourly to daily average tem-perature is presented in Figure 1. The daily average temtem-perature data for the period of January 2002 to December 2010 is used for comparing the performance of the two models.

Time(hours) ×104 0 2 4 6 8 10 12 14 Temperature [°C] -30 -20 -10 0 10 20 30 40 Time(days) 0 1000 2000 3000 4000 5000 6000 Temperature [°C] -25 -20 -15 -10 -5 0 5 10 15 20 25

Figure 1: Daily average temperature data for Malmslätt (right) obtained from

hourly data (left) from the same measurement station for the period of 1stJanuary,

1998 to 31st December, 2012. 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 1stJanuary, 1998 to 31stDecember, 2001.

1Visit http://opendata-download-metobs.smhi.se/explore/ for access to a large range of weather

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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 a0 a1 a2 a3

Estimates 9.5786 78.6415 −4.4 × 10−5 6.9360

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 2001. Time(days) 0 1000 2000 3000 4000 5000 6000 Temperature[° C] -25 -20 -15 -10 -5 0 5 10 15 20 25 Time(days) 0 1000 2000 3000 4000 5000 6000 Deseasonalized Temperature -25 -20 -15 -10 -5 0 5 10 15

Figure 2: Daily average temperature data plot for Malmslätt with seasonal cy-cle (left) and deseasonalized temperature data (right) after removing the seasonal

cycle from daily average temperature for the period of 1st January, 1998 to 31st

December, 2012.

4

Results and Discussions

Using the results of parameter estimates in Table 2, simulated daily average perature values were generated using the two models. The simulated daily

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tem-Parameter p1 p2 µ1 β σ1 µ2 σ2

Estimates 0.9630 0.0370 3.0104 6.4060 2.2420 0.0185 0.0651

Parameter p1 p2 L K σ1 µ2 σ2

Estimates 0.9693 0.0307 0.0921 0.7087 2.2208 5.0185 4.1056

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 2001. Time 0 500 1000 1500 2000 2500 3000 3500 4000 Deseasonalized Temperature -30 -20 -10 0 10 20 30 Time 0 500 1000 1500 2000 2500 3000 3500 4000 Deseasonalized Temperature -10 -5 0 5 10 15 20 25 30 35 40

Figure 3: Simulations of new model (left) and Elias model (right) for deseasonal-ized daily temperature values for the first 1200 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 Octo-ber, NovemOcto-ber, DecemOcto-ber, January, February, March and April. In these months

temperature is less than the reference temperature Tref 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 2002 to December 2010, see Figure 5 for temper-ature data from Malmsätt and Table 3 for tempertemper-ature 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

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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.

Days

0 500 1000 1500 2000 2500 3000 3500

Daily Average Temperature

-30 -20 -10 0 10 20 30 40 Real data New Model Days 0 500 1000 1500 2000 2500 3000 3500

Daily Average Temperature

-30 -20 -10 0 10 20 30 40 50 Real data Elias Model

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 2002 to December 2010.

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 Tref. 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.

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Years 1 2 3 4 5 6 7 8 9 Sum of HDDs 0 100 200 300 400 500 600 700 800 December Years 1 2 3 4 5 6 7 8 9 Sum of HDDs 0 100 200 300 400 500 600 700 800 January Years 0 1 2 3 4 5 6 7 8 9 10 Sum of HDDs 0 100 200 300 400 500 600 700 800 February

Figure 5: Comparison of sum of Heating Degree days for the months (Decem-ber, January and February) for the period January 2002 to December 2010. 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 Tref with

very small margin for some years or completely less than Tref for other years.

This implies that there were cool summer for areas around Malmslätt. This re-sults justifies the use cumulative average temperature (CAT) instead of CDDs for European cities in summer for pricing weather derivatives contracts based on tem-perature.

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 tem-perature data from Malmslätt and Table 3 for historical temtem-perature 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 dy-namics 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

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Year 2005 2006 2007 2008 2009 2010 Bromma January Real 528.8583 628.3250 571.6208 452.1583 553.7500 718.0167 New 474.5424 586.7842 616.7580 544.8745 619.6845 591.8758 Elias 530.9621 431.9143 309.2512 264.1144 267.9123 289.8013 February Real 669.9917 665.5396 594.4875 461.2333 586.9333 727.7563 New 628.5344 591.4825 563.5050 548.7692 566.6199 579.3780 Elias 418.9181 213.2231 425.9431 352.4516 317.7124 220.1828 July Real 526.0417 593.7250 531.1625 552.2000 546.5625 595.1750 New 526.2811 516.6843 526.0414 533.3943 525.5319 534.0315 Elias 895.4957 900.8121 750.9305 762.9610 814.5187 877.3199 August Real 492.3146 513.2500 462.0979 452.7500 482.8458 450.9333 New 496.4368 477.0538 522.3635 499.4363 516.7369 548.9360 Elias 699.4653 995.3949 677.9366 897.6459 886.4392 773.1009 T arf ala January Real 638.6958 543.2167 318.2563 331.3500 268.4000 288.3312 New 776.2425 902.7058 949.3571 901.8355 970.4183 956.5738 Elias 855.8026 766.1469 658.1817 625.0899 640.9328 674.8668 February Real 468.8500 433.7188 281.1792 330.8167 376.8813 427.4500 New 926.0711 907.5581 890.2130 873.9234 920.8313 943.3380 Elias 736.2952 542.3220 767.4102 705.9637 682.7097 597.7849 July Real -92.1458 -224.2708 -275.9292 -280.7375 -365.3667 148.3083 New 149.8269 128.1850 125.4971 120.8051 100.8976 97.3522 Elias 519.0414 512.3128 350.3862 350.3718 389.8844 440.6406 August Real -277.9417 -170.3208 -239.3125 -280.6854 -412.6625 279.4861 New 110.9061 79.4781 112.7428 77.7706 83.0262 103.1803 Elias 313.9346 597.8192 268.3159 475.9802 452.7285 327.3452 Rynge January Real 589.4000 584.5729 488.3438 419.9937 539.5458 578.8375 New 454.3665 571.8572 592.0687 487.8539 572.8983 539.9201 Elias 445.0115 344.9808 210.1606 158.4538 158.2954 171.0007 February Real 457.6333 528.0896 354.2875 444.7458 427.4687 393.8208 New 504.1896 443.2037 430.6282 458.8833 418.9252 430.9198 Elias 284.3673 101.9151 278.2523 203.6967 168.3619 55.7667 July Real 491.9521 479.2167 462.0625 481.3979 494.7479 438.0042 New 498.9187 495.8919 511.8190 525.7419 524.4495 539.5191 Elias 868.1333 880.0197 736.7081 755.3086 813.4363 882.8075 August Real 384.7188 454.8250 350.3569 334.4500 322.8458 320.9292 New 422.6735 409.8605 461.7402 445.3830 469.2536 508.0227 Elias 625.7020 928.2016 617.3133 843.5926 838.9559 732.1876

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 2005 to Decem-ber 2010. The bold values from the models are closest values to the real data.

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Years 0 1 2 3 4 5 6 7 8 9 10 Sum of CDDs 0 50 100 150 200 250 300 350 400 June Years 0 1 2 3 4 5 6 7 8 9 10 Sum of CDDs 0 50 100 150 200 250 300 350 400 450 July Years 0 1 2 3 4 5 6 7 8 9 10 Sum of CDDs 0 50 100 150 200 250 300 350 August

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 2001 to December 2010. The colours blue, green and yellow represent real data, new model and Elias model accumulated CDDs.

Years 1 2 3 4 5 6 7 8 9 CAT 0 100 200 300 400 500 600 700 800 900 1000 June Years 0 1 2 3 4 5 6 7 8 9 10 CAT 0 100 200 300 400 500 600 700 800 900 1000 July Years 0 1 2 3 4 5 6 7 8 9 10 CAT 0 100 200 300 400 500 600 700 800 900 August

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 2001 to Decem-ber 2010. 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.

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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 time-dependent parameters in the model to capture the extreme temperature variations.

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