Resource Economy

Pricing Weather Derivatives Index based on Temperature: The Case of Bahir Dar, Ethiopia

  • Tesfahun BERHANE , * ,
  • Aemiro SHIBABAW ,
  • Gurju AWGICHEW
  • Department of mathematics, Bahir Dar University, Bahir Dar, P.O. Box79, Ethiopia
*Corresponding author: Tesfahun Berhane, E-mail:

Received date: 2018-11-16

  Accepted date: 2019-03-22

  Online published: 2019-07-30


All rights reserved


In this paper we present a stochastic model for daily average temperature to calculate the temperature indices upon which temperature-based derivatives are written. We propose a seasonal mean and volatility model that describes the daily average temperature behavior using the mean-reverting Ornstein-Uhlenbeck process. We also use higher order continuous-time autoregressive process with lag 3 for modeling the time evolution of the temperatures after removing trend and seasonality. Our model is fitted to 11 years of data recorded, in the period 1 January 2005 to 31 December 2015, Bahir Dar, Ethiopia, obtained from Ethiopia National Meteorological Services Agency. The analytical approximation formulas are used to price heating degree days (HDD) and cooling degree days (CDD) futures. The suggested model is analytically tractable for derivation of explicit prices for CDD and HDD futures and option. The price of the CDD future is calculated, using analytical approximation formulas. Numerical examples are presented to indicate the accuracy of the method. The results show that our model performs better to predict CDD indices.

Cite this article

Tesfahun BERHANE , Aemiro SHIBABAW , Gurju AWGICHEW . Pricing Weather Derivatives Index based on Temperature: The Case of Bahir Dar, Ethiopia[J]. Journal of Resources and Ecology, 2019 , 10(4) : 415 -423 . DOI: 10.5814/j.issn.1674-764X.2019.04.008

1 Introduction

Weather has a significant effect on different economic activities. Now days, it becomes one of the most uncontrollable and main sources of risk for business activities like energy, agricultural, tourism, leisure, construction industries and others. Weather derivative is now using as an important risk management tool to reduce the financial loss of businesses sectors that is caused by unexpected fluctuation in weather condition. It is usually designed to protect the companies from non-catastrophic weather events like warmer or colder than the usual periods, rainy or dry periods, etc. It works just like most other derivative contracts except that the asset in weather derivative is a weather variable such as rainfall, temperature or snowfall has no value with which to price the derivative contract.
The idea of weather derivative market was first introduced by US energy sectors in 1997 to hedge their seasonal weather risk due to the risk of significant earnings decline,as a result of the severe weather events of EI Nino that occurred during winter of 1997-1998 (Alaton et al., 2002). Over the last two decades, weather derivative has been expanded more rapidly and at this moment, countries like United States, Australia, Canada, Europe and Japan use weather derivative as a risk management strategy to hedge various business activities (Benth and Saltyte-Benth, 2007). Though the market is well established in developed countries, weather derivatives in Africa countries are still underdeveloped. So far, South Africa, Morocco, Ghana and Kenya are countries where there are weather derivative deals taking place in Africa (Mraoua and Bari, 2007).
In this paper we model the dynamics of temperature using Ornstein-Uhlenbeck process approach and then we calculate the price of weather derivative based on different temperature indices. We know that the payoff temperature contract depends on the cumulative temperature indices over the contract period. So, to calculate reasonable pricefor different type of temperature contract like future and option, we need to have a model that accurately characterizes the dynamics of temperature. When we develop a mathematical model describing temperature movements, the basic features of temperature such as trend due to green house and urbanization, seasonality of mean temperature, autoregressive properties for temperature changes, and seasonal variations in volatility should be taken into consideration in order to get precise model. In the last two decades several researches have been conducted for studying temperature processes, constructing and pricing weather derivatives written on temperature. Dornier and Querel (2000) suggested Ornstein-Uhlenbeck (OU) process to model Chicago temperature data and the volatility has been considered as a constant. Alaton et al. (2002) applied the Ornstein-Uhlenbeck process to model temperature data the Stockholm, Sweden. The volatility of the temperature process varies through different months and it has been modeled as a piece-wise function with constant value each month. They price different temperature indices based on the resulting model. Zhu et al. (2012) used the same model as Alaton P’ model (Alaton et al., 2002) for modelling daily temperature in a dry region of China in order to price drought option through stochastic simulation. (Mraoua and Bari, 2007) presented a price model for weather derivatives with payouts depending on temperature indices. They have used a mean-reverting stochastic process to characterize dynamics of the temperature data recorded for 44 years on the area of Casablanca, Morocco. Their model is easily tractable to simulate a temperature swap contract. Fred E. Benth and Saltyte-Benth (2005) suggested a mean- reverting Ornstein-Uhlenbeck process driven by generalized hyperbolic Levy process and having seasonal mean and volatility to model the daily average temperature variations. Benth and Saltyte-Benth (2007) proposed Ornstein-Uhlenbeck process with seasonal volatility to model the time dynamics of daily average temperature for Stockholm. They used a truncated Fourier series to model the seasonal volatility, and validate their model on more than 40 years of daily data collected from Stockholm, Sweden. They also provide explicit formulas for options written on heating, cooling degree-days and cumulative average temperature (CAT) futures.
Campbell and Diebold (2005) apply a time series ap-proach to model temperature in U.S, including trend sea-sonality represented by a low-ordered Fourier series andcyclical patterns represented by autoregressive lags. Thecontributions to conditional variance dynamics are comingfrom seasonal and cyclical components. Zapranis andAlexandridis (2009) used neural networks to model theseasonal component of the residual variance of a mean- reverting Ornstein-Uhlenbeck temperature process. In addi-tion, they have used wavelet analysis to determine the exactstructure of the models representing both the seasonalitycomponent and the volatility of the temperature data. Themodel was estimated on daily average temperature data col-lected from Paris Ahmet Goncu (2011) proposed a mean-reverting Ornstein-Uhlenbeck (O-U) process with seasonal-ity in long-run mean and volatility to model the daily aver-age temperature of Beijing, Shanghai and Shenzhen, China.To calculate HDD and CDD options prices, he used both theanalytical approximation formula and Monte Carlo methods.Wang et al. (2015) proposed a seasonal mean and volatilitymodel that describes the daily average temperature behaviorusing the mean-reverting Ornstein-Uhlenbeck process. Theyused a daily average temperature data recorded for 62 yearsfrom Zhengzhou meteorological station, China to estimatethe values of parameters involving the proposed model andthe Monte Carlo method for the HDD call pricing. Birhanand Azize (2017) proposed a temperature model which isdesigned as a mean-reverting process driven by a Levyprocess to represent jumps and other features of temperature.The temperature model is a mean reverting Levy process.The Levy part contains a Brownian motion and two meanreverting jump processes driven by compound Poissonprocesses.
In this work we propose a seasonal volatility model that estimates daily average temperature measurements using the mean-reverting Ornstein-Uhlenbeck model and we derive a pricing dynamic for future. The paper is organized as follows. In section two we present our proposed stochastic model for the daily temperature variation, and analyze it empirically based on our temperature data. Section three is focused on the derivation of futures prices for different temperature indices such as HDD and CDD and then price these options using analytic approximation formula. Section four discusses findings of our research and concluding remarks.

2 Model of temperature process

Several different models have been developed to characterize the dynamics of a temperature process. Usually temperatures exhibit seasonality in all of the mean, variance, distribution, and autocorrelation, and there is evidence of long memory in the autocorrelation. So, when we modeling temperature data, we have to give more attention to all possible features of temperature because small miss-specification in the models can lead to large mispricing of the temperature contracts (Jewson S and Brix A, 2005). The stochastic process on which we build our analysis is the mean-reverting process with seasonality in the level and volatility as suggested by Benth and Saltyte-Benth (2005) and to fit our stochastic model to the data, we consider higher order continuous-time autoregressive process to capture the autoregressive properties for temperature change. To model temperature we need to decompose the time series into different components like trend, seasonality, an AR process and residual term, and then we estimate the parameters of each of these components step-by-step that is one after the others. This approach helps us to understand basic feature of the data structure and is likely able to come up with a precise model. We propose the following a mean reverting Ornstein-Uhlenbeck model for the time evolution of temperatures (Fred E. Benth and Saltyte-Benth, 2005).
$\text{d}T\left( t \right)=\text{d}S\left( t \right)+\alpha \left( S\left( t \right)-T\left( t \right) \right)\text{d}t+\sigma \left( t \right)\text{d}B\left( t \right)$ (1)
Where T(t) is the daily average temperature, B(t) is a noise deriving process, S(t) is a deterministic function modeling the trend and seasonality of the average temperature, and σ(t) is the daily volatility of temperature variations, $\alpha $ is a speed of mean reversion. The seasonal component S(t) and the volatility σ(t) in equation (1) can be formulated as a truncated Fourier series, that is:
$S\left( t \right)=a+bt+{{a}_{0}}+\underset{i=1}{\overset{{{I}_{1}}}{\mathop \sum }}\,{{a}_{i}}\sin \left( \frac{2i\pi t}{365} \right)+\underset{j=1}{\overset{{{J}_{1}}}{\mathop \sum }}\,{{b}_{j}}\cos \left( \frac{2i\pi t}{365} \right)$ (2)
$\sigma \left( t \right)=c+\underset{i=1}{\overset{{{I}_{1}}}{\mathop \sum }}\,{{c}_{i}}\sin \left( \frac{2i\pi t}{365} \right)+\underset{j=1}{\overset{{{J}_{1}}}{\mathop \sum }}\,{{d}_{j}}\cos \left( \frac{2i\pi t}{365} \right)$ (3)
Where a and b are the coefficients of the trend component, I1 and J1 represent the number of sine and cosine terms. ai are the coefficient of sine terms and bj are coefficient of cosine terms in equation (2) and ci are the coefficient of sine terms and dj are coefficient of cosine terms in equation (3).
The optimal order of truncated Fourier series in equation (2) and equation (3) will be specified later based on statistical analysis of historical temperature.

2.1 Analysis of temperature data

In this section we present the characteristics and dynamics of the daily average temperature data recorded for last 11 years, in the period 1 January 2005 to 31 December 2015 obtained from Ethiopia Meteorological Station Bahir Dar branch. The data contains 4015 values after the observations made on February 29 in all leap years are discarded. We have partitioned the data into two, one for estimating the parameters of the model, from 1 January 2005 to 31 December 2014, the other one for model validation that is recorded over the period from 1 January 2015 to 31 December 2015. Therefore, we use 3650 observations for estimating the parameters in the required model. In Fig. 1, we plot daily average temperature data recorded from 2005 to 2015and the ACF plot for the total dataset in Fig. 2.
Fig. 1 Daily average temperature data in Bahir Dar
Fig. 2 Autocorrelation function (ACF) of daily average temperature data in Bahir Dar
We observe a clear seasonal pattern in both the time series and ACF plots for the temperatures. So, to capture such seasonal pattern we use a truncated Fourier series specially to model the deterministic seasonal part (component) of the unconditional temperature fluctuation.

2.2 Estimation of mean temperature

From the temperature data in Fig. 1 we see that there is a seasonal variation in the temperature and a gentle, positive trend due to global warming and urban effects. Therefore, we propose a linear function to capture the trend and truncated Fourier series with four terms that is, we set I1=J1=2, to incorporate the seasonal mean of the temperature movements. Adding up, a deterministic function S(t) modeling the trend and seasonality of temperature takes the following form:
$\begin{align}& S\left( t \right)=a+bt+{{a}_{1}}\sin \left( \frac{2\pi t}{365} \right)+{{a}_{2}}\sin \left( \frac{4\pi t}{365} \right)+ \\& \ \ \ \ \ \ \ \ \ {{b}_{1}}\cos \left( \frac{2\pi t}{365} \right)+{{b}_{2}}\cos \left( \frac{4\pi t}{365} \right) \\\end{align}$ (4)
The parameters in equation (4) are estimated using the ordinary least squares regression and the estimated values of parameters of the seasonal function are given in Table 1. In Fig. 2, we plot the observed daily average temperature together with the seasonal function S(t).
Table 1 Estimated value of seasonal parameters
Parameter a b a1 a2 b1 b2
Value 19.3628 0.0002 1.2086 -0.7071 1.3632 -1.2230
We now need to remove the linear trend and seasonal components by subtracting the estimated mean temperature S(t) from the original observations, and we have the de- seasonalized temperature:
${{X}_{t}}={{T}_{t}}-{{S}_{t}}$ (5)
According to Cao and Wei (2004) the de-seasonalized temperature Xt have an autoregressive structure on a daily scale. To incorporate this property in our model, we apply AR process for de-seasonalized data. By examining the partial autocorrelation (PACF) for Xt plotted in Fig. 3, we can conclude that the AR (3) is suitable for modeling residuals temperature which is obtained after removing the trend and seasonality. The choice p=3 is also confirmed by the Akaike and Schwarz information criteria. The coefficients of the fitted autoregressive process are presented in Table 2.
${{X}_{t+3}}={{\beta }_{1}}{{X}_{t+2}}+{{\beta }_{2}}{{X}_{t+1}}+{{\beta }_{3}}{{X}_{t}}+{{\sigma }_{t}}{{\varepsilon }_{t}}$ (6)
Where ${{\beta }_{1}},{{\beta }_{2}}$ and ${{\beta }_{3}}$ are the coefficients of the autoregressive process.
Table 2 Estimates of parameters of AR (3) process
Parameters ${{\beta }_{1}}$ ${{\beta }_{2}}$ ${{\beta }_{3}}$
Values $0.56352$ $0.12533$ $0.067029$
Fig. 3 Daily average temperature together with the fitted seasonal function
Fig. 4 The PACF of the residuals of DATs after removing linear trend and seasonal component

2.3 Modeling the residual process

The autocorrelation function (ACF) of the residuals and the squared residuals obtained after eliminating different components such as trend, seasonal and AR (3) from the historical temperature data are presented in Fig. 5 and Fig. 6 respectively. The autocorrelation of the residuals indicates a time dependency in the variance of the residual. In Fig. 6, we can clearly observe a seasonal variation. To capture seasonality in the volatility of temperature, we propose a lower order truncated Fourier series (Benth F. E. and Saltyte- Benth 2007). We choose I1=J1=1 in equation (3) and the expansion is of the form:
${{\sigma }^{2}}\left( t \right)=c+{{c}_{1}}\sin \left( \frac{2\pi t}{365} \right)+{{d}_{1}}\cos \left( \frac{2\pi t}{365} \right)$ (7)
Fig. 5 The ACF for the residuals of the AR (3) model of the de-trended and de-seasonalized Bahir Dar average daily data
Fig. 6 The ACF of the squared residuals of the AR (3) model of the de-trended and de-seasonalized Bahir Dar average daily data
In order to get the seasonal volatility σ(t), we first group the residuals in 365 groups by averaging the values of the squared residuals of the particular day overall years. So, we get 365 variances of data, one for each day of the year. Here we put the assumption that σ2(t) to be a periodic function such that σt = σt+365k for $t~=1,\cdots ,365$ and k=1, 2, 3,…, that means, we assume that the seasonal variance is repeated every year. Through least squares (nonlinear regression), the estimation of the parameters is given in Table 3.
Table 3 Estimated value of volatility parameters
Parameters c c1 d1
Values 0.9686 0.4372 0.0403
In Fig. 7, we plot the empirical values of daily variance of the residuals together with the fitted truncated Fourier function. We observe that the variance takes its highest values in winter season, while spring has the lowest variations than other seasons.
Fig. 7 Daily squared volatility together with the fitted volatility function σ2(t)
The autocorrelation function of the squared residual obtained after dividing out the seasonal variance function σ(t) from the regression residual is presented in Fig. 8. As we can see, the seasonality in theautocorrelation function for squared residual is completely removed.
Fig. 8 The ACF of the squared residuals of the AR (3) model after dividing out the volatility function σ(t) from the regression residuals

3 Continuous-time model and weather derivatives pricing

Most of the models that have been proposed by different scholars for the temperature dynamics are stated as continuous-time stochastic process, with the exception of the model of Campbell and Diebold (Campbell and Diebold,2005). For instance, in Benth F.B. and Saltyte-Benth’ model (Benth F.B. and Saltyte-Benth, 2007) and Zapranis and Alexandridis’ model (Zapranis and Alexandridis, 2009), the dynamics of the de-seasonalized temperature is assumed to follow an Ornstein-Uhlenbeck process of the form:
$\text{d}{{\bar{T}}_{t}}=-\alpha {{\bar{T}}_{t}}\text{d}t+{{\sigma }_{t}}\text{d}{{B}_{t}}$ (8)
Where ${{\bar{T}}_{t}}={{T}_{t}}-{{S}_{t}}$, α is a speed of mean reversion and Bt is a Brownian motion. Later, Fred al., (2007) generalized the continuous-time autoregressive process to higher-order continuous-time autoregressive models with seasonal variance for temperature. In this paper we propose a continuous-time models for the temperature dynamics that allows for the derivation of futures prices.

3.1 Continuous-time AR-models

In this section, we consider a higher-order continuous autoregressive model to describe the dynamics of the de-seasonalized temperature as proposed by Fred E.B., Jurate S. B. and Koekebakker S. (Fred al., 2007). Let Xt be a stochastic process in ${{\mathbb{R}}^{p}}$ for $p>1$ and further assume that Bt denotes the winner process. We now define the vectorial Ornstein-Uhlenbeck equation
$\text{d}{{X}_{t}}=A{{X}_{t}}\text{ d}t+{{e}_{p}}{{\sigma }_{t}}\text{ d}{{B}_{t}}$ (9)
where, ${{e}_{k}}$ is the kth unit vector in ${{\mathbb{R}}^{p}},k=1,2,\cdots ,p,{{\sigma }_{t}}$ is temperature volatility and A is the square matrix of order p of the type:
$A=\left( \begin{align}& \begin{matrix}0 & \ \ \ \ \ \ \ \ 1 & \begin{matrix}\ \ \ \ \ \ 0 & \begin{matrix}\ \ \ \ldots & \ \ 0 \\\end{matrix} \\\end{matrix} \\\end{matrix} \\& \begin{matrix}0 & \ \ \ \ \ \ \ \ 0 & \begin{matrix}\ \ \ \ \ \ 1 & \begin{matrix}\ \ \ \ldots & \ \ 0 \\\end{matrix} \\\end{matrix} \\\end{matrix} \\& \begin{matrix}\vdots & \ \ \ \ \ \ \ \ \ \vdots & \begin{matrix}\ \ \ \ \ \ \ \vdots & \begin{matrix}\ \ \ \ \ \vdots & \ \ \ \vdots \\\end{matrix} \\\end{matrix} \\\end{matrix} \\& \begin{matrix}0 & \ \ \ \ \ \ \ 0 & \begin{matrix}\ \ \ \ \ \ \ 0 & \begin{matrix}\ \ \ \ 0 & \ \ \ 1 \\\end{matrix} \\\end{matrix} \\\end{matrix} \\& \begin{matrix}-{{\alpha }_{p}} & -{{\alpha }_{p-1}} & \begin{matrix}-{{\alpha }_{p-2}} & \begin{matrix}\ldots & -{{\alpha }_{1}} \\\end{matrix} \\\end{matrix} \\\end{matrix} \\\end{align} \right)$ (10)
The constants ${{\alpha }_{q}},q=1,2,\cdots ,p$are assumed to be non- negative and also assume that Xqt represents the qth coordinate of the vector Xt with $q~=1,\cdots p$. Here it is assumed that Xt is the de-seasonal temperature at times, $t-1,\begin{matrix}{} \\\end{matrix}t-2,$ $t-3,\cdots $, and St is a deterministic seasonal function, CAR(p) model for the temperaturetime series at time $t(q=1)$ is:
${{T}_{t}}={{S}_{t}}+{{X}_{1t}}$ (11)
By applying the multi-dimensional Ito Formula (Fred al., 2007), the stochastic process in equation (9) has the explicit form:
${{X}_{s}}=\exp \left( A\left( s-t \right) \right)X+\int_{\ t}^{\ s}{\exp \left( A\left( s-u \right) \right){{e}_{p}}\sigma \left( u \right)dB\left( u \right)}$ (12)
For $s\ge t\ge 0$ and ${{X}_{t}}=X\in {{\mathbb{R}}^{p}}$ and stationary holds when the eigenvalues of A have negative real part.

3.2 Estimation of the parameters in the CAR-model

Here we need to estimate the parameters of CAR (3)-model that corresponds to the fitted AR (3)-model for our data in the earlier section. The recursive scheme for CAR (3) is obtained by using the idea of Euler discretization. By iterating the Euler approximation, we have discrete-time process for $t=0,1,2,\cdots $
$\begin{align}& {{X}_{1}}\left( t+3 \right)=\left( 3-{{\alpha }_{1}} \right){{X}_{1}}\left( t+2 \right)+\left( 2{{\alpha }_{1}}-{{\alpha }_{2}}-3 \right){{X}_{1}}\left( t+1 \right)+ \\& \ \ \ \ \left( {{\alpha }_{2}}+1-\left( {{\alpha }_{1}}+{{\alpha }_{3}} \right) \right){{X}_{1}}\left( t \right)+\sigma \left( t \right)\varepsilon \left( t \right) \\\end{align}$(13)
Where $\varepsilon (t)=B(t+1)-B(t)$ is white noise. Using the estimated values of ${{\beta }_{1}},{{\beta }_{2}},$and ${{\beta }_{3}}$displayed in Table 2 and equating with the parameters in equation (13), we can obtain values of the parameters ${{\alpha }_{1}},{{\alpha }_{2}}$ and ${{\alpha }_{3}}$ and the estimated values are given in Table 4.
Table 4 Estimates of parameters of CAR (3) process volatility parameters
Parameters ${{\alpha }_{1}}$ ${{\alpha }_{2}}$ ${{\alpha }_{3}}$
Values $2.43648$ $1.74763$ $0.24086$
The stationarity condition of the CAR (3)-model is fulfilled since the eigenvalues of the fitted matrix.
$A=\left( \begin{matrix}0 & 1 & 0 \\0 & 0 & 1 \\-0.24086 & -1.74763 & -2.43648 \\\end{matrix} \right)$ (14)
${{\lambda }_{1}}=-0.1794$and$~{{\lambda }_{2,3}}=-1.1286\pm 0.2629i$ have negative real parts. The element components of the matrix A in equation (10) do not change over time and this makes the process stable.

3.3 Weather derivatives pricing

Weather derivatives involving the temperature are usually written on heating degree days (HDD) and cooling degree days (CDD). The reference temperature that we use to calculate different temperature indices highly depends on the location and possibly a temperature which is closer to the expected mean temperature for the period can be considered as reference level (Alaton et al., 2002). In this work, we use 19.7915°C as reference temperature in the calculation of HDD and CDD because the mean of the temperature data used is 19.7915°C.
In this section we present how to price a heating degree days and cooling degree days. Since the underlying temperature cannot be traded directly, the temperature derivative market is incomplete, and thus there is no single fixed price for the derivative. Due to this incompleteness, the market is a risky. To derive an expression for future temperature price, we need to take into account risk preferences of investors. This is usually given by a market price of risk (MPR) $\lambda $ charged for issuing the derivative, which can be calculated from historical data or by looking at the market price of contracts. The MPR is an important parameter that is used to find a unique risk-neutral probability measure QP such that all tradable assets in the market are martingales after discounting. The change of measure from the real world to the risk-neutral world can be performed using the Girsanov theorem (or the Esscher transform for a jump process). The Girsanov theorem tells us how a stochastic process changes under changes in the measure. Then, the discounted expected payoff of the various weather contracts can be estimated. Using Girsanov’s theorem, under the equivalent measure $Q$, we have
$\text{d}w_{t}^{\lambda }=\text{d}{{w}_{t}}-\lambda \text{d}t$ (15)
Therefore, by combining equation (9) and equation (15), the dynamics of X(t) under the risk-neutral probability Q is
$\text{d}{{X}_{t}}=\left( A{{X}_{t}}+{{e}_{p}}{{\sigma }_{t}}{{\lambda }_{t}} \right)\text{d}t+{{e}_{p}}{{\sigma }_{t}}\text{ d}{{B}_{t}}$ (16)
Using the multi-dimensional Ito formula, the solution of equation (16) under Q is:
$\begin{align}& {{X}_{s}}=\exp \left( A\left( s-t \right) \right)X+ \\& \ \ \ \ \int_{\ t}^{\ s}{\exp \left( A\left( s-u \right) \right){{e}_{p}}\sigma \left( u \right)\lambda \left( u \right)\text{d}u+} \\& \ \ \ \ \int_{\ t}^{\ s}{\exp \left( A\left( s-u \right) \right){{e}_{p}}\sigma \left( u \right)\text{d}B\left( u \right)} \\\end{align}$ (17)
for $s\ge t\ge 0$.
In the new probability measure Q, the temperature (price) process has the following explicit dynamics:
$\begin{align}& ~~T\left( s \right)=S\left( s \right)+{{{{e}'}}_{1}}\exp \left( A\left( s-t \right) \right)X+ \\& \ \ \ \ \int_{\ t}^{\ s}{{{{{e}'}}_{1}}\exp \left( A\left( s-u \right) \right){{e}_{p}}\sigma \left( u \right)\lambda \left( u \right)\text{d}u+} \\& \ \ \ \ \int_{\ t}^{\ s}{{{{{e}'}}_{1}}\exp \left( A\left( s-u \right) \right){{e}_{p}}\sigma \left( u \right)\text{d}B\left( u \right)} \\\end{align}$ (18)
Since the price of a derivative is expressed as a discounted expected value under martingale measure Q, we have to know the distributional properties of daily temperature Tt under the martingale measure Q and we start by computing the conditional expected value and the variance of Tt under the measure Q. Since a Girsanov transformation only changes the drift term, the variance of Tt is the same under both measures. Therefore:
$Var\left[ {{T}_{s}}\text{ }\!\!|\!\!\text{ }{{T}_{t}} \right]=\int_{\ t}^{\ s}{\sigma _{u}^{2}{{\left( {{{{e}'}}_{1}}\exp \left( A\left( s-u \right) \right){{e}_{p}} \right)}^{2}}\text{d}u}$ (19)
Hence, from equation (18), we get
$\begin{align}& {{E}^{Q}}\left[ {{T}_{s}}\text{ }\!\!|\!\!\text{ }{{T}_{t}} \right]=S\left( s \right)+{{{{e}'}}_{1}}\exp \left( A\left( s-t \right) \right)X+ \\& \ \ \ \ \int_{\ t}^{\ s}{{{{{e}'}}_{1}}\exp \left( A\left( s-u \right) \right){{e}_{p}}\sigma \left( u \right)\lambda \left( u \right)\text{d}u} \\\end{align}$ (20)
Therefore, under the martingale measure Q and given information at time $s<t,{{T}_{t}}$ is a Gaussian process:
${{T}_{t}}\tilde{\ }N\left( {{\mu }_{t}},\sigma _{t}^{2} \right)$ (21)
where mean ${{\mu }_{t}}$ given by equation (20) and variance $\sigma _{t}^{2}$by equation (19)

3.4 Pricing CDD: futures

Let be t1 and t2 the first and the last calendar day of the contract month respectively. As stated before, the cumulative CDD and cumulative HDD indices over a period $[{{t}_{1}},{{t}_{2}}]$ are given by
$HDD=\underset{{{t}_{1}}}{\overset{{{t}_{2}}}{\mathop \sum }}\,\text{max}\left\{ {{T}_{ref}}-{{T}_{t}},0 \right\}$ (22)
$CDD=\underset{{{t}_{1}}}{\overset{{{t}_{2}}}{\mathop \sum }}\,\text{max}\left\{ {{T}_{t}}-{{T}_{ref}},0 \right\}$ (23)
According to Antonis K.A. and Achilleas D.Z. (Antonis et al., 2013), if Q is the risk-neutral probability measure, then the arbitrage free future price of a $CDD$contract at time ${{t}_{s}}<{{t}_{1}}<{{t}_{2}}$ is given by:
${{F}_{CDD}}\left( {{t}_{s}} \right)={{E}_{Q}}\left[ \underset{{{t}_{1}}}{\overset{{{t}_{2}}}{\mathop \sum }}\,\text{max}\{{{T}_{t}}-{{T}_{ref}},0\}|{{T}_{{{t}_{s}}}} \right]$ (24)
and using Ito isometry, we can interchange the expectation and the sum
$\begin{align}& {{F}_{HDD}}\left( {{t}_{s}} \right)=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{E}_{Q}}\left[ \max \left( T\left( {{t}_{i}} \right)-c,0 \right)|{{T}_{{{t}_{s}}}} \right]= \\& \ \ \ \ \underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ \int_{\ c}^{\ \infty }{\left( T-c \right)f_{T\left( {{t}_{i}} \right)}^{Q}\left( T \right)\text{d}T} \right] \\\end{align}$ (25)
${{F}_{CDD}}\left( {{t}_{s}} \right)=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ \left( \mu \left( i \right)-c \right)\text{ }\!\!\Phi\!\!\text{ }\left( -\alpha \left( i \right) \right)+\frac{\sigma \left( i \right)}{\sqrt{2\pi }}{{e}^{\frac{-{{\alpha }^{2}}\left( i \right)}{2}}} \right]$ (26)
where, $\alpha \left( i \right)=\frac{c-\mu \left( i \right)}{\sigma \left( i \right)},\begin{matrix}{} \\\end{matrix}{{f}_{T}}$is the probability density function for the normal distribution and Φ denotes the cumulative distribution function for a standard normal distribution.
In the same way the price of an$HD{{D}_{n}}$future contract at time ${{t}_{s}}<{{t}_{1}}$ is computed as:
$\begin{align}& {{F}_{HDD}}\left( {{t}_{s}} \right)=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{E}_{Q}}\left[ \max \left( c-T\left( {{t}_{i}} \right),0 \right)|{{T}_{{{t}_{s}}}} \right]= \\& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ \int_{\ 0}^{\ c}{\left( c-T \right)f_{T\left( {{t}_{i}} \right)}^{Q}\left( T \right)\text{d}T} \right] \\\end{align}$ (27)
$\begin{align}& {{F}_{CDD}}\left( {{t}_{s}} \right)=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ \left( \mu \left( i \right)-c \right)\left[ \text{ }\!\!\Phi\!\!\text{ }\left( -\alpha \left( i \right) \right)-\text{ }\!\!\Phi\!\!\text{ }\left( \frac{-\mu \left( i \right)}{\sigma \left( i \right)} \right) \right]+ \right. \\& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. \frac{\sigma \left( i \right)}{\sqrt{2\pi }}({{e}^{\frac{-{{\alpha }^{2}}\left( i \right)}{2}}}-{{e}^{\frac{-1}{2}{{\left( \frac{\mu \left( i \right)}{\sigma \left( i \right)} \right)}^{2}}}}) \right] \\\end{align}$ (28)
To derive the above analytic approximation formulas for pricing HDD and CDD futures, we put the assumption that probabilities of $\max \{{{T}_{ref}}-{{T}_{t}},0\}=0$ and $\max \{{{T}_{t}}-$${{T}_{ref}},$ $0\}=0$ are extremely small during winter and summer, respectively. This assumption may account some problems in estimating the distribution of HDD or CDD in real data. Because the temperature outside the reference temperature is not counted in the cumulative index as a result of truncation.

3.5 The warm and the cold seasons

In this section we need to identify the warm and cold periods during the year. In Fig. 9, we plot the monthly mean of temperature data that was recorded over the period from 2005 to 2015. As we have seen in Fig. 9, the contract months of CDD are March, April, May and June and that of HDD are months from July to December.
By using equation (25), (26), we calculate CDD future prices for the months April, May and June 2015 shown in Table 5, assuming that MPR=0, with the trading date is 31 January, 2015.
Table 5 Observed and simulated values of CDD
Month Observed Simulated
April $99.655$ $102.7946$
May $102.4135$ $107.4860$
June $70.005$ $71.2982$
Fig. 9 The monthly evolution of the monthly mean temperature

4 Conclusions

In this paper, we have proposed Ornstein-Uhlenbeck model driven by Brownian process to characterize the stochastic dynamics of temperature, with seasonality in the mean level and volatility. We use truncated Fourier series to model the seasonal mean and volatility, and we fit the model to 11 years of daily data obtained from Ethiopia Meteorological Agency. To represent the non-seasonal effects in the data not captured by the seasonal mean function, we use autoregressive model of order three AR (3) on de-trended and de-seasonalized data. Our proposed model is simple for deriving futures prices based on typical temperature indices like heating and cooling degree-days (HDD and CDD), and we derive analytic approximation formulas for pricing temperature derivatives. We calculate price of CDD index using this formula. We find that our model performs better when forecasting CDD indices. For pricing more complex deriva-tive structures, the performance of the model should be improved using other mathematical tools. In future work, tools like wavelet function and B-spline functions, can be used for a more accurate modeling of dynamics of temperature.
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