International Journal of Stochastic Analysis

Volume 2018, Article ID 8534131, 15 pages

https://doi.org/10.1155/2018/8534131

## Regime-Switching Temperature Dynamics Model for Weather Derivatives

^{1}Pan African University, Institute of Basic Sciences, Technology, and Innovation, Kenya^{2}University of Nairobi, Kenya^{3}University of South Africa, South Africa

Correspondence should be addressed to Samuel Asante Gyamerah; moc.liamg@maygsaas

Received 7 March 2018; Revised 14 May 2018; Accepted 24 May 2018; Published 10 July 2018

Academic Editor: Huyên Pham

Copyright © 2018 Samuel Asante Gyamerah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Weather is a key production factor in agricultural crop production and at the same time the most significant and least controllable source of peril in agriculture. These effects of weather on agricultural crop production have triggered a widespread support for weather derivatives as a means of mitigating the risk associated with climate change on agriculture. However, these products are faced with basis risk as a result of poor design and modelling of the underlying weather variable (temperature). In order to circumvent these problems, a novel time-varying mean-reversion Lévy regime-switching model is used to model the dynamics of the deseasonalized temperature dynamics. Using plots and test statistics, it is observed that the residuals of the deseasonalized temperature data are not normally distributed. To model the nonnormality in the residuals, we propose using the hyperbolic distribution to capture the semiheavy tails and skewness in the empirical distributions of the residuals for the shifted regime. The proposed regime-switching model has a mean-reverting heteroskedastic process in the base regime and a Lévy process in the shifted regime. By using the Expectation-Maximization algorithm, the parameters of the proposed model are estimated. The proposed model is flexible as it modelled the deseasonalized temperature data accurately.

#### 1. Introduction

From tilling of the farmland to selling of the output of the crop yield, farmers around the world make countless decisions that affect their performance. Yet, there is one very important factor that they cannot control, climate. The world’s climate keeps on changing and this change will persist at rates that are projected to be out of the ordinary for some centuries [1]. Africa is no exception of these extreme climate changes across the world. Extreme climate events cause strain on food security, water resources, and human health in Africa. Ordinarily, it is the cause of limited economic growth and obstructs poverty reduction efforts for most countries in Africa [2]. With agriculture being the major contributing factor of the gross domestic product (GDP) growth of most countries in Africa [3] and climate conditions having extensive and causal correlation with the production variables [4], there should be an effective management technique to hedge against agricultural production risk. Most agricultural producers have encountered crop failures because of extreme weather conditions due to changes in climate. As a result, most farmers in Africa have developed their own traditional ways to improve the effect of extreme weather changes.

As an institutional response to weather changes, the Chicago Mercantile Exchange (CME) introduced the weather derivative (WD). WD has been in existence in most developed countries (Canada, Europe, USA, and Japan). However, most farmers in Africa have rarely heard about this effective hedging tool. WD, if introduced in Africa, will be more viable, reliable, and efficient to the agricultural industry and can hedge against the increasing weather changes that affect agriculture since it is devoid of factors like loss adjustments, moral hazards, adverse selections, high premiums, and complex information requirements. To avoid basis risk associated with WD, there should be an efficient model for the underlying weather variable used in pricing WD. The weather variable considered in this study is temperature. Temperature controls and influences other elements of weather like clouds, humidity, air pressure, and precipitation that affect crops during and after crop production (https://en.wikibooks.org/wiki/Basic Geography/Climate/Climate Elements, accessed on 26/01/2018.)

In the last decade, there has been empirical literatures on modelling the dynamics of temperature. Dischel [5] was the first to propose a continuous stochastic model for temperature. He modelled temperature as a mean-reverting process by adapting directly the Hull-White model. The noise process in his model was driven by two Wiener processes corresponding to the distribution of the temperature and the distribution of the changes in temperature. Thereafter, McIntyre and Doherty [6] proposed a mean-reverting SDE with a constant volatility daily average temperature at Heathrow airport in the United Kingdom (UK). Dornier and Queruel [7] disagreed with the direct use of the Hull-White model adopted by Dischel. They rather used a conventional Autoregressive Moving Average model rather than the AR model proposed by Dischel. By replacing Brownian motion with fractional Brownian motion, Brody et al. [8] modelled the evolution of temperature that allowed the integration of a long memory effect. Other researchers [9–11] used different kinds of mean-reverting OU model driven by Brownian motion. In contrast to other researchers using Brownian motion to capture the residuals, Benth and Šaltyt-Benth [12] proposed an OU model that incorporate seasonal volatility and mean. In the model of Benth and Šaltyt-Benth, the residuals were driven by the generalized hyperbolic Lévy process rather than Brownian motion. The process they employed was an adjustable class of Lévy process that captured the skewness and semiheavy tails properties of the residuals.

Clearly, it can be observed that most of these researchers assumed no changes in state of the dynamics of temperature and hence modelled temperature dynamics as a single regime. The above methods of modelling temperature may lead to intractable pricing techniques for temperature derivatives. As noted by Brockett et al. [13], temperature time-series data shows sudden changes due to artificial and natural factors. By employing regime-switching models, the researcher can capture such sudden and discrete shifts in the temperature dynamics. Regime-switching models do capture most of the stylized facts of temperature accurately more than the single stochastic differential equation model, hence the need for different stochastic model for each switching state.

With a mean-reverting process as their base regime and a Brownian motion with mean different from zero as their shifted regime, Elias et al. [14] presented a constant volatility two-state MRS model for temperature dynamics at the city of Toronto, Canada. The model of Elias et al. (from hence we will call the model developed by Elias et al. as Elias’ Model) failed to capture the fact that volatility of temperature varies with varying temperature as it goes through discrete changes between the states of the regime process. Evarest et al. [15] improved on Elias’ model by capturing the fact that volatility of temperature varies as temperature goes through discrete changes between the states of the regime. They priced weather derivatives contracts based on the daily temperature dynamics. They used their model to calculate the future contract of HDD, CDD, and CAT indices. The introduction of the local volatility in the base regime helped in capturing well the dynamics of the underlying process. This led to a better pricing process as compared to Elias’ model. However, they failed to capture the extreme and fat tail characteristics of temperature data in their model. In his seminal thesis, Cui [16] modelled and priced temperature derivative. He modelled the dynamics of temperature by a standard mean-reverting Ornstein-Uhlenbeck process with a general Lévy process as the driving noise. He extended his model by proposing a continuous-time autoregressive (CAR) model driven by a general Lévy process which he calibrated to the Canadian data. The two models he proposed were used in deriving futures price on HDD, CDD, and CAT. He later developed a two-state MRS model with a “normal” regime and a “jump” regime. The “normal” regime depended on a standard OU process. For the “jump” regime, he used different noise process (Brownian motion with more extreme drift and volatility) to drive the abnormal positive or negative “jumps” in the temperature dynamics. However, he failed to capture the changes in volatility of temperature during the MRS model but rather assumed a constant volatility in both regimes.

Several models have been formulated over time to capture the stylized facts of temperature; however these models proposed in literature have failed to capture well the stylized features of temperature, thus affecting the pricing models of WD. Inaccurate representation of the dynamics of temperature affects the pricing of WD. WD also relies on accurate extensive long-term time-series data [17]. However, there is lack of accessible, accurate, complete, and usable weather data in most African countries. Calibrating the MRS model is not trivial because the regimes are not clearly observable but latent. To outwit these problems, we use Expectation-Maximization (EM) algorithm to estimate the parameters in the model.

From the above literatures presented, Brownian motion has been replaced with a fractional Brownian motion and subsequently by a generalized hyperbolic Lévy processes. Nevertheless, it will be interesting to explore both Brownian motion and Lévy processes in a MRS model that incorporates “normal” temperatures and “extremes” in temperature. The contribution of this paper is twofold; firstly we developed a mathematically tractable temperature dynamics model for the African farmer by using regime-switching model and secondly we showed that Gaussian distribution cannot capture the dynamics of real-life temperature. To the best of our knowledge, the two-state regime-switching model developed is the first kind of model that can be used to price futures and options on futures.

#### 2. Daily Temperature Dynamics

The most widely used temperature indices in most industries (energy consumers, energy industry, travel, transportation, agriculture, government, retailing, and construction) are the cumulative average temperature (CAT), cooling degree days (CDD), and heating degree days (HDD). Nevertheless, in this research, we use the CAT and growing degree days (GDD) since they are the dominant indices that affect agriculture in Africa [18, 19]. GDD is the measure of the suitability for a crop to grow in relation to the standard temperature.

*Definition 1. *For a given single temperature weather station, let and represent the daily maximum and minimum temperature (the temperatures used in this research are measured in degrees Celsius) recorded at day , respectively. We define the daily average temperature at day as

*Definition 2. *Assume the daily average temperature (DAT) at time , and then the and generated at a specific location over a specific measurement period are defined as

##### 2.1. Stylized Facts of Temperature

Temperature has clear characteristics which differs largely from commodities and other financial assets. The most palpable characteristics of temperature are the following.

*(i) Seasonality Feature*. Temperature exhibits annual (365 days) seasonal movements. The DAT at time is defined as the sum of the deseasonalized temperature and deterministic seasonal component given asTo model the variations of temperature without the deterministic seasonality, the seasonal component in (4) will be removed to obtain the deseasonalized temperature . The deterministic seasonal model at time , , is defined aswhere and represent the constant and coefficient in the linear seasonal trend of the raw data, respectively, captures the amplitude of the variation, and is the phase angle.

*(ii) Mean-Reverting Feature*. It is practically impossible for daily temperature to deviate from the mean temperature over a long period. Daily temperature reverts toward the mean, a feature that is common to other commodities. As observed by Alaton et al. [9], long-term changes may be as a result different factors which includes but are not limited to global warming, green-house effects, and urbanization.

*(iii) Extreme Feature*. Temperature data have extremal data points. These extremal data points are “abnormal” movement caused by abrupt changes in temperature. In contrast to stocks which usually exhibit jumps in their price movements, daily temperature can show some signs of spikes which are normally short-lived and of very extreme size.

*(iv) Locality Feature*. Temperature has a strongly localized response in temperature modelling and as such requires caution in making generalization, hence the need for different models to capture these different characteristics at different locations.

*(v) Volatility*. In their two-state regime switching model formulation, Elias et al. [14] considered a constant volatility in either sate of his model. But this assumption might not be a reality since a shift in temperature residuals from one state to the other causes a change in the volatility from one state to the other. Extremal data points in temperature residues have greater volatility effects than is the case when there are no spikes or sudden increase in temperature. This is ascertained in the Engle test performed to check for heteroscedasticity in the temperature residuals (see Table 2). In model (9), the volatility is assumed to be dependent on the current deseasonalized temperature . More precisely, the higher the deseasonalized temperature level, the larger the changes in the deseasonalized daily average temperature. Hence, in this study we will propose a model whose volatility differ with each regime and underlying process.

#### 3. Markov Regime-Switching (MRS) Model

The Markov switching model developed by Hamilton [20] and Hamilton [21] inferred that the distribution of a variable is known, conditional on the occurrence of a specific regime/state. The switching process between the regimes is Markovian and is determined by an unobserved random variable. The underlying regimes, however, do not necessarily have to be Markovian but should be independent. The daily temperatures do change from day to day and these changes are not directly observable but latent. Therefore, statistical inference with regard to the likelihood of occurrence of each of the regimes at any time should be drawn.

MRS has been used effectively in modelling the behaviour of the stock market and spot price of electricity [22–25]. Chevallier and Goutte [26] used sixteen international stock markets to compare the performance of regime-switching Lévy models. Chevallier and Goutte [27] developed an estimation methodology that provided a better fit for electricity and market prices by using mean-reverting Lévy jump processes.

In temperature modelling, it is typical to assume that there are different regimes that can capture distinct principal weather condition or the localized weather behaviour. In our study, the daily temperature is assumed to be latent with two possible regimes, either in the base regime (“normal or mean-reverting regime” ) or in the shifted regime (“extreme” regime ). Suppose that each regime in the regime-switching model undergoes discrete shifts between the regimes of the process, and then follows a first-order Markov process with the transition matrix:The transition probabilities of our temperature process in (6) is given asDue to the Markov property of the states at any given time , the future state of the underlying process (temperature) is independent of the past state of the underlying process given the present state of the underlying process.

##### 3.1. Modelling Daily Temperature Dynamics

To efficiently model the dynamics of temperature, it is assumed that the deseasonalized temperature is either under base regime or shifted regime and each regime is independent and parallel to the other regimes. The deseasonalized temperature is assumed to be driven by two sources of randomness: a Markov process and Lévy process. We assume a constant mean-reversion rate in the base regime. Based on the stylized facts of temperature, a regime-switching stochastic model that describes the dynamics of temperature is formulated. This model can be used to price weather derivatives. The base regime model is assumed to follow a mean-reverting stochastic process with a time-varying volatility. The residuals of the base regime are assumed to be generated by a Brownian process.

To effectively capture the nonnormality of the temperature residuals (see Figures 3, 5, 6, and 7 and Table 5), the residuals of the shifted regime are captured by a Lévy process. By comparing the generalized hyperbolic distribution to its subclasses (normal-inverse Gaussian, Hyperbolic, and Variance-Gamma), we were able to find the best distribution that can model the asymmetry and heavy tails of the residuals data. As our first regime-switching model, we call it time-varying mean-reversion Lévy (TML) regime-switching model. This proposed model is distinctly appropriate to capture the dynamics of temperature. In sequel, the propose TML model for the deseasonalized temperature dynamics is given aswhere is deseasonalized daily volatility of the base through time and is the volatility of the shifted regimes and is the mean-reversion rate of the deseasonalized temperature in the base regime which reverses the deseasonalized temperature to the long-term equilibrium level after the deseasonalized temperature has drifted from this equilibrium. is the standard Brownian motion. is a Lévy process which is cádlág, adapted, real-valued general Lévy process with independent, stationary increments and stochastically continuous, and is the deseasonalized temperature at time .

Proposition 3. *If the deseasonalized daily average temperature follows model (9), then the explicit solution is given by*

*Proof. *Determining the stochastic integral of the base regime process demands a variation of parameters approach to spell out a new function . By Itô’s lemma, the derivative of the new function can be found. For the shifted regime

#### 4. Analysis of Temperature Data

The daily maximum and minimum surface temperature data were taken from the weather measurement stations at Bole and Tamale. Bole and Tamale are located in the Northern region (the hottest region in Ghana) of Ghana. Bole and Tamale are the district capital of Bole and Tamale, respectively. In Ghana, the main source of weather data is from the Ghana Meteorological Agency. The sample period expands from 01/01/1987 to 31/08/2012 and consists of a total of 9375 observations. The average of the daily maximum and minimum temperature is calculated according to Definition 1. The raw data is checked for missing data to avoid gaps in the historical data. Depending on the size of the missing data (the proportion of the missing data should not be more than 10%), the missing data is filled using the method of combined average. The combined average is calculated using two distinct averages: the average of 7 days (d) after and before the missing day,and the average of that missing day across previous years,The missing values in the dataset are filled by averaging the calculated value in (13) and (14).

In Table 1, the descriptive statistics for the daily average temperature of the two measurement stations (Bole and Tamale) are presented. The values of the median, mean, maximum, and minimum temperature for both towns are consistent and this can be attributed to the fact that the geographical locations of these two measurement stations are not distant apart. The amount of variation (std) is relatively small but vary between two measurement stations. With a skewness value of 0.41 and 0.31 for Bole and Tamale, respectively, the empirical distribution of these two towns are asymmetrical. With a negative excess kurtosis for both towns, it can be explained that the distribution of the DAT data is more outlier-prone than the normal distribution. We present the values -statistics of Pearson’s criteria of goodness-of-fit with its P values of the DAT time-series data (see Table 1). From the values of the goodness-of-fit test and at a -level of significance, the null hypothesis (DAT data is normally distributed) can be rejected. With a Hurst exponent (H) greater than 0.5 for the two towns, there is a strong trend in the DAT data. However, the trend in Bole DAT data is more predictable than that of Tamale.