Advances in Meteorology

Volume 2017, Article ID 3913817, 14 pages

https://doi.org/10.1155/2017/3913817

## Modeling Temperature and Pricing Weather Derivatives Based on Temperature

^{1}Department of Banking and Finance, Faculty of Management, Girne American University, 99428 Karmi Campus, University Drive, P.O. Box 5, Karaoglanoglu, Kyrenia, Northern Cyprus, Mersin 10, Turkey^{2}Department of Financial Mathematics, Institute of Applied Mathematics, Middle East Technical University, Üniversiteler Mahallesi, Dumlupınar Bulvarı No. 1, Çankaya, 06800 Ankara, Turkey

Correspondence should be addressed to Birhan Taştan; rt.ude.uag@natsatnahrib

Received 29 December 2016; Accepted 16 February 2017; Published 19 March 2017

Academic Editor: Eduardo García-Ortega

Copyright © 2017 Birhan Taştan and Azize Hayfavi. 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

This study first proposes a temperature model to calculate the temperature indices upon which temperature-based derivatives are written. The model is designed as a mean-reverting process driven by a Levy process to represent jumps and other features of temperature. Temperature indices are mainly measured as deviations from a base temperature, and, hence, the proposed model includes jumps because they may constitute an important part of this deviation for some locations. The estimated value of a temperature index and its distribution in this model apply an inversion formula to the temperature model. Second, this study develops a pricing process over calculated index values, which returns a customized price for temperature-based derivatives considering that temperature has unique effects on every economic entity. This personalized price is also used to reveal the trading behavior of a hypothesized entity in a temperature-based derivative trade with profit maximization as the objective. Thus, this study presents a new method that does not need to evaluate the risk-aversion behavior of any economic entity.

#### 1. Introduction

Temperature-based derivatives represent a new financial tool to buy and sell a natural phenomenon. Doing so requires two things: a unit of measurement for the natural phenomenon that everyone agrees upon and a price that may facilitate a transaction. This study is designed to evaluate these two requirements.

Some preexisting measures already appear in the form of indices to meet the first requirement. To find values for these indices, the literature contains several temperature models using mean-reverting processes as the main tool. The most cited study develops an Ornstein-Uhlenbeck (OU) process to model temperature [1]. Using the equivalent martingale measures approach, the authors determine the price of an option. Benth and Šaltytė-Benth [2] model temperature as a continuous time autoregressive process for Stockholm and report a clear seasonal variation in regression residuals. They propose a model using a higher-order continuous time autoregressive process, driven by a Wiener process with seasonal standard deviation. While pricing futures and options, they consider a Gaussian structure in the temperature dynamics. In another study [3], they model temperature with an OU process driven by a generalized Levy process. The model contains seasonal mean and volatility. Instead of dynamic models, some authors offer time-series models to represent temperature. Campbell and Diebold [4] apply a time series approach to model temperature, including trend seasonality represented by a low-ordered Fourier series and cyclical patterns represented by autoregressive lags. The contributions to conditional variance dynamics are coming from seasonal and cyclical components. The authors used Fourier series and GARCH processes to represent seasonal volatility components and cyclical volatility components, respectively. Jewson and Caballero [5] discuss the use of weather forecasts in pricing weather derivatives, presenting two methods for strong seasonality in probability distributions and the autocorrelation structure of temperature anomalies. Elias et al. [6] develop four regime-switching models of temperature for pricing temperature based derivatives and find that a two-state model governed by a mean-reverting process as the first state and by a Brownian motion as the second state was superior to the others. Schiller et al. [7] and Oetomo and Stevenson [8] provide a comparison of different models.

To comply with the first requirement, the current study offers a temperature model based on Alaton et al. [1], which was defined after analyzing temperature data from different locations. The temperature model in this study is a mean-reverting Levy process. The Levy part contains a Brownian motion and two mean reverting jump processes driven by compound Poisson processes. For some flexibility, the jumps are designed as slow and fast mean-reverting processes, which are independent. The main difference with the model proposed here is its inclusion of jumps. Because temperature indices are mainly calculated as deviation of temperature from a base temperature, the model assumes that jumps are inevitable, at least for certain locations. The numerical estimates in this study contain test results related to this issue. The solution to the proposed temperature model is applied inversion formula to obtain approximated expected value of a specific index type and to obtain the approximated distribution of the same index.

Notably, temperature has unique behavior for any location in which it is measured. Therefore, it is not possible to develop a single model that explains every temperature behavior in every location. In addition, more than 100,000 weather stations worldwide measure temperature for different periods. It may even be difficult to develop a temperature model that is valid for all time at a single location. Thus, this study aims to cover more locations and periods by simply using a flexible model that can include or exclude jumps.

The second requirement, temperature-based derivatives pricing, is more complicated. Because the underlying commodity is not a traded asset, weather derivatives based on temperature have an incomplete market [9]. Carr et al. [10], Magill and Quinzii [11], and El Karoui and Quenez [12] provide a general discussion of incomplete markets. Pricing temperature-based derivatives is mainly based on two approaches: dynamic valuation and equilibrium asset pricing. The dynamic pricing approaches [1, 2] were discussed above. For equilibrium pricing, Cao and Wei [13] use a generalization of the Jr. Lucas model [14], which considers weather as another source of uncertainty. Richards et al. [15] suggest another equilibrium model. Davis [16] uses the marginal substitution value approach for pricing in incomplete markets. In addition, Xu et al. [17] use another classification for pricing temperature-based derivatives and add actuarial pricing and extended risk-neutral valuation in addition to equilibrium asset pricing, where the former is based on Jewson and Brix [18] and the latter on Hull [19] and Turvey [20]. In addition, some researchers used Monte-Carlo simulations in pricing temperature-based derivatives [21].

This study bases its pricing on the monetary effect of the natural phenomenon on economic entities. Further, this study shows that temperature has different effects on different entities. The same temperature may have a positive effect on one entity and a negative effect on another and is therefore personal, ceteris paribus. Thus, the study develops a personal price, which may require a determination of the entity’s risk aversion behavior. To address this problem, this study focuses on entity-specific trading behavior rather than the entity’s risk aversion behavior in order to develop a more realistic approach by avoiding an inconclusive debate over the risk premiums and utility functions used to calculate risk premiums. Moreover, a benefit of the proposed pricing model is that it is independent from how researchers measure temperature.

Critics may object to the move from a stochastic temperature model to some form of actuarial pricing model. There are several reasons for this move: first, this study demonstrates that risk-neutral pricing ends up with super-hedging; second, the discussion about risk premiums in the literature is unclear; and, finally, the calculations of jump processes needed approximations to obtain certain results. These considerations led to this study’s development of a more appropriate and practical method.

The second section of the paper provides the approximated index calculation and distribution of temperature after presenting a temperature model. Third section develops individualized prices and discusses the trading behavior of a hypothetical entity. The paper then presents the study’s conclusions.

#### 2. Model

Some basic terminology is defined in the following:where is the daily average temperature, represents a certain day, and and are the maximum and minimum temperatures of the given day, respectively.

For the Heating Degree Day (HDD) temperature index used in temperature-based derivatives, for a given day,where is a predetermined temperature level and is the average temperature calculated as in (1) for a given day .

Cumulative HDD (CHDD):where is calculated as in (2) and is the time horizon, which is generally a month or a season.

##### 2.1. The Temperature Model

Based on Alaton et al. [1], the temperature model is an OU process driven by a Levy process that contains independent processes as Brownian motions and two mean-reverting compound Poisson processes. The model is represented as follows:where is a cyclical process of temperature and represented in (5). Additionally, is the mean-reversion parameter, and subscript represents time. The differential of the driving Levy process is defined as follows:

The Brownian component of will be approximated by the ARCH (1) model. To represent the different jump structures in temperature in the form of a single jump and a series of jumps, and are defined as fast and slow mean-reverting OU processes driven by compound Poisson processes with intensities of and , and and being mean-reversion parameters, respectively. Hayfavi and Talasli [22] use a somewhat similar mean-reverting jump process combination in their model of spot electricity prices. The solutions to these non-Gaussian processes are [23] the following:The solution to (4) is given as

To find the value of a temperature-based derivative, one needs the distribution of the underlying temperature given in (9). However, this does not have a closed-form solution. One way to address this problem is to use a characteristic function of the temperature and apply inversion techniques to find the value of an HDD, an approximated distribution of CHDD, and an approximated distribution for temperature itself.

##### 2.2. Characteristic Function of Temperature

This study follows Cont and Tankov [23] to find the characteristic function given in (9). First, using , the characteristic exponent of (6) will be determined, where characteristic exponent is defined as . The solution to is Then, the characteristic exponents of the Levy components will be and for jump processes can be written similarly. It is not possible to evaluate the integral in (12). Consequently, the following approximation method was developed.

Let and . Let . Then, .

Further, let , where .

Then, by using linear approximation, Again, can be written similarly. Finally, the characteristic function of the temperature model can be written explicitly. Referring to Cont and Tankov [23], In explicit form,

##### 2.3. HDD and Distribution Function

This part of the study focuses on measuring HDDs. It is easy to apply the calculations into other types of indices. In the current case, inversion techniques will be used to find the value of an HDD and its distribution and hence the CHDD values.

###### 2.3.1. Approximating Density Function of Temperature

To find an approximating density function of temperature, inversion formula will be applied to the characteristic function of the temperature defined in (15). Before applying the inversion formula, the following shortcuts are derived from (15). Let and be the density function and characteristic function of temperature, respectively.Then, by inversion formula , the result is Because weather derivatives are defined on CHDDs, their distribution will be defined. Clearly, HDD values may show autocorrelation. In addition, due to the nature of the proposed temperature model in terms of the independence of the included processes and motivation to keep the process simple, the model assumes the independence of the HDDs. With this assumption, the approximated distribution of CHDD can be found using

###### 2.3.2. Measuring HDD

HDDs are clearly contingent claims on how temperature deviates from a base temperature. As one way to find the expected value of an HDD, this study will first find its Fourier transform. Then, the inverse Fourier transform will be applied to both the HDD’s Fourier transform and the characteristic function of temperature [24].

Let , is HDD’s payoff function given in (2), Base = , and is its generalized Fourier transform. Then, .

Then,Now, the inversion will be applied to , where is defined in (22) and is the characteristic function defined in (15). Let temperature in (9) be defined in shorthand notation as , where is defined as in (16) and .

The characteristic function of can be obtained from (15) and written as + − + .

Then, , where represents expectationsHowever, it was not possible to evaluate this integral analytically, and thus the elliptic package of the statistical software package [25] was applied to evaluate the integral numerically. The results indicated that the integral is equal to ; therefore,

###### 2.3.3. Numerical Estimates

The success of the proposed temperature model and (24) were tested in terms of forecasting Cooling Degree Day (CDD), which is another index based on temperature, and HDD values for the 12 cities listed in Tables 1 and 2. CDD is calculated as , where is a predetermined temperature level and is the average temperature calculated as in (1). Cumulative CDD (CCDD) is calculated using , where is calculated as in the previous sentence and is the time horizon, which is generally a month or season. The test assumes a base temperature of 18 degrees Celsius and proceeds in the following manner: