یک دیدگاه انتقادی درباره مدل سازی درجه حرارت برای استفاده در بازارهای مشتقات آب و هوا

In this paper we present a stochastic model for daily average temperature. The model contains seasonality, a low-order autoregressive component and a variance describing the heteroskedastic residuals. The model is estimated on daily average temperature records from Stockholm (Sweden). By comparing the proposed model with the popular model of Campbell and Diebold (2005), we point out some important issues to be addressed when modelling the temperature for application in weather derivatives market.

In recent years there has been a growing interest in modelling the dynamics of surface air temperature with application in pricing weather derivatives. We follow up this stream of research with a critical discussion on model building and estimation, contrasting two stochastic models proposed by Campbell and Diebold, 2005 and Benth and Šaltytė-Benth, 2007. Both models are based on a seasonal autoregressive (AR) process, but with significant differences in structure which influences their applicability in relation to temperature derivatives. The two models are widely used in the field, and are similar to or nest a number of related models, see, for example, Dornier and Querel, 2000, Alaton et al., 2002 and Cao and Wei, 2004 to mention a few. The performance (in terms of forecasting weather indices) of various models for temperature dynamics, including the two considered here, was compared by Oetoma and Stevenson, 2005, Svec and Stevenson, 2007, Papazian and Skiadopoulos, 2010, Zapranis and Alexandridis, 2008 and Schiller et al., forthcoming. Our main goal is to point out the principle differences between the models of Campbell and Diebold, 2005 and Benth and Šaltytė-Benth, 2007. At the Chicago Mercentile Exchange (CME) there is an organized trade in weather futures and options. In particular, the CME offers trade in futures contracts written on temperature indices measured at various locations world wide, providing financial instruments to hedge weather risk exposure. The locations are major cities in the US, Canada, Europe and Asia. The temperature indices measure the daily cumulative average temperature (CAT), the cumulative heating-degree days (HDD) or the cumulative cooling-degree days (CDD). The reference temperature is taken as the average of the daily minimum and maximum temperature, which we further refer to as temperature. More specifically, if we denote the temperature on day t by Z(t), then the CAT index over a measurement period [T1, T2] is defined as equation(1) View the MathML sourceCATT1,T2=∑t=T1T2Zt, Turn MathJax on where the measurement period is typically a given month or season. At CME, CAT futures are traded on European cities for measurement periods in warm season. The HDD index measures the demand for heating in the cold period of the year, and is defined as the cumulative amount of average temperatures below a threshold over a measurement period. That is, one aggregates max(c − Z(t), 0), where the threshold c is 65°F or 18 °C. The CDD index analogously aggregates max(Z(t) − c, 0) and measures the demand for air-conditioning cooling. The CDD and HDD futures are traded for US cities. The temperature futures contracts are financially settled proportionally to the underlying index at the end of the measurement period. To assess the risk in trading such contracts and to be able to settle reasonable futures prices, one needs precise models for the temperature dynamics. A model should incorporate such properties as possible trend due to global warming and/or urbanisation, seasonal component describing periodic temperature variations related to cold and warm seasons, AR properties for temperature changes, and seasonal variations in residuals. In this study, we follow up the analysis from papers Benth and Šaltytė-Benth, 2007, Benth et al., 2007 and Šaltytė et al., 2007, and contrast it to the approach of Campbell and Diebold (2005). As observed by Campbell and Diebold (2005) for US temperature data, and later confirmed for Swedish, Norwegian and Lithuanian temperatures (see above-cited papers), there is a clear seasonality in the temperature variations after removing trend, seasonal variations and AR effects from the data. The model for temperature proposed by Benth and Šaltytė Benth contains (linear) trend, seasonal component, low-order AR process and a seasonal variance component describing the remaining heteroskedasticity in temperature residuals. It differs from the model of Campbell and Diebold (2005) first of all in its simplicity. Beside trend and seasonal component, Campbell and Diebold propose to use a high-order AR time series model with seasonal generalized AR conditional heteroskedesticity (GARCH) model for the residuals. We are able to show that the model by Benth and Šaltytė Benth is sophisticated enough to explain the basic stylized facts of temperatures just as well as the parameter-intensive alternative proposed by Campbell and Diebold (2005). The contribution of this paper is threefold. First, we critically review the process of modelling the temperatures. In this paper we promote a stepwise procedure used in Benth and Šaltytė-Benth (2007), Benth et al. (2007) and Šaltytė et al. (2007), where one models and estimates each component step-by-step. This turns out to be advantageous in order to build a confident model explaining the various stylized facts of temperature. In particular, such an approach leads to a very low-order AR structure in the temperature dynamics, in contrast to the approach of Campbell and Diebold (2005) suggesting to use an AR model with 25 lags. We argue that one can model temperature dynamics equally good using AR(3) and show that a simpler model explains the temperature evolution very well. A bigger empirical analysis also shows that the model by Benth and Šaltytė Benth explains extremely good the historical index values of CAT, HDD and CDD. As a second contribution, we discuss the major role played by the mean of temperature in the context of weather derivatives. The main factor explaining the indices CAT, HDD and CDD turns out to be the seasonal mean temperature, as we demonstrate later on. This is not surprising, taking into consideration the relatively strong mean reversion of temperatures along with indices averaging temperatures over intervals like months. In the model proposed by Benth and Šaltytė Benth, the seasonal mean is modelled explicitly, and is directly estimated fromtemperature observations. In this way one obtains a confident model for the seasonality of temperature. Campbell and Diebold (2005) choose to model it indirectly, estimated together with all the other parameters in the model. In the model by Benth and Šaltytė Benth, one regresses the deseasonalized temperatures on deseasonalized temperatures, that is, the AR structure is modelled after removing the seasonal mean. Campbell and Diebold (2005) choose to regress today's deseasonalized temperature on the temperature in previous days. Their seasonal function will then not be the seasonal mean, but merely a seasonal component. We demonstrate how one can compute the seasonal mean from the model of Campbell and Diebold, involving the AR parameters and thus leading to potentially increased uncertainty in parameter estimates. The third contribution of the paper is a multiplicative seasonal stochastic volatility model. Instead of using an additive GARCH process in modelling the seasonal heteroskedastic residuals as Campbell and Diebold (2005) do, we suggest using a product between a seasonal deterministic function and a classical GARCH process instead. With a multiplicative structure one avoids potential problems related to the positivity of variance. Moreover, no new estimation procedure is required to estimate the GARCH component, leading to a model which is simpler to fit and therefore more practically applicable. The paper is organized as follows. First, we state the model for the daily temperature variations and discuss in detail the different components of it. Then we describe the data and estimate the model. Next, we validate the proposed model on out-of-sample data and apply it for forecasting different temperature indices. We end our paper by discussing continuous-time models and weather derivatives pricing.