Weather Forecasting for Weather Derivatives - Research Proposal Example

Weather Forecasting for Weather Derivatives This research proposal examines three different approaches in pricing of weather derivatives through forecasting daily temperature based on the historical readings. Three methods are fundamentally different one to another and include ARMA, causal band-limited approximation that drops the time invariance constraint and another approximation for temperature signals with well-defined spectral characteristic. These three methods are reviewed in details and their forecasting power is compared based on the daily temperature reading from Sydney Hill Observatory weather station. The goal is to determine which method leads to the smallest forecasting error. The forecast is performed only for the one time step ahead and is not required to provide long lasting forecast. Introduction Since their inception in 1996, weather derivatives have grown in large quantities. Today, weather derivatives are being used for hedging purposes by companies and industries, whose profits can be adversely affected by unseasonal weather or, for speculative purposes by hedge funds and others interested in capitalizing on those volatile markets. A weather derivative is a financial instrument that has a payoff derived from variables such as temperature, snowfall, humidity and rainfall. However, it is estimated that 98-99% of the weather derivatives now traded are based on temperature. Temperature contracts have as an underlying variable, temperature indices such as Heating Degree Days (HDD) or Cooling Degree Days (CDD) defined on average daily temperatures. The list of traded contracts is extensive and constantly evolving. In the Chicago Mercantile Exchange (CME) there are traded weather contracts based on an index of Cumulative Average Temperature (CAT) for European cities for May to September (Zapranis). Many weather derivatives are traded long before the start of the contract and long before there are any useful forecasts which can indicate the likely weather during the contract period. For instance, contracts for the winter period may be traded in the preceding spring and early summer. In this case, only historical observational data are required for derivative valuation. It is also common for weather derivatives to be priced just before and during the period of the contract. There are two main reasons for this. The first is that weather derivatives are traded at these times. This can be for economic hedging reasons, or purely for speculation. The second is that companies that have traded a weather derivative often need to track the value of the derivative as the weather during the contract period progresses: this is known as ‘marking’ the contract. However, pricing weather derivatives is far from a straightforward task, since the underlying weather index (HDD, CDD, CAT, etc.) cannot be traded. Furthermore, the corresponding market is relatively illiquid. Consequently, since weather derivatives cannot be cost-efficiently replicated with other weather derivatives, arbitrage pricing cannot directly apply to them. The weather derivatives market is a classic incomplete market, meaning that prices cannot be derived from the no-arbitrage condition, since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities. This research focuses entirely on the temperature forecast without well-defined connection to the derivatives pricing. Once a model for temperature forecast is established it can be used a predictor for any temperature derived weather security. However, the number of possible securities is quite large and therefore the focus remains on the predictability of the temperature. As mentioned before, the weather derivatives are very localized, therefore, for simplicity this study is limited to Sydney area. The daily temperature readings are taken from Sydney Observatory Hill weather station. The simulation are modelled in Matlab using functions from statistical toolbox. Objectives The primary objectives of this research is to select one out of three forecasting methods that leads to the smallest cumulative forecasting error. The forecasting error is defined as a standard deviation of the time series: Where is the forecasted temperature for the period and is the actual observed temperature. The daily data is taken for the last 10 years which would provide sufficient statistics for the forecasting power. The first 100 days will be used for a warm up when the forecasting is not executed, but the data is already used to compute model related quantities. For example for ARMA based models these 100 days would provide enough horizon to produce moving averages of the temperature deviations. Each model is implemented in MATLAB environment, which offers comprehensive tools for rapid development of research applications. MATLAB is centered on vector and matrix based datasets that are efficiently analyzed in a descriptive matrix based manner. The evaluation Methodology The following three forecasting methods are selected for this study: ARMA (Härdle, 2012) (Jewson, 2004) Causal band-limited approximation (Dokuchaev, 2011) Pathwise predictability (Dokuchaev, 2012) Below each method is presented and key points relevant to this study are discussed. Each method has well described associated theory that proves its optimality for specific conditions. Each method yields profound results only in specific conditions, therefore, it is important to evaluate the data set before each forecast whether it satisfy the basic requirements of the method. The next step is to run the forecast method according to the original recipe and look at the forecasting error. ARMA ARMA is a classical tool that is widely used for forecasting time series. The simplicity of the model provided wide acceptance and implementation in many mathematical software packages. Its applications in weather derivatives is well described in (Jewson, 2004) (Zapranis) and (Campbell, 2000). Traditional time-series models are parametric models, and the simplest example is probably a first order autoregressive (AR(1)) process. For such a process the temperature at one day is given by a linear dependency on the temperature the previous day with some random noise added Here denotes the anomaly at day i and the are independent and identically distributed random variables following a mean zero normal distribution. Despite its simplicity an AR(1) process can be a useful model for a variety of problems, but is unfortunately too simple for daily temperature anomalies. A simple extension of the AR(1) process provides us with a flexible class of models, known as ARMA models, which can be used to approximate any stationary time-series model. The definition of an ARMA( p, q) process is: The interpretation of the model is that the temperature today depends in a linear way on the temperatures on the previous p days through the parameters . Just as in the AR(1) case random perturbations are added to reflect the fact that we do not expect the temperature today to be a perfect linear function of the past p days’ temperatures. The are parameters used to express linear dependence between the random perturbations. The expectation is that ARMA would not be able to adequately forecast daily temperature deviations as the perturbations in temperature are short lived compared with the seasonal effects. Causal band-limited Approximation Dokuchaev presented his casual approximation of a non-limited discrete process by a band-limited on in (Dokuchaev, 2011). The main idea is to find a spectral function that would generate minimum fitting errors in a form of linear combination of sinc functions. Once there is an approximation that a non-band limited process can be replaced with a band-limited Dokuchaev easily finds an optimal solution in a form: (1) The details of how exactly find the optimal solution are omitted here and easily can be found in the original work. The most important property of this solution is it is not time invariant. Therefore, strictly speaking it is not a mathematically sound solution that fully resolves original question. However, the practicality of this approximation outweighs the approximate nature of this solution. This extrapolation of the time series also can be used as an optimal forecast for the next discrete step. This model fits well with the original requirements of the research to find a good model for one step ahead forecast. It is interesting to measure the predictive power of such a simplistic, but powerful model. In order to find coefficients for the sinc series that describe the original data one has to use, for example, MATLAB signal processing toolbox. There are several ways to determine the coefficients of the series depending on the problem nature and size. For a finite time series from problem (1) the coefficients can be found from (Dokuchaev, 2012): Where components of the matrix R can be found numerically from the equality: Pathwise Predictability Another paper by (Dokuchaev, Predictors for time series with energy decay on higher frequencies, 2012) offers yet another approach in finding a forecasting method. The idea is similar to well-known Nyquist theorem, which considering signal band restrictions defines conditions of a spectra that can fully describe the original signal. This work (Dokuchaev, Predictors for time series with energy decay on higher frequencies, 2012) makes the following assumptions about the input signal: Signal is time discrete Band limited Most important – the spectral density function rapidly decays for higher frequencies The author concludes that the last condition is very tough and imposes very tight constraint on signals that can fit in. However, this method offers high quality predictors that are far better than conventional models. The difficulty here is come up with a dataset that would satisfy all the requirements. It is possible that, for example, a temperature data set would require a previous data treatment to get rid of higher harmonics in the spectra. One has to be careful in pre-analysis data treatment in order to keep aligned the goals of the research and the selected data transformation. This method requires a construction of predictive kernels that can be constructed in the following way (Dokuchaev, 2012): , Where and the required sequence of kernels is given by: And the original time series can be calculated as: This transformation can be implemented directly in MATLAB with the help of the signal processing toolbox. Before the analysis each temperature time series is analyzed with the spectral analysis to ensure that indeed on higher harmonics it experiences sufficient energy decay. Conclusions Three temperature forecasting methods were presented in order to select a model that can be used in weather derivatives pricing. The first, ARMA, is well studied and well explained. It adequately deals with seasonality and long-term trends if they are present in the data. However, in the most interesting cases ARMA model does not work. There is hope that other presented methods can fill the gap. The second method is the most promising as it is easy to implement and was originally designed as applied method for real life data time series. The last method has high predictability for a specific sort if input data, which has to be examined prior applying the method. There is a possibility the historical temperature data might be that kind of a data set that has rapidly decaying higher frequencies in the spectra. In this case it might be the most optimal solution. It is too early to tell which method is able to produce results that would lead to minimum forecasting errors and the more detailed analysis with real life historical data is required. References Campbell, S. D. (2000). Weather Forecasting for Weather. The Wharton Financial Institutions Center, 02-42. Dokuchaev, N. (2011). On predictors for band-limited and high-frequency time series. Signal Processing, 2571-2575. Dokuchaev, N. (2012). Predictors for time series with energy decay on higher frequencies. IEEE Transactions on Signal Processing, 6027-6030. Härdle, W. K. (2012). Forecast based Pricing of Weather Derivatives. SFB 649 Discussion Paper. Berlin. James W. Taylor, R. B. (2006). Density Forecasting for Weather Derivative Pricing. International Journal of Forecasting, 22, 29-42. Jewson, S. (2004, 6 21). Introduction to Weather Derivative Pricing. Retrieved from Social Science Research Network: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=557831 Massimiliano Caporin, J. P. (2012). Modelling and forecasting wind speed intensity for weather. Computational Statistics and Data Analysis, 56, 3459–3476. Moses, R. L. (1994). An efficient linear method for ARMA spectral estimation. International Journal of Control, 59(2), 337-356. Zapranis, A. (n.d.). Weather Derivatives Pricing: Modeling the Seasonal Residual Variance of an Ornstein-Uhlenbeck Temperature Process with Neural Networks. Retrieved from http://ceur-ws.org/Vol-284/page178.pdf Read More