Modelling and Pricing Temperature Derivatives Using Wavelet Networks and Wavelet Analysis

Weather derivatives are financial instruments that can be used by organizations or individuals as part of a risk management strategy to reduce risk associated with adverse or unexpected weather conditions. Just as traditional contingent claims, whose payoffs depend upon the price of some fundamental, a weather derivative has an underlying measure such as: rainfall, temperature, humidity or snowfall. In this thesis the problem of pricing weather futures written on various temperature indices, as well as weather options on weather futures is addressed. In order to accurately price weather derivatives based on temperature indices, first, a model that describes the evolution of the daily average temperature was developed. This thesis provides a concise and rigorous treatment of the stochastic modelling of weather market. The Ornstein-Uhlenbeck process is described as the basic modelling tool for daily average temperature dynamics, while the innovations are driven by a Brownian motion. We emphasize in the accurate estimation of the seasonal component in the mean and variance using wavelet analysis. A modelling approach that efficiently extracts all the seasonalities from the temperature was developed. In addition we use wavelet networks in order to examine the time dependence of the speed of the mean reversion parameter of the process, α. We estimate nonparametrically with a wavelet network a model of the temperature process and then compute the derivative of the network output with respect to the network input, in order to obtain a series of daily values for α. To our knowledge, this is the first time that this has been done, and it gives us a much better insight into the temperature dynamics and temperature derivative pricing. Our results indicate strong time dependence in the daily values of α , and no seasonal patterns. This is important, since in all relevant studies performed thus far, α was assumed to be constant. Our analysis is based on seven cities that weather derivatives are actively traded on the Chicago Mercantile Exchange. Comparing our method with alternative approaches, our results indicate that our model significantly outperforms them both in-sample and out-of-sample. Furthermore, the residuals of the wavelet neural network provide a better fit to the normal distribution when compared with the residuals of the classic linear models used in the context of temperature modelling. Our model captured efficiently and successfully all the seasonal components of the temperature and completely removed the autocorrelation in the residuals. Finally, our results indicate greater accuracy of our model in forecasting the temperature indices in contrast to alternative models. In order to obtain a better understanding of the distributions of the residuals we expanded our analysis by fitting additional distributions besides the classical Brownian motion. More precisely, a Levy family distribution was fitted to the residuals. Our results indicate that a hyperbolic distribution provides a better fit. Finally, we provide the pricing equations for temperature futures and options on futures written on the most common temperature indices, when κ is time dependent, under the assumption of both a Brownian motion and a Levy process. Our results are very promising and suggest that the proposed method significantly outperforms other methods previously proposed in literature