Developing analytical distributions for temperature indices for the purposes of pricing temperature-based weather derivatives

Temperature-based weather derivatives are written on an index which is normally defined to be a nonlinear function of average daily temperatures. Recent empirical work has demonstrated the usefulness of simple time-series models of temperature for estimating the payoffs to these instruments. This paper develops analytical distributions of temperature indices on which temperature derivatives are written. If deviations of daily temperature from its expected value is modelled as an Ornstein-Uhlenbeck process with time-varying variance, then the distributions of the temperature index on which the derivative is written is the sum of truncated, correlated Gaussian deviates. The key result of this paper is to provide an analytical approximation to the distribution of this sum, thus allowing the accurate computation of payoffs without the need for any simulation. A data set comprising average daily temperature spanning over a hundred years for four Australian cities is used to demonstrate the efficacy of this approach for estimating the payoffs to temperature derivatives. It is demonstrated that expected payoffs computed directly from historical records is a particulary poor approach to the problem when there are trends in underlying average daily temperature. It is shown that the proposed analytical approach is superior to historical pricing.