Weather Derivative Pricing and the Normal Distribution: Fitting the Variance to Maximise Expected Predictive Log-Likelihood

The normal distribution is commonly used to predict weather indices when pricing weather derivatives. The standard method for making such predictions involves calculating an unbiased estimator for the population variance. The variance of the prediction (the predictive variance) is then the unbiased estimator for the population variance with an adjustment to account for sampling error on the mean. This is not, however, the only way to model the predictive variance, and it is not necessarily the best way. We investigate an alternative method, based on adjusting the predictive variance so as to maximise the expected predictive log-likelihood. For the small sample sizes often used in weather derivative pricing the resulting predictive variances are significantly larger than those calculated using the standard method.