Statistical analysis of max-stable processes used to model spatial extremes has been limited by the difficulty in calculating the joint likelihood function. This precludes all standard likelihood-based approaches, including Bayesian approaches. Here we present a Bayesian approach through the use of approximate Bayesian computing. This circumvents the need for a joint likelihood function and instead relies on simulations from the (unavailable) likelihood. This method is compared with an alternative approach based on the composite likelihood. When estimating the spatial dependence of extremes, we demonstrate that approximate Bayesian computing can provide estimates with a lower mean square error than the composite likelihood approach, though at an appreciably higher computational cost. As this approach very naturally incorporates parameter uncertainty into predictions, it is well suited for use in pricing weather derivatives to manage environmental risks. We discuss the construction and pricing of such weather derivatives. The method described utilizes results from spatial statistics and extreme value theory to first model extremes in the weather as a max-stable process, and then use these models to simulate payments for a general collection of weather derivatives. These simulations capture the spatial dependence of payments. Incorporating results from catastrophe ratemaking, we show how this method can be used to compute risk loads and premiums for weather derivatives which are renewal-additive. We illustrate the performance of the approximate Bayesian computing method and weather derivative pricing with applications to United States temperature data. The first application considers pricing weather derivatives for temperature extremes in the Midwestern United States. The second application demonstrates the use of the approximate Bayesian computing method in estimating the risk of crop loss due to an unlikely freeze event in northern Texas
