The dynamics of temperature can be modelled by means of a stochastic process known as fractional Brownian motion. Based on this empirical observation, we characterize temperature dynamics by a fractional Ornstein-Uhlenbeck process. This model is used to price two types of contingent claims: one based on heating and cooling degree days, and one based on cumulative temperature. We derive analytic expressions for the expected discounted payoffs of such derivatives, and discuss the dependence of the results on the fractionality of the temperature dynamics.