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Weather derivatives are financial instruments that can be used by farmers to hedge against adverse weather conditions, such as drought or frost. The difference to other derivatives is that the underlying asset (rain/temperature/snow) has no direct value to price the weather derivative.
Weather derivatives can be used by people and institutions other than farmers. For example, power companies may use HDD (Heating Degree Days)* contracts to smooth earnings. Theme parks may want to insure against rainy weekends during peak summer seasons.
There are other, more specific examples listed on the web, but I'm not sure if I can repeat the details here.
From personal experience, farmers are not the most common users of these instruments.