In this appendix we derive exact expressions for the pay-off distributions and the expected pay-offs of weather derivatives on a normally distributed index. Specific examples of these expressions have been given by a number of authors, such as McIntyre (1999), Jewson (2003t) and Brix et al. (2002). The derivations given below come from Jewson (2003a).

In section D.1 we define an eighth contract type in addition to the seven defined in chapter 1. This new contract type has a general piecewise linear pay-off function. In section D.2 we give the closed-form expressions for the pay-off distributions of each of these types of contracts in terms of the index distribution. In section D.3 we derive various relations that greatly simplify the subsequent algebra, and in section D.4 we derive the expected pay-off for each of the eight contract types. Finally we give some numerical examples of each expression.

Pay-off definitions

In addition to the pay-off functions defined in chapter 1 we will also consider the general form, given below.