Description: Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max[S K,0] also arises in excess-of-loss of stop-loss insurance. This is the abstract of a paper that shows that Fourier methods may be used to compute them. The authors take the idea of using Parseval’s theorem further: 1. formulas requiring weaker assumptions, 2. relationship with classical inversion theorems for probability distributions, 3. formulas for pay-offs which occur in insurance. Numerical examples are provided.