Fourier Inversion Formulas in Option Pricing and Insurance

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 or stop-loss insurance, and we show that Fourier methods may be used to compute them. In this paper, we 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 payoffs which occur in insurance. Numerical examples are provided.