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.
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Dufresne, D., Garrido, J. & Morales, M. Fourier Inversion Formulas in Option Pricing and Insurance. Methodol Comput Appl Probab 11, 359–383 (2009). https://doi.org/10.1007/s11009-007-9049-z