The quintessential option pricing formula under Lévy processes

The basic digital method for option pricing developed in Ingersoll [J. Ingersoll, Digital contracts: Simple tools for pricing complex derivatives, Journal of Business 73 (1) (2000) 67–88] and Buchen and Skipper [P. Buchen, M. Skipper, The quintessential option pricing formula, School of Mathematics and Statistics, University of Sydney, 2003, pp. 1–31] is generalized to a Lévy environment. The approach is combined with the mathematical methodology of Boyarchenko and Levendorskiĭ [S.I. Boyarchenko, S.Z. Levendorskiĭ, Non-Gaussian Merton–Black–Scholes theory, World Scientific, 2002] that employs pseudo-differential operators whose symbol is expressed in terms of the characteristic exponent of the underlying Lévy process. Some new valuation formulas are obtained.