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A general class of truncated Lévy processes is introduced, and possible ways of fitting parameters of the constructed family of truncated Lévy processes to data are discussed. For a market of a riskless bond and a stock whose log-price follows a truncated Lévy process, TLP-analogs of the Black–Scholes equation, the Black–Scholes formula, the Dynkin derivative and the Leland's model are obtained, a locally risk-minimizing portfolio is constructed, and an optimal exercise price for a perpetual American put is computed.
Boyarchenko et al. (Sat,) studied this question.