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This study shows that in particular cases, the minimal martingale measure coincides with the Esscher martingale measure. Using the martingale approach can produce an exact solution for the price of a European call option on an asset modeled as an exponential Lévy process when a closed-form expression exists for the Lévy measure under some integrability conditions. If the jump component vanishes, the solution reduces to the Black–Scholes formula. To compute the option price accurately and quickly, this study uses polynomial interpolation with divided differences. A numerical analysis compares the accuracy and CPU time of the latter method with those of three Fourier-based formulas described by Lewis (2001). © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:469-491, 2013