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We study the numerical approximation of the implied volatility for models in option pricing that generalize the Black–Scholes equation when the processes which generate the underlying stock returns may contain both continuous part and jumps. Here we propose weighted positive difference schemes with implicit-explicit iterations to solve the integral terms explicitly. We aim to identify the volatility from extra point measurement. First, we linearize in time the diffusion quadratic nonlinear term. Next, we employ a special decomposition of the approximate solution. Numerical tests are presented to show the computational efficiency of the algorithms. [ABSTRACT FROM AUTHOR]