Before revision:  Approximations formulas for pricing European type option are derivedand a robust, stable algorithm for its implementation is proposed. An error analysis on the option pricing provides an estimate for the rate of convergence, which depends essentially on the smoothness of the density function and not on the payoff function .
After revision:  Approximations formulas for pricing European type options are derived. To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function. Some numerical instabilities arise because of serious subtractive cancellations in its formulation (96) in proposition 7.1. To overcome this difficulty, we rewrite this quantity as solution of a second-order linear difference equation and solve it using a robust and stable algorithm from Olver. Derivation of the pricing method has been accompanied by an error analysis. Errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff? functions, but rather by the density function of the density function and not on the payoff function .