Here we develop an option pricing method based on Legendre series expansion of the density function. The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Approximations formulas for pricing European type option are derived and 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. The numerical experiments show exponential convergence.
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