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 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 . The numerical experiments  show exponential convergence.
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. Based on this representation for the density function, 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 underlying stochastic models. This is particularly relevant for options pricing where the payoff of the contract are generally not smooth functions . The numerical experiments on a class of models widely used in quantitative finance show exponential convergence.
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Approximations
Based on this representation for the density function, approximations
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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
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
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density function and not on the payoff function
underlying stochastic models. This is particularly relevant for options pricing where the payoff of the contract are generally not smooth functions
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on a class of models widely used in quantitative finance