Stochastic development and closed-form pricing for European options.

Summary This thesis develops a new methodology to establish analytical approximations for European option prices. Our approach cleverly combines stochastic developments and Malliavin calculus to obtain explicit formulas and accurate error estimates. The interest of these formulas lies in their computation time which is as fast as that of the Black-Scholes formula. Our motivation comes from the growing need for real-time calculations and calibration procedures, while controlling the numerical errors related to the model parameters. We treat four categories of models, performing specific parameterizations for each model in order to better target the right proxy model and thus obtain easy to evaluate correction terms. The four parts treated are: diffusions with jumps, local volatilities or Dupire models, stochastic volatilities and finally hybrid models (rate-share). It should also be noted that our approximation error is expressed as a function of all the parameters of the model in question and is also analyzed in terms of the regularity of the payoff.