Numerical methods by optimal quantization in finance.

Summary This thesis is divided into four parts that can be read independently. In this manuscript, we make some contributions to the theoretical study and to the applications in finance of optimal quantization. In the first part, we recall the theoretical foundations of optimal quantization as well as the classical numerical methods to construct optimal quantizers. The second part focuses on the numerical integration problem in dimension 1, which arises when one wishes to compute numerically expectations, such as in the valuation of derivatives. We recall the existing strong and weak error results and extend the results of the second order convergences to other classes of less regular functions. In a second part, we present a weak error result in dimension 1 and a second development in higher dimension for a product quantizer. In the third part, we focus on a first numerical application. We introduce a stationary Heston model in which the initial condition of the volatility is assumed to be random with the stationary distribution of the EDS of the CIR governing the volatility. This variant of the original Heston model produces a more pronounced implied volatility smile for European options on short maturities than the standard model. We then develop a numerical method based on recursive quantization produced for the evaluation of Bermudian and barrier options. The fourth and last part deals with a second numerical application, the valuation of Bermudian options on exchange rates in a 3-factor model. These products are known in the markets as PRDCs. We propose two schemes to evaluate this type of options, both based on optimal product quantization and establish a priori error estimates.