Derivative securities, when used correctly, allow investors to increase their expected profits and minimize their exposure to risk. Options offer leverage and insurance for risk-averse investors while they can be used as ways of speculation for the more risky investors. When an option is issued, we face the problem of determining the price of a product at the same time we must make sure to eliminate arbitrage opportunities. In this thesis, we introduce a robust, accurate, and highly efficient financial option valuation technique, the so-called SWIFT method (Shannon wavelets inverse Fourier technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. This method is adapted to the pricing of European options and Discrete Lookback options. Numerical experiments show exponential convergence and confirm the robustness, efficiency and versatility of the method
