The book focuses on numerical methods for derivative pricing with an emphasis on their mathematical foundations. It offers the most frequently explored numerical methods of derivative pricing and covers the material of standard courses in computational finance. The book presents the best-known methods of multinomial trees, Monte Carlo simulations for European, American, and exotic options, and finite difference and finite element methods for PDEs. However, unlike many textbooks on computational finance, it also presents rigorous results on analyzed numerical algorithms with a focus on the mathematical content — including theorems with possibly complete proofs.The book gives the reader the necessary tools for analyzing algorithm consistency and offers an efficient approach to assessing the stability and convergence of numerical methods. It consolidates mathematical results previously dispersed across different monographs into a single volume, while tailoring the presentation to the specific needs of computational finance.Each chapter includes a set of exercises designed to help readers apply theoretical concepts to specific algorithms and enhance their computational skills in derivative pricing.