| This thesis explores the mathematical techniques used in the pricing and analysis of complex financial derivatives. We study the Geometric Brownian Model of stock price behavior, stochastic behavior of assets and contingent claims, Ito's lemma in its most general form, deduce Black-Scholes differential equation in the very general case of non-tradable parameters and the especial solution for European options. We also analyze the more complex interest rate-sensitive derivatives, study the concept of duration and convexity, and develop a powerful model of the yield curve. We generalize these results to include derivatives dependent on n state variables, and give practical numerical methods, such as Monte Carlo Simulations, Multi-period binomial tree approaches, finite difference methods, and several Runge-Kuta path approximation methods for solving the pricing problems. |
