Aplicação do método de Crank Nicholson para a modelação de preço de um derivado financeiro

Abstract

Partial differential equations play a fundamental role in various areas of science due to their wide applicability. In Financial Mathematics, for example, they are used to model economic growth and financial risk. However, most real-world problems involving partial differential equations do not have a known or feasible analytical solution. To overcome this limitation, numerical approximation methods are employed, among which the finite difference method stands out as one of the most widely used techniques. This method approximates the solution by replacing continuous derivatives with differences between discrete values on a grid of points. In this work, a theoretical summary of the mathematical and financial concepts to be used in the development of the topic is first presented. Next, a literature review on differential equations is con- ducted, addressing aspects related to their classification and the analytical and numerical methods for their resolution. An application example of the Crank-Nicholson method is considered for solving a pricing model of a European call option. The convergence of the developed numerical method is analyzed, and the source term methodology is used to verify the correct implementation of the method. Additionally, an algorithm for the method is developed and implemented using MATLAB software. Through the numerical tests performed, it was found that the Crank-Nicholson method exhibits a spatial and temporal convergence order close to 2, confirming the theory underlying this method. The numerical simulations demonstrated that the spatial error is more dominant in less refined grids, and the accuracy of the results improves significantly with grid refinement.