### Application of numerical methods, derivatives theory and Monte Carlo simulation in evaluating BM&F BOVESPA's POP (Protected and Participative Investment)

Giuliano Carrozza Uzêda Iorio de Souza, Carlos Patrício Samanez, Gustavo Santos Raposo, Antonio Carlos Gonçalves
2011 Pesquisa Operacional
This article presents a practical case in which two of the most efficient numerical procedures developed for derivative analysis are applied to evaluate the POP (Investment Protection with Participation), a structured operation created by São Paulo Stock Exchange -BM&FBOVESPA. The first procedure solves the differential equation through the use of implicit finite differences method. Due to its characteristics, the approach makes it possible to run sensitivity analysis as well as price
more » ... . In the second, the problem is solved by Monte Carlo simulation, which facilitates the identification of the probability related to the exercise of the embedded options. others, are kept present in the most recent pricing models, renewed efforts are required in the extension of some of these works and in the improvement of the proper numerical methods for the solution of several issues related to the topic, such as the Finite Difference method and the Monte Carlo simulation method, applied to this work. Black & Scholes (1973) established the bases of the modern financial options theory, when they developed an equilibrium model that did not need any restrictive assumption on the individual preferences regarding risk, or on market price formation in equilibrium. They achieved that through the establishment of a "risk-free" portfolio, whose pricing arises out of non-arbitration conditions. The model defines a differential equation that describes price behavior for the derivative that, in conjunction with proper boundary conditions, is used under certain circumstances to derive an expression that allows the determination of the value of an European-type call option. One of the characteristics of the Black & Scholes model -also present in Merton (1973 ), Black (1974 and Cox et al. (1979) models -, is the evaluation based on the so-called risk-neutral world. In the risk-neutral evaluation, it is not assumed that the investors' preferences before risk are neutral, and it does not use actual probabilities, but the risk-neutral probabilities or also called martingale measures. According to the Black & Scholes line, Merton (1973a) extended the model so that it envisaged the possibility that the underlying actions might distribute dividends, what would eventually also enable applications in the evaluation of future options, stock indexes and currencies. Additionally, in another important article, Merton (1972b) showed the main relationships between call and put options.