Modelo binomial para precificação de opções

Abstract

In this work we start by introducing some basic definitions of probability and finance. The definition from finance includes terms such as asset, inflation, interest rate, etc. Next we defined what derivatives are, that is, financial instruments derived from assets. In this work, we focused on a specific type of derivative: the options. More precisely, we dealt with traditional buy options and sell options, also known as “vanilla” options. We didn’t cover exotic options. Financial options, as the name says, form a financial instrument which the parties agree that the buyer of the option gains the option of exercising the terms of the contract they agreed upon. Consider, for example, options for buying some financial asset. In the case of a buy option, the buyer of the option gains the right to purchase the asset in question at the price that has been agreed until maturity (in the case of American options) or exactly at maturity (in the case of European options). A difficult question is this: if we know the price history of the asset in question, what should be the “fair” price of the options? There are several strategies for pricing options. The most famous was obtained by Black and Scholes (1973), in his seminal article. The model we presented in this work, that is, the binomial model, can be seen as an approximation of the Black and Scholes model. The advantage of the binomial model is that it manages to be very useful at the same time and its understanding is very accessible. We will also see some strategies of “portfolio preservation”, known in the middle of finance as hedging. We provied an example on how to use the binomial model to perform the famous “delta hedging”. In which it is determined at each moment of time what should be the position in the asset in question (related asset ’the option) so that the expected equity is constant (and thus seeking to avoid significant losses). Finally, we illustrated the pricing and hedging models through computational simulation.