Black-Scholes Option Pricing Model

Abstract

Simple generally accepted economic assumptions are insufficient to develop a rational option pricing theory. Assuming a perfect financial market in Section 2.1 leads to elementary arbitrage relations which options have to fulfill. While these relations can be used as a verification tool for sophisticated mathematical models, they do not provide an explicit option pricing function depending on parameters such as time and the stock price as well as the options underlying parameters K, T. To obtain such a pricing function the value of the underlying financial instrument (stock, currency, ...) has to be modelled. In general, the underlying instrument is assumed to follow a stochastic process either in discrete or in continuous time. While the latter are analytically easier to handle, the former, which we will consider as approximations of continuous time processes for the time being, are particularly useful for numerical computations. In the second part of this text, the discrete time version will be discussed as financial time series models.