Pricing derivatives: Black-Scholes

In this paper, we present and prove the validity of an extension of the original Black-Scholes option pricing model. We\ud extend the work done in the initial 1973 paper by pricing a European call option, but allowing for the volatility and drift processes of the underlying stock to vary over time. The model is considered through the perspective of an arbitrage construction strategy. First, the discrete case is used to simplify the fundamental argument. Then, the calculus of stochastic processes is developed, and we prove the existence of arbitrage opportunities that force the price to converge to its value given by the model. This paper is designed to be easier to understand and more stand-alone than most\ud other works in the field. Some knowledge of basic probability and measure theory is assumed