In this thesis we introduce the concept of the Wiener process and stochastic integration in order to analyze and derive pricing models for derivatives pricing. We present a construction and analysis of stochastic integrals and Itō’s lemma. These analytical tools are then used to derive the Black-Scholes equation for the purpose of options pricing. A numerical treatment of the Black-Scholes equation is presented, along with some drawbacks of the simplified model. We then introduce stochastic volatility and describe techniques in options pricing with stochastic volatility models.
