Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: José Manuel Corcuera Valverde[en] The main goal of this work is to introduce the stochastic volatility models in mathematical finance and to develop a closed-form solution to option pricing in Heston’s stochastic volatiltiy model, following the arguments in Heston 1993. No background in mathematical finance will be assumed, so another main goal of this work is to develop the theory of stochastic integration and to introduce the Black-Scholes market model, the benchmark model in mathematical finance. Standard topics in the framework of market models, such as trading strategies, completeness and replication, and the notion of arbitrage, will also be reviewed.