[en] The authors Fischer Black, Myron Scholes and Robert Merton, developed a theory for the evaluation of European options, which is currently known as the Black-Scholes-Merton model. It should be noted that in 1997 the importance of the model was recognized by receiving the Nobel Prize in Economics. The fundamental idea of the Black-Scholes model is to build a portfolio formed by the option and its underlying asset. Then with a specif hedging strategy, the portfolio can be converted into risk-free. However, the hypotheses that are assumed in the construction of the model are not entirely correct when the model is translated into practice. We emphasize that the assumption of the continuity of hedged is impossible in practice. As a result of the hedging in discrete time, the portfolio is not free of risk as the model assumes. This fenomenon was studied by Phelim P. Boyle and David Emanuel in 1980 for a European vanilla option. In this memory we will study the process of deduction of the basic Black-Scholes model and their extension for options with several underlying assets. Afterwards, we will examine the evaluation of the Margrabe option and the Quanto option. Finally, we will analyze the discrete time hedges error for these two types of options.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Josep Vives i Santa Eulàlia i Oriol Roch