In mathematical finance, tools from probability, and more specifically from stochastic analysis, are applied to tackle problems arising in financial markets. Therefore, one needs to build suitable models to describe the phenomena that appear in the market. 

In the first part of the talk, I am going to present some basic probability preliminaries and some intuitive concepts in finance. 

Then, I will move on to consider the more famous models in financial mathematics, describing their main properties. To be accessible to a mathematical audience of non-experts, I will extensively use examples, graphics and intuitive definitions. 

In the very last part of the talk, the more technical one, I will mention some result of my research, more precisely, I will introduce a joint calibration problem via stochastic-local volatility model along with an exercise based on real data.