In this project we looked into financial markets. The goal of this project was to find out if a bubble is forming and when the most probable time of bursting would be. that is what we call the critical time. In order to do this we studied the work of Professor Didier Sornette, who is an expert in this field of mathematics. In this bachelor thesis we use the Johansen-Ledoit-Sornette (JLS) model and the Levenberg-Marquardt algorithm to predict the critical time of bubbles. By critical time we mean the most probable time of a bubble to burst, but not for certain: there is always a probability to attain the end of the bubble without bursting. \\
We looked at the results Didier Sornette got in his work on Black Monday and tried to obtain the same results. Besides that we investigated the sensitivity of the JLS model and differences between variants of it, also when simulating our own data. In the end we looked at applying the model at real data. For the data we chose the Amsterdam Exchange Index and Bitcoin.