This thesis captures the calibration of a FX hybrid model: The FX Black-Scholes Hull-White model. The main focus is on the calibration of the parameters in the Hull-White process: The mean reversion and the volatility parameter. The latter is commonly calibrated as a time-dependent parameter, whilst the mean reversion parameter is not. This thesis covers the calibration of the mean reversion as a time-dependent parameter. A known method from the Literature is researched, where we calibrate the mean reversion independently from the volatility parameter to the ratio of two swaptions with the same expiry but different tenor. In our research this method is extended to the negative interest rate environment by assuming that the swap rate follows a shifted lognormal distribution instead of a lognormal distribution. We show that a specific set of swaptions can be chosen, so that the calibration problem is simplified. This choice leads to sequential calibration of convex optimization problems. Numerical results of calibration to artificial and market data are presented, where we compare our method to picking the mean reversion parameter arbitrarily. The findings suggest that the choice of mean reversion parameter affects the calibration procedure. Therefore, we argue that a calibration method for the mean reversion would be appropriate. Besides the focus on the Hull-White process, the calibration of the volatility parameter in the FX Black-Scholes process of the hybrid model is investigated. For the calibration of the FX volatility parameter ATM options on FX rates are used. Numerical results of calibration to artificial and market data are shown. A typical problem of calibration in the industry is discussed: Precision for late maturities. The results in this thesis suggest that this problem could be solved. For research on homogeneity constraints on the time-dependent parameters, the performance of delta-hedging is investigated. The delta-hedging is performed in a simple Black-Scholes world with time-dependent volatility. We show that given a fixed amount hedges beforehand and interest rate zero, there exists an optimal distribution of time points that will lead to equal variance on the profit and loss for every volatility function that has equal implied volatility from t0 to maturity T. Using this strategy, the homogeneity of the volatility parameter has no impact on the performance of delta-hedging. Therefore, in this thesis no homogeneity constraints are used for the calibration of the time-dependent parameters.

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.