Oberseminar Finanz- und Versicherungsmathematik

We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in extensions of the Black-Merton-Scholes model under progressively enlarged filtrations. It is assumed that the information available from the market is modelled by Brownian filtrations progressively enlarged with the random times at which the underlying process attains its global maximum or minimum, that is, the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval, which are supposed to be progressively observed by the holders of the contracts. We show that the optimal exercise times are the first times at which the asset price process reaches certain lower or upper stochastic boundaries depending on the current values of its running maximum or minimum depending on the occurrence of the random times of the global maximum or minimum of the risky asset price process. The proof is based on the reduction of the original necessarily three-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and either normal-reflection or normal-entrance conditions for the value functions at the optimal exercise boundaries and the edges of the state spaces of the processes, respectively.

This is a joint work with Libo Li (Sydney).

We show that established performance metrics in binary classification, such as the F-score, the Jaccard similarity coefficient or Matthews' correlation coefficient (MCC), are not robust to class imbalance in the sense that if the proportion of the minority class tends to 0, the true positive rate (TPR) of the Bayes classifier under these metrics tends to 0 as well. Thus, in imbalanced classification problems, these metrics favour classifiers which ignore the minority class. To alleviate this issue we introduce robust modifications of the F-score and the MCC for which, even in strongly imbalanced settings, the TPR is bounded away from 0. We numerically illustrate the behaviour of the various performance metrics in simulations as well as on a credit default data set. We also discuss connections to the receiver operating characteristic and precision-recall curves and give recommendations on how to combine their usage with performance metrics.