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).

This paper develops a framework for cost-sensitive training of probabilistic machine learning models that predict the direction of aggregate stock returns. We introduce a bi-objective loss function that augments conventional log-loss optimization with an expected error-cost objective, using the option-implied conditional value-at-risk as a forward-looking measure of misclassification cost. We illustrate how cost-sensitive learning produces economically sensible predictions. Market timing strategies constructed using forecasts from elastic-net logistic regression and gradient boosted decision trees trained using our framework improve out-of-sample risk-adjusted returns and substantially reduce downside risk.