We study a set of optimal stopping problems arising from three branches from within the field of Behavioural Finance. We first consider a problem of an investor having S-shaped reference-dependent preferences who wishes to liquidate a divisible asset position at times of their choosing. We prove that it may be optimal for the investor to partially liquidate the asset at distinct price thresholds above the reference level rather than liquidate all the position in one block sale.

In the second part of our study we consider problems describing the behaviour of an investor who experiences realisation utility whenever they realise gains or losses after liquidating an asset. We build upon the work of Barberis and Xiong [2012] and propose two problems, which we solve by applying the methodology of Dayanik and Karatzas [2003]. The first part considers an agent whose preferences are described by the classical Cumulative Prospect Theory S-shaped Utility proposed by Tversky and Kahneman [1992]. The second problem extends upon the first, and we propose a new utility function under which the agent does not only compare their gains relative to the reference level linearly but also proportionally. As part of the solutions presented for these two problems, we provide explicit conditions differentiating between the optimal strategies arising under different parameter cases.

In the final part of our study, we consider models of optimal stopping with regret. We provide a continuous time re-formulation and extension to the dynamic model presented in Strack and Viefers [2015]. This model describes an agent whose preference structure incorporates a Regret term, where Regret is defined in the context of the work of Loomes and Sugden [1982].