Optimally Stopping at a Given Distance from the Ultimate Supremum of a Spectrally Negative Lévy Process

We consider the optimal prediction problem of stopping a spectrally negative Lévy process as close as possible to a given distance $b \geq 0$ from its ultimate supremum, under a squared-error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We find that the solution has an interesting non-trivial structure: if b is larger than a certain threshold then it is optimal to stop as soon as the difference between the running supremum and the position of the process exceeds a certain level (less than b), while if b is smaller than this threshold then it is optimal to stop immediately (independent of the running supremum and position of the process). We also present some examples.