Valuation of optional stopping problems with HOST feedback structure

We extend the Poisson optional stopping times option pricing models from Hobson (2021) and Lange et al. (2020) to a broader class of jump-diffusions in which feedback between the optional stopping times and jumps in the value of the underlying asset is modelled by embedding both processes in a multidimensional Hawkes process. For such valuation problems, we derive the corresponding value function equation, which is a partial integro-differential equation with a free boundary. To circumvent having to numerically solve a free boundary problem, we provide an approximation scheme that requires solving a sequence of fixed boundary partial integro-differential equations where the resulting functions convergence uniformly on certain sets. We derive several analytical properties of the value functions, along with a novel result on the inclusion of Hawkes processes that is of independent interest. We illustrate the flexibility of the model by providing several examples to which a HOST feedback structure can apply. Furthermore, we present some numerical examples to show the effects of various types of feedback on asset pricing; neglecting the feedback structure is found to potentially lead to substantial pricing differences.