Executive stock option exercise with full and partial information on a drift change point

We analyse the optimal exercise of an executive stock option (ESO) written on a stock whose drift parameter falls to a lower value at a change point, an exponentially distributed random time independent of the Brownian motion driving the stock. Two agents, who do not trade the stock, have differing information on the change point, and seek to optimally exercise the option by maximising its discounted payoff under the physical measure. The first agent has full information, and observes the change point. The second agent has partial information and filters the change point from price observations. This scenario is designed to mimic the positions of two employees of varying seniority, a fully informed executive and a partially informed less senior employee, each of whom receives an ESO. The partial information scenario yields a model under the observation filtration $\widehat{\mathbb{F}}$ in which the stock drift becomes a diffusion driven by the innovations process, an $\widehat{\mathbb{F}}$-Brownian motion also driving the stock under $\widehat{\mathbb{F}}$, and the partial information optimal stopping value function has two spatial dimensions. We rigorously characterise the free boundary PDEs for both agents, establish shape and regularity properties of the associated optimal exercise boundaries, and prove the smooth pasting property in both information scenarios, exploiting some stochastic flow ideas to do so in the partial information case. We develop finite difference algorithms to numerically solve both agents' exercise and valuation problems and illustrate that the additional information of the fully informed agent can result in exercise patterns which exploit the information on the change point, lending credence to empirical studies which suggest that privileged information of bad news is a factor leading to early exercise of ESOs prior to poor stock price performance.