Executive Stock Option Exercise with Full and Partial Information on a Drift Change Point

We analyse the valuation and exercise of an American executive call option written on a stock whose drift parameter falls to a lower value at a change point given by an exponential 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. Our setup captures the position of an executive (insider) and employee (outsider), who receive executive stock options. The latter yields a model under the observation filtration $\widehat{\mathbb F}$ where the drift process 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 problem has two spatial dimensions. We analyse and numerically solve to value the option for both agents and illustrate that the additional information of the insider can result in exercise patterns which exploit the information on the change point.