Perpetual American knock-out barrier options in a random inspection scheme

Karatzas and Wang (2000) study an extension of the perpetual American put option by introducing a knock-out barrier for the price of the underlying asset that deactivates the option. We revisit the problem of pricing perpetual American barrier-type options studied by Karatzas, I., & Wang, H. (2000), but this time with the additional assumption that the holder has the opportunity to exercise the option only at random epochs occurring according to an exogenous Poisson process with constant intensity. In contrast, the barrier-triggered deactivation of the option occurs under continuous monitoring, e.g., by an automated system. By employing a constant optimal exercise boundary strategy and assuming geometric Brownian motion dynamics for the underlying asset price process, we provide closed form formulas for the expected present value of the options payoff, the probability of exercising the option and the Laplace transform of the time until exercising the option for both put and call option cases. The determination of the optimal exercise boundary and the fair price of the option is achieved by maximizing the option’s expected present value. The impact of the random exercise opportunities on the optimal expected present value, the optimal exercise threshold and the probability of the optimal exercise before the option is knocked-out is numerically investigated for the put and call option case.