Real-Option Valuation in Multiple Dimensions Using Poisson Optional Stopping Times

Abstract

We provide a new framework for valuing multidimensional real options where opportunities to exercise the option are generated by an exogenous Poisson process, which can be viewed as a liquidity constraint on decision times. This approach, which we call the Poisson optional stopping times (POST) method, finds the value function as a monotone sequence of lower bounds. In a case study, we demonstrate that the frequently used quasi-analytic method yields a suboptimal policy and an inaccurate value function. The proposed method is demonstrably correct, straightforward to implement, reliable in computation, and broadly applicable in analyzing multidimensional option-valuation problems.

Constrained optimal stopping, liquidity and effort

“…. Lange et al [14] consider a multidimensional version of the constrained optimal stopping problem (with constant stopping rate) and consider the sequence {V (n) λ } n≥0 . They observe that V (n) λ is increasing in n and show, under an assumption that a certain iterated expectation is finite, that V (n) λ converges to V (∞) λ = V λ geometrically fast.…”

“…where T λ is the set of event times of a Poisson process with rate λ. The constrained optimal stopping problem has been extended in many ways and to many settings, for example to allow for regime switching (Liang and Wei [16]), non-exponential inter-arrival times (Menaldi and Robin [18]) and running costs and multi-dimensions (Lange et al [14]). A related work in which actions are constrained to occur only at event times of a Poisson process is Rogers and Zane [22] who model portfolio optimisation.…”

“…Solving (4), even numerically, may be challenging as the unknown V θ appears on both sides. One strategy, as described in Lange et al [14] is as follows. Let V = V θ , and hence deduces monotonicity and convexity results for V θ .…”

Randomised rules for stopping problems

“…For each choice of c we have three alternative representations of the value function, via (6), (8), and (9). In the next section we concentrate on the existence and uniqueness of solutions to (8) and (9) and the extent to which solutions of the stochastic integral equation or of the differential equation can be identified with solutions of the problem (6) with randomised stopping. Then, in Sections 4 and 5, we consider solutions to the problem for particular choices of payoff function.…”

“…Optimal stopping problems in which stopping is only possible at event times of a Poisson process have been studied previously by Dupuis and Wang [5] and Lempa [10]. In corporate finance, Lange, Ralph, and Støre [9] considered a problem in real options of this form. They interpreted the fact that agents cannot stop, or in their context exercise an option, as a liquidity constraint.…”