Optimal Multiple Stopping with Negative Discount Rate and Random Refraction Times under Lévy Models

Abstract

This paper studies a class of optimal multiple stopping problems driven by Lévy processes. Our model allows for a negative effective discount rate, which arises in a number of financial applications, including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. Moreover, successive exercise opportunities are separated by i.i.d. random refraction times. Under a wide class of two-sided Lévy models with a general random refraction time, we rigorously show that the optimal strategy to exercise successive call options is uniquely characterized by a sequence of up-crossing times. The corresponding optimal thresholds are determined explicitly in the single stopping case and recursively in the multiple stopping case.

An Analytic Recursive Method for Optimal Multiple Stopping: Canadization and Phase-Type Fitting

“…The optimality of the threshold strategy for the single stopping problem has been shown by Mordecki [46]. The same characterization holds for the multi-stage problem, and we refer the reader to [13,51] and the authors' companion paper [41] for the proof.…”

“…It can be shown that the threshold levels are bounded from below by log K and increase as the number of remaining stopping opportunities decreases, i.e., log K < a * N ≤ · · · ≤ a * 1 . It has been shown in [41] that this monotonicity also holds when the refraction times δ's are generalized to be independent, identically distributed random variables provided that they are independent of X, and X δ admits a density. They also show that there exists a limit a * ∞ := lim N →∞ a * N ≥ log K.…”

“…This is a critical condition so that the solution of the problem is nontrivial (see, e.g., [13,41,51]). In fact, we can slightly weaken the condition for the case α < 0 (where exp(−αt)K grows to infinity) to accommodate the case ψ(1) = α given ψ ′ (1) < 0 (see [41] for a proof).…”

“…In fact, we can slightly weaken the condition for the case α < 0 (where exp(−αt)K grows to infinity) to accommodate the case ψ(1) = α given ψ ′ (1) < 0 (see [41] for a proof).…”

“…For related optimal multiple stopping problems under spectrally negative models, we mention [51] for a swing put option with constant refraction times, and [50] with a more general payoff function without refraction times. For models with more general processes, Leung et al [41] study a refracted optimal multiple stopping problem driven by a two-sided Lévy process with general random refraction times, and Christensen and Lempa [16] consider a similar problem driven by a general Markov process with exponential refraction times.…”