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

TL;DR

This paper analyzes optimal multiple stopping problems driven by Lévy processes with negative discount rates and random refraction times, providing explicit and recursive solutions for optimal exercise strategies.

Contribution

It introduces a novel framework for multiple stopping under Lévy models with negative discount rates and random refraction times, characterizing optimal strategies explicitly and recursively.

Findings

Optimal strategies are characterized by a sequence of up-crossing times.

Explicit thresholds are derived for single stopping cases.

Recursive thresholds are determined for multiple stopping cases.

Abstract

This paper studies a class of optimal multiple stopping problems driven by L\'evy 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 L\'evy 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.