Double continuation regions for American and Swing options with negative discount rate in Lévy models

Abstract

In this paper we study perpetual American call and put options in an exponential Lévy model. We consider 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. We show that in this case a double continuation region arises and we identify the two critical prices. We also generalize this result to multiple stopping problems of Swing type, that is, when successive exercise opportunities are separated by i.i.d. random refraction times. We conduct an extensive numerical analysis for the Black-Scholes model and the jump-diffusion model with exponentially distributed jumps.Keywords. American option negative rate optimal stopping Lévy process 15 6. Numerical analysis 17 6.1. Black-Scholes model 17 6.2. Jump-diffusion model 18 7. Conclusions and future work 22 Acknowledgments 23 References 23

American step options

“…First, it complements a literature dealing with unusual features of exercise regions associated with real options or financial derivatives. Battauz, De Donno, & Sbuelz (2012 and De Donno, Palmowski, & Tumilewicz (2019), for instance, examine economic settings leading to negative discount rates and show that a double continuation region can then arise. 5 Under such circumstances, it is optimal to wait if the underlying asset price is either high or low.…”

Double continuation regions for American options under Poisson exercise opportunities

“…Before concluding this section, we show the convergence to the classical case (De Donno et al, 2020) as the rate of observation goes to infinity. Solely in this subsection, in order to spell out the dependence on the rate of observation, for = , , let , ( ) be the value function, * , and * , the optimal barriers and  , the stopping region when the rate of observation is > 0.…”

“…(2015), and De Donno et al. (2020) for a detailed literature review on the American option problem with a negative discount rate.…”

Solving the dual Russian option problem by using change‐of‐measure arguments

“…The idea of writing this paper was to draw the attention of the readership to the article of Shepp and Shiryaev () in which an optimal stopping problem with positive exponential discounting rates was studied for one of the first times to the best of our knowledge. Other optimal stopping problems with positive exponential discounting rates were recently considered by Xia and Zhou (), Battauz, De Donno, and Sbuelz (, ), and De Donno, Palmowski, and Tumilewicz () among others. The introduction of positive exponential discounting rates into the optimal stopping problems implied the appearance of disconnected continuation regions or so‐called double continuation regions for the underlying processes.…”