Perpetual American options with asset-dependent discounting

TL;DR

This paper analyzes perpetual American options with asset-dependent discounting, providing conditions for convexity, explicit solutions in certain cases, and exploring properties like symmetry and smooth fit.

Contribution

It introduces a framework for pricing perpetual American options with asset-dependent discount rates, including explicit solutions and conditions for key properties.

Findings

Convexity conditions for the value function are established.

Explicit solutions are derived for geometric Lévy processes.

The paper proves a put-call symmetry and smooth fit conditions.

Abstract

In this paper we consider the following optimal stopping problem $$V^{\omega}_{\rm A}(s) = \sup_{\tau\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^\tau \omega(S_w) dw} g(S_\tau)],$$ where the process $S_t$ is a jump-diffusion process, $\mathcal{T}$ is a family of stopping times while $g$ and $\omega$ are fixed payoff function and discount function, respectively. In a financial market context, if $g(s)=(K-s)^+$ or $g(s)=(s-K)^+$ and $\mathbb{E}$ is the expectation taken with respect to a martingale measure, $V^{\omega}_{\rm A}(s)$ describes the price of a perpetual American option with a discount rate depending on the value of the asset process $S_t$. If $\omega$ is a constant, the above problem produces the standard case of pricing perpetual American options. In the first part of this paper we find sufficient conditions for the convexity of the value function $V^{\omega}_{\rm A}(s)$. This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of $V^{\omega}_{\rm A}(s)$. We also prove a put-call symmetry for American options with asset-dependent discounting. In the case when $S_t$ is a geometric L\'evy process we give exact expressions using the so-called omega scale functions introduced in Li and Palmowski (2018). We prove that the analysed value function satisfies the HJB equation and we give sufficient conditions for the smooth fit property as well. Finally, we present a few examples for which we obtain the analytical form of the value function $V^{\omega}_{\rm A}(s)$.