Pricing Perpetual American put options with asset-dependent discounting

TL;DR

This paper develops an algorithm for pricing perpetual American put options with asset-dependent discounting, generalizing classic models by incorporating a variable discount function and providing exact solutions under certain conditions.

Contribution

It introduces a novel algorithm for pricing options with asset-dependent discounting and derives closed-form solutions for specific discount functions, extending existing models.

Findings

The value function is convex under certain conditions.

Closed-form solutions are obtained for specific discount functions.

The approach generalizes classic constant discounting models.

Abstract

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as \begin{equation*} V^{\omega}_{\text{A}^{\text{Put}}}(s) = \sup_{\tau\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^\tau \omega(S_w) dw} (K-S_\tau)^{+}], \end{equation*} where $\mathcal{T}$ is a family of stopping times, $\omega$ is a discount function and $\mathbb{E}$ is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process $S_t$ is a geometric L\'evy process with negative exponential jumps, i.e. $S_t = s e^{\zeta t + \sigma B_t - \sum_{i=1}^{N_t} Y_i}$. The asset-dependent discounting is reflected in the $\omega$ function, so this approach is a generalisation of the classic case when $\omega$ is constant. It turns out that under certain conditions on the $\omega$ function, the value function $V^{\omega}_{\text{A}^{\text{Put}}}(s)$ is convex and can be represented in a closed form; see Al-Hadad and Palmowski (2021). We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of $\omega$ such that $V^{\omega}_{\text{A}^{\text{Put}}}(s)$ takes a simplified form.