Pricing Perpetual American put options with asset-dependent discounting

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.