  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.
