Pricing Perpetual American put options with asset-dependent discounting

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^ω_{\text{A}^{\text{Put}}}(s) = \sup_{τ\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^τω(S_w) dw} (K-S_τ)^{+}], \end{equation*} where $\mathcal{T}$ is a family of stopping times, $ω$ 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 Lévy process with negative exponential jumps, i.e. $S_t = s e^{ζt + σB_t - \sum_{i=1}^{N_t} Y_i}$. The asset-dependent discounting is reflected in the $ω$ function, so this approach is a generalisation of the classic case when $ω$ is constant. It turns out that under certain conditions on the $ω$ function, the value function $V^ω_{\text{A}^{\text{Put}}}(s)$ 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 $ω$ such that $V^ω_{\text{A}^{\text{Put}}}(s)$ takes a simplified form.

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