In this paper we consider the following optimal stopping problem V^{\omega}_{ \rm A }(s) = \tau\in\mathcal{T} E_{s}[e^{-\int_0^\tau \omega(S_w) dw} g(S_\tau)], where the process S_t is a jump-diffusion process, T is a family of stopping times and g and \omega are fixed payoff function and discounting function, respectively. In a financial market context, if g(s)=(K-s)^+ or g(s)=(s-K)^+ and 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 HJB and we give sufficient conditions for the smooth fit property as well. Finally, we analyse few cases in detail performing extensive numerical analysis_{\rm A}(s)} .
In this paper we consider the following optimal stopping problem V^{\omega}_{ \rm A }(s) = \tau\in\mathcal{T} E_{s}[e^{-\int_0^\tau \omega(S_w) dw} g(S_\tau)], where the process S_t is a jump-diffusion process, 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 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)} .
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