Perpetual American Options with Asset-Dependent Discounting

Abstract

In this paper we consider the following optimal stopping problem $$\begin{aligned} V^{\omega }_{\text {A}}(s) = \sup _{\tau \in {\mathcal {T}}} {\mathbb {E}}_{s}[e^{-\int _0^\tau \omega (S_w) dw} g(S_\tau )], \end{aligned}$$ V A ω ( s ) = sup τ ∈ T E s [ e - ∫ 0 τ ω ( S w ) d w g ( S τ ) ] , where the process $$S_t$$ S t is a jump-diffusion process, $${\mathcal {T}}$$ T is a family of stopping times while g and $$\omega $$ ω are fixed payoff function and discount function, respectively. In a financial market context, if $$g(s)=(K-s)^+$$ g ( s ) = ( K - s ) + or $$g(s)=(s-K)^+$$ g ( s ) = ( s - K ) + and $${\mathbb {E}}$$ E is the expectation taken with respect to a martingale measure, $$V^{\omega }_{\text {A}}(s)$$ V A ω ( s ) describes the price of a perpetual American option with a discount rate depending on the value of the asset process $$S_t$$ S t . If $$\omega $$ ω 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 $$V^{\omega }_{\text {A}}(s)$$ V A ω ( s ) . This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of $$V^{\omega }_{\text {A}}(s)$$ V A ω ( s ) . We also present a put-call symmetry for American options with asset-dependent discounting. In the case when $$S_t$$ S t is a spectrally negative geometric Lévy process we give exact expressions using the so-called omega scale functions introduced in [30]. We show 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 $${V}^{\omega }_{\text {A}}(s)$$ V A ω ( s ) .