论文标题
与资产依赖折扣的永久选择
Perpetual American options with asset-dependent discounting
论文作者
论文摘要
在本文中，我们考虑以下最佳停止问题$$ v^ω_ {\ rm a}（s）= \ sup_ {τ{τ\ in \ Mathcal {t}} \ Mathbb {e} _ {e} _ {s} _ {s}跳转过程，$ \ MATHCAL {T} $是停止时间的家庭，而$ G $和$ω$分别是固定的回报功能和折扣功能。 In a financial market context, if $g(s)=(K-s)^+$ or $g(s)=(s-K)^+$ and $\mathbb{E}$ is the expectation taken with respect to a martingale measure, $V^ω_{\rm A}(s)$ describes the price of a perpetual American option with a discount rate depending on the value of the asset process $S_t$.如果$ω$是一个常数，则以上问题会产生定价永久选择的标准案例。在本文的第一部分中，我们找到了足够的条件，可以使值函数的凸度$ v^ω_ {\ rm a}（s）$。这使我们能够确定停止区域作为一定间隔，因此我们能够确定$ v^ω_ {\ rm a}（s）$的形式。我们还证明了与资产相关折扣的美国选择的呼叫对称性。在$ s_t $是几何lévy过程的情况下，我们使用Li and Palmowski（2018）中引入的所谓欧米茄量表功能提供了精确表达式。我们证明，分析的值函数满足HJB方程，并为光滑拟合属性提供了足够的条件。最后，我们提出了一些示例，其中我们获得了值函数的分析形式$ v^ω_ {\ rm a}（s）$。
In this paper we consider the following optimal stopping problem $$V^ω_{\rm A}(s) = \sup_{τ\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^τω(S_w) dw} g(S_τ)],$$ where the process $S_t$ is a jump-diffusion process, $\mathcal{T}$ is a family of stopping times while $g$ and $ω$ are fixed payoff function and discount function, respectively. In a financial market context, if $g(s)=(K-s)^+$ or $g(s)=(s-K)^+$ and $\mathbb{E}$ is the expectation taken with respect to a martingale measure, $V^ω_{\rm A}(s)$ describes the price of a perpetual American option with a discount rate depending on the value of the asset process $S_t$. If $ω$ 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 $V^ω_{\rm A}(s)$. This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of $V^ω_{\rm A}(s)$. We also prove a put-call symmetry for American options with asset-dependent discounting. In the case when $S_t$ is a geometric Lévy 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 $V^ω_{\rm A}(s)$.
