双侧走廊

Two Sided Corridor With Rates

专题: Finance / 金融
难度: L4
来源: QuantQuestion

题目详情

Let $B_{t}$ be a Brownian Motion and $u$ and $d$ two positive real numbers, We consider an option which pays 1 if $B_{t}$ reaches $u$ and remained greater then $- d$ since inception

\exists t_0: B_{t_0} = u; \forall t \in [0, t_0], B_t > - d

payment is made when the barrier is touched. Calculate the price of this option with rates $r > 0$ .

解析

设 $\tau$ 为首次命中 $u$ 或 $-d$ 的时间，payoff 为 $e^{-r\tau}\mathbf{1}_{\{\text{先到 }u\}}$ 。

用指数鞅并在停时应用可选停止，可得

\boxed{\text{价格}=\mathbb{E}[e^{-r\tau}\mathbf{1}_{\{\tau_u<\tau_{-d}\}}]= \frac{\sinh(\sqrt{2r}\,d)}{\sinh(\sqrt{2r}(u+d))}}.

当 $r\to 0$ 时， $\sinh(x)\sim x$ ，退化为 $d/(u+d)$ 。