单侧走廊：命中上界的贴现概率

One Sided Corridor

Let $B_{t}$ be a Brownian Motion and $u$ a positive real number. We consider an option which pays 1 if $B_{t}$ reaches $u$

\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 when rates are zero and with rates $r > 0$

解析

设 $\tau_u$ 为布朗运动首次命中 $u>0$ 的时间。

用指数鞅 $\exp\left(aB_t-\tfrac12a^2t\right)$ 在停时 $\tau_u$ 应用可选停止，得

\mathbb{E}\left[e^{-\frac12a^2\tau_u}\right]=e^{-au}.

取 $a=\sqrt{2r}$ ，得到

\boxed{\mathbb{E}[e^{-r\tau_u}]=e^{-\sqrt{2r}\,u}}.