双侧走廊：先碰上界且不碰下界

Two Sided Corridor

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 when rates are zero. Calculate the average exit time i.e. the average time before touching $u$ or $- d$ .

解析

令 $\tau$ 为首次命中 $u$ 或 $-d$ 的时间。

(1) $r=0$ 时，价格就是先碰 $u$ 的概率 $p$ 。对停时用鞅 $B_{t\wedge\tau}$ ：

0=\mathbb{E}[B_\tau]=pu+(1-p)(-d)\Rightarrow p=\frac{d}{u+d}.

所以

\boxed{\text{价格}=\frac{d}{u+d}}.

(2) 平均退出时间：对鞅 $B_t^2-t$ 在停时应用可选停止，得

\mathbb{E}[\tau]=\mathbb{E}[B_\tau^2]=pu^2+(1-p)d^2=ud.

因此

\boxed{\mathbb{E}[\tau]=ud}.