反弹走廊

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 touched the down barrier before. The option is knocked out and pays zero if it touches the up barrier first.

$$\mathrm{pays~1~if~}\exists t_{0}:B_{t_{0}} = u;\exists t_{1}\in [0,t_{0}]:B_{t_{1}} = -d$$ $$\mathrm{pays~0~if~}\exists t_{0}:B_{t_{0}} = u;\forall t\in [0,t_{0}],B_{t} > - d$$

payment is made when the barrier is touched. Calculate the price of this option with rates $r > 0$ . Generalize to an option paying 1 after $n$ bounces (pays 1 if the

option touches $u$ then $u$ then $- d$ etc... n times, pays 0 if $u$ is touched first).