# What are the boundary conditions for an up-and-out binary call option of Bermudan type?

I know that an up-and-out binary call option of American type will never knock out if barrier is greater than the strike since the option stops immediately if the stock price touches the strike due to unit payoff and thus it will never go up to barrier. But how about finite knock-out observation dates?

I am wondering the boundary conditions for the up-and-out call option of Bermudan type. The structure is as follows. The final payoff is 1 if $$S_T > K$$ and 0 else. The option will only knock out if $$S_t > K_{out}$$ observed in finite observation dates which are decided in initiation, say $$t_1, t_2, t_3, ..., t_n$$, where the last knock out observation date $$t_n$$ is exactly before the maturity date.

The right boundary condition for all prices at maturity is just its payoff $$1_{S_T > K}$$. What are the upper and lower condition for all time points when $$S_t = 0$$ and $$S_t = \infty$$? Are they all 0?

• You mean to find the bounds when $t \to 0$ or when $t \to +\infty$ ( not when $S_t \to 0$ or when $S_t \to +\infty$ ), right? And how are these $(t_i)_{i=1,..,n}$ distributed in $(0,T)$?
– NN2
Oct 20, 2023 at 12:17
• @NN2 Thanks for your comments. I am confused with the bounds when $S_t \rightarrow 0$ and when $S_t \rightarrow +\infty$. For the finite difference method, I already know the right boundary condition when $t = T$ but I am not sure about the upper and lower boundaries. And the knock out observation dates are distributed customarily in between $(0,T)$ you may think in this way. To put them in reality, they can be monthly knock out observation dates where the last one is exactly before the maturity date. In the image, the upper black lines mean the value $V(i,j) = 0$ in the knock out date. Oct 23, 2023 at 2:11
• I updated my image. Hope it is more straight and useful. Oct 23, 2023 at 2:38