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In the appendix A of this paper, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.227.5073&rep=rep1&type=pdf, a finite element method is demonstrated to price a straddle. The same method can be used for an up and out call barrier option.

My question is regarding the boundary condition. In (A.6), the boundary conditions vector involves the quantities $u_b^n - u_b^{n-1}$, with $u_b^n$ the option price at the time step $n$ and the barrier $b$. For an up and out call option, wouldn't $u_b^n - u_b^{n-1} = 0$ for every $n$ ? Or in general, wouldn't $u_b^n - u_b^{n-1} = 0$ for all $n$ since the time is not relevant as long as the option is at the boundary?

Thanks

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