Boundary Values in Finite-Element Barrier Option Pricing

Summary

The document raises a boundary-condition question about using a finite element method to price an up-and-out call. It refers to an appendix that demonstrates the method for a straddle and focuses on a boundary vector containing the difference between option prices at successive time steps. The question is whether that difference should be zero at the barrier, and whether boundary option values generally remain constant through time.

This is a question about how boundary conditions enter a numerical option-pricing scheme, especially for a barrier contract whose value at the barrier may be constrained by its payoff definition. The document supplies no answer, derivation, or numerical evidence, so it does not resolve whether the proposed zero difference is appropriate. The correct condition depends on the option specification and the method’s treatment of the boundary; readers would need the referenced formulation and a clear statement of the barrier payoff to assess it.

Key ideas

• The document asks how time-step boundary values should be represented in a finite-element option-pricing method.
• It considers an up-and-out call and a boundary vector built from successive option values.
• The question is whether the boundary value remains constant across time steps.
• No derivation or answer is provided, so the appropriate condition remains unresolved.

Tags

# boundary conditions in finite element method


# boundary conditions in finite element method












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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