A first-touch approach to American option valuation with piecewise linear boundaries

초록

We develop a tractable first-touch framework for valuing American options when the exercise boundary is specified as piecewise linear on a monitoring grid. Our main result delivers closed-form valuation formulas for first-touch digital and rebate options under time-varying piecewise-linear boundaries. This resolves a practical bottleneck: boundary approximation is often feasible, but evaluating the payoff at the first boundary crossing is typically difficult. We also propose a simple plug-in refinement that takes an externally obtained boundary and computes the induced boundary-hitting value using our formulas. Numerical experiments show that the plug-in valuation improves accuracy relative to premium-based approximations and yields values closer to benchmarks under the same boundary inputs.

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