Piecewise linear boundary crossing probabilities, barrier options, and variable annuities
提出分段线性障碍期权并给出定价公式，通过解析分段线性边界穿越概率来近似任意边界穿越概率，并证明含提前行权的金融工具可分解为敲出障碍期权与立即回扣，以可变年金为例验证了分解的有效性。
Abstract Barrier options have been instrumental in satisfying various market demands. This paper introduces piecewise linear barrier options and provides their pricing formulas. To this end, we establish the analytical piecewise linear boundary crossing probability and explain how to approximate arbitrary boundary crossing probabilities. In addition, we show that a financial instrument with early exercise is decomposable into a knock‐out barrier option and immediate rebate, which casts a new illumination of the value of early exercise. We consider a variable annuity with guaranteed minimum accumulation benefit rider and surrender option to illustrate the decomposition. Extensive numerical experiments validate theoretical findings.