Piecewise linear boundary crossing probabilities, barrier options, and variable annuities

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.