In-out parity relations for American-style barrier options

The classic in-out parity relation establishes that the sum of European-style knock-in and knock-out barrier option prices is equal to the price of a standard European-style option. In the literature, however, it is commonly argued that such a barrier parity relationship cannot be applied to American-style barrier options. In this article, we derive novel in-out parity relations for American-style single-barrier options and show that the missing link between knock-in, knock-out, and standard options is simply the value of a non-deferrable rebate. Our results are tested through an extensive numerical analysis run under both the geometric Brownian motion and constant elasticity of variance architecture modeling setups