The Mathematical Foundations of Barrier Option-Pricing Theory

Abstract

Barrier options are one of the most popular first-generation exotic options, yet little theoretical research exists. This paper begins by presenting, in an intuitive and unified framework, the mathematical foundation necessary to value barrier options. Closed-form solutions are derived and comparative statics are presented for European barrier options with a fixed rebate and barrier level. From both a theoretical and hedging perspective, barrier options are shown to be more complex than standard options. Further, it is shown that under certain conditions barrier options are identical in value to standard options, that all valuation solutions are linearly homogeneous of degree one with respect to the underlying asset price, exercise price, barrier level, and rebate, and that barrier options have quite different hedging properties than standard options. Closed-form solutions are also derived for European barrier options with barriers that depend exponentially on time to expiration. Applications discussed in the paper include covered calls, portfolio insurance, options embedded in bonds, and when barrier option prices are more likely to correspond to observed market prices than Black-Scholes option prices.

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