Closed form valuation of barrier options with stochastic barriers

Abstract

This article deals with the computation of the probability, for a GBM (geometric Brownian motion) process, to hit sequences of one-sided stochastic boundaries defined as GBM processes, over a closed time interval. Explicit formulae are obtained, allowing the analytical valuation of all the main kinds of barrier options in a much more general setting than the usual one assuming constant or time-dependent, deterministic barriers. The numerical implementation of all stated formulae is shown to be easy, fast and accurate. The practical applications are potentially substantial, since barrier options play a major role in quantitative finance, not only as intensively traded contracts on their own, but also as the building blocks of a large variety of structured products. Barrier options are also an important tool in financial modelling, used to measure default risk in the so-called “structural” models.