Pricing Bounds on Barrier Options

Abstract

This article proposes the optimal pricing bounds on barrier options in an environment where plain-vanilla options and no-touch options can be used as hedging instruments. Super-and sub-hedging portfolios are derived without specifying any underlying processes, which are static ones consisting of not only plain-vanilla options but also cash-paying no-touch options and/or asset paying no-touch options that pay one cash or one underlying asset, respectively, if the barrier has not been hit. Moreover, the prices of these portfolios turn out to be the optimal pricing bounds through finding risk-neutral measures under which the barrier option price is equal to the hedging portfolio's value. The model-independent pricing bounds are useful because the price of a barrier option is significantly dependent on a model. It is demonstrated through numerical examples that prices outside the pricing bounds can be produced by models that are calibrated to market prices of plain-vanilla options, but not to that of a no-touch option.

A model‐free approximation for barrier options in a general stochastic volatility framework

“…The exact pricing of barrier options under realistic assumptions can therefore only be achieved by numerical techniques which will include some calibration errors. Even though Brown et al (2001) and Tsuzuki (2014) showed that in the most general case model‐free bounds for prices of barrier options can be found, it is still useful to have analytical approximations where possible.…”