Pricing Parisian Options

Abstract

Introduction Parisian options are barrier options for which the knock-in/knock-out feature is only activated after the price process has spent a certain prescribed, consecutive time beyond the barrier. This specification has two motivations: First, there is the need to make the option more robust against short-term movements of the share price. This is achieved in Parisian options where it is ensured that a single outlier cannot trigger the barrier. In particular, it is far harder to a#ect the triggering of the barrier by manipulation of the underlying (see Taleb [4]). Second, classical barrier options present hedging problems close to the barrier because their Gamma becomes very large. To some extent, these problems are reduced, or at least `smoothed', in the Parisian contract. We present a flexible approach to valuing such options using the numerical solution of a partial di#erential equation. This approach can price a variety of modifications of the basic Parisian contract