An American in Paris

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 is motivated by the need to make the option more robust against short-term movements of the share price, a single outlier cannot trigger the barrier. In particular, it is far harder to affect the triggering of the barrier by manipulation of the underlying. 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 differential equation. This approach can price a variety of modifications of the basic Parisian contract including Parasian options (activation of the barrier conditional on the total time spent above the barrier), American early exercise rights and general payoffs. The approach readily accommodates features, such as early exercise, that render the traditional Monte Carlo approach impractical.