Encyclopedia of Quantitative Finance, IV Volume Set

Parisian Option

Parisian options are barrier options that are activated or canceled—depending on the type of option—if the underlying asset has been continuously traded above or below the barrier level long enough. A down-and-out Parisian option denotes a contract that expires worthless if the underlying asset reaches a prespecified level L and remains constantly below this level for a time interval longer than a fixed number D, called the window. Its price (for a call option) at time 0 is given by

where is the first time the asset S makes an excursion longer than D below L. Parisian-style options are mostly encountered in convertible bonds with “soft-call” provision for conversion. For example, the bond’s specifications may be such that conversion will be allowed if and only if the share price remains above a theoretical price for a given amount of time, for example, 20 business days prior to the conversion date (this is Parisian option). Other covenants stipulate that the average share price trades for n days above the trigger level. While the latter does not correspond sensu stricto to a Parisian option, the motivation is similar: to render the conversion rule more stable—and ...