Parasian Over Parisian, How Much Earlier Should One Exercise?

Abstract

In this paper, an integral equation approach used for pricing American-style Parisian options is extended to pricing Parasian options, after overcoming an additional difficulty of losing the “reset” feature of the latter in addition to still dealing with the high nonlinearity associated with American-style exercises. More specifically, compared with an American Parisian option, the accumulative feature of the “tracking clock” time results in a pair of coupled three-dimensional (3D) Partial Differential Equation (PDE) systems instead of coupled two-dimensional (2D) and 3D PDE systems. Upon successfully establishing the couple integral equation systems, we have successfully computed the option price and the optimal exercise price and compared the results with those of its Parisian counterpart. Through the comparison, we can directly observe how the accumulativeness of the “tracking clock” time affects the option and the optimal exercise prices and provide some meaningful financial explanations.