Pricing American-style Parisian down-and-out call options

Abstract

We propose an integral equation approach for pricing American-style Parisian down-and-out call options under the Black–Scholes framework. For this type of options, the knock-out feature is activated only if the underlying asset price continuously remains below a pre-determined barrier for a sufficiently long period of time. As such, the corresponding pricing problem becomes a three-dimensional (3-D) free boundary problem, instead of a two-dimensional (2-D) one as is the case of “one-touch” barrier options, and this poses a computational challenge. In our approach , we first reduce the 3-D problem to a 2-D one, and then, by applying the Fourier sine transform to the resulting 2-D problem, we can derive a pair of coupled integral equations governing the option price at any given time in terms of (i) the option price at the barrier and (ii) the optimal exercise boundary at that time. This pair of coupled integral equations can be solved using the Newton–Raphson iterative procedure, after which, the option price, the optimal exercise boundary, and the hedging parameters can be obtained in a straightforward manner. A complexity analysis of the method, together with numerical results, show that the proposed approach is robust and significantly more efficient than existing uniform finite difference methods with Crank–Nicolson timestepping, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight into several interesting properties of the option price and the optimal exercise boundary.