Pricing Algorithms for Options with Exotic Path-Dependence

Abstract

Path-dependent derivatives present a major challenge to standard valuation algorithms. Closed-form solutions are available for a handful of simple cases like “out” options with continuously monitored barriers, or odd special cases like Asian options based on the geometric average. Basic lattice techniques can work for a somewhat larger subset of instruments, such as discretely monitored barrier options. But as the variety of options traded in the marketplace expands, seemingly without bound, the path-dependency problem quickly becomes formidable. For some important cases, lattice-based solutions have been devised that involve carrying an auxiliary variable through the asset price tree to keep track of the current state of the path at each price node, a technique known as a “Forward Shooting Grid.” In this article, Kwok and Lau demonstrate how to set one up and provide several examples to illustrate its use, pricing Parisian options, both standard and exotic (e.g., a “cumulative Parisian” contract), with complex dependence on the asset price path. The convergence rate in the examples is shown to be proportional to the square root of the number of time steps.