Pricing Parisian down-and-in options

TL;DR

This paper develops a new analytical pricing formula for American-style Parisian down-and-in call options within the Black-Scholes model, simplifying computation by expressing the solution as four double integrals.

Contribution

It introduces a novel analytical solution for American Parisian down-and-in options, leveraging the moving window technique for easier numerical evaluation.

Findings

Derived a simple analytical formula involving four double integrals

Demonstrated the applicability of the moving window technique to American Parisian options

Provided a computationally efficient method for pricing these options

Abstract

In this paper, we price American-style Parisian down-and-in call options under the Black-Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exercise boundary associated with an American-style Parisian knock-in option only appears implicitly in its pricing partial differential equation (PDE) systems, instead of explicitly as in the case of an American-style Parisian knock-out option. We also recognize that the "moving window" technique developed for pricing European-style Parisian up-and-out options can be adopted to price American-style Parisian knock-in options as well. In particular, we obtain a simple analytical solution for American-style Parisian down-and-in call options and our new formula is written in terms of four double integrals, which can be easily computed numerically.