Pricing American-style Parisian options

Barrier options are the most common path-dependent options traded in financial markets. They are particularly attractive to investors, because not only are they cheaper than vanilla options but they also offer different choices of investment, which allow investors to bet their views on the movement of the underlying asset prices. The “one-touch” breaching barrier however is prone to market manipulations which can be made by influential agents in order to free them from their liabilities to the option holders. Aiming to prevent such market manipulations, Parisian options were introduced, with an extended trigger clause, which makes the knock-in or knock-out feature much harder to be activated. Pricing Parisian options has become an increasingly important problem from both financial and mathematical perspectives. Financially, the introduction of Parisian options, which makes the market fairer in the sense that it protects the holder of Parisian options from deliberate action taken by the writer, requires an efficient way to precisely evaluate the option prices. On the other hand, due to the presence of the newly-added trigger clause, the valuation of Parisian options becomes a three-dimensional problem, a challenging problem, which has hindered the application of various mathematical methods. In this thesis, we explore the integral equation method and the “moving window” technique to price different types of Parisian options under the Black-Scholes framework