Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from
the value of some underlying security. Most options can be grouped into either of
the two categories: European options which can be exercised only on the expiration
date, and American options which can be exercised on or before the expiration date.
American options are much harder to deal with than European ones. The reason being
the optimal exercise policy of these options which led to free boundary problems. Ever
since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659],
the differential equation approach in pricing options has attracted many researchers.
Recently, numerical singular perturbation techniques have been used extensively for
solving many differential equation models of sciences and engineering. In this thesis,
we explore some of those methods which are based on spline approximations to solve
the option pricing problems. We show a systematic construction and analysis of these
methods to solve some European option problems and then extend the approach to
solve problems of pricing American options as well as some exotic options. Proposed
methods are analyzed for stability and convergence. Thorough numerical results are
presented and compared with those seen in the literature
