ABSTRACT
In this chapter we extend the previously developed options pricing techniques to handle more exotic, path dependent option structures. Two main approaches for doing this are discussed: simulation and numerical solutions to PDEs. We begin with an overview of the theoretical foundation of simulation and then provide several examples using stochastic volatility and jump models. We then proceed to discuss how to solve partial differential equations numerically, beginning with a review of how PDEs can be derived from a given SDE via Ito's Lemma. We then introduce the concept of finite differences and derive the most common finite difference estimates. Setting boundary conditions and a time and space grid for a PDE algorithm is also considered. Additionally, the implicit and explicit PDE schemes are presented, as well as Crank-Nicolson. Stability analysis and the complications that arise from multi-dimensional PDEs and the integral that arises when using a jump-process are presented.
