Numerical methods for option pricing: Finite differences and multigrid techniques

The main topic of this thesis is the analysis of finite differences and multigrid methods for the solution of the Black-Scholes equation in the context of option pricing. Particular attention is addressed to some generalizations of this problem, including exchange and spread option pricing. The description and the numerical analysis of the main algorithms for this problem are carried out together with some numerical experiments, which point out the main features and drawbacks of the different solution techniques. The thesis is organized as follows. In Chapter 1, we briefly introduce the Black-Scholes model and the options which we will use in the numerical experiments. In Chapter 2, we describe classical methods for option pricing and we focus on finite difference methods. In Chapter 3, we show how classical finite difference methods, such as the Crank-Nicolson scheme, provide spurious oscillations, when the equation becomes convection-dominated. We discuss how to overcome this problem with the exponentially fitted scheme. In Chapter 4, we deal with the American option pricing and we present the front-fixing technique. Later on, in Chapter 5, we introduce the multigrid technique and we apply it to the exchange and spread option pricing as an example of multi-asset option pricing. We numerically compare these methods through numerical experiments, which are presented in Chapter 6. Finally, we conclude and we discuss some possible future developments