• option pricing
• multigrid
• finite difference methods
• Black-Scholes
• Monte Carlo
• Binomial

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.