Numerical Methods for Pricing Path-Dependent Options

Numerical methods for pricing derivatives are of the utmost importance in 
Quantitative Finance since analytical solutions are provided only in few cases. 
In this talk we consider path-dependent derivatives (as, for example, barrier, 
lookback and asian options) in the framework of Lévy models and we deal with numerical 
methods based on the maturity randomization procedure.
More precisely, for the discretely monitored case, once the pricing problem is written in a 
recursive way, randomizing the derivative expiry according to a Geometric distribution, we 
obtain a set of independent integral equation. In order to solve these equations, we consider 
different quadrature procedures that transform each integral equation into a linear system and 
we study the benefits of suitable preconditioning techniques. Moreover, since the integral equations 
are independent, we exploit the computational benefits of grid computing.
Finally, since the pricing problem with continuous monitoring can be formulated in terms of 
an integro-differential parabolic partial differential equation with suitable boundary conditions, 
we introduce the finite element approach for general derivative contracts.