Abstract
The core target of this chapter is numerical analysis and computing of novel finite difference methods related to several different option pricing models, including jump-diffusion, regime switching and multi-asset options. A special attention is paid to positivity, consistency and stability of the proposed methods. The consideration of jump processes leads to partial integro-differential equation (PIDE) for the European option pricing problem. The problem is solved by using quadrature formulas for the approximation of the integrals and matching the discretization of the integral and differential part of the PIDE problem. More complicated model under assumption that the volatility is a stochastic process derives to a PIDE problem where the volatility is also an independent variable. Such a problem is solved by introducing appropriate change of variables. Moreover, American options are considered proposing various front-fixing transformations to treat a free boundary. This free boundary challenge can be treated also by a recent rationality parameter approach that takes into account the irrational behavior of the market. Dealing with multidimensional problems the core difficulty is the appearance of the cross derivative terms. Appropriate transformations allow eliminating the cross derivative terms and reduce of the computational cost and the numerical instabilities. After using a semidiscretization approach, the time exponential integration method and appropriate quadrature integration formulas, the stability of the proposed method is studied independent to the problem dimension.