Two-Dimensional Diffusion Problems in Finance

All the Black Scholes type problems treated in the preceding chapters have multi-dimensional analogs which, in principle, can be solved approximately with a variety of numerical methods. Not surprisingly, the numerical difficulties discussed above also have their multi-dimensional analogs which need to be resolved by whatever numerical technique is chosen for the problem at hand. The present chapter deals with two-dimensional problems for some partial differential equations in finance. It thus avoids the dominant difficulty brought on by the so-called “curse of dimensionality” which, for sufficiently high dimensionality, may make the PDE approach infeasible. But even in two space dimensions the lack of smoothness and nonlinear features cause complications which far exceed those encountered up to now. We shall again choose a variety of models to illustrate these complications. Our numerical results for these problems will be obtained with the line iterative approach outlined in Section 2.3. They may serve as a benchmark for readers developing or adapting their own codes for such problems…