Ali Hamdi: PDE-regularization for pricing high-dimensional Bermudan options with Monte Carlo

We consider the problem of pricing high-dimensional Bermudan derivatives by using Monte Carlo simulation. A non-parametric method for estimating conditional expectations is proposed, which combined with dynamic programming provides a way of pricing the derivatives.

The method is based on a non-parametric projection, which is ill-posed due to the overfitting introduced. This overfitting is handled by adding a regularization part in the formulation.

It is known that the estimated conditional expectations solve a PDE which characterizes the underlying process. This information is used by letting the regularization part be a discretization of the squared norm of the PDE. The point is that it is less costly to calculate the norm of the PDE than to actually solve it, thus avoiding the curse of dimensionality.

Using a simple model and a benchmark, the method is shown to produce accurate prices in high-dimensional settings, given a good choice of regularization parameter. However, there are a few issues which has to be handled, of which choosing this parameter systematically is an important one.

We discuss a number of possible solutions to the issues, and if the issues can be resolved then the method might become competitive for pricing high-dimensional Bermudan derivatives.