An irregular grid method for high-dimensional free-boundary problems in finance
We propose and test a new method for pricing American options in a high-dimensional setting. The method is centred around the approximation of the associated variational inequality on an irregular grid. We approximate the effect of the partial differential operator on this grid by appealing to the stochastic differential equation (SDE) representation of the stock process and computing a root of the transition probability matrix of an approximating Markov chain. The results of numerical tests in five dimensions are promising.
free-boundary problems american options variational inequalities high-dimensional problems optimal stopping numerical methods