Journal of Computational Finance

Risk.net

The two-dimensional tree–grid method

Igor Kossaczký, Matthias Ehrhardt and Michael Günther

  • The two-dimensional Tree-Grid method is an unconditionally stable, convergent explicit method for solving the two-dimensional  stochastic control problems and Hamilton-Jacobi-Bellman equation.
  • The method can be used on an arbitrary (equidistant or non-equidistant) rectangular grid, and no interpolation is needed in the stencil construction.
  • We present the solution of the two-factor uncertain volatility model using the two-dimensional Tree-Grid method.

In this paper, we introduce a novel, explicit, wide-stencil, two-dimensional (2D) tree–grid method for solving stochastic control problems (SCPs) with two space dimensions and one time dimension, or, equivalently, the corresponding Hamilton– Jacobi–Bellman equation. This new method can be seen as a generalization of the tree–grid method for SCPs with one space dimension that was recently developed by the authors. The method is unconditionally stable and no 2D interpolation is needed in the stencil construction. We prove the convergence of the method and exemplify it in our application to a two-factor uncertain volatility model.

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