American Options Pricing by Using Stochastic Optimal Control Problems

Stochastic optimal control problems frequently occur in Economics and Finance. Dynamicprogramming method represents the most known method for solving optimal control prob-lems analytically. As analytical solutions for problems of optimal control are not alwaysavailable, finding an approximate solution is at least the most logical way to solve them.In this paper, we present some of the basic ideas which are in current use for the solutionof the dynamic programming equations. Also, based on the Markov chain approximationtechniques, a numerical procedure is constructed for solution of stochastic optimal controlproblems. We focus on the approximation in value space method. And the Jacobi andGauss-Seidel relaxation (iterative) methods are discussed. These are fundamental iterativemethods which are used in value space approach. Finally, American options pricing arepresented as simplest control problem which is called optimal stopping problem.

Keywords:

Stochastic optimal control problems , American options Pricing , Dynamic programming method , Markov chain approximation , Jacobi and Gauss-Seidelmethods