Optimal Stopping, Multi-Asset American/Bermudan Options

The main aim of this chapter is to present and analyze two methods for the pricing of multi-asset American—in practice Bermudan—options: the regression methods “à la Longstaff-Schwarz” and the quantization methods. The pricing of American options is a typical example of an optimal stopping problem. So, we start from such a general optimal stopping problem but for a diffusion model, we introduce the Snell envelope of a payoff or reward process and its value function in ou Markov framework. We give some rate results for various “levels” of time discretization of such Snell envelope. Then we consider a discrete time optimal stopping problem in a Markov framework. We introduce the backward dynamic programming principle and its variant based on optimal stopping times. We describe in detail the algorithmic aspects of both the regression and the quantization method. The Monte Carlo error (Central Limit Theorem) induced by the regression method is described. Finally, a theoretical analysis of the convergence rate of the quantization method is carried out.