Optimal dual martingales, their analysis and application to new algorithms for Bermudan products

Abstract

In this paper we introduce and study the concept of optimal and surely optimal dual martingales in the context of dual valuation of Bermudan options and outline the development of new algorithms in this context. We provide a characterization theorem, a theorem which gives conditions for a martingale to be surely optimal, and a stability theorem concerning martingales which are nearly surely optimal in a sense. Guided by these results we develop a framework of backward algorithms for constructing such a martingale which can be utilized for computing an upper bound of the Bermudan product. The methodology is purely dual in the sense that it does not require certain input approximations to the Snell envelope. In an Itô--Lévy environment we outline a particular regression-based backward algorithm which allows for computing dual upper bounds without nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the product, which in turn determine a stopping policy. We hence obtain lower bounds at the same time. We finally supplement our presentation with numerical experiments