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 concern- ing martingales which are near to be surely optimal in a sense. Guided by these results we develop a framework of backward algorithms for con- structing such a martingale. In turn this martingale may then be utilized for computing an upper bound of the Bermudan product. The method- ology is purely dual in the sense that it doesn’t require certain input approximations to the Snell envelope. In an Itˆo-L´evy environment we outline a particular regression based backward algorithm which allows for computing dual upper bounds with- out nested Monte Carlo simulation. Moreover, as a by-product this al- gorithm also provides approximations to the continuation values of the product, which in turn determine a stopping policy. Hence, we may ob- tain lower bounds at the same time. In a first numerical study we demonstrate the backward dual regres- sion algorithm in a Wiener environment at well known benchmark ex- amples. It turns out that the method is at least comparable to the one in Belomestny et. al. (2009) regarding accuracy, but regarding computa- tional robustness there are even several advantages.