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

TL;DR

This paper introduces optimal dual martingales for Bermudan options, providing new algorithms that improve upper bound computation without nested simulations, and demonstrates their effectiveness through numerical experiments.

Contribution

It develops a framework for constructing optimal dual martingales and introduces a regression-based backward algorithm that enhances computational robustness.

Findings

The new algorithm provides accurate upper bounds comparable to existing methods.

It avoids nested Monte Carlo simulations, reducing computational complexity.

Numerical results show improved robustness and efficiency in benchmark tests.

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 near to be surely optimal in a sense. Guided by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn't require certain input approximations to the Snell envelope. In an It\^o-L\'evy 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. Hence, we may obtain lower bounds at the same time. In a first numerical study we demonstrate the backward dual regression algorithm in a Wiener environment at well known benchmark examples. It turns out that the method is at least comparable to the one in Belomestny et. al. (2009) regarding accuracy, but regarding computational robustness there are even several advantages.