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

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.Comment: This paper is an extended version of Schoenmakers and Huang, "Optimal dual martingales and their stability; fast evaluation of Bermudan products via dual backward regression", WIAS Preprint 157

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

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. We provide a theorem which give 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 theorems we develop a regression based backward construction of such a martingale in a Wiener environment. In turn this martingale may be utilized for computing upper bounds by non-nested Monte Carlo. As a by-product, the algorithm also provides approximations to continuation values of the product, which in turn determine a stopping policy. Hence, we obtain lower bounds at the same time. The proposed algorithm is pure dual in the sense that it doesn't require an (input) approximation to the Snell envelope, is quite easy to implement, and in a numerical study we show that, regarding the computed upper bounds, it is comparable with the method of Belomestny, et. al. (2009)

Multilevel dual approach for pricing American style derivatives

In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example

From optimal martingales to randomized dual optimal stopping

In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that doesn't require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm efficiently selects a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance

A primal-dual algorithm for BSDEs

We generalize the primal-dual methodology, which is popular in the pricing of early-exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was pre-computed, e.g., by least-squares Monte Carlo, our methodology allows to construct a confidence interval for the unknown true solution of the time discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two five-dimensional nonlinear pricing problems where tight price bounds were previously unavailable