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

Please always quote using this URN:urn:nbn:de:0296-matheon-8416
  • 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 Ito-Levy 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 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 a backward dual regression algorithm in a Wiener environment that is easy to implement and is regarding accuracy comparable with the method of Belomestny et. al. (2009).

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Metadaten
Author:John Schoenmakers, Junbo Huang
URN:urn:nbn:de:0296-matheon-8416
Referee:Dirk Becherer
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2012/01/27
Release Date:2012/01/27
Tag:
Institute:Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
MSC-Classification:62-XX STATISTICS / 62Lxx Sequential methods / 62L15 Optimal stopping [See also 60G40, 91A60]
65-XX NUMERICAL ANALYSIS / 65Cxx Probabilistic methods, simulation and stochastic differential equations (For theoretical aspects, see 68U20 and 60H35) / 65C05 Monte Carlo methods
Preprint Number:884
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