WIAS Preprint No. 1574, (2010)

Optimal dual martingales and their stability; fast evaluation of Bermudan products via dual backward regression



Authors

  • Schoenmakers, John G. M.
  • Huang, Junbo

2010 Mathematics Subject Classification

  • 62L15 65C05 91B28

Keywords

  • Bermudan options, duality, Monte Carlo simulation, linear regression

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. 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).

Appeared in

  • SIAM J. Financial Math., 4 (2013) pp. 86--116 as: Optimal dual martingales, their analysis and application to new algorithms for Bermudan products, by J.G.M Schoenmakers, J. Zhang, J. Huang

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