Iterative construction of the optimal Bermudan stopping time

Abstract

We present an iterative procedure for computing the optimal Bermudan stopping time, hence the Bermudan Snell envelope. The method produces an increasing sequence of approximations of the Snell envelope from below, which coincide with the Snell envelope after finitely many steps. Then, by duality, the method induces a convergent sequence of upper bounds as well. In a Markovian setting the presented procedure allows to calculate approximative solutions with only a few nestings of conditional expectations and is therefore tailor-made for a plain Monte Carlo implementation. The method may be considered generic for all discrete optimal stopping problems. The power of the procedure is demonstrated for Bermudan swaptions in a full factor LIBOR market model. Copyright Springer-Verlag Berlin/Heidelberg 2006Bermudan options, optimal stopping, Monte Carlo simulation, LIBOR market model,

Tight bounds for American options via multilevel Monte Carlo

“…For more general versions of policy iteration and their analysis, see Kolodko and Schoenmakers 2006. Next, we introduce the (F j )-martingale…”

“…While the first approach relies on the dual method leading to a multilevel version of the Andersen and Broadie 2004 algorithm, the second one leads to a multilevel version of the policy iteration approach presented in Kolodko and Schoenmakers 2006. Regarding the latter part only standard (Howard) policy iteration is considered, but, with no doubt the method may be applied successfully to the more refined policy iteration procedure in Kolodko and Schoenmakers 2006 as well.…”

Regression Methods for Stochastic Control Problems and Their Convergence Analysis

“…{belomest,kolodko,schoenma}@wias-berlin.de. to problems in stochastic optimal control. In fact, an active interplay between stochastic control and financial mathematics has been emerged in the last decades: While stochastic control has been a powerful tool for studying problems in finance on the one hand side, financial applications have been stimulating the development of several new methods in optimal stopping and control on the other hand, see for example besides the works mentioned above, Rogers (2002), Andersen and Broadie (2004), Broadie and Glasserman (2004), Haugh and Kogan (2004), Ibáñez (2004), Meinshausen and Hambly (2004), , Bender and Schoenmakers (2006), Belomestny et al (2007), Chen and Glasserman (2007), Kolodko and Schoenmakers (2006), Jamshidian (2007), Rogers (2007), and Carmona and Touzi (2008), and many others.…”

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