Abstract
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 (Manag. Sci. 50:1222–1234, 2004) that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example.
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Acknowledgements
This research was partially supported by the Deutsche Forschungsgemeinschaft through the SPP 1324 “Mathematical methods for extracting quantifiable information from complex systems” and through the Research Center Matheon “Mathematics for Key Technologies”, and by the Laboratory for Structural Methods of Data Analysis in Predictive Modeling, MIPT, RF government grant, ag. 11.G34.31.0073.
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Belomestny, D., Schoenmakers, J. & Dickmann, F. Multilevel dual approach for pricing American style derivatives. Finance Stoch 17, 717–742 (2013). https://doi.org/10.1007/s00780-013-0208-5
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DOI: https://doi.org/10.1007/s00780-013-0208-5