WIAS Preprint No. 1186, (2006)

True upper bounds for Bermudan products via non-nested Monte Carlo


  • Belomestny, Denis
  • Bender, Christian
  • Schoenmakers, John G. M.
    ORCID: 0000-0002-4389-8266

2010 Mathematics Subject Classification

  • 60H30 65C05 91B28


  • Bermudan options, Monte Carlo method, primal-dual method, martingale representation theorem, regression




We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic ``true'' stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale may be employed for computing dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Anderson & Broadie (2004)) and nested consumption based (Belomestny & Milstein (2006)) methods . Numerical experiments indicate the efficiency of the non-nested Monte Carlo algorithm and the variance reduced nested one.

Appeared in

  • Math. Finance, 19 (2009) pp. 53--71.

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