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Improving the Monte Carlo estimation of boundary crossing probabilities by control variables

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  • Pötzelberger Klaus

    (Institute for Statistics and Mathematics, Augasse 2-6, A-1090 Vienna, Austria)

Abstract

We propose an efficient Monte Carlo approach to compute boundary crossing probabilities (BCP) for Brownian motion and a large class of diffusion processes, the method of adaptive control variables. For the Brownian motion the boundary b (or the boundaries in case of two-sided boundary crossing probabilities) is approximated by a piecewise linear boundary , which is linear on m intervals. Monte Carlo estimators of the corresponding BCP are based on an m-dimensional Gaussian distribution. Let N denote the number of (univariate) Gaussian variables used. The mean squared error for the boundary is of order , leading to a mean squared error for the boundary b of order with , if the difference of the (exact) BCP's for b and is . Typically, for infinite-dimensional Monte Carlo methods, the convergence rate is less than the finite-dimensional . Let be a further approximating boundary which is linear on k intervals. If k is small compared to m, the corresponding BCP may be estimated with high accuracy. The BCP for as control variable improves the convergence rate of the Monte Carlo estimator to with . The constant depends on the correlation of the estimators for and . We show that this method of adaptive control variable improves the convergence rate considerably. Iterating control variables leads to a rate of convergence (of the mean squared error) of order , reducing the problem of estimating the BCP to an essentially finite-dimensional problem.

Suggested Citation

  • Pötzelberger Klaus, 2012. "Improving the Monte Carlo estimation of boundary crossing probabilities by control variables," Monte Carlo Methods and Applications, De Gruyter, vol. 18(4), pages 353-377, December.
    • Handle: RePEc:bpj:mcmeap:v:18:y:2012:i:4:p:353-377:n:4
    DOI: 10.1515/mcma-2012-0013
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    References listed on IDEAS

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    1. K. Borovkov & Alexander Novikov, 2004. "Explicit Bounds for Approximation Rates for Boundary Crossing Probabilities for the Wiener Process," Research Paper Series 115, Quantitative Finance Research Centre, University of Technology, Sydney.
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