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The Transport-based Mesh-free Method (TMM) and its applications in finance: a review

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  • Philippe G. LeFloch
  • Jean-Marc Mercier

Abstract

We review a numerical technique, referred to as the Transport-based Mesh-free Method (TMM), and we discuss its applications to mathematical finance. We recently introduced this method from a numerical standpoint and investigated the accuracy of integration formulas based on the Monte-Carlo methodology: quantitative error bounds were discussed and, in this short note, we outline the main ideas of our approach. The techniques of transportation and reproducing kernels lead us to a very efficient methodology for numerical simulations in many practical applications, and provide some light on the methods used by the artificial intelligence community. For applications in the finance industry, our method allows us to compute many types of risk measures with an accurate and fast algorithm. We propose theoretical arguments as well as extensive numerical tests in order to justify sharp convergence rates, leading to rather optimal computational times. Cases of direct interest in finance support our claims and the importance of the problem of the curse of dimensionality in finance applications is briefly discussed.

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  • Philippe G. LeFloch & Jean-Marc Mercier, 2019. "The Transport-based Mesh-free Method (TMM) and its applications in finance: a review," Papers 1911.00992, arXiv.org, revised Nov 2019.
    • Handle: RePEc:arx:papers:1911.00992
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    1. Alan Brace & Dariusz G¸atarek & Marek Musiela, 1997. "The Market Model of Interest Rate Dynamics," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 127-155, April.
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