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Monte Carlo Euler approximations of HJM term structure financial models


We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

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This work has been partially supported by: The Swedish National Network in Applied Mathematics (NTM) ‘Numerical approximation of stochastic differential equations’ (NADA, KTH), The EU-TMR project HCL # ERBFMRXCT960033, UdelaR and UdeM in Uruguay, The Swedish Research Council for Engineering Science (TFR) Grant#222-148, The VR project ‘Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar’ (NADA, KTH), the European Union’s Seventh Framework Programme (FP7-REGPOT-2009-1) under grant agreement no. 245749 ‘Archimedes Center for Modeling, Analysis and Computation’ (University of Crete, Greece), The University of Crete (Sabbatical Leave of the fourth author), and The King Abdullah University of Science and Technology (KAUST). The third author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.

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Correspondence to R. Tempone.

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Communicated by Desmond Higham.

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Björk, T., Szepessy, A., Tempone, R. et al. Monte Carlo Euler approximations of HJM term structure financial models. Bit Numer Math 53, 341–383 (2013).

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  • HJM model
  • Option price
  • Bond market
  • Stochastic differential equations
  • Monte Carlo methods
  • A priori error estimates
  • A posteriori error estimates

Mathematics Subject Classification

  • 65C05
  • 65C30
  • 65C20
  • 91B28
  • 91B70