Monte Carlo Euler approximations of HJM term structure financial models

Tomas Björk; Anders Szepessy; Raul Tempone; Georgios E. Zouraris

doi:10.1007/s10543-012-0410-4

Monte Carlo Euler approximations of HJM term structure financial models

Tomas Björk, Anders Szepessy, Raul Tempone, Georgios E. Zouraris

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

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. © 2012 Springer Science+Business Media Dordrecht.

Original language	English (US)
Pages (from-to)	341-383
Number of pages	43
Journal	BIT Numerical Mathematics
Volume	53
Issue number	2
DOIs	https://doi.org/10.1007/s10543-012-0410-4
State	Published - Nov 22 2012

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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 for stokastiska differentialekvationer med tillampningar' (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.

ASJC Scopus subject areas

Computational Mathematics
Software
Applied Mathematics
Computer Networks and Communications

Access to Document

10.1007/s10543-012-0410-4

Cite this

@article{8c8ac512e88b4164993dbf3af8e0ffe5,

title = "Monte Carlo Euler approximations of HJM term structure financial models",

abstract = "We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on It{\^o} 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. {\textcopyright} 2012 Springer Science+Business Media Dordrecht.",

author = "Tomas Bj{\"o}rk and Anders Szepessy and Raul Tempone and Zouraris, {Georgios E.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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 for stokastiska differentialekvationer med tillampningar' (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.",

year = "2012",

month = nov,

day = "22",

doi = "10.1007/s10543-012-0410-4",

language = "English (US)",

volume = "53",

pages = "341--383",

journal = "BIT Numerical Mathematics",

issn = "0006-3835",

publisher = "Springer Verlag",

number = "2",

}

TY - JOUR

T1 - Monte Carlo Euler approximations of HJM term structure financial models

AU - Björk, Tomas

AU - Szepessy, Anders

AU - Tempone, Raul

AU - Zouraris, Georgios E.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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 for stokastiska differentialekvationer med tillampningar' (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.

PY - 2012/11/22

Y1 - 2012/11/22

N2 - 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. © 2012 Springer Science+Business Media Dordrecht.

AB - 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. © 2012 Springer Science+Business Media Dordrecht.

UR - http://hdl.handle.net/10754/562421

UR - http://arxiv.org/abs/arXiv:1204.1733v1

UR - http://www.scopus.com/inward/record.url?scp=84878807291&partnerID=8YFLogxK

U2 - 10.1007/s10543-012-0410-4

DO - 10.1007/s10543-012-0410-4

M3 - Article

SN - 0006-3835

VL - 53

SP - 341

EP - 383

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

IS - 2

ER -

Monte Carlo Euler approximations of HJM term structure financial models

Abstract

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this