Explosive behavior in the black–derman–toy model

Dan Pirjol

doi:10.1007/978-3-319-12307-3_52

Explosive behavior in the black–derman–toy model

Dan Pirjol

School of Business

Stevens Institute of Technology

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We consider the simulation of the Black–Derman–Toy (BDT) model with log-normally distributed rates in the spot measure, in discrete time and with a continuous state variable. We note an explosive behavior in the Eurodollar futures convexity adjustment at a critical value of the volatility, which depends on maturity, rate tenor, and simulation time step size. In the limit of a very small time step, this singularity appears for any volatility, and reproduces the Hogan–Weintraub singularity, which is generic for short-rate interest rate models with lognormally distributed rates. The singular behavior arises from a region in the state space which is usually truncated off in finite difference and tree methods, or is very poorly sampled in Monte Carlo methods, and thus is not observed under usual simulation methods.

Original language	English
Title of host publication	Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science
Editors	Roman N. Makarov, Roderick V. N. Melnik, Ilias S. Kotsireas, Hasan Shodiev, Monica G. Cojocaru
Publisher	Springer New York LLC
Pages	361-366
Number of pages	6
ISBN (Print)	9783319123066
DOIs	https://doi.org/10.1007/978-3-319-12307-3_52
State	Published - 2015
Event	International Conference on Applied Mathematics, Modelling and Computational Science, AMMCS 2013 - Waterloo, Canada Duration: Aug 26 2013 → Aug 30 2013

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	117

Conference

Conference	International Conference on Applied Mathematics, Modelling and Computational Science, AMMCS 2013
Country/Territory	Canada
City	Waterloo
Period	8/26/13 → 8/30/13

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-3-319-12307-3_52

Cite this

Pirjol, D. (2015). Explosive behavior in the black–derman–toy model. In R. N. Makarov, R. V. N. Melnik, I. S. Kotsireas, H. Shodiev, & M. G. Cojocaru (Eds.), Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science (pp. 361-366). (Springer Proceedings in Mathematics and Statistics; Vol. 117). Springer New York LLC. https://doi.org/10.1007/978-3-319-12307-3_52

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title = "Explosive behavior in the black–derman–toy model",

abstract = "We consider the simulation of the Black–Derman–Toy (BDT) model with log-normally distributed rates in the spot measure, in discrete time and with a continuous state variable. We note an explosive behavior in the Eurodollar futures convexity adjustment at a critical value of the volatility, which depends on maturity, rate tenor, and simulation time step size. In the limit of a very small time step, this singularity appears for any volatility, and reproduces the Hogan–Weintraub singularity, which is generic for short-rate interest rate models with lognormally distributed rates. The singular behavior arises from a region in the state space which is usually truncated off in finite difference and tree methods, or is very poorly sampled in Monte Carlo methods, and thus is not observed under usual simulation methods.",

author = "Dan Pirjol",

note = "Publisher Copyright: {\textcopyright} Springer International Publishing Switzerland 2015.; International Conference on Applied Mathematics, Modelling and Computational Science, AMMCS 2013 ; Conference date: 26-08-2013 Through 30-08-2013",

year = "2015",

doi = "10.1007/978-3-319-12307-3\_52",

language = "English",

isbn = "9783319123066",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer New York LLC",

pages = "361--366",

editor = "Makarov, \{Roman N.\} and Melnik, \{Roderick V. N.\} and Kotsireas, \{Ilias S.\} and Hasan Shodiev and Cojocaru, \{Monica G.\}",

booktitle = "Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science",

}

Pirjol, D 2015, Explosive behavior in the black–derman–toy model. in RN Makarov, RVN Melnik, IS Kotsireas, H Shodiev & MG Cojocaru (eds), Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science. Springer Proceedings in Mathematics and Statistics, vol. 117, Springer New York LLC, pp. 361-366, International Conference on Applied Mathematics, Modelling and Computational Science, AMMCS 2013, Waterloo, Canada, 8/26/13. https://doi.org/10.1007/978-3-319-12307-3_52

Explosive behavior in the black–derman–toy model. / Pirjol, Dan.
Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science. ed. / Roman N. Makarov; Roderick V. N. Melnik; Ilias S. Kotsireas; Hasan Shodiev; Monica G. Cojocaru. Springer New York LLC, 2015. p. 361-366 (Springer Proceedings in Mathematics and Statistics; Vol. 117).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We consider the simulation of the Black–Derman–Toy (BDT) model with log-normally distributed rates in the spot measure, in discrete time and with a continuous state variable. We note an explosive behavior in the Eurodollar futures convexity adjustment at a critical value of the volatility, which depends on maturity, rate tenor, and simulation time step size. In the limit of a very small time step, this singularity appears for any volatility, and reproduces the Hogan–Weintraub singularity, which is generic for short-rate interest rate models with lognormally distributed rates. The singular behavior arises from a region in the state space which is usually truncated off in finite difference and tree methods, or is very poorly sampled in Monte Carlo methods, and thus is not observed under usual simulation methods.

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Explosive behavior in the black–derman–toy model

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