World Library  
Flag as Inappropriate
Email this Article

Black–Derman–Toy model

Article Id: WHEBN0003152504
Reproduction Date:

Title: Black–Derman–Toy model  
Author: World Heritage Encyclopedia
Language: English
Subject: Emanuel Derman, Fischer Black, Ho–Lee model, Vasicek model, Mathematical finance
Collection: Financial Economics, Fixed Income Analysis, Mathematical Finance, Short-Rate Models
Publisher: World Heritage Encyclopedia

Black–Derman–Toy model

Short-rate tree calibration under BDT:

0. Set the Risk-neutral probability of an up move, p, = 50%
1. For each input spot rate, iteratively:

  • adjust the rate at the top-most node at the current time-step, i;
  • find all other nodes in the time-step, where these are linked to the node immediately above (ru; rd being the node in question) via 0.5×ln(ru/rd) = σi×√Δt (this node-spacing being consistent with p = 50%; Δt being the length of the time-step);
  • discount recursively through the tree using the rate at each node, i.e. via "backwards induction", from the time-step in question to the first node in the tree (i.e. i=0);
  • repeat until the discounted value at the first node in the tree equals the zero-price corresponding to the given spot interest rate.

2. Once solved, retain these known short rates, and proceed to the next time-step (i.e. input spot-rate), "growing" the tree until it incorporates the full input yield-curve.

In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) #Interest rate derivatives. It is a one-factor model; that is, a single stochastic factor – the short rate – determines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the lognormal distribution, [1] and is still widely used. [2][3]

The model was introduced by Fischer Black, Emanuel Derman, and Bill Toy. It was first developed for in-house use by Goldman Sachs in the 1980s and was published in the Financial Analysts Journal in 1990. A personal account of the development of the model is provided in one of the chapters in Emanuel Derman's memoir "My Life as a Quant."[4]

Under BDT, using a binomial lattice, one calibrates the model parameters to fit both the current term structure of interest rates (yield curve), and the volatility structure for interest rate caps (usually as implied by the Black-76-prices for each component caplet); see aside. Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and interest rate derivatives.

Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous stochastic differential equation:[5][6]

d\ln(r) = [\theta_t + \frac{\sigma'_t}{\sigma_t}\ln(r)]dt + \sigma_t\, dW_t
r\, = the instantaneous short rate at time t
\theta_t\, = value of the underlying asset at option expiry
\sigma_t\, = instant short rate volatility
W_t\, = a standard Brownian motion under a risk-neutral probability measure; dW_t\, its differential.

For constant (time independent) short rate volatility, \sigma\,, the model is:

d\ln(r) = \theta_t\, dt + \sigma \, dW_t

One reason that the model remains popular, is that the "standard" Root-finding algorithms – such as Newton's method (the secant method) or bisection – are very easily applied to the calibration.[7] Relatedly, the model was originally described in algorithmic language, and not using stochastic calculus or martingales. [8]


  • Benninga, S.; Wiener, Z. (1998). "Binomial Term Structure Models" (PDF). Mathematica in Education and Research: vol.7 No. 3. 
  • Black, F.; Derman, E.; Toy, W. (January–February 1990). "A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options" (PDF). Financial Analysts Journal: 24–32. 
  • Klose, C.; Li C. Y. (2003). "Implementation of the Black, Derman and Toy Model" (PDF). Seminar Financial Engineering, University of Vienna. 

External links

  • Online: Black-Derman-Toy short rate tree generator Dr. Shing Hing Man, Thomson-Reuters' Risk Management
  • Online: Pricing A Bond Using the BDT Model Dr. Shing Hing Man, Thomson-Reuters' Risk Management
  • Calculator for BDT Model QuantCalc, Online Financial Math Calculator
  • Excel BDT calculator and tree generator, Serkan Gur
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.