Black Derman Toy model / BDT model explained plus formula

Black Derman Toy model - Toolshero

Black Derman Toy model: This article explains the Black Derman Toy model in a practical way. Next to what this model is, the development, the Nobel Prize and the formula are also highlighted. Enjoy reading!

The Black Derman Toy model exlained

The Black Derman Toy model, or BDT model for short, is a financial model that is used in the pricing of bond options and other interest rate derivatives.

It is a 1-factor model, which means that a single stochastic factor is used. This factor determines the future evolution of interest rates.

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The BDT model was one of the first models of its kind and is still widely used.

Development BDT model

The model was introduced by Fischer Black, Emmanuel Derman and Bill Toy.

Initially, the model was only used internally at Goldman Sachs, where the gentlemen were employed. In 1990 it was published in the Financial Analyst Journal. Emmanuel Derman writes about the development of the model in his memoir My Life as a Quant.

The BDT model is different from the more well-known Black-Scholes model in that it focuses on equity derivatives, while the former focuses on fixed income derivatives. While Black-Scholes was developed before Fischer Black joined Goldman Sachs in 1984, the BDT model was developed during his career with the company.

Nobel Prize

Fischer Black is known as the genius of the finance world and financial analysis history. Two years after his death at age 57, the Nobel Prize in Economics was awarded to Myron S. Scholes and Robert C. Merton. They had further developed the model.

Fischer Black was not eligible for the award as it is not awarded posthumously.

The formula

The formula of the Black Derman Toy model is very complex to the untrained eye. The terminology is also experienced as complex. The basis of the formula is explained below.

Black Derman Toy model formula - Toolshero

Figure 1 – the basic formula (Blank, Derman & Toy).

Using BDT and the binomial grid, the user calibrates the model parameters with the model to fit both the current term structure of interest rates and the volatility structure for interest rate caps.

Using this grid, a variety of more complex and interest-rate sensitive securities and interest rate derivatives are then valued.

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Now it is your turn

What do you think? ? Do you recognize the explanation about the Black Derman Toy model? Is this specialist model used in your work environment? What other financial management models do you know? Do you find financial analyzes interesting and do you want to know more about it? Leave a message in the comments.

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More information

  1. Izgi, B., & Bakkaloglu, A. (2018). Invariant approaches for the analytic solution of the stochastic Black-Derman toy model. Thermal Science, 22(Suppl. 1), 265-275.
  2. Tung, H. K., Lai, D. C., Wong, M. C., & NG, S. (2010). The Black–Derman–Toy Model. Professional Financial Computing Using Excel and VBA, 95.
  3. Boyle, P. P., Tan, K. S., & Tian, W. (2001). Calibrating the Black-Derman-Toy model: some theoretical results. Applied Mathematical Finance, 8(1), 27-48.

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Published on: 12/01/2022 | Last update: 12/01/2022

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Ben Janse
Article by:

Ben Janse

Ben Janse is a young professional working at ToolsHero as Content Manager. He is also an International Business student at Rotterdam Business School where he focusses on analyzing and developing management models. Thanks to his theoretical and practical knowledge, he knows how to distinguish main- and side issues and to make the essence of each article clearly visible.

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