# Black–Derman–Toy model

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Published: 2026-10-24T23:57:26.997Z

Updated: 2026-10-24T23:57:26.997Z

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 com

## Key facts

## Overview

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 log-normal distribution, and is still widely used.

## History

The model was introduced by Fischer Black, Emanuel Derman, and William (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 Emanuel Derman's memoir My Life as a Quant.

## Formulae

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:

d         ln         ⁡         (         r         )         =

\[

θ

t

+

σ

t

′

σ

t

ln             ⁡             (             r             )

\]

d         t         +

σ

t

d

W

t

\{\\displaystyle d\\ln(r)=\\left\[\\theta \_\{t\}+\{\\frac \{\\sigma '\_\{t\}\}\{\\sigma \_\{t\}\}\}\\ln(r)\\right\]dt+\\sigma \_\{t\}\\,dW\_\{t\}\}

## References

Notes

## External links

R function for computing the Black–Derman–Toy short rate tree, Andrea Ruberto Excel BDT calculator and tree generator, Serkan Gur

## Sources

- Black–Derman–Toy model · Wikipedia: https://en.wikipedia.org/?curid=3152504

Source: Wikipedia contributors — https://en.wikipedia.org/?curid=3152504&oldid=1379420883

Licence: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/

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