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 derivative In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of diff ...

s; see . 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 In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...

, and is still widely used.

History

The model was introduced by

Fischer Black Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American economist, best known as one of the authors of the Black–Scholes equation. Background Fischer Sheffey Black was born on January 11, 1938. He graduated from Harvard ...

Emanuel Derman Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''. He is a co-author of Black–Derm ...

, 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 Emanuel Derman's

memoir A memoir (; , ) is any nonfiction narrative writing based in the author's personal memories. The assertions made in the work are thus understood to be factual. While memoir has historically been defined as a subcategory of biography or autobiog ...

'' 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 In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or ye ...

), 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

s. Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous

stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...

: :

t + \sigma_t\, dW_t

::where, ::

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 Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

under a

risk-neutral In economics and finance, risk neutral preferences are preferences that are neither risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indif ...

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 algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...

s—such as Newton's method (the

secant method In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function ''f''. The secant method can be thought of as a finite-difference approximation o ...

) or bisection—are very easily applied to the calibration. Relatedly, the model was originally described in

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

ic language, and not using

stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...

or martingales.

References

Notes Articles * * * * *

External links

R function for computing the Black–Derman–Toy short rate tree
Andrea Ruberto
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
Excel BDT calculator and tree generator
Serkan Gur {{DEFAULTSORT:Black-Derman-Toy Model Fixed income analysis Short-rate models Financial models Options (finance)