mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field. In general, there exist two separate branches of finance that req ...

, the Black–Derman–Toy model (BDT) is a popular

short-rate model A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,. The short rate Under a sh ...

used in the pricing of

bond option In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC. *A European bond option is an option to buy or sell a bond at a certain date in fu ...

s, 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 dif ...

s; see . It is a one-factor model; that is, a single

stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...

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 normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...

, 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. Working variously at the University of Chicago, the Massachusetts Institute of Technology, ...

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–D ...

, and Bill Toy. It was first developed for in-house use by

Goldman Sachs The Goldman Sachs Group, Inc. ( ) is an American multinational investment bank and financial services company. Founded in 1869, Goldman Sachs is headquartered in Lower Manhattan in New York City, with regional headquarters in many internationa ...

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 on 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 autob ...

'' 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 Yield to maturity, yields on debt instruments – such as bonds – vary as a function of their years remaining to Maturity (finance), maturity. Typically, the graph's horizontal ...

), and the volatility structure for

interest rate cap In finance, an interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive ...

s (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 have many applications throughout pure mathematics an ...

: :

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 is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

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 in ...

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 numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function is a number such that . As, generally, the zeros of a function cannot be computed exactly nor ...

s—such as

Newton's method In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...

(the secant method) or

bisection In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''s ...

—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 Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

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
Excel BDT calculator and tree generator
Serkan Gur {{DEFAULTSORT:Black-Derman-Toy Model Fixed income analysis Short-rate models Financial models Options (finance)