black-dermnnan-toy.docx

8/10/2019 black-dermnnan-toy.docx

1/4

Libo-on, Joannalyn M. FINOPVA K32 June 11, 2014

11130040

Black Derman Toy Model

Interest rates are one of the major drivers in the bond market, stock market

and the foreign exchange market due to the fact they influence the investment

decisions of the investors. In the world of finance, experts are continuing to introduce

models that would value interest rate sensitivity security. In 1980s, Black-Derman

Toy Model was introduced by Fischer Black, Emanuel Derman, and Bill Toy, by

establishing the expectations of all interest rates using a one-factor model which astochastic factor known as the short rate. Even though, Black, Derman and Toy

introduced the model, it was first improved for in-house use by Goldman Sachs and

at some point published in the Financial Analysts Journal in 1990. The Black-

Derman-Toy model is mostly dependent to only one factor, which is the short rate,

since the model used the short rate in constructing a tree that can be used to value

securities that are interest rate sensitive. Using the current structure of long rates

and their estimated volatilities, the model can construct a tree of probable future

short rates. The Black-Derman-Toy model is important in valuing bond options

because the model can determine the future prices of the bonds at various points

and in return, the model can also be used to determine the options value at

expiration. The Black-Derman Toy model was derived from the evolution of the yield

curve model that was proven to be favored by the practitioners for valuing interest

rate derivatives. The single factor used in the model known as the short rate is an

annualized one-period interest rate that was applied in the valuation of security

prices. Some of the underlying assumptions provided by the Black-Derman-Toy

model are:

Expected returns on all securities over one period are equal.

Changes in all bond yields are perfectly correlated.

There exists no taxes or transaction costs.
http://en.wikipedia.org/wiki/Fischer_Blackhttp://en.wikipedia.org/wiki/Emanuel_Dermanhttp://en.wikipedia.org/wiki/William_Toyhttp://en.wikipedia.org/wiki/William_Toyhttp://en.wikipedia.org/wiki/Emanuel_Dermanhttp://en.wikipedia.org/wiki/Fischer_Black


2/4

The assumption in the model that the short are log-normally distributed is

advantageous because the results of model for the interest rates will always be

positive. In one of the assumptions provided by the model that the changes in all

bond yields are expected to be perfectly correlated due to the reason that the model

is only using a single factor that it is presume that the uncertainty in the short rates is

the only thing assumed to affect the rates of different maturities.

The short-rate volatility is prospectively time dependent, and the continuous process

of the short-term interest rate given by the Black-Derman Toy Model is:

wherein:

r = the instantaneous short rate at time t

= value of the underlying asset at option expiry

= instant short rate volatility

Wt = a standardBrownian motion under aRisk-neutral probability measure;itsdifferential.

Applying the Black-Derman-Toy model in valuing options has many

advantages. One of its advantages is that the volatility used in the model is a

percentage unit, which conforms to the market convention. This popular model is

favored by the experts because of its good estimate of predicting the future interestrates. The model surmises that the short-term rates revert to their mean and has a
http://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Risk-neutral_measurehttp://en.wikipedia.org/wiki/Differential_(mathematics)http://en.wikipedia.org/wiki/Differential_(mathematics)http://en.wikipedia.org/wiki/Risk-neutral_measurehttp://en.wikipedia.org/wiki/Brownian_motion


3/4

normal logarithmic distribution. The Black-Derman-Toy model integrates two

independents of time in order for the model to fit the term structure of spot interest

rates and the term structure of spot rate volatilities. The no-arbitrage model was

formulated so that it is exactly constant with the term structure of interest rates that is

observed in the market. The model has an advantage that it can be easily be

represented in the form of a binomial tree but in order for the model to be able to

constitute the formation of a binomial tree, it was presumed that the interest rates

that would be using are compounded annually.

The Black Derman Toy model has 3 essential features:

The model fluctuates an array of means and an array of volatility

changes, the future volatility changes, and the future mean reversion

changes.

The structural variable of the model is the short rate, which is the

annualized one-period interest rate that acquires all the security prices.

The mode derives model takes as inputs an array of long rates, which

is the yield on zero-coupon bonds for diverse maturities, and array of

yield volatilities for the same bonds. The first array is known as the

yield curve while the second array is called as the volatility curve and

jointly these curves form the term structure.

REFERENCE:

Black-Derman-Toy Model. (n.d.). TheFreeDictionary.com. Retrieved June 11,2014, from http://financial-dictionary.thefreedictionary.com/Black-Derman-

Toy+Model


4/4

BlackDermanToy model. (2014, May 26). Wikipedia. Retrieved June 11,

2014, from http://en.wikipedia.org/wiki/BlackDermanToy_model

. (n.d.). . Retrieved June 11, 2014, from

http://belkcollegeofbusiness.uncc.edu/wtian1/bdt.pdf


http://savage.wharton.upenn.edu/FNCE-

934/syllabus/papers/Black_Derman_Toy_FAJ_90.pdf


http://prosoftware.se/stud/II2008/BDT.pdf


http://www.seasholes.com/files/Klose_Yuan_2003_WP.pdf


http://financetrainingcourse.com/education/2010/10/interest-rate-models-black-

derman-and-toy-bdt-building-bdt-in-excel-introduction/

. (n.d.). . Retrieved June 11, 2014, from http://janroman.dhis.org/calc/BDT.pdf

. (n.d.). . Retrieved June 11, 2014, from http://www-

2.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote23.pdf

black-dermnnan-toy.docx

Documents