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“A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options,”

by Fischer Black, Emanuel Derman, and William Toy

Reference:  Black F., Derman E., Toy W. (1990) ‘A one-factor model of interest rates and its

application to Treasury bond options,’ Financial Analysts’ Journal, 46, 33-39

Article describes model of interest rates that can be used to value any interest-rate sensitive security.  In demonstrating, authors concentrate mostly on the case of Treasury bond options.

Observation by Ken:  Can it be used to handle the case of negative interest rates?  As I understand it, rates are lognormal in this model, which disallows neg. in. rates.

Second question:  Does this matter?  Supposedly there were neg. interest rates in Switzerland in the 1960’s, and also in Japan during the Asian crisis.

The three key features of the model:


Fundamental variable is the short rate.  Changes in short rate dictate all changes in security prices.


Model takes as inputs an array of long rates and their associated volatilities.  Together these are called the term structure.


Model varies an array of mean interest rates and vols to match the inputs.  This is the calibration.

Further Observations/Assumptions:

Because this is a one-factor model with the short rate as the factor, changes in all bond yields will be perfectly correlated.

Furthermore, expected returns on all securities over one period are equal.

Short rates at any time are assumed to be lognormally distributed.

No taxes or trading costs.

Valuing Securities

Authors use binomial distribution approach.  If the security’s price today is S, then one period from now, it can be either Su or Sd, each with probability ½.  So the expected price one year from today is ½(Su + Sd).

Now, due to the assumption that all securities should have the same one-period (expected) return in this one-factor model, and because we can lend at the short rate r, then the security’s price today should satisfy the relation

S = [½(Su + Sd)]  /  (1 + r)

Getting Today’s Prices from Future Prices

The above relation tells how to relate today’s prices to prices one step away.  We can extend this to obtain prices one step in the future from prices two steps in the future.  The following numerical example is given:


The two step-tree is given, with the current short rate at 10%.  For one step into the future, we expect it to either rise to 11% or fall to 9%, each with equal probability.  (Ken:  Don’t think too much about how they got these numbers.  At this point they’re just saying, we have these numbers for the way the short rate may move, let’s see how we can obtain prices from this tree.)

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