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1 January 1990 Financial Analysts Journal Volume 46, Issue 1

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

  1. Fischer Black
  2. Emanuel Derman
  3. William Toy

In one simple and versatile model of interest rates, all security prices and rates depend on only one factor—the short rate. The current structure of long rates and their estimated volatilities are used to construct a tree of possible future short rates. This tree can then be used to value interest-rate-sensitive securities.

For example, a two-year, zero-coupon bond has a known price at the end of the second year, no matter what short rate prevails. Its possible prices after one year can be obtained by discounting the expected two-year price by the possible short rates one year out. An iterative process is used to find the rates that will be consistent with the current market term structure. The price today is then determined by discounting the one-year price (in a binomial tree, the average of the two possible one-year prices) by the current short rate.

Given a market term structure and resulting tree of short rates, the model can be used to value a bond option. First the future prices of a Treasury bond at various points in time are found. These prices are used to determine the option’s value at expiration. Given the values of a call or put at expiration, their possible values before expiration can be found by the same discounting procedure used to value the bond. The model can also be used to determine option hedge ratios.

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