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“A One-Factor Model of Interest Rates and Its Application to Treasury - page 5 / 7





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ruu / rud  =  rud / rdd

rud2  =  ruurdd

Question:  What does this equation tell us?  What it says is that, in truth, there really aren’t 3 unknown quantities at this time step.  Because, rud is really determined by the relation given above, once ruu and rdd are known.  So, in the end, we’re using the quantities ruu and rdd to fit for the 2 market quantities, the 3-year yield and vol. So, 2 equations in 2 unknowns, you most likely will be OK.  (Ken:  The situations where you don’t get a solution are probably pretty pathologic.)

Next, the authors give the answer for this tree as ruu = 19.42%, rud = 13.77%, and rdd = 9.76%.  With these three answers, you get a yield for the three-year zero of 12%, and a vol of 18%, which agrees with the market numbers.

Continuing in this manner, you can calibrate to the yield and vol for all the rest of the time steps.  Because, at each step, you will be able to solve for all the internal short rates in the same manner as given above.  The short rates at the top and bottom of the tree nodes for the time step will determine all the internal nodes.  So you will have two unknowns to solve for, to calibrate to the two market constants, the market yield and vol.

Valuing Options on Treasury Bonds

Now, given the calibration procedure followed from above, and the resulting short rate tree, then various bond options may be valued.

Coupon Bonds as Collection of Zeroes

In this section, the authors use the constructed tree of short rates, to value a coupon-paying Treasury bond.  Specifically, consider a Treasury with:

10 percent (annual) coupon,

$100 face value,

3 years left to maturity.

The easiest way to value this bond is by considering it as a portfolio of 3 discount bonds.  The first such bond has a face value of $10 and a maturity of 1 year, the second is also $10, with a maturity of 2 years, and lastly, there is a 3-year zero with a face value of $110.

Now, each of these three zeros can be easily valued by averaging and discounting back through the short rate tree previously constructed.  The author show figures which actually give the numbers thus obtained, though they don’t show the actual computations and formulas.

And, the value of the coupon-paying Treasury is obtained at each node, by adding together all the values of the zeros, at that node, that haven’t yet matured.

With the data of Table 1 given previously, then the price of this 3-year coupon-paying Treasury comes out as $95.51.

Puts and Calls on Treasuries

In this section, the authors give a sample valuation of a European call and put on the Treasury considered in the previous section.  To do so, they employ the lattices of short rates and interest rates worked out previously.

Giving some numbers, they found previously that the 3-year, 10% percent Treasury should sell for $95.51, which is below par.  (Ken:  Just to include as a side-note, so I can keep this stuff straight:  The sub-par price comes from the fact that rates are generally higher than

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