Finally, we solve for the short rate volatility for the second step of the tree. Given that the possible short rates here are ru = 14.32 and rd = 9.79, and the rates have been assumed to be distributed lognormally, then we get the relation
2 = (1/2) ln(14.32/9.79) = 19%
which again agrees with the market vol. Question: Where did I get this formula for the volatility? This comes from the fact that if you have a sample of size two from any distribution, then the sample variance comes out as:
(1/2) /Sample1 – Sample2/
This is a general probability fact, and I work out this relation in detail in my Rebanato, Chapter 8 Notes. Now, here, r is lognormal, so ln r is normal, with the same variance. So, if we take ln 14.32 and ln 9.79 to be the two samples from the distribution of ln r, then we get the formula for 2 given above.
With the numbers worked out in the procedure given above, we have matched the yields and volatilities of the tree to those of the market.
In this section the authors spell out how to calibrate to the short rate one further period into the future. This case is a little more difficult than the last one, and some assumptions about the interest rate and its vol must be called into play in order to work the calibration at this step. Proceeding:
We wish to find the short rate r two years into the future. Since the tree is re-combinant, then there are 3 possibilities which the short rate may migrate to in the meantime. Label these possibilities as ruu, rud, and rdd. Recall that at this point, we know the initial short rate as 10%, and the two possible short rates one year in the future as 14.32% and 9.79%.
In finding the three values ruu, rud, and rdd, we must calibrate to the yield and the volatility of the three-year zero. Authors say that these values must be found by guessing, or by trial-and-error. Read as, use a bisection or Newton’s method.)
There’s a problem here though: we’re using three variables here, ruu, rud, and rdd, in order to solve for only two unknowns, the yield and vol of the 3-year zero. So, in general, the answer will not be unique. In fact, there will be an infinity of possible answers. To get around this, we must invoke the assumptions on the short rate and the vol.
Specifically, recall that the short rate was assumed to be distributed lognormally. Also recall that the vol is a function of . Spelling out what this gives us, we have
One year in the future, if the short rate is 14.32%, then the vol at that node on the tree will come out to be (1/2)ln(ruu / rud). This specific formula for the vol at a given node was worked out in the last section.
And, one year in the future, if the short rate has come out to be 9.79%, then the vol at that node will be given by (1/2)ln(rud / rdd).
Now, because the vol is a function of time alone, we must have
(1/2)ln(ruu / rud) = (1/2)ln(rud / rdd)