The Binomial Interest Rate Tree

A binomial interest rate tree is created using a lognormal random walk, where each node in the tree that is not the bottom node for the time period is the rate below it multiplied by $$ e^{2 \sigma} $$. The tree is calibrated with a sigma of 12%. If the $$ i_{2,HL} $$ node is 1.897%, what is the difference in basis points between $$ i_{2,LL} $$ and $$ i_{2,HH} $$?

Right! The other two nodes for time 2 can be calculated as $$\displaystyle i_{2,HH} = 0.01897e^{2(0.12)} = 0.0241 $$ and $$\displaystyle i_{2,LL} = \frac{0.01897}{e^{2(0.12)}} = 0.0149 $$. The difference between these two values is $$\displaystyle 0.0241 - 0.0149 = 0.0092 $$, or about 92 basis points.

Actually, no. This is possibly the result of adding and subtracting twice the standard deviation, but then taking the difference of 0.48 to mean 48 basis points. This ignores the continual compounding necessary in this calculation.

Not quite. This can be calculated by using the exponential function to move 4 standard deviations away from the center value. But this is too much, since the exponential function creates a much bigger difference when calculating the change this way. Instead, calculate both the higher and lower rates separately.

92 basis points

48 basis points

117 basis points