Binomial Interest Rate Tree

solebreadwinner · December 19, 2018, 10:36pm

Are forward rates derived from spot curves never used in the binomial interest rate tree?

My understanding is that only the par curve and the spot curves are used in calibrating a binomial interest rate tree and during the calibration, “hypothetical” forward rates are generated using those par and spot curves. These hypothetical forward rates would not match the forward rates calculated using just the spot curve?

Am I right on this? Please help!

S2000magician · December 19, 2018, 10:47pm

The 1-period forward rates in the tree generally _ will not _ match any of the forward rates on the (1-period) forward curve.

solebreadwinner · December 19, 2018, 10:48pm

Thank you! Would you mind explaining why that is?

solebreadwinner · December 19, 2018, 10:53pm

I guess I’m trying to ask what the relationship is between the actual forward rates and the forward rates from the forward rate curve. Is it possible to derive one from the other since they both use the same spot and par curves?

S2000magician · December 19, 2018, 11:22pm

To derive the forward rates in the tree from the forward curve involves solving polynomial equations: a linear (1^st-order) equation for the forward rates at time t = 1, a quadratic (2^nd-order) equation for the forward rates at time t = 2, a cubic (3^rd-order) equation for the forward rates at time t = 3, and so on. Theoretically, it’s possible (though equations of order higher than 4 cannot be solved using elementary functions (i.e., addition, subtraction, multiplication, division, raising to integer powers, and extracting integer roots)), but, as a practical matter, they’re usually solved numerically (i.e., you find a rate that’s accurate to however many decimal places you need).

At any particular time, the “center” forward rate (e.g., the HL or LH rate at time t = 3, or the geometric average of the H and L rates at time t = 2) will be close to the rate on the forward curve, but will generally differ from it by some non-zero amount.

solebreadwinner · December 20, 2018, 3:51am

Awesome, thank you so much!

S2000magician · December 20, 2018, 4:05am

You’re quite welcome.

If you set up a binomial tree in Excel, you can use Goal Seek or Solver iteratively to get the forward rates.

solebreadwinner · December 20, 2018, 4:15am

I’ve been doing that to construct the binomial tree but was confused as to why it didn’t match the actual forward curve rates, but you cleared it up!

S2000magician · December 20, 2018, 4:48am

Glad to help.