Arbitrage Interest Rate model

[hellmaster]

Hi, Guys,

I have a question about arbitrage interest rate model and am hoping to get some help from some of our experts here. :-)

For arbitrage interest rate model, it is mentioned in the V-C125-07 that the entire yield curve is assumed to move randomly (compared to equilibrium model). But in the numerical examples, I only see them modeling one year spot rate using interest lattice. I am thinking in order to model the yield curve probably, shouldn't construct one interest lattice for each spot rate with a particular term. However, this will lead to constructing infinite amount of interest lattices as there are infinite amount of spot rates with different terms embedded in a yield curve.

Or are they assuming modeling one year term spot rate will somehow be the same as modeling the whole yield curve?

Many thanks in advance.

[Laurelinda]

Or are they assuming modeling one year term spot rate will somehow be the same as modeling the whole yield curve?

Many thanks in advance.

I think you pretty much hit the nail on the head there. The Equilibrium models actually model how the term structure develops through time. The Arbitrage-free models don't. They match the current yield curve exactly, and what happens after may not be realistic. In the Ho-Lee model, for example, the yield curve is assumed only to move up or down in parallel at each node. That's my understanding, at least.

[CDD]

You can get any rate out of a lattice even though you are only modeling the 1-year rate. Just model a zero coupon bond from any point.

As an example, randomly pick a node on the lattice. If you want to know the 3-year year rate at that node, move three years to the right from that node and fill in "100" where applicable (in this example where we are modeling the 1-year rate in 1-year increments, you would write "100" on four nodes). Now work backwards through the lattice to the node you picked and you have the 3-year zero price at that node. It's then easy to convert to the 3-year zero rate at that node.

You could do the above for any term at any of the nodes, so implicitly you have the entire term structure at every node in the lattice.

[Eroboy]

Most of the interest models presented in the paper are the one-factor model, which models yield curve by either short rate or forward rate with only one diffusion factor. It models the entire yield curve not one specific short rate or forward rate. What you mentioned as 3 year rate can be derived based on the yield curve generated by the model by the non arbitrage argument. In fact if you model 3 year rate using the same assumption you should get the same results from 1 year rate model based on same diffusion process and same initial yield curve. The lattice should be recombined. The one factor referred here to one diffusion process not a specific rate.

The drawback for the one-factor model is that the rates at different maturity are perfectly correlated.

I think the paper mentions the principle component analysis and said that most of the yield curve changes can be captured by 3 factors.