The Binomial Interest Rate Tree

If you have a par curve, you can bootstrap the spot rates. If you have the spot rates, you can calculate the arbitrage-free forward rates. That means that the par curve, the spot curve, and the forward curve essentially all provide the same information. With some patience and accurate calculations, you can start with one and derive the other two.

But if you're working with a bond with an embedded option, the cash flows depend on the rate, and that's when you need to model future interest rates. Thankfully, there's the interest rate tree. Specifically, a __binomial interest rate tree__ will allow an up or down movement in each time period to create a distribution of potential future rates. What do you think _binomial_ really refers to here?

Exactly!

Not exactly. Depending on the period, there are more options than just a pair. Instead, _binomial_ refers to the up or down movement in each time period.

Now think about taking one time step forward, moving up or down in Time 1, and having two nodes there (each point in the tree when there's a rate is called a __node__). From there, you move forward to Time 2, where you could have up to four nodes, then eight in Time 3. But how many nodes would you expect in Time 3 if the branches merged, meaning that "up then down" was the same as "down then up"?

No. Just four, actually.

Excellent!

This describes a __recombining tree__, and it helps to keep the possibilities from getting out of hand. The number of nodes grows linearly when you do it this way, and it looks like this: 

Now you're ready to take a lovely walk from node to node with each annual time step. There are multiple ways to set up the values of the steps, but start here with the lognormal random walk. Using this creates a lognormal tree, which has some nice properties: The interest rates won't go negative, and there's greater volatility when rates get higher. Informally, you're going to set up this tree by taking every node that isn't on the very bottom and making it equal to the node below it multiplied by _e_^2σ. What formal notation for Time 1 is consistent with this idea?

Absolutely! That's the idea. So if the "low" interest rate in Time 1 were 2%, and you assumed an annual standard deviation of 10%, then you can complete the calculation. $$\displaystyle i_{1,H} = i_{1,L} e^{2 \sigma} = 2 \% \times e^{2 (0.10)} = 2.443 \% $$

Not quite. This is moving in the wrong direction. Note that the higher rate is _H_, and the lower rate is _L_.

No. This isn't consistent with the statement. The exponential function is multiplied by a rate to find another rate, but this would require division.

Now take it a step further to Time 2 and then Time 3. Once you have the bottom interest rate determined in Time 3 ($$i_{3,LLL}$$), what will you need to multiply this by if you want to get $$i_{3,HHH}$$ next?

No. That's too far. Remember that you're using just three "upward steps" to get to the highest rate.

Yes! You can use the same multiplier. $$\displaystyle i_{3,HHH} = i_{3,HHL} e^{2 \sigma} = i_{3,LLH} e^{4 \sigma} = i_{3,LLL} e^{6 \sigma} $$ Obviously, there are a few more questions to be answered in putting this all together, but once you have the estimated volatility and the lowest nodes, calculating a lognormal tree is pretty straightforward.

No. That will only move you one step up. You'll need to go all the way up to the highest rate.

To summarize: [[summary]]

The up or down movement each in time period

The pair of possible rates derived in each time period

Four

Six

$$ i_{1,H} = i_{1,L} e^{2 \sigma} $$

$$ i_{1,L} = i_{1,H} e^{2 \sigma} $$

$$ e^{2 \sigma} = i_{1,H} \times i_{1,L} $$

_e_^2σ

_e_^6σ

_e_^8σ

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