Zero Coupon Bonds

[ndaka26]

I have been going through ASM Leesion 15. I understand how to get prices and volatility given a BDT tree and converting it to one year zero coupon prices. How do you interpret the prices if you are given an n-year zero coupon bond tree instead of a 1 yera zero coupon bond. I am having problems relating.

[jason.]

I have been going through ASM Leesion 15. I understand how to get prices and volatility given a BDT tree and converting it to one year zero coupon prices. How do you interpret the prices if you are given an n-year zero coupon bond tree instead of a 1 yera zero coupon bond. I am having problems relating.

I'm having a little trouble understanding your specific question. Could you provide a little more detail about what you're asking?

[ndaka26]

I'm having a little trouble understanding your specific question. Could you provide a little more detail about what you're asking?

J.a.s.o.n it looks like you are my guardian angel. You respond to everything that I ask. Thanks. My problem started when I was doing CAS 3 Spring Exam 2007 question 39, to calsulate volatility. I had problems getting the right Ru and Rd because I tried to used the trre given and tried to discount the prices from t=2 at t=1. It's wasn't the case because the bond was 3 zero copon bond. It would have been true if the tree was a 1 year zero coupon bond tree. Is there a way of relating the prices we are looking for to the bond trees(both 1 year zero coupon and n-year zero coupon)?

[KingWithoutACrown]

Ndaka26,

Do you mean find the yield for an n-year zero coupon bond tree following the BDT model? If so, here is a technical explanation, i think its correct, someone would have to back me up on it.

1. Find the interest rate at each node/ intermediate node.
2. Starting from the ending nodes, pull back by discounting at the previous node rate and use the risk neutral, p* = 0.5.
3. Once pulled back to the initial node, you have the bond price, now you have to find the yield.
4. Exponential to the (-1/x), where n is a n-year zero coupon bond, and subtract one to find the yield.

[satogaeru]

Take a look at Prof. Broverman's solution to this problem; I think he addresses your specific concern in the first two or three sentences of the solution.

http://www.sambroverman.com/07s-cas3sol.pdf

[jason.]

J.a.s.o.n it looks like you are my guardian angel. You respond to everything that I ask. Thanks. My problem started when I was doing CAS 3 Spring Exam 2007 question 39, to calsulate volatility. I had problems getting the right Ru and Rd because I tried to used the trre given and tried to discount the prices from t=2 at t=1. It's wasn't the case because the bond was 3 zero copon bond. It would have been true if the tree was a 1 year zero coupon bond tree. Is there a way of relating the prices we are looking for to the bond trees(both 1 year zero coupon and n-year zero coupon)?

The two nodes in year one of the tree are 0.8133 and 0.8537. In year one (labeled t=1) in the problem statement, what was initially a 3-year bond is now a 2-year bond. Its price is either 0.8133 or 0.8537 so that the annualized yield is either \sqrt{{1 \over 0.8133}}-1=0.108854629 or \sqrt{{1 \over 0.8537}}-1=0.082299254. The volatility is then 0.5 \cdot \ln \left({0.108854629 \over 0.0822992544}\right) = 0.139825633. Therefore the answer is "B".

Does that answer your question?