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  #1  
Old 01-06-2018, 11:15 AM
Peter T.W. Chen Peter T.W. Chen is offline
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Default Black-Derman-Toy tree model question

I have a question for BDT tree.
The question is from ASM practice exam 4 #27 (p.519)

#27
You are given the following Black-Derman-Toy tree with effective annual interest rates:

Year 0........Year 1............Year 2
.............................../.....r(uu)
.......... / ....6.351%...\
5%.................................5.939%
...........\.....5.200%... /
................................\.....r(dd)

A 3-year cap of 6% on a notional loan amount of $1000 costs $4.80.
Determine r(uu).

Anyone help? I cannot understand the solution that manual provides. Other better ways?

Last edited by Peter T.W. Chen; 01-06-2018 at 11:18 AM..
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  #2  
Old 01-06-2018, 11:41 AM
Academic Actuary Academic Actuary is offline
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Quote:
Originally Posted by Peter T.W. Chen View Post
I have a question for BDT tree.
The question is from ASM practice exam 4 #27 (p.519)

#27
You are given the following Black-Derman-Toy tree with effective annual interest rates:

Year 0........Year 1............Year 2
.............................../.....r(uu)
.......... / ....6.351%...\
5%.................................5.939%
...........\.....5.200%... /
................................\.....r(dd)

A 3-year cap of 6% on a notional loan amount of $1000 costs $4.80.
Determine r(uu).

Anyone help? I cannot understand the solution that manual provides. Other better ways?
It seems that there is missing information. You need the one year volatility from time 2 to time 3.
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  #3  
Old 01-11-2018, 12:31 PM
BartimaeusOfUruk BartimaeusOfUruk is offline
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Join Date: Nov 2017
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From the chart you can see the cap will only pay at nodes ru and ruu.

At ru cap pays: (.06315-.06)/(1.06315)
At ruu cap pays: (ruu-.06)/(1+ruu)

Add in probabilities at each node
At ru: (.5)*[(.06315-.06)/(1.06315)]
At ruu:[(.5)(.5)]*[(ruu-.06)/(1+ruu)]

Discounting payoffs at these nodes leaves:
At ru: (.5)*[(.06315-.06)/(1.06315)] * [1/1.05] = .001410902
At ruu:[(.5)(.5)]*[(ruu-.06)/(1+ruu)] *[1/1.06315]* [1/1.05]
= .22395263* [(ruu-.06)/(1+ruu)]

We know the value of this cap is 4.80
The notional amt is 1000

Therefore: 4.80/1000 = .001410902 + .22395263* [(ruu-.06)/(1+ruu)] <- sum of the two discounted caps.

.0048 = .001410902 + .22395263* [(ruu-.06)/(1+ruu)]
.01513302 = [(ruu-.06)/(1+ruu)]
.01513302(1+ruu) = ruu -.06
.075133102 = 1.01513302 ruu
ruu = .074

Overall:
1. Note where cap pays off
2. Find value of caps at their node
3. Add in the probabilities
4. Discount
5. Set equal to the cost


This is my best guess! I haven't passed the exam yet though- so don't take my word for it!

Last edited by BartimaeusOfUruk; 01-11-2018 at 12:35 PM..
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