Actuarial Outpost Black-Derman-Toy tree model question
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#1
01-06-2018, 12:15 PM
 Peter T.W. Chen CAS SOA Non-Actuary Join Date: Oct 2017 Posts: 15
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 12:18 PM..
#2
01-06-2018, 12:41 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,225

Quote:
 Originally Posted by Peter T.W. Chen 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.
#3
01-11-2018, 01:31 PM
 BartimaeusOfUruk SOA Join Date: Nov 2017 Location: Des Moines, Iowa Studying for FAP Posts: 8 Blog Entries: 3

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
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 01:35 PM..

 Tags asm 10, mfe asm manual