To make sure that I’m on the right track, in a binomial tree, we usual assume that the probability of the upward and the downward path is 50% each unless the question is stating other probabilities, right?
For a binomial _ interest rate _ tree, it’s 50/50.
For a binomial _ price _ tree, you have to calculate the weights (probabilities).
In CAFI Exam 1 morning session question #46, the answer assumed a probability of 50%, why is that?
Krishnan then analyzes Bond D, which pays an annual 3.20% coupon rate and matures 3 years from now. The bond is putable at 98 one year and two years from now. She assumes 15% interest rate volatility and, using yields on par bonds, constructs the binomial interest rate tree found in Exhibit 2.
EXHIBIT 2
BINOMIAL INTEREST RATE TREE
Q. Using the interest rate information found in Exhibit 2, the value of the three-year putable bond analyzed by Krishnan is closest to:
- 101.072.
- 99.727.
- 99.206.
Solution
B is correct. The value is calculated using the interest rate tree, starting with final cash flow (par plus coupon payment) in Year 3.
Vuu = 103.2001.0621 = 97.166, so the bond is put at 98.
Vud = 103.2001.0460 = 98.662 and Vdd = 103.2001.0341 = 99.797, therefore
Vu = 0.5×(98+3.2001.0431+98.662+3.2001.0431)= 97.336, so the bond is put at 98.
Vd = 0.5×(98.662+3.2001.0319+99.797+3.2001.0319)= 99.263, therefore
Vd =0.5×(98+3.2001.0211+99.263+3.2001.0211)= 99.727
As S2000magician mentioned, for the purposes of the CFA exams, binomial interest rate trees move up and down with probability 0.5. That is the understanding you need for the exams. This assumption makes it easier to work on problems with interest rate trees.
Note in the equations for the values at the nodes, a 0.5 coefficient is used. Another key item: since the bond is putable at 98, the value at the node V_u is 98 rather than 97.336.
So in general, we calculate the probabilities for prices binomial tree but for the sake of the exam we will assume it is 50%, is that correct?
Yes, except for options, where the probability of going up = (1+r-d)/(u-d), where:
r = risk-free
d = down factor
u = up factor
obviously, the probability of going down = 1 - probability of going up = 1- (1+r-d)/(u-d)
No, that’s not correct.
Didn’t you read what I wrote:
I even bolded the important parts. If it’s a price tree, you have to calculate them. Period. If it’s an interest rate tree, they’re 50% up and 50% down. Period.
They can be used for options, but they can be used for anything else as well.
Did you read the example I posted, why did the CAFI used 50% probability in that sense?
I did, and I answered your question.
What kind of tree is it?
Price tree
Try again… What does 6.21% mean? Is it $6.21 USD?
What does the text under the exhibit 2 say?
Dem sure don’t look like prices to me.
I was under the impression that its a price tree because we are valuing the bond, my bad.