Fabozzi ... complete tard.

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cjones65

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Does Frank Fabozzi upload his lectures anywhere?
Reading 55 is a formidable nut to crack. I feel like a complete tard.
I need to see a demonstration of how arbitrage-free bond pricing binomial trees are created.
Anyone care to walk through a problem? Perhaps one of the in-chapter questions from Volume 5?
I can offer you protection from the unwanted advances of WillyR’s mom.
 
Maybe if I were more specific. Start simple with the Treasury’s on-the-run yield curve.
@ = assumed volatility of the 1-year rate
1-year spot rate can take on two possible values the next period.
That rate can evolve over time based on a lognormal random walk with certain volatility.
R0 = 1-year spot rate
R1,H = (e^2@)R1,L
R2,HH = (e^4@)R2,LL; R2,HL = (e^2@)R2
R3,HHH = (e^6@)R2,LLL; R3HHL = (e^4@)R3,LLL; R3HHL = (e2@)R2,LLL
In the third year there are four possible values for the 1-year rate (i.e. HHH, HHL, HLL, LLL).
Okay. So I guess this is where first get lost. RT is the 1-year rate T years from now for the lower rate since all the other short rates T years from now depend on that rate.
VH + c / (1 + r*) = present value for the higher 1-year rate
VL + c / (1 + r*) = present value for the lower 1-year rate
Value at a node = 1/2 [ VH + C / (1 + r*) + VL + C / (1 + r*) ]
Some volatility assumption and the yield on the 2-year on-the-run then tell us the rates at the 1-year nodes? And you pull a lower rate from where? Then it iterates until it equals the market value of the on-the-run … not giving up on this.
 
Fine. I’ll figure it out all on my own. But don’t come crying to me after WillyR’s mom violates your no-no special places.
 
I will when I get home. Taking a half day today…
 
Okay… Volume 5 (page 309), Reading 55, #5
An on-the-run issue for the Inc.Net Company is shown below:
Maturity, YTM, Market price
1, 7.5, 100
2, 7.6, 100
3, 7.7, 100
Using the bootstrapping methodology, the spot rates are:
Maturity (years), Spot rate (%)
1, 7.500
2, 7.604
3, 7.710
Assuming an interest rate volatility of 10% for the 1-year rate, the binomial interest rate tree for valuing a bond with a maturity of up to three years is shown below:
_____________7.5%____________
_______6.944%_____8.481%_____
6.437%______7.862%______9.603%
A. Demonstrate using the 2-year on-the-run issue that the binomial interest rate tree above is in fact an arbitrage-free tree.
B. Deomonstrate using the 3-year on-the-run issue that the binomial interest rate tree above is in fact an arbitrage-free tree.
C. Using the spot rates given above, what is the arbitrage-free value of a 30year 8.5% coupon issue of Inc.Net Company?
D. Using the binomial tree, determine the value of an 8.5% coupon 3-year option-free bond.
E. Suppose that the 3-year 8.5% coupon issue is callable starting in Year 1 at par (100) (that is, the call price is 100). Also assume that the following call rule is used: if the price exceeds 100, the issue will be called. What is the value of this 30year 8.5% coupon callable issue?
F. What is the value of the embedded call option for the 3-year 8.5% coupon callable issue?
 
A. Demonstrate using the 2-year on-the-run issue that the binomial interest rate tree above is in fact an arbitrage-free tree.
Easy.
R1L = 6.944%, so R1H = 6.944e^2*.1 = 8.481421% (I know it’s given, but I’ve had rounding problems, so I calculate it myself)
2-year rate is 7.6%, so
[((107.6/1.06944)+7.6)/1.075]+[((107.6/1.08481421)+7.6)/1.075] =
100.663633 + 99.337201, and the average is 100.000417
So since that equals market price, there’s no arbitrage
 
B. Deomonstrate using the 3-year on-the-run issue that the binomial interest rate tree above is in fact an arbitrage-free tree.
Not really sure how this is done.
R2LL = 6.437%, so R2LH = 7.86217%, and R2HH = 9.602876%
3-year rate is 7.7%, so
[((107.7/1.06437)+7.7)/1.075]+[((107.7/1.0786217)+7.7)/1.075]+[((107.7/1.09602876)+7.7)/1.075] =
101.289877 + 100.46187 + 98.571018, and the average is 99.969027
I guess there’s no arbitrage (?)
 
C. Using the spot rates given above, what is the arbitrage-free value of a 30year 8.5% coupon issue of Inc.Net Company?
This is my bread and butter
8.5/1.075 + 8.5/1.07604^2 + 108.5/1.0771^3 =
7.906997 + 7.341116 + 86.858372 = 102.076
 
D. Using the binomial tree, determine the value of an 8.5% coupon 3-year option-free bond.
Impossible. Can you relate?.
 
I’m missing something, cjones. Why are you having problems with D? If you can do A and B, then D is simply extending the binomial tree out an extra year. The question does not ask you to prove that it is an arbitrage-free scenario so the value at t=0 is not par (which it is not; the value is 102.08.
 
I have no idea. I’m usually much better at getting a handle on topics quickly.
So for D, which rates do I use?
8.5/1+ ? + 8.5/(1 + ?)^2 + 108.5/(1 + ?)^3 = 102.08
Goal seek gives me 0.07697294
I know this must be wrong.
Zizoubleu, please tell me how you got from A&B to D. I understand if you want me to turn in my glock.
 
Please don’t bring your glock to the exam site….or I’ll have to bring a tommy gun if we are going to play Mafia Wars.
Sorry I’m not much help…too many numbers..LOL
 
No worries. Me and the other interns share one glock, and I’m not scheduled to carry it on the sixth. Besides, Mr. Z will probably kick me off the team now that he’s seen me get Al Capwned by a simple binomial interest rate tree problem.
 
When you become a big time mobster, you can outsource this kind of dirty nitty gritty work to sub mob members, sometimes called juniors analysts…hahaha
I think the real tree will be a lot simpler. Mr. Z will be delighted with any arbitrage loot…
 
cjones,
Check appendix A, page 20 for the binomial tree graph.
For C, you are using the graph from the question at the top of page 310 of the CFAI text. You simply extend another time period. Given a coupon rate of 8.5%, you will discount 108.5 by the three interest rates given, 9.603%, 7.862%, and 6.437%. Your discounted values at year two, working down the tree are: 98.994, 100.591, 101.938.
From there, you discount again:
98.994 + 8.5=107.494/1.08481 =99.090
100.591 +8.5=109.091/1.08481=100.562
100.591+ 8.5=109.091/1.06944=102.008
101.938+8.5=110.438/1.06944=103.267
Add 99.090 + 100.562/2=99.826 This is the value of the top node of year 1
102.008+103.267/2=102.638
One last time to calculate value at t=0. Discount at interest rate ‘Today’=7.5%
99.286 +8.5=108.326/1.075=100.768
102.836+8.5=111.138/1.075=103.384
Add these up and divide by two for you answer:
100.768 + 103.384=204.152=102.076
I hope you can work your way through my explanation.
 
Beautiful. I’m crystal on this now. Thank you.
(FREEEEEEEEEEEEEEEDOOOOOOOOOOOOOOOM!!!!!!!!!!!!!!)
 
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