39
CHAPTER 8
THE ARBITRAGE-FREE
VALUATION FRAMEWORK
PROBLEMS
is item set was developed by Karen O’Connor Rubsam, CFA (Phoenix, AZ, USA). Copy-
e following information relates to Questions 1–6
Katrina Black, portfolio manager at Coral Bond Management, Ltd., is conducting a training
session with Alex Sun, a junior analyst in the fixed income department. Black wants to explain
to Sun the arbitrage-free valuation framework used by the firm. Black presents Sun with Ex-
hibit 1, showing a fictitious bond being traded on three exchanges, and asks Sun to identify
the arbitrage opportunity of the bond. Sun agrees to ignore transaction costs in his analysis.
EXHIBIT 1 ree-Year, €100 par, 3.00% Coupon, Annual Pay Option-Free Bond
Eurex NYSE Euronext Frankfurt
Price €103.7956 €103.7815 €103.7565
Black shows Sun some exhibits that were part of a recent presentation. Exhibit 3 presents
most of the data of a binomial lognormal interest rate tree fit to the yield curve shown in Ex-
hibit 2. Exhibit 4 presents most of the data of the implied values for a four-year, option-free,
annual pay bond with a 2.5% coupon based on the information in Exhibit 3.
40 Part I: Learning Objectives, Summary Overview, and Problems
EXHIBIT 2 Yield to Maturity Par Rates for One-, Two-, and ree-Year Annual
Pay Option-Free Bonds
One-Year Two-Year ree-Year
1.25% 1.50% 1.70%
EXHIBIT 3 Binomial Interest Rate Tree Fit to the Yield Curve (Volatility = 10%)
Current Year 1 Year 2 Year 3 Year 4
1.2500% 1.8229% 1.8280% 2.6241% Node 4–1
1.4925% Node 2–2 Node 3–2 4.2009%
1.2254% 1.7590% 3.4394%
Node 3–4 2.8159%
Node 4–5
EXHIBIT 4 Implied Values (in Euros) for a 2.5%, Four-Year, Option-Free, Annual Pay Bond
Based on Exhibit 3
Year 0 Year 1 Year 2 Year 3 Year 4
103.4960 104.2876 103.2695 102.3791 102.5000
Node 1–2 104.0168 102.8442 102.5000
104.6350 103.2282 102.5000
103.5448 102.5000
102.5000
Black asks about the missing data in Exhibits 3 and 4 and directs Sun to complete the
following tasks related to those exhibits:
Task 1 Test that the binomial interest tree has been properly calibrated to be
arbitrage-free.
Task 2 Develop a spreadsheet model to calculate pathwise valuations. To test the ac-
curacy of the spreadsheet, use the data in Exhibit 3 and calculate the value of
the bond if it takes a path of lowest rates in Year 1 and Year 2 and the second
lowest rate in Year 3.
Task 3 Identify a type of bond where the Monte Carlo calibration method should
be used in place of the binomial interest rate method.
Task 4 Update Exhibit 3 to reflect the current volatility, which is now 15%.
1. Based on Exhibit 1, the best action that an investor should take to profit from the arbitrage
opportunity is to:
A. buy on Frankfurt, sell on Eurex.
B. buy on NYSE Euronext, sell on Eurex.
C. buy on Frankfurt, sell on NYSE Euronext.
Chapter 8 e Arbitrage-Free Valuation Framework 41
2. Based on Exhibits 1 and 2, the exchange that reflects the arbitrage-free price of the bond
is:
A. Eurex.
B. Frankfurt.
C. NYSE Euronext.
3. Which of the following statements about the missing data in Exhibit 3 is correct?
A. Node 3–2 can be derived from Node 2–2.
B. Node 4–1 should be equal to Node 4–5 multiplied by e0.4.
C. Node 2–2 approximates the implied one-year forward rate one year from now.
4. Based on the information in Exhibits 3 and 4, the bond price in euros at Node 1–2 in
Exhibit 4 is closest to:
A. 102.7917.
B. 104.8640.
C. 105.2917.
5. A benefit of performing Task 1 is that it:
A. enables the model to price bonds with embedded options.
B. identifies benchmark bonds that have been mispriced by the market.
C. allows investors to realize arbitrage profits through stripping and reconstitution.
6. If the assumed volatility is changed as Black requested in Task 4, the forward rates shown
in Exhibit 3 will most likely:
A. spread out.
B. remain unchanged.
C. converge to the spot rates.
e following information relates to Questions 7–101
Betty Tatton is a fixed income analyst with the hedge fund Sailboat Asset Management (SAM).
SAM invests in a variety of global fixed-income strategies, including fixed-income arbitrage.
Tatton is responsible for pricing individual investments and analyzing market data to assess the
opportunity for arbitrage. She uses two methods to value bonds:
Method 1 Discount each year’s cash flow separately using the appropriate interest rate
curve.
Method 2 Build and use a binomial interest rate tree.
Tatton compiles pricing data for a list of annual pay bonds (Exhibit 1). Each of the bonds
will mature in two years, and Tatton considers the bonds as being risk-free; both the one-year
and two-year benchmark spot rates are 2%. Tatton calculates the arbitrage-free prices and
identifies an arbitrage opportunity to recommend to her team.
EXHIBIT 1 Market Data for Selected Bonds
Asset Coupon Market Price
Bond A 1% 98.0584
Bond B 3% 100.9641
Bond C 5% 105.8247
1
is question set was developed by Jennie I. Sanders, CFA (Brooklyn, NY, USA).
42 Part I: Learning Objectives, Summary Overview, and Problems
Next, Tatton uses the benchmark yield curve provided in Exhibit 2 to consider arbitrage
opportunities of both option-free corporate bonds and corporate bonds with embedded op-
tions. e benchmark bonds in Exhibit 2 pay coupons annually, and the bonds are priced
at par.
EXHIBIT 2 Benchmark Par Curve
Maturity (years) Yield to Maturity (YTM)
1 3.0%
2 4.0%
3 5.0%
Tatton then identifies three mispriced three-year annual-pay bonds and compiles data on
the bonds (see Exhibit 3).
EXHIBIT 3 Market Data of Annual-Pay Corporate Bonds
Company Coupon Market Price Yield Embedded Option?
Hutto-Barkley Inc. 3% 94.9984 5.6% No
Luna y Estrellas Intl. 0% 88.8996 4.0% Yes
Peaton Scorpio Motors 0% 83.9619 6.0% No
Lastly, Tatton identifies two mispriced Swiss bonds, Bond X, a three-year bond, and
BondY, a five-year bond. Both are annual-pay bonds with a coupon rate of 6%. To calculate
the bonds’ values, Tatton devises the first three years of the interest rate lognormal tree pre-
sented in Exhibit 4 using historical interest rate volatility data. Tatton considers how this data
would change if implied volatility, which is higher than historical volatility, were used instead.
EXHIBIT 4 Interest Rate Tree; Forward Rates Based on Swiss Market
Year 1 Year 2 Year 3
4% 6%
1% 5%
2% 3%
7. Based on Exhibit 1, which of the following bonds most likely includes an arbitrage oppor-
tunity?
A. Bond A
B. Bond B
C. Bond C
8. Based on Exhibits 2 and 3 and using Method 1, the amount (in absolute terms) by which
the Hutto-Barkley corporate bond is mispriced is closest to:
A. 0.3368 per 100 of par value.
B. 0.4682 per 100 of par value.
C. 0.5156 per 100 of par value.
Chapter 8 e Arbitrage-Free Valuation Framework 43
9. Method 1 would most likely not be an appropriate valuation technique for the bond issued
by:
A. Hutto-Barkley Inc.
B. Luna y Estrellas Intl.
C. Peaton Scorpio Motors.
10. Based on Exhibit 4 and using Method 2, the correct price for Bond X is closest to:
A. 97.2998.
B. 109.0085.
C. 115.0085.