141
CHAPTER 8
THE ARBITRAGE-FREE
VALUATION FRAMEWORK
SOLUTIONS
Frankfurt exchange (which has the lowest price of €103.7565) and selling it on the Eurex
exchange (which has the highest price of €103.7956) to generate a risk-free pro t of
€0.0391 (as mentioned, ignoring transaction costs) per €100 par.
purchased in Frankfurt and sold on Eurex.
Eurex.
ow. e value of this bond, 103.7815, is calculated as follows:
Year 1 Year 2 Year 3 Total PV
Yield to maturity 1.2500% 1.500% 1.700%
(continued)
142 Part II: Solutions
Eurex NYSE Euronext Frankfurt
Notes:
ow, FV = cash ow amount, and r = spot rate.
the Year 3 spot rate to discount all the cash ows.
from Node 2–2; it can be derived from any other Year 3 node; for example, Node 3–2 can
be derived from Node 3–4 (equal to the product of Node 3–4 and
e 4 σ
).
node in Year 2, equal to 104.6350. e bond value at Node 1–2 is calculated as follows:
Once its accuracy is con rmed, the interest rate tree can then be used to value bonds
with embedded options. While discounting with spot rates will produce arbitrage-free
tree, as they are assumed to be correctly priced by the market.
tionship multiple between nodes ( e
x σ , where x = 2 times the number of nodes above the
Chapter 8 e Arbitrage-Free Valuation Framework 143
lowest node in a given year in the interest rate tree). Conversely, using a lower estimate of
volatility would cause the forward rates to narrow or converge to the implied forward rates
from the prevailing yield curve.
of volatility are assumed.
(zeros), capturing the excess value. e arbitrage-free prices of Bond A and Bond C are
equal to the market prices of the respective bonds, so there is no arbitrage opportunity for
these two bonds:
Bond A:
1.02
101
1.02 98.0584
2
Bond C:
105
the one-year issue has only one cash ow remaining, the YTM equals the spot rate of 3%
3
e correct arbitrage-free price for the Hutto-Barkley Inc. bond is:
1.0300
3
1.0402
103
1.0507 94.4828
023
P
()
()()
=+ + =
erefore, the bond is mispriced by 94.4828 – 94.9984 = –0.5156 per 100 of par value.
A is incorrect because the correct spot rates are not calculated and instead the
144 Part II: Solutions
leads to an incorrect mispricing of 94.5302 – 94.9984 = –0.4682 per 100 of par value.
values bonds with embedded options, interest rates must be allowed to take on di erent
potential values in the future based on some assumed level of volatility (Method 2).
Method 2 to produce an arbitrage-free valuation.
Time 0 Time 1 Time 2 Time 3
106
6
6 106
06
6 106
6
106
Next, calculate the cash ows for each year beginning with Year 3 and move back-
wards to Year 1:
Year 3:
106
⎡
1.05
1.05 6 106.9524×⎛
⎝
⎠
⎝
⎠
⎣
⎦
0.5 106
1.03
106
1.03 6 108.9126×⎛
⎝
⎜⎞
⎠
⎟+⎛
⎝
⎜⎞
⎠
⎟
⎡
⎣
⎢⎤
⎦
⎥+=
Year 2:
106.9524
⎡
1.02
1.02 6 111.8162×⎛
⎝
⎠
⎝
⎠
⎣
⎦
Year 1:
0.5 108.3810
1.01
111.8162
1.01 109.0085×⎛
⎝
⎜⎞
⎠
⎟+⎛
⎝
⎜⎞
⎠
⎟
⎡
⎣
⎢⎤
⎦
⎥=
A is incorrect because the coupon payment is not accounted for at each node calcu-