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Unformatted text preview: hedge error is signiﬁcantly smaller (in absolute terms) in both scenarios.
142 Chapter 23 Interest Rate Models √
c)
The one day standard deviation for the CIR model is σCI R r/365 = 5. 234 2 × 10−3 which
is, by design, the same as part b). The duration hedge is
Nduration = − .89776
2 P (0, 2)
= −.2
= −.29401.
10 P (0, 10)
.6107 (21) which has a total cost of .89776 − .29401 (.6107) = .71821. The delta hedge is
Ndelta = − 1.3374
Pr (0, 2)
=−
= −.94723.
Pr (0, 10)
1.4119 (22) which has a total cost of .89776 − .94723 (.6107) = .3193. If r rises by the one day standard
deviation, the bond prices will be P2 = .89092 and P10 = .603436. This leads to “up” returns of
returnduration = .89092 − .29401 (.6034) − .71821e.05/365 = −.0048. (23) returndelta = .89092 − .94723 (.6034) − .3193e.05/365 = −6.42 × 10−6 (24) and If the short term rate falls by the one day standard deviation, the bond prices will be P2 = .9049
and P10 = .6182, leading to “down” returns of
returnduration = .9049 − .2940 (.6182) − .71821e.05/365 = .0048. (25) returndelta = .9049 − .94723 (.6182) − .3193e.05/365 = −2.13 × 10−5 . (26) and Without rounding errors the return is closer to −6 × 10−6 .
Question 23.8.
Instead of one long equation we will work backwards. In year 3, the four year bond is worth
the same value as the 1year bond in the terminal nodes of Figure 23.6. In year two the bond
will be worth three possible values, .8321 .8331+.8644 = .70624, .8798 .8644+.8906 = .77202,
2
2
and .9153 (.8906+.9123) = .8251. In year one, the bond will be worth two possible values,
2
+
.8832 .70624+.77202 = .6528 or .9023 .772022 .8251 = .72054. Finally, the current value is the
2
discounted expected value
P (0, 4) = .9091 .6528 + .72054
2
143 = .6243. (27) Part 5 Advanced Pricing Theory Question 23.10.
The value of the year2 cap payment has been shown to be V2 = 1.958. We must add to this the value
of the year3 cap payment and the value of the year1 cap payment. In year 2, the year3 cap payment will be worth three possible values: .8321 6.689+3.184 = 4.1077, .8798 3.184+.25 = 1.5106,
2
2
or .9153 (.125) = .11441. In year 1, the year3 cap payment will be worth two possible values:
.8832 4.1077+1.5106 = 2.481 or .9023 1.5106+.11441 = .73312. Hence the year3 cap payment
2
2
has a current value of
V3 = .9091 2.481 + .73312
2 The year1 cap payment has a value of V1 = .9091 1.078
2 = 1.461. (28) = .49. Summing the three we have V1 + V2 + V3 = .49 + 1.958 + 1.461 = 3.909. (29) Question 23.12.
See Table Two for the bond prices which are the same for the two trees. The one year bonds are
simply 1/ (1 + r) where r is the short rate from the given trees. For the two year bonds we can Table Two (Problem 23.12) Tree #1
0.08 0.07676
0.10362 0.0817
0.10635
0.13843 0.07943
0.09953
0.12473
0.1563 Tree #2
0.08 0.07552
0.09084
0.10927
0.13143
0.15809 One Year Bond Prices
0.925926
0.928712 0.924471 0.926415 0.929783
0.906109 0.903873 0.90948 0.916725
0.878403 0.889102 0.901494
0.864827 0.883837
0.863491 0.925926 Two Year Bond Prices
0.849454
0.849002 0.848615 0.855316
0.807468 0.812845 0.826816
0.770328 0.793671
0.755569 0.849453 Three Year Bond Prices
0.766885
0.771509 0.777541
0.717264 0.732357
0.680428 0.766884 Four Year Bond Prices
0.689247
0.70113
0.640069 0.689246 Five Year Bond Price
0.620926 0.620921 144 0.08112
0.09908 0.08749
0.10689
0.1306 0.08261
0.10096
0.12338
0.15078 0.07284
0.08907
0.10891
0.13317
0.16283 0.924967 0.919549 0.923694 0.932105
0.909852 0.903432 0.908298 0.918215
0.884486 0.890171 0.901786
0.868976 0.88248
0.859971
0.843098 0.842303 0.854564
0.81337 0.812397 0.826552
0.77797 0.794151
0.757074
0.765271 0.772934
0.7235 0.732097
0.686018
0.696052
0.645138 Chapter 23 Interest Rate Models solve recursively with formulas such as B (0, 2) = B (0, 1) × B (0,1)u +B(0,1)d where B (0, 1) is the
2
node’s 1 year bond and B (0, 1)u and B (0, 1)d are the one year bond prices at the next node. Once we
have two year bonds, three year bond values can be given by B (0, 3) = B (0, 1) × B (0,2)u +B(0,2)d
2
and similarly for the four and 5 year bonds.
Question 23.14.
See Table Four for the numerical answers to parts a) and b). Let rf (i) and re (i) be the one period
forward rate for borrowing at time i . Table Four (Problem 23.14) 1 year
forward rate American European Difference
Year 2
9.002% 9.019% 0.017%
Year 3
10.767% 10.803% 0.036%
Year 4
11.264% 11.308% 0.044%
Year 5
11.003% 11.041% 0.037%
Year 3 European Calculations
0.0917689 0.087322
0.110899 1 year
forward rate American European Difference
Year 2
9.003% 9.003% 0.000%
Year 3
10.767% 10.788% 0.021%
Year 4
11.264% 11.299% 0.035%
Year 5
11.004% 11.048% 0.044%
Year 3 European Calculations
0.0916379 0.089898
0.10804 0.10803279 0.10787872 Year 4 European Calculations
0.08671898 0.085475 0.082722
0.101838 0.101351
0.123429 Year 4 European Calculations
0.08664816 0.085901 0.084401
0.101259 0.101338
0.121245 Year 5 European Calculations
0.07609656 0.0...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.
 Spring '14
 NguyenThiMaiLan
 Derivatives

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