Initial condition :r0,0=5%,u=1.1,d=0.9andq=1q=1/2.

question 1a:Compute the initial value of a forward-starting swap that begins att=1, with maturityt=10and a fixed rate of 4.5%. (The first payment then takes place att=2and the final payment takes place att=11as we are assuming, as usual, that payments take place in arrears.) You should assume a swap notional of 1 million and assume that you receive floating and pay fixed.)

question 1b:Compute the initial price of a swaption that matures at timet=5and has a strike of 0. The underlying swap is the same swap as described in the previous question with a notional of 1 million. To be clear, you should assume that if the swaption is exercised att=5then the owner of the swaption will receive all cash-flows from the underlying swap from timest=6tot=11inclusive. (The swaption strike of 0 should also not be confused with the fixed rate of 4.5% on the underlying swap.)

The attachment (on the swap+swaptions page) is my work for the questions. The model should be half correct. I think the problem of my model is that it cannot handle "The first payment then takes place att=2" Something has to be changed on my model but I don't know how.

Can someone help me fix the model? thanks a lot !

Term Structure Lattice r(0,0) 5.00% u 1.10 d 0.9 q 0.50 1-q 0.50 Short-Rate Lattice 0 5 4 3 2 1 0 5.00% 4-Year Zero-Coupon Bond 0 5 4 3 2 1 0 82.29 1 3 4 5.50% 4.50% 6.05% 4.95% 4.05% 6.66% 5.45% 4.46% 3.65% 7.32% 5.99% 4.90% 4.01% 3.28% 1 2 3 88.92 90.79 92.37 93.76 94.84 95.74 96.48 9 5 8.05% 6.59% 5.39% 4.41% 3.61% 2.95% 100.00 100.00 100.00 100.00 100.00 American Zero Option Value Expiration 6 Strike 80.00 Option type 1 Expiration Strike Option type 0 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0 13.18 13.29 14.38 13.48 14.56 15.48 3 13.76 14.84 15.74 16.48 0 1 2 3 1 2 0 4 0.05 0.055 0.045 0.0605 0.0495 0.0405 0.06655 0.05445 0.04455 0.03645 0 1 2 3 4 63.8381111744 62.6764023 69.0985381117 61.9650824238 68.0699216 73.7802273489 61.6219581175 67.441029623 72.8818303 77.8868615086 European Zero Option Value 2 84.00 1 2 1 0 0.073205 0.059895 0.049005 0.040095 0.032805 74.8848449384 10 4 85.17 87.64 2 5 6 0.0805255 0.0658845 0.0539055 0.0441045 0.0360855 0.0295245 0.08857805 0.07247295 0.05929605 0.04851495 0.03969405 0.03247695 0.02657205 5 6 67.9685611 65.555982012 72.7414542 70.617092934 76.9519532 75.1048140897 80.6186978 79.029458865 83.7775923 82.4222163561 86.4745621 71.2606292518 75.622897786 79.4450792973 82.7550941889 85.5935960835 88.00790104 90.0474595179 7 8 9 0.097435855 0.079720245 0.065225655 0.053366445 0.043663455 0.035724645 0.029229255 0.023914845 0.107179441 0.08769227 0.071748221 0.05870309 0.048029801 0.03929711 0.032152181 0.02630633 0.021523361 0.1178973846 0.0964614965 0.0789230426 0.0645733985 0.0528327806 0.0432268205 0.0353673986 0.0289369625 0.0236756966 0.0193710245 7 8 9 75.6832238607 79.4622898046 82.7447347477 85.5669826355 87.9729242559 90.0093808764 91.7228776069 93.1575326222 81.58393985 84.53102713 87.06305891 89.22056963 91.04620698 92.58204517 93.86782294 94.93991503 95.83084612 89.4536487714 91.2024729767 92.6850164991 93.9343404087 94.9818450255 95.8564312571 96.5840726104 97.1876836477 97.6871877852 98.0997081548 1 6.10 5.55 7.25 10 0.129687123 0.1061076461 0.0868153468 0.0710307383 0.0581160586 0.0475495025 0.0389041384 0.0318306587 0.0260432662 0.0213081269 0.017433922 10 100 100 100 100 100 100 100 100 100 100 100 2 4.92 6.79 8.37 8 7 6 5 4 3 2 1 0 2.3572151638 0 0.2890684737 0.83945526 1.4914161229 1.556651755 2.44507995 3.640806687 3.3935000891 4.64733524 6.0302979445 1 2 3 0 0 0 0 0 0 0.6166119612 1.30708787 2.7550941889 2.5286355004 3.9980147 5.5935960835 5.0773737494 6.5638874 8.00790104 7.4228308597 8.76878625 10.0474595179 4 5 6 Term Structure Lattice r(0,0) 6.00% u 1.25 d 0.9 q 0.50 1-q 0.50 Short-Rate Lattice 0 5 4 3 2 1 0 6.00% 1 2 3 4 14.65% 10.55% 7.59% 5.47% 3.94% 4-Year Zero-Coupon Bond 0 5 18.31% 13.18% 9.49% 6.83% 4.92% 3.54% 7.50% 5.40% 9.38% 6.75% 4.86% 11.72% 8.44% 6.08% 4.37% 1 2 3 4 5 104.03 112.49 119.27 124.57 101.66 108.44 113.83 118.00 121.16 102.98 107.19 110.46 112.96 114.84 116.24 2 3 4 77.22 79.27 84.43 83.08 87.35 90.64 89.51 92.22 94.27 95.81 100.00 100.00 100.00 100.00 100.00 0 5 4 3 2 1 0 1 1 2 3 100.81 105.64 98.09 103.52 107.75 95.05 101.14 105.91 109.58 6-Year 10% Coupon Bond 0 6 5 4 3 2 1 0 Coupon Maturity 124.14 115.83 126.14 108.98 118.55 126.27 A Bond Forward 10.0% 4 Coupon Maturity 0 4 3 2 1 0 Bond Forward Price 6 110.00 110.00 110.00 110.00 110.00 110.00 110.00 79.83 1 79.99 89.24 103.38 2 3 81.53 90.45 97.67 85.08 93.27 99.85 104.99 4 91.66 98.44 103.83 108.00 111.16 A Bond Future 10.0% 4 4 3 2 1 0 Bond Futures Price 103.22 103.22 4 91.66 98.44 103.83 108.00 111.16 Term Structure Lattice r(0,0) 6.00% u 1.25 d 0.9 q 0.50 1-q 0.50 Short-Rate Lattice 0 5 4 3 2 1 0 6.00% Fixed Rate Caplet With Expiration t = 6 7.50% 5.40% 2 3 4 9.38% 6.75% 4.86% 11.72% 8.44% 6.08% 4.37% 14.65% 10.55% 7.59% 5.47% 3.94% 2 3 4 0.0637 0.0471 0.0323 0.0800 0.0592 0.0412 0.0264 0.1032 0.0756 0.0528 0.0346 0.0206 5 18.31% 13.18% 9.49% 6.83% 4.92% 3.54% 2.0% 0 5 4 3 2 1 0 1 0.0420 1 0.0515 0.0376 5 0.1379 0.0988 0.0684 0.0453 0.0278 0.0149 Term Structure Lattice r(0,0) 5.00% u 1.10 d 0.9 q 0.50 1-q 0.50 0 Short-Rate Lattice 0 5 4 3 2 1 0 5.00% Fixed Rate Swap With Expiration t = 6 1 5.50% 4.50% 0.0247 Swaption Strike Swaption: Expiration t = 3 4 6.66% 5.45% 4.46% 3.65% 7.32% 5.99% 4.90% 4.01% 3.28% 1 0.0423 -0.0003 2 0.0534 0.0158 -0.0165 3 4 0.0568 0.0254 -0.0013 -0.0239 0.0507 0.0273 0.0074 -0.0093 -0.0233 0% 0 3 2 1 0 6.05% 4.95% 4.05% 3 5 8.05% 6.59% 5.39% 4.41% 3.61% 2.95% 4.5% 0 5 4 3 2 1 0 2 0.0142 1 0.0241 0.0058 2 0.0388 0.0121 0.0000 3 0.0568 0.0254 0.0000 0.0000 5 0.0329 0.0196 0.0084 -0.0009 -0.0086 -0.0150 1000.00% 900.00% 800.00% 700.00% 600.00% 500.00% 400.00% 300.00% 200.00% 100.00% 0.00% 10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0 1 2 5 6 0.1025781026 0.0300897214 -0.0348309086 2 0.148555036 0.1282231199 0.081857701 0.0583450358 0.022286145 -0.0041867106 -0.0301326 -0.0592964102 -0.07568293 3 4 0.1626273078 0.0998227071 0.0439084288 -0.0051618738 -0.0477096636 -0.0842317465 5 0.1691907567 0.1112041494 0.059825803 0.0149138662 -0.0239019376 -0.0571316436 -0.0853546498 6 0.1868424185 0.1524387122 0.048140327 0.122273943 0.254226597 0.209286081 0 0 0.162627308 0.099822707 0.43908429 0 0 0 1 2 122270.274524916 0.0805255 0.0658845 0.0539055 0.0441045 0.0360855 0.0295245 0.08857805 0.07247295 0.05929605 0.04851495 0.03969405 0.03247695 0.02657205 4 0 0.1607967444 0.0959708321 6 3 0.055 0.045 0.1222702745 5 0.06655 0.05445 0.04455 0.03645 0.05 0.0723544189 -0.0022685106 1 4 0.073205 0.059895 0.049005 0.040095 0.032805 0.0605 0.0495 0.0405 0.0381361468 0 3 0.1765039332 0.2197888364 0.1001800204 0 7 8 9 0.097435855 0.079720245 0.065225655 0.053366445 0.043663455 0.035724645 0.029229255 0.023914845 0.1071794405 0.0876922695 0.0717482205 0.0587030895 0.0480298005 0.0392971095 0.0321521805 0.0263063295 0.0215233605 0.1178973846 0.0964614965 0.0789230426 0.0645733985 0.0528327806 0.0432268205 0.0353673986 0.0289369625 0.0236756966 0.0193710245 7 8 9 0.1665589806 0.1146396076 0.0689413767 0.0292129969 -0.0049680735 -0.0341214311 -0.0588066792 -0.0795828164 7 0.1524050088 0.1082988757 0.0698180446 0.0366068917 0.0082041998 -0.0159011033 -0.036229001 -0.0532806182 -0.067521126 8 0.1234485138 0.0896719899 0.0605351709 0.0356231183 0.0144823614 -0.0033454708 -0.0183006897 -0.030791356 -0.0411857745 -0.0498097616 9 10 0.129687123 0.1061076461 0.0868153468 0.0710307383 0.0581160586 0.0475495025 0.0389041384 0.0318306587 0.0260432662 0.0213081269 0.017433922 10 0.0749651132 0.0552456592 0.0384751162 0.0243043802 0.0123956711 0.0024337776 -0.0058675881 -0.0127630839 -0.0184755696 -0.0231975762 -0.0270937281 10 Term Structure Lattice r(0,0) 6.00% u 1.25 d 0.9 q 0.50 1-q 0.50 Short-Rate Lattice 0 2 3 4 6.00% 7.50% 5.40% 9.38% 6.75% 4.86% 11.72% 8.44% 6.08% 4.37% 14.65% 10.55% 7.59% 5.47% 3.94% 5 18.31% 13.18% 9.49% 6.83% 4.92% 3.54% 0 5 4 3 2 1 0 1 1 2 3 4 5 0.0449 0.1868 0.2901 0.1992 0.0511 0.0196 0.1041 0.2193 0.2293 0.1190 0.0246 6 0.0083 0.0543 0.1461 0.2075 0.1640 0.0686 0.0119 77.22 6.68% 71.59 6.91% 66.06 7.15% Elementary Prices 6 5 4 3 2 1 0 Zero Coupon Bond Prices Spot Rates 1.0000 0.4717 0.4717 0.2194 0.4432 0.2238 0.1003 0.3079 0.3143 0.1067 94.34 6.00% 88.63 6.22% 82.91 6.45%