This is #2 question in the attachment.

2. The data for this question is in the sheet called F inancialCalls:xls and it contains call prices for options on BAC, GS, JPM and MS on September10 2008 as calibrated using an arbitrage free model with spot prices set to 100; zero rates and dividend yields, with strikes ranging in two dollar intervals from 50 to 150 for the two maturities of one and three months respectively.(a) Use the call option prices at successive strikes to determine the price of a contract paying a dollar if the stock at maturity is above thelower of the two successive strikes.(b) Next determine the price of a contract paying a dollar if the stock at maturity is between two successive strikes.(c) Let si denote the mid points of your successive strikes and let I determined in part b, be the price that the stock at maturity is si:Graph the prices qi against the stock prices si for all four underlies and their two maturities.(d) For a continuously compounded interest rate of 3:5% per annum for the three month maturity, determine the spot price of the contract that pays in three months the dierence between the squared return at three months and the squared return at one month times a notional1of 10 million dollars on one of the four underliers. The returns aredened byRi =si100100and the contract pays10; 000; 000R2i(:25)R2i(:0833):

BMGT 444 Project 1 Fall 2016 Dilip B. Madan Robert H. Smith School of Business October 2, 2016 The ...rst project consists of the following 3 questions. The assignment is due by October 31, 2016. 1. Consider the contract paying 100 ln(S(t)) at time t = 0:25: Suppose the current price of the stock S(0) is 125: Observe that the tangent to the log contract constructed at the point S(0) lies above the log contract. Use this observation to develop an upper bound for the price of the contract paying 100 ln(S(t)): The continuously compounded interest rate for the three month maturity is currently 5:6%: Describe the arbitrage strategy to be conducted if the upper bound was violated. 2. The data for this question is in the sheet called F inancialCalls:xls and it contains call prices for options on BAC, GS, JPM and MS on September 10 2008 as calibrated using an arbitrage free model with spot prices set to 100; zero rates and dividend yields, with strikes ranging in two dollar intervals from 50 to 150 for the two maturities of one and three months respectively. (a) Use the call option prices at successive strikes to determine the price of a contract paying a dollar if the stock at maturity is above the lower of the two successive strikes. (b) Next determine the price of a contract paying a dollar if the stock at maturity is between two successive strikes. (c) Let si denote the mid points of your successive strikes and let qi determined in part b, be the price that the stock at maturity is si : Graph the prices qi against the stock prices si for all four underliers and their two maturities. (d) For a continuously compounded interest rate of 3:5% per annum for the three month maturity, determine the spot price of the contract that pays in three months the dierence between the squared return at three months and the squared return at one month times a notional 1 of 10 million dollars on one of the four underliers. The returns are de...ned by si 100 Ri = 100 and the contract pays Ri2 (:25) 10; 000; 000 Ri2 (:0833 ): 3. The data for this question is in the ...le SP XOptionData20160311.xls. For the four pairs of strikes of (1950; 2100); (1980; 2100); (1950; 2150) and (1980; 2150) consider a position of x options in the lower strike ...nanced by a position in the upper strike. If you sell the lower strike then you buy the upper strike and the other way around if you buy the lower strike. For a range of stock prices you are given their probabilities. You are also given the initial level of the index S: For an expected utility maximizing investor with exponential utility u(C) = 1 exp( 0:75 C); S determine graphically the expected utility maximizing levels for x in the four option trades. You may evaluate the expected utility for each of level of x from say 10 to 10 in steps of :05 to then ...nd the maximum value. 2 Call Prices for Spot at 100 and zero rates and divs Maturity one month Strike BAC GS 50 50.16928 50.00024 52 48.2138 48.00832 54 46.26519 46.0163 56 44.32499 44.02538 58 42.39475 42.03674 60 40.47602 40.05159 62 38.57033 38.07113 64 36.67927 36.09662 66 34.80453 34.12956 68 32.94794 32.17175 70 31.11146 30.2253 72 29.2972 28.29274 74 27.50739 26.37699 76 25.74446 24.48144 78 24.01095 22.61001 80 22.3096 20.76715 82 20.64328 18.95792 84 19.01507 17.18797 86 17.42818 15.46362 88 15.88604 13.79182 90 14.39222 12.18015 92 12.95049 10.63678 94 11.56482 9.170404 96 10.23932 7.790126 98 8.978317 6.50529 100 7.786294 5.325247 102 6.667907 4.25903 104 5.627973 3.314906 106 4.671448 2.499782 108 3.803407 1.818379 110 3.029003 1.272082 112 2.353403 0.85724 114 1.781678 0.562186 116 1.318558 0.36334 118 0.96671 0.23292 120 0.710511 0.148545 122 0.524077 0.094344 124 0.387987 0.059638 126 0.288272 0.037433 128 0.214915 0.023214 130 0.160727 0.014091 132 0.120533 0.008223 134 0.090597 0.004435 136 0.068211 0.001982 138 0.051402 0.000387 JPM 50.0233 48.03825 46.05486 44.07458 42.09884 40.12909 38.16678 36.21341 34.27067 32.34053 30.42519 28.52716 26.64926 24.79464 22.96681 21.16967 19.40749 17.68499 16.00727 14.37989 12.80882 11.30045 9.86154 8.499223 7.220898 6.034162 4.946667 3.965934 3.099075 2.352389 1.730721 1.236344 0.866273 0.603094 0.419168 0.29138 0.202746 0.14124 0.098484 0.068687 0.047857 0.033246 0.022959 0.015691 0.010534 MS 50.06562 48.09169 46.1218 44.15767 42.201 40.25351 38.31691 36.39296 34.48362 32.59101 30.71748 28.86561 27.03819 25.23825 23.46903 21.73405 20.03699 18.38182 16.77268 15.21391 13.71003 12.2657 10.88569 9.574826 8.337932 7.179769 6.104933 5.117751 4.222138 3.421424 2.718133 2.113678 1.607958 1.198736 0.880529 0.641924 0.466331 0.338249 0.245239 0.177828 0.128989 0.09358 0.067873 0.049173 0.03554 Maturity 3 months BAC 50.99682 49.15105 47.3215 45.5093 43.71556 41.94141 40.18797 38.45635 36.74771 35.0632 33.40401 31.77133 30.16638 28.59039 27.04461 25.5303 24.04875 22.60123 21.18907 19.81356 18.47605 17.17787 15.92035 14.70487 13.53276 12.4054 11.32416 10.29038 9.305449 8.370708 7.487508 6.657184 5.881049 5.160391 4.496464 3.890468 3.343536 2.856696 2.430806 2.066287 1.759186 1.500623 1.282604 1.098443 0.94258 140 142 144 146 148 150 0.038733 0.029147 0.021867 0.016317 0.012072 0.008813 0 0 0 0 0 0 0.006862 0.004237 0.002353 0.000995 1.36E-005 0 0.025575 0.018272 0.012904 0.008947 0.006022 0.003852 0.810405 0.698091 0.602462 0.520876 0.451133 0.391398 Maturity 3 months GS 50.22586 48.28435 46.35233 44.43163 42.52407 40.63147 38.75567 36.89852 35.06198 33.24809 31.45903 29.69704 27.96448 26.26379 24.59746 22.96808 21.37829 19.83076 18.3282 16.87332 15.46884 14.11745 12.82176 11.58435 10.40766 9.294002 8.245515 7.26412 6.351475 5.508925 4.737441 4.037553 3.409267 2.851976 2.364336 1.94412 1.58801 1.29125 1.047168 0.848003 0.686282 0.555363 0.449565 0.364141 0.295184 JPM 50.41317 48.49952 46.59811 44.71053 42.83838 40.98323 39.14669 37.33036 35.5359 33.76505 32.0196 30.30139 28.61234 26.95441 25.32959 23.73995 22.18756 20.67455 19.20306 17.77525 16.3933 15.05938 13.77565 12.54425 11.36728 10.24678 9.184746 8.183048 7.243458 6.367596 5.556898 4.812572 4.135541 3.526367 2.985146 2.511347 2.103534 1.758559 1.46977 1.228961 1.028497 0.861706 0.722906 0.607323 0.51098 MS 50.62667 48.74429 46.87723 45.02695 43.19492 41.38263 39.59154 37.82314 36.07893 34.36045 32.66925 31.00693 29.37506 27.77526 26.20914 24.67831 23.18437 21.72892 20.31353 18.93977 17.60916 16.32318 15.08327 13.89082 12.74715 11.6535 10.61103 9.620792 8.683744 7.800701 6.97234 6.199175 5.481536 4.819542 4.213075 3.661744 3.164836 2.721265 2.329491 1.987398 1.692059 1.439255 1.223883 1.040899 0.885676 0.239504 0.194517 0.158136 0.128682 0.104804 0.08542 0.430578 0.363387 0.307154 0.260017 0.220442 0.187159 0.754108 0.642616 0.548125 0.468008 0.400032 0.34231 Data for SPX 20160311 Level of Spot Probability 680 4.99750124937531E-006 681.32875 4.99750124937531E-006 682.6575 4.99750124937531E-006 683.98625 4.99750124937531E-006 685.315 4.99750124937531E-006 686.64375 4.99750124937531E-006 687.9725 4.99750124937531E-006 689.30125 4.99750124937531E-006 690.63 4.99750124937531E-006 691.95875 4.99750124937531E-006 693.2875 4.99750124937531E-006 694.61625 4.99750124937531E-006 695.945 4.99750124937531E-006 697.27375 4.99750124937531E-006 698.6025 4.99750124937531E-006 699.93125 4.99750124937531E-006 701.26 4.99750124937531E-006 702.58875 4.99750124937531E-006 703.9175 4.99750124937531E-006 705.24625 4.99750124937531E-006 706.575 4.99750124937531E-006 707.90375 4.99750124937531E-006 709.2325 4.99750124937531E-006 710.56125 4.99750124937531E-006 711.89 4.99750124937531E-006 713.21875 4.99750124937531E-006 714.5475 4.99750124937531E-006 715.87625 4.99750124937531E-006 717.205 4.99750124937531E-006 718.53375 4.99750124937531E-006 719.8625 4.99750124937531E-006 721.19125 4.99750124937531E-006 722.52 4.99750124937531E-006 723.84875 4.99750124937531E-006 725.1775 4.99750124937531E-006 726.50625 4.99750124937531E-006 727.835 4.99750124937531E-006 729.16375 4.99750124937531E-006 730.4925 4.99750124937531E-006 731.82125 4.99750124937531E-006 733.15 4.99750124937531E-006 734.47875 4.99750124937531E-006 735.8075 4.99750124937531E-006 737.13625 4.99750124937531E-006 738.465 4.99750124937531E-006 739.79375 741.1225 742.45125 743.78 745.10875 746.4375 747.76625 749.095 750.42375 751.7525 753.08125 754.41 755.73875 757.0675 758.39625 759.725 761.05375 762.3825 763.71125 765.04 766.36875 767.6975 769.02625 770.355 771.68375 773.0125 774.34125 775.67 776.99875 778.3275 779.65625 780.985 782.31375 783.6425 784.97125 786.3 787.62875 788.9575 790.28625 791.615 792.94375 794.2725 795.60125 796.93 798.25875 799.5875 800.91625 802.245 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 803.57375 804.9025 806.23125 807.56 808.88875 810.2175 811.54625 812.875 814.20375 815.5325 816.86125 818.19 819.51875 820.8475 822.17625 823.505 824.83375 826.1625 827.49125 828.82 830.14875 831.4775 832.80625 834.135 835.46375 836.7925 838.12125 839.45 840.77875 842.1075 843.43625 844.765 846.09375 847.4225 848.75125 850.08 851.40875 852.7375 854.06625 855.395 856.72375 858.0525 859.38125 860.71 862.03875 863.3675 864.69625 866.025 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 867.35375 868.6825 870.01125 871.34 872.66875 873.9975 875.32625 876.655 877.98375 879.3125 880.64125 881.97 883.29875 884.6275 885.95625 887.285 888.61375 889.9425 891.27125 892.6 893.92875 895.2575 896.58625 897.915 899.24375 900.5725 901.90125 903.23 904.55875 905.8875 907.21625 908.545 909.87375 911.2025 912.53125 913.86 915.18875 916.5175 917.84625 919.175 920.50375 921.8325 923.16125 924.49 925.81875 927.1475 928.47625 929.805 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937531E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937532E-006 4.99750124937533E-006 4.99750124937533E-006 4.99750124937533E-006 4.99750124937533E-006 4.99750124937534E-006 4.99750124937534E-006 4.99750124937535E-006 4.99750124937535E-006 4.99750124937536E-006 4.99750124937537E-006 4.99750124937538E-006 4.99750124937538E-006 4.9975012493754E-006 931.13375 932.4625 933.79125 935.12 936.44875 937.7775 939.10625 940.435 941.76375 943.0925 944.42125 945.75 947.07875 948.4075 949.73625 951.065 952.39375 953.7225 955.05125 956.38 957.70875 959.0375 960.36625 961.695 963.02375 964.3525 965.68125 967.01 968.33875 969.6675 970.99625 972.325 973.65375 974.9825 976.31125 977.64 978.96875 980.2975 981.62625 982.955 984.28375 985.6125 986.94125 988.27 989.59875 990.9275 992.25625 993.585 4.99750124937541E-006 4.99750124937542E-006 4.99750124937544E-006 4.99750124937546E-006 4.99750124937548E-006 4.99750124937551E-006 4.99750124937554E-006 4.99750124937557E-006 4.99750124937561E-006 4.99750124937566E-006 4.99750124937571E-006 4.99750124937577E-006 4.99750124937584E-006 4.99750124937592E-006 4.99750124937601E-006 4.99750124937611E-006 4.99750124937623E-006 4.99750124937636E-006 4.99750124937652E-006 4.9975012493767E-006 4.9975012493769E-006 4.99750124937713E-006 4.99750124937739E-006 4.99750124937769E-006 4.99750124937803E-006 4.99750124937841E-006 4.99750124937886E-006 4.99750124937936E-006 4.99750124937993E-006 4.99750124938058E-006 4.99750124938132E-006 4.99750124938216E-006 4.99750124938312E-006 4.9975012493842E-006 4.99750124938544E-006 4.99750124938683E-006 4.99750124938842E-006 4.99750124939022E-006 4.99750124939225E-006 4.99750124939456E-006 4.99750124939718E-006 4.99750124940013E-006 4.99750124940348E-006 4.99750124940726E-006 4.99750124941154E-006 4.99750124941638E-006 4.99750124942184E-006 4.99750124942801E-006 994.91375 996.2425 997.57125 998.9 1000.22875 1001.5575 1002.88625 1004.215 1005.54375 1006.8725 1008.20125 1009.53 1010.85875 1012.1875 1013.51625 1014.845 1016.17375 1017.5025 1018.83125 1020.16 1021.48875 1022.8175 1024.14625 1025.475 1026.80375 1028.1325 1029.46125 1030.79 1032.11875 1033.4475 1034.77625 1036.105 1037.43375 1038.7625 1040.09125 1041.42 1042.74875 1044.0775 1045.40625 1046.735 1048.06375 1049.3925 1050.72125 1052.05 1053.37875 1054.7075 1056.03625 1057.365 4.99750124943497E-006 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5.03851173227192E-006 5.03785078918449E-006 5.03720037269476E-006 5.03656031761479E-006 5.03593046130113E-006 5.0353106436164E-006 5.0347007068917E-006 5.03410049588939E-006 5.03350985776646E-006 5.03292864203843E-006 5.03235670054384E-006 5.03179388740911E-006 5.03124005901416E-006 5.03069507395821E-006 5.03015879302641E-006 5.0296310791567E-006 5.02911179740732E-006 5.02860081492472E-006 Current Spot Price option strike is call 2021.3 1360 2021.3 1370 2021.3 1375 2021.3 1380 2021.3 1390 2021.3 1400 2021.3 1410 2021.3 1420 2021.3 1425 2021.3 1430 2021.3 1440 2021.3 1450 2021.3 1460 2021.3 1470 2021.3 1475 2021.3 1480 2021.3 1490 2021.3 1500 2021.3 1510 2021.3 1520 2021.3 1525 2021.3 1530 2021.3 1540 2021.3 1550 2021.3 1560 2021.3 1570 2021.3 1575 2021.3 1580 2021.3 1590 2021.3 1600 2021.3 1610 2021.3 1620 2021.3 1625 2021.3 1630 2021.3 1640 2021.3 1650 2021.3 1660 2021.3 1670 2021.3 1675 2021.3 1680 2021.3 1690 2021.3 1700 2021.3 1705 2021.3 1710 2021.3 1715 option price 0 1.95 0 2.075 0 2.275 0 2.2 0 2.3 0 2.375 0 2.55 0 2.7 0 2.675 0 2.875 0 3 0 3.075 0 3.3 0 3.5 0 3.6 0 3.7 0 3.9 0 4.25 0 4.35 0 4.65 0 4.75 0 4.85 0 5.15 0 5.45 0 5.75 0 6.1 0 6.3 0 6.45 0 6.85 0 7.25 0 7.7 0 8.15 0 8.35 0 8.55 0 9.05 0 9.55 0 10.1 0 10.65 0 10.95 0 11.25 0 11.95 0 12.55 0 12.95 0 13.25 0 13.65 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 2021.3 1720 1725 1730 1735 1740 1