solutions to derivative markets 3ed by mcdonald

Solution to Derivatives Markets: 3rd editionSOA Exam MFE and CAS Exam 3 FE

Yufeng Guo

May 31, 2015

Contents

9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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10.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6510.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6910.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7110.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7710.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8210.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8811.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8911.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9111.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9311.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9411.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9611.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9611.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9711.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9711.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9711.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9711.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9711.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9711.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9811.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9811.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10011.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10011.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10011.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10012.1 skip this spreadsheet problem . . . . . . . . . . . . . . . . . . . . 10412.2 skip this spreadsheet problem . . . . . . . . . . . . . . . . . . . . 10412.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10412.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10912.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11012.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11012.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11212.12 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11212.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11212.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11412.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11512.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11512.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11512.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11512.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11712.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11912.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11913.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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13.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12213.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12713.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12913.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13013.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13213.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13213.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13413.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13513.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13513.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13613.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13713.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13713.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13813.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13813.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14013.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14113.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14213.19-13.20 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14314.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14314.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14414.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14414.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14614.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14714.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14814.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14914.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14914.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15014.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15014.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15014.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15214.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15214.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15314.15 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15314.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15314.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15314.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15414.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15414.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15414.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15514.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15718.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15818.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15818.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15818.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15818.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15918.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

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18.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16018.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16118.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16318.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16418.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16518.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16918.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17118.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17218.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17219.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17319.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17419.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17519.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17619.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17819.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17919.7-19.17 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18120.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18120.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18320.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18320.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18420.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18420.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18520.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18520.8-20.14 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18621.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18621.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18621.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18721.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18721.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18821.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19021.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19221.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19221.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19321.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19321.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19421.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19521.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19621.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19623.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19723.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19823.3-23.17 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19824.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19824.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20024.3-24.5 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20024.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20024.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

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24.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20124.9-24.20 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20125.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20125.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20325.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20525.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20625.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20825.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21025.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21725.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21925.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21925.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21925.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22125.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22225.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22425.14 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22525.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22525.19-25.20 Skip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227Recommendations on using this solution manual:

1. Obviously, you�ll need to buy Derivatives Markets (3rd edition) to see theproblems.

2. Make sure you download the textbook errata from http://derivatives.kellogg.northwestern.edu/typos3e.html

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http://derivatives.kellogg.northwestern.edu/typos3e.html

http://derivatives.kellogg.northwestern.edu/typos3e.html

9.1 CONTENTS

9.1

S0 = 32 T = 6=12 = 0:5 K = 35C = 2:27 r = 0:04 � = 0:06

C + PV (K) = P + S0e��T

2:27 + 35e�0:04(0:5) = P + 32e�0:06(0:5) P = 5: 522 7

9.2

S0 = 32 T = 6=12 = 0:5 K = 30C = 4:29 P = 2:64 r = 0:04C + PV (K) = P + S0 � PV (Div)4:29 + 30e�0:04(0:5) = 2:64 + 32� PV (Div)PV (Div) = 0:944

9.3

S0 = 800 r = 0:05 � = 0T = 1 K = 815 C = 75 P = 45a. Buy stock+ sell call+buy put=buy PV (K)C + PV (K) = P + S0! PV (K = 815) = S0|{z}

buy stock

+ �C|{z}sell call

+ P|{z}buy put

= 800 + (�75) + 45 = 770

So the position is equivalent to depositing 770 in a savings account (or buy-ing a bond with present value equal to 770) and receiving 815 one year later.770eR = 815 R = 0:056 8So we earn 5:68%.

b. Buying a stock, selling a call, and buying a put is the same as depositingPV (K) in the savings account. As a result, we should just earn the risk freeinterest rate r = 0:05. However, we actually earn R = 0:056 8 > r. To arbitrage,we "borrow low and earn high." We borrow 770 from a bank at 0.05%. We usethe borrowed 770 to �nance buying a stock, selling a call, and buying a put.Notice that the net cost of buying a stock, selling a call, and buying a put is770.One year later, we receive 770eR = 815. We pay the bank 770e0:05 = 809:

48. Our pro�t is 815� 809: 48 = 5: 52 per transaction.If we do n such transactions, we�ll earn 5: 52n pro�t.

Alternative answer: we can burrow at 5% (continuously compounding) andlend at 5:6 8% (continuously compounding), earning a risk free 0.68%. So ifwe borrow $1 at time zero, our risk free pro�t at time one is e0:0568 � e0:05 =0:00717 3; if we borrow $770 at time zero, our risk free pro�t at time one is

deeperunderstandingfastercalc.com 1

9.4 CONTENTS

0:00717 3�770 = 5: 52. If we borrow n dollars at time zero, we�ll earn 0:00717 3ndollars at time one.c. To avoid arbitrage, we need to have:PV (K = 815) = S0|{z}

buy stock

+ �C|{z}sell call

+ P|{z}buy put

= 815e�0:05 = 775: 25

! C � P = S0 � PV (K) = 800� 775: 25 = 24: 75

d. C � P = S0 � PV (K) = 800�Ke�rT = 800�Ke�0:05If K = 780 C � P = 800� 780e�0:05 = 58: 041If K = 800 C � P = 800� 800e�0:05 = 39: 016If K = 820 C � P = 800� 820e�0:05 = 19: 992If K = 840 C � P = 800� 840e�0:05 = 0:967

9.4

To solve this type of problems, just use the standard put-call parity.To avoid calculation errors, clearly identify the underlying asset.The underlying asset is e1. We want to �nd the dollar cost of a put option

on this underlying.The typical put-call parity:C + PV (K) = P + S0e

��T

C, K, P , and S0 should all be expressed in dollars. S0 is the current (dollarprice) of the underlying. So S0 = $0:95.

C = $0:0571 K = $0:93� is the internal growth rate of the underlying asset (i.e. e1). Hence � = 0:04Since K is expressed in dollars, PV (K) needs to be calculated using the

dollar risk free interest r = 0:06.0:0571 + 0:93e�0:06(1) = P + 0:95e�0:04(1) P = $0:02 02

9.5

As I explained in my study guide, don�t bother memorizing the following com-plex formula:

C$ (x0;K; T ) = x0KPf

�1

x0;1

K;T

�Just use my approach to solve this type of problems.Convert information to symbols:


9.5 CONTENTS

The exchange rate is 95 yen per euro. Y 95 =e1 or Y 1 =e1

95

Yen-denominated put on 1 euro with strike price Y100 has a premium Y8:763! (e1! Y 100)0 =Y8:763

What�s the strike price of a euro-denominated call on 1 yen? eK ! 1Y

Calculate the price of a euro-denominated call on 1 yen with strike price eK(eK ! 1Y )0 = e?

e1! Y 100 ! e1

100! Y 1

The strike price of the corresponding euro-denominated yen call isK =e1

100=e0:01�

e1

100! Y 1

�0

=1

100� (e1! Y 100)0 =

1

100(Y 8:763)

Since Y 1 =e1

95, we have:

1

100(Y 8:763) =

1

100(8:763)

�e1

95

�=e9: 224 2� 10�4

!�e1

100! Y 1

�0

=e9: 224 2� 10�4

So the price of a euro-denominated call on 1 yen with strike price K =e1

100is e9: 224 2� 10�4


9.6 CONTENTS

9.6

The underlying asset is e1. The standard put-call parity is:C + PV (K) = P + S0e

��T

C, K, P , and S0 should all be expressed in dollars. S0 is the current (dollarprice) of the underlying.� is the internal growth rate of the underlying asset (i.e. e1).We�ll solve Part b �rst.

b. 0:0404 + 0:9e�0:05(0:5) = 0:0141 + S0e�0:035(0:5) S0 = $0:920 04So the current price of the underlying (i.e. e1) is S0 = $0:920 04. In other

words, the currency exchange rate is $0:920 04 =e1

a. According to the textbook Equation 5.7, the forward price is:F0;T = S0e

��T erT = 0:920 04e�0:035(0:5)e0:05(0:5) = $0:926 97

9.7

The underlying asset is one yen.a. C +Ke�rT = P + S0e��T

0:0006 + 0:009e�0:05(1) = P + 0:009e�0:01(1)

0:0006 + 0:008561 = P + 0:008 91 P = $0:00025

b. There are two puts out there. One is the synthetically created put usingthe formula:

P = C +Ke�rT � S0e��TThe other is the put in the market selling for the price for $0:0004.

To arbitrage, build a put a low cost and sell it at a high price. At t = 0, we:

� Sell the expensive put for $0:0004

� Build a cheap put for $0:00025. To build a put, we buy a call, depositKe�rT in a savings account, and sell e��T unit of Yen.

T = 1 T = 1t = 0 ST < 0:009 ST � 0:009

Sell expensive put 0:0004 ST � 0:009 0Buy call �0:0006 0 ST � 0:009Deposit Ke�rT in savings �0:009e�0:05(1) 0:009 0:009

Short sell e��T unit of Yen 0:009e�0:01(1) ST STTotal $0:00015 0 0

0:0004� 0:0006� 0:009e�0:05(1) + 0:009e�0:01(1) = $0:00015


9.7 CONTENTS

At t = 0, we receive $0:00015 yet we don�t incur any liabilities at T = 1 (sowe receive $0:00015 free money at t = 0).

c. At-the-money means K = S0 (i.e. the strike price is equal to the currentexchange rate).Dollar-denominated at-the-money yen call sells for $0.0006. To translate this

into symbols, notice that under the call option, the call holder can give $0:009and get Y 1.

"Give $0:009 and get Y 1" is represented by ($0:009! Y 1). This option�spremium at time zero is $0.0006. Hence we have:($0:009! Y 1)0 = $0:0006We are asked to �nd the yen denominated at the money call for $1. Here

the call holder can give c yen and get $1. "Give c yen and get $1" is representedby (Y c! $1). This option�s premium at time zero is (Y c! $1)0.First, we need to calculate c, the strike price of the yen denominated dollar

call. Since at time zero $0:009 = Y 1, we have $1 = Y1

0:009. So the at-the-

money yen denominated call on $1 is c =1

0:009. Our task is to �nd this option�s

premium:�Y

1

0:009! $1

�0

=?

We�ll �nd the premium for Y 1 !$0:009, the option of "give 1 yen and get$0:009." Once we �nd this premium, we�ll scale it and �nd the premium of "give1

0:009yen and get $1."

We�ll use the general put-call parity:(AT ! BT )0 + PV (AT ) = (BT ! AT )0 + PV (BT )

($0:009! Y 1)0 + PV ($0:009) = (Y 1! $0:009)0 + PV (Y 1)

PV ($0:009) = $0:009e�0:05(1)

Since we are discounting $0.009 at T = 1 to time zero, we use the dollarinterest rate 5%.

PV (Y 1) = $0:009e�0:01(1)

If we discount Y1 from T = 1 to time zero, we get e�0:01(1) yen, which isequal to $0:009e�0:01(1).

So we have:$0:0006+$0:009e�0:05 = (Y 1! $0:009)0 + $0:009e

�0:01(1)

(Y 1! $0:009)0 = $2: 506 16� 10�4�1

0:009Y 1! $1

�0

=1

0:009(Y 1! $0:009)0 = $

2: 506 16� 10�40:009

= $2:

784 62� 10�2 = Y 2: 784 62� 10�2

0:009= Y 3: 094


9.8 CONTENTS

So the yen denominated at the money call for $1 is worth $2: 784 62� 10�2or Y 3: 094.

We are also asked to identify the relationship between the yen denominatedat the money call for $1 and the dollar-denominated yen put. The relationshipis that we use the premium of the latter option to calculate the premium of theformer option.Next, we calculate the premium for the yen denominated at-the-money put

for $1:�$! Y

1

0:009

�0

=1

0:009($0:009! Y 1)0

=1

0:009� $0:0006 = $ 0:0 666 7

= Y 0:0 666 7� 1

0:009= Y 7: 407 8

So the yen denominated at-the-money put for $1 is worth $ 0:0 666 7 or Y7: 407 8.I recommend that you use my solution approach, which is less prone to errors

than using complex notations and formulas in the textbook.

9.8

The textbook Equations 9.13 and 9.14 are violated.This is how to arbitrage on the calls. We have two otherwise identical

calls, one with $50 strike price and the other $55. The $50 strike call is morevaluable than the $55 strike call, but the former is selling less than the latter.To arbitrage, buy low and sell high.We use T to represent the common exercise date. This de�nition works

whether the two options are American or European. If the two options areAmerican, we�ll �nd arbitrage opportunities if two American options are ex-ercised simultaneously. If the two options are European, T is the commonexpiration date.

The payo¤ is:T T T

Transaction t = 0 ST < 50 50 � ST < 55 ST � 55Buy 50 strike call �9 0 ST � 50 ST � 50Sell 55 strike call 10 0 0 � (ST � 55)Total 1 0 ST � 50 � 0 5

At t = 0, we receive $1 free money.At T , we get non negative cash �ows (so we may get some free money, but

we certainly don�t owe anybody anything at T ). This is clearly an arbitrage.


9.9 CONTENTS

This is how to arbitrage on the two puts. We have two otherwise identicalputs, one with $50 strike price and the other $55. The $55 strike put is morevaluable than the $50 strike put, but the former is selling less than the latter.To arbitrage, buy low and sell high.The payo¤ is:

T T TTransaction t = 0 ST < 50 50 � ST < 55 ST � 55Buy 55 strike put �6 55� ST 55� ST 0Sell 50 strike put 7 � (50� ST ) 0 0Total 1 5 55� ST > 0 0At t = 0, we receive $1 free money.At T , we get non negative cash �ows (so we may get some free money, but

we certainly don�t owe anybody anything at T ). This is clearly an arbitrage.

9.9

The textbook Equation 9.15 and 9.16 are violated.We use T to represent the common exercise date. This de�nition works

whether the two options are American or European. If the two options areAmerican, we�ll �nd arbitrage opportunities if two American options are exer-cised simultaneously at T . If the two options are European, T is the commonexpiration date.This is how to arbitrage on the calls. We have two otherwise identical calls,

one with $50 strike price and the other $55. The premium di¤erence betweenthese two options should not exceed the strike di¤erence 15� 10 = 5. In otherwords, the 50-strike call should sell no more than 10+5. However, the 50-strikecall is currently selling for 16 in the market. To arbitrage, buy low (the 55-strikecall) and sell high (the 50-strike call).

The $50 strike call is more valuable than the $55 strike call, but the formeris selling less than the latter.The payo¤ is:

T T TTransaction t = 0 ST < 50 50 � ST < 55 ST � 55Buy 55 strike call �10 0 0 ST � 55Sell 50 strike call 16 0 � (ST � 50) � (ST � 50)Total 6 0 � (ST � 50) � �5 �5

So we receive $6 at t = 0. Then at T , our maximum liability is $5. So makeat least $1 free money.

This is how to arbitrage on the puts. We have two otherwise identical puts,one with $50 strike price and the other $55. The premium di¤erence betweenthese two options should not exceed the strike di¤erence 15� 10 = 5. In other


9.10 CONTENTS

words, the 55-strike put should sell no more than 7 + 5 = 12. However, the55-strike put is currently selling for 14 in the market. To arbitrage, buy low(the 50-strike put) and sell high (the 55-strike put).The payo¤ is:

T T TTransaction t = 0 ST < 50 50 � ST < 55 ST � 55Buy 50 strike put �7 50� ST 0 0Sell 55 strike put 14 � (55� ST ) � (55� ST ) 0Total 7 �5 � (55� ST ) < �5 0

So we receive $7 at t = 0. Then at T , our maximum liability is $5. So makeat least $2 free money.

9.10

Suppose there are 3 options otherwise identical but with di¤erent strike priceK1 < K2 < K3 where K2 = �K1 + (1� �)K3 and 0 < � < 1.Then the price of the middle strike price K2 must not exceed the price of a

diversi�ed portfolio consisting of � units of K1-strike option and (1� �) unitsof K3-strike option:

C [�K1 + (1� �)K3] � �C (K1) + (1� �)C (K3)P [�K1 + (1� �)K3] � �P (K1) + (1� �)P (K3)

The above conditions are called the convexity of the option price with respectto the strike price. They are equivalent to the textbook Equation 9.17 and 9.18.If the above conditions are violated, arbitrage opportunities exist.

We are given the following 3 calls:Strike K1 = 50 K2 = 55 K3 = 60Call premium 18 14 9:50

�50 + (1� �) 60 = 55! � = 0:5 0:5 (50) + 0:5 (60) = 55

Let�s check:C [0:5 (50) + 0:5 (60)] = C (55) = 14

0:5C (50) + 0:5C (60) = 0:5 (18) + 0:5 (9:50) = 13: 75C [0:5 (50) + 0:5 (60)] > 0:5C (50) + 0:5C (60)So arbitrage opportunities exist. To arbitrage, we buy low and sell high.The cheap asset is the diversi�ed portfolio consisting of � units of K1-strike

option and (1� �) units of K3-strike option. In this problem, the diversi�edportfolio consists of half a 50-strike call and half a 60-strike call.The expensive asset is the 55-strike call.


9.10 CONTENTS

Since we can�t buy half a call option, we�ll buy 2 units of the portfolio (i.e.buy one 50-strike call and one 60-strike call). Simultaneously,we sell two 55-strike call options.We use T to represent the common exercise date. This de�nition works

whether the options are American or European. If the options are American,we�ll �nd arbitrage opportunities if the American options are exercised simulta-neously. If the options are European, T is the common expiration date.The payo¤ is:

T T T TTransaction t = 0 ST < 50 50 � ST < 55 55 � ST < 60 ST � 60buy two portfoliosbuy a 50-strike call �18 0 ST � 50 ST � 50 ST � 50buy a 60-strike call �9:5 0 0 0 ST � 60Portfolio total �27: 5 0 ST � 50 ST � 50 2ST � 110

Sell two 55-strike calls 2 (14) = 28 0 0 �2 (ST � 55) �2 (ST � 55)Total 0:5 0 ST � 50 � 0 60� ST > 0 0

�27: 5 + 28 = 0:5ST � 50� 2 (ST � 55) = 60� ST2ST � 110� 2 (ST � 55) = 0So we get $0:5 at t = 0, yet we have non negative cash �ows at the expiration

date T . This is arbitrage.

The above strategy of buying � units of K1-strike call, buying (1� �) unitsof K3-strike call, and selling one unit of K2-strike call is called the butter�yspread.

We are given the following 3 puts:Strike K1 = 50 K2 = 55 K3 = 60Put premium 7 10:75 14:45

�50 + (1� �) 60 = 55! � = 0:5 0:5 (50) + 0:5 (60) = 55

Let�s check:P [0:5 (50) + 0:5 (60)] = P (55) = 10:75

0:5P (50) + 0:5P (60) = 0:5 (7) + 0:5 (14:45) = 10: 725P [0:5 (50) + 0:5 (60)] > :5P (50) + 0:5P (60)So arbitrage opportunities exist. To arbitrage, we buy low and sell high.The cheap asset is the diversi�ed portfolio consisting of � units of K1-strike

put and (1� �) units of K3-strike put. In this problem, the diversi�ed portfolioconsists of half a 50-strike put and half a 60-strike put.The expensive asset is the 55-strike put.


9.11 CONTENTS

Since we can�t buy half a option, we�ll buy 2 units of the portfolio (i.e. buyone 50-strike put and one 60-strike put). Simultaneously,we sell two 55-strikeput options.The payo¤ is:

T T T TTransaction t = 0 ST < 50 50 � ST < 55 55 � ST < 60 ST � 60buy two portfoliosbuy a 50-strike put �7 50� ST 0 0 0buy a 60-strike put �14:45 60� ST 60� ST 60� ST 0Portfolio total �21: 45 110� 2ST 60� ST 60� ST 0

Sell two 55-strike puts 2 (10:75) �2 (55� ST ) �2 (55� ST ) 0 0Total 0:05 0 ST � 50 � 0 60� ST > 0 0�21: 45 + 2 (10:75) = 0:0550� ST + 60� ST = 110� 2ST

�21: 45 + 2 (10:75) = 0:05110� 2ST � 2 (55� ST ) = 060� ST � 2 (55� ST ) = ST � 50So we get $0:05 at t = 0, yet we have non negative cash �ows at the expiration

date T . This is arbitrage.The above strategy of buying � units of K1-strike put, buying (1� �) units

of K3-strike put, and selling one unit of K2-strike put is also called the butter�yspread.

9.11

This is similar to Problem 9.10.We are given the following 3 calls:Strike K1 = 80 K2 = 100 K3 = 105Call premium 22 9 5

80�+ 105 (1� �) = 100! � = 0:2 0:2 (80) + 0:8 (105) = 100C [0:2 (80) + 0:8 (105)] = C (100) = 9

0:2C (80) + 0:8C (105) = 0:2 (22) + 0:8 (5) = 8: 4C [0:2 (80) + 0:8 (105)] > 0:2C (80) + 0:8C (105)So arbitrage opportunities exist. To arbitrage, we buy low and sell high.The cheap asset is the diversi�ed portfolio consisting of � units of K1-strike

option and (1� �) units of K3-strike option. In this problem, the diversi�edportfolio consists of 0.2 unit of 80-strike call and 0.8 unit of 105-strike call.The expensive asset is the 100-strike call.


9.11 CONTENTS

Since we can�t buy a fraction of a call option, we�ll buy 10 units of the port-folio (i.e. buy two 80-strike calls and eight 105-strike calls). Simultaneously,wesell ten 100-strike call options.We use T to represent the common exercise date. This de�nition works

whether the options are American or European. If the options are American,we�ll �nd arbitrage opportunities if the American options are exercised simulta-neously. If the options are European, T is the common expiration date.The payo¤ is:

T TTransaction t = 0 ST < 80 80 � ST < 100buy ten portfoliosbuy two 80-strike calls �2 (22) 0 2 (ST � 80)buy eight 105-strike calls �8 (5) 0 0Portfolio total �84 0 2 (ST � 80)

Sell ten 100-strike calls 10 (9) 0 0Total 6 0 2 (ST � 80) � 0

T TTransaction t = 0 100 � ST < 105 ST � 105buy ten portfoliosbuy two 80-strike calls �2 (22) 2 (ST � 80) 2 (ST � 80)buy eight 105-strike calls �8 (5) 0 8 (ST � 105)Portfolio total �84 2 (ST � 80) 10ST � 1000

Sell ten 100-strike calls 10 (9) �10 (ST � 100) �10 (ST � 100)Total 6 8 (105� ST ) > 0 0

�2 (22)� 8 (5) = �44� 40 = �84

�84 + 10 (9) = �84 + 90 = 62 (ST � 80) + 8 (ST � 105) = 10ST � 10002 (ST � 80)� 10 (ST � 100) = 840� 8ST = 8 (105� ST )10ST � 1000� 10 (ST � 100) = 0So we receive $6 at t = 0, yet we don�t incur any negative cash �ows at

expiration T . So we make at least $6 free money.

We are given the following 3 put:Strike K1 = 80 K2 = 100 K3 = 105Put premium 4 21 24:8

80�+ 105 (1� �) = 100! � = 0:2 0:2 (80) + 0:8 (105) = 100P [0:2 (80) + 0:8 (105)] = P (100) = 21

0:2P (80) + 0:8P (105) = 0:2 (4) + 0:8 (24:8) = 20: 64


9.12 CONTENTS

P [0:2 (80) + 0:8 (105)] > 0:2P (80) + 0:8P (105)So arbitrage opportunities exist. To arbitrage, we buy low and sell high.The cheap asset is the diversi�ed portfolio consisting of � units of K1-strike

option and (1� �) units of K3-strike option. In this problem, the diversi�edportfolio consists of 0.2 unit of 80-strike put and 0.8 unit of 105-strike put.The expensive asset is the 100-strike put.Since we can�t buy half a fraction of an option, we�ll buy 10 units of the port-

folio (i.e. buy two 80-strike puts and eight 105-strike puts). Simultaneously,wesell ten 100-strike put options.The payo¤ is:

T TTransaction t = 0 ST < 80 80 � ST < 100buy ten portfoliosbuy two 80-strike puts �2 (4) 2 (80� ST ) 0buy eight 105-strike puts �8 (24:8) 8 (105� ST ) 8 (105� ST )Portfolio total �84 1000� 10ST 8 (105� ST )

Sell ten 100-strike puts 10 (21) �10 (100� ST ) �10 (100� ST )Total 3: 6 0 2 (ST � 80) � 0

T TTransaction t = 0 100 � ST < 105 ST � 105buy ten portfoliosbuy two 80-strike puts �2 (4) 0 0buy eight 105-strike puts �8 (24:8) 8 (105� ST ) 0Portfolio total �84 8 (105� ST ) 0

Sell ten 100-strike puts 10 (21) 0 0Total 3: 6 8 (105� ST ) > 0 0

�2 (4)� 8 (24:8) = �206: 42 (80� ST ) + 8 (105� ST ) = 1000� 10ST�206: 4 + 10 (21) = 3: 61000� 10ST � 10 (100� ST ) = 08 (105� ST )� 10 (100� ST ) = 2 (ST � 80)

We receive $3: 6 at t = 0, but we don�t incur any negative cash �ows at T .So we make at least $3: 6 free money.

9.12

For two European options di¤ering only in strike price, the following conditionsmust be met to avoid arbitrage (see my study guide for explanation):0 � CEur (K1; T )� CEur (K2; T ) � PV (K2 �K1) if K1 < K2


9.12 CONTENTS

0 � PEur (K2; T )� PEur (K1; T ) � PV (K2 �K1) if K1 < K2

a.Strike K1 = 90 K2 = 95Call premium 10 4

C (K1)� C (K2) = 10� 4 = 6K2 �K1 = 95� 90 = 5C (K1)� C (K2) > K2 �K1 � PV (K2 �K1)Arbitrage opportunities exist.To arbitrage, we buy low and sell high. The cheap call is the 95-strike call;

the expensive call is the 90-strike call.We use T to represent the common exercise date. This de�nition works

whether the two options are American or European. If the two options areAmerican, we�ll �nd arbitrage opportunities if two American options are ex-ercised simultaneously. If the two options are European, T is the commonexpiration date.The payo¤ is:

T T TTransaction t = 0 ST < 90 90 � ST < 95 ST � 95Buy 95 strike call �4 0 0 ST � 95Sell 90 strike call 10 0 � (ST � 90) � (ST � 90)Total 6 0 � (ST � 90) � �5 �5

We receive $6 at t = 0, yet our max liability at T is �5. So we�ll make atleast $1 free money.

b.T = 2 r = 0:1Strike K1 = 90 K2 = 95Call premium 10 5:25

C (K1)� C (K2) = 10� 5:25 = 4: 75K2 �K1 = 95� 90 = 5PV (K2 �K1) = 5e

�0:1(2) = 4: 094C (K1)� C (K2) > PV (K2 �K1)Arbitrage opportunities exist.

Once again, we buy low and sell high. The cheap call is the 95-strike call;the expensive call is the 90-strike call.The payo¤ is:

T T TTransaction t = 0 ST < 90 90 � ST < 95 ST � 95Buy 95 strike call �5:25 0 0 ST � 95Sell 90 strike call 10 0 � (ST � 90) � (ST � 90)Deposit 4: 75 in savings �4: 75 4: 75e0:1(2) 4: 75e0:1(2) 4: 75e0:1(2)

Total 0 5: 80 95: 80� ST > 0 0:80


9.12 CONTENTS

4: 75e0:1(2) = 5: 80� (ST � 90) + 4: 75e0:1(2) = 95: 80� STST � 95� (ST � 90) + 4: 75e0:1(2) = 0:80Our initial cost is zero. However, our payo¤ is always non-negative. So we

never lose money. This is clearly an arbitrage.It�s important that the two calls are European options. If they are American,

they can be exercised at di¤erent dates. Hence the following non-arbitrageconditions work only for European options:0 � CEur (K1; T )� CEur (K2; T ) � PV (K2 �K1) if K1 < K2

0 � PEur (K2; T )� PEur (K1; T ) � PV (K2 �K1) if K1 < K2

c.We are given the following 3 calls:Strike K1 = 90 K2 = 100 K3 = 105Call premium 15 10 6

�90 + (1� �) 105 = 100 � =1

3

! 1

3(90) +

2

3(105) = 100

C

�1

3(90) +

2

3(105)

�= C (100) = 10

1

3C (90) +

2

3C (105) =

1

3(15) +

2

3(6) = 9

C

�1

3(90) +

2

3(105)

�>1

3C (90) +

2

3C (105)

Hence arbitrage opportunities exist. To arbitrage, we buy low and sell high.

The cheap asset is the diversi�ed portfolio consisting of1

3unit of 90-strike

call and2

3unit of 105-strike call.

The expensive asset is the 100-strike call.Since we can�t buy a partial option, we�ll buy 3 units of the portfolio (i.e.

buy one 90-strike call and two 105-strike calls). Simultaneously,we sell three100-strike calls.

The payo¤ at expiration T :

T T T TTransaction t = 0 ST < 90 90 � ST < 100 100 � ST < 105 ST � 105buy 3 portfoliosbuy one 90-strike call �15 0 ST � 90 ST � 90 ST � 90buy two 105-strike calls 2 (�6) 0 0 0 2 (ST � 105)Portfolios total �27 0 ST � 90 ST � 90 3ST � 300

Sell three 100-strike calls 3 (10) 0 0 �3 (ST � 100) �3 (ST � 100)Total 3 0 ST � 90 � 0 2 (105� ST ) > 0 0


9.13 CONTENTS

�15 + 2 (�6) = �27ST � 90 + 2 (ST � 105) = 3ST � 300�27 + 3 (10) = 3ST � 90� 3 (ST � 100) = 210� 2ST = 2 (105� ST )3ST � 300� 3 (ST � 100) = 0

So we receive $3 at t = 0, but we incur no negative payo¤ at T . So we�llmake at least $3 free money.

9.13

a. If the stock pays dividend, then early exercise of an American call optionmay be optimal.Suppose the stock pays dividend at tD.Time 0 ... ... tD ... ... T

Pro and con for exercising the call early at tD.

� +. If you exercise the call immediately before tD, you�ll receive dividendand earn interest during [tD; T ]

� �. You�ll pay the strike price K at tD, losing interest you could haveearned during [tD; T ]. If the interest rate, however, is zero, you won�t loseany interest.

� �. You throw away the remaining call option during [tD; T ]. Had youwaited, you would have the call option during [tD; T ]

If the accumulated value of the dividend exceeds the value of the remainingcall option, then it�s optimal to exercise the stock at tD:As explained in my study guide, it�s never optimal to exercise an American

put early if the interest rate is zero.

9.14

a. The only reason that early exercise might be optimal is that the underlyingasset pays a dividend. If the underlying asset doesn�t pay dividend, then it�snever optimal to exercise an American call early. Since Apple doesn�t paydividend, it�s never optimal to exercise early.

b. The only reason to exercise an American put early is to earn interest onthe strike price. The strike price in this example is one share of AOL stock.Since AOL stocks won�t pay any dividends, there�s no bene�t for owning anAOL stock early. Thus it�s never optimal to exercise the put.If the Apple stock price goes to zero and will always stay zero, then there�s

no bene�t for delaying exercising the put; there�s no bene�t for exercising the


9.15 CONTENTS

put early either (since AOL stocks won�t pay dividend). Exercising the putearly and exercising the put at maturity have the same value.If, however, the Apple stock price goes to zero now but may go up in the

future, then it�s never optimal to exercise the put early. If you don�t exerciseearly, you leave the door open that in the future the Apple stock price mayexceed the AOL stock price, in which case you just let your put expire worthless.If the Apple stock price won�t exceed the AOL stock price, you can alwaysexercise the put and exchange one Apple stock for one AOL stock. There�s nohurry to exercise the put early.

c. If Apple is expected to pay dividend, then it might be optimal to exercisethe American call early and exchange one AOL stock for one Apple stock.However, as long as the AOL stock won�t pay any dividend, it�s never optimal

to exercise the American put early to exchange one Apple stock for one AOLstock.

9.15

This is an example where the strike price grows over time.

If the strike price grows over time, the longer-lived option is at least asvaluable as the shorter lived option. Refer to Derivatives Markets Page 298.We have two European calls:Call #1 K1 = 100e

0:05(1:5) = 107: 788 T1 = 1:5 C1 = 11:50Call #2 K2 = 100e

0:05 = 105: 127 T2 = 1 C2 = 11:924

The longer-lived call is cheaper than the shorter-lived call, leading to arbi-trage opportunities. To arbitrage, we buy low (Call #1) and sell high (Call#2).The payo¤ at expiration T1 = 1:5 if ST2 < 100e

0:05 = 105: 127

T1 T1Transaction t = 0 T2 ST1 < 100e

0:05(1:5) ST1 � 100e0:05(1:5)Sell Call #2 11:924 0 0 0

buy Call #1 �11:50 0 ST1 � 100e0:05(1:5)Total 0:424 0 ST1 � 100e0:05(1:5) � 0

We receive $0:424 at t = 0, yet our payo¤ at T1 is always non-negative. Thisis clearly an arbitrage.

The payo¤ at expiration T1 = 1:5 if ST2 � 100e0:05 = 105: 127

T1 T1Transaction t = 0 T2 ST1 < 100e

0:05(1:5) ST1 � 100e0:05(1:5)Sell Call #2 11:924 100e0:05 � ST2 100e0:05(1:5) � ST1 100e0:05(1:5) � ST1buy Call #1 �11:50 0 ST1 � 100e0:05(1:5)Total 0:424 100e0:05(1:5) � ST1 < 0 0


9.16 CONTENTS

If ST2 � 100e0:05, then payo¤ of the sold Call #2 at T2 is 100e0:05 � ST2 .From T2 to T1,

� 100e0:05 grows into�100e0:05

�e0:05(T1�T2) =

�100e0:05

�e0:05(0:5) = 100e0:05(1:5)

� ST2 becomes ST1 (i.e. the stock price changes from ST2 to ST1)

We receive $0:424 at t = 0, yet our payo¤ at T1 can be negative. This is notan arbitrage.

So as long as ST2 < 100e0:05 = 105: 127 , there�ll be arbitrage opportunities.

9.16

Suppose we do the following at t = 0:

1. Pay Ca to buy a call

2. Lend PV (K) = Ke�rL at rL

3. Sell a put, receiving P b

4. Short sell one stock, receiving Sb0

The net cost is P b + Sb0 � (Ca +Ke�rL).The payo¤ at T is:

If ST < K If ST � KTransactions t = 0Buy a call �Ca 0 ST �KLend Ke�rL at rL �KerL K KSell a put P b ST �K 0Short sell one stock Sb0 �ST �STTotal P b + Sb0 � (Ca +Ke�rL) 0 0

The payo¤ is always zero. To avoid arbitrage, we need to haveP b + Sb0 � (Ca +Ke�rL) � 0

Similarly, we can do the following at t = 0:

1. Sell a call, receiving Cb

2. Borrow PV (K) = Ke�rB at rB

3. Buy a put, paying P a

4. Buy one stock, paying Sa0


9.17 CONTENTS

The net cost is�Cb +Ke�rB

��P b + Sb0

�.

The payo¤ at T is:If ST < K If ST � K

Transactions t = 0Sell a call Cb 0 K � STBorrow Ke�rB at rB Ke�rB �K �KBuy a put �P a K� ST 0Buy one stock �Sa0 ST STTotal

�Cb +Ke�rB

��P b + Sb0

�0 0

The payo¤ is always zero. To avoid arbitrage, we need to have�Cb +Ke�rB

��P b + Sb0

�� 0

9.17

a. According to the put-call parity, the payo¤ of the following position is alwayszero:

1. Buy the call

2. Sell the put

3. Short the stock

4. Lend the present value of the strike price plus dividend

The existence of the bid-ask spread and the borrowing-lending rate di¤erencedoesn�t change the zero payo¤ of the above position. The above position alwayshas a zero payo¤ whether there�s a bid-ask spread or a di¤erence between theborrowing rate and the lending rate.If there is no transaction cost such as a bid-ask spread, the initial gain of

the above position is zero. However, if there is a bid-ask spread, then to avoidarbitrage, the initial gain of the above position should be zero or negative.The initial gain of the position is:�P b + Sb0

�� [Ca + PVrL (K) + PVrL (Div)]

There�s no arbitrage if�P b + Sb0

�� [Ca + PVrL (K) + PVrL (Div)] � 0

In this problem, we are given

� rL = 0:003

� rB = 0:004

� Sb0 = 168:89. We are told to ignore the transaction cost. In addition, weare given that the current stock price is 168:89. So Sb0 = 168:89.

� The dividend is 0:75 on August 8, 2011.


9.17 CONTENTS

To �nd the expiration date, you need to know this detail. Puts and calls arecalled equity options at the Chicago Board of Exchange (CBOE). The CBOEwebsite http://www.cboe.com/learncenter/concepts/basics/expiration.aspx tells us the following:

When do options expire?

Expiration day for equity and index options is the Saturday im-mediately following the third Friday of the expiration month untilFebruary 15, 2015. On and after February 15, 2015, the expirationdate will be the third Friday of the expiration month.

When is the last day to trade or exercise an equity option?

The day expiring equity options last trade is the third Friday ofthe month. This is also generally the last day an investor may no-tify his brokerage �rm of his intent to exercise an expiring equity callor put. If this third Friday happens to be an exchange holiday, thenthe last day of trading for expiring equity options is the day before,or the third Thursday of the month. Check with your brokerage �rmabout its procedures and deadlines for instruction to exercise equityoptions.

In 2011, the 3rd Friday of June is June 17; the 3rd Friday of October isOctober 21. The option is issued on May 6, 2011.The time to expiration tillJune 17 is

T =6=17=2011� 5=6=2011

365=42

365

The time to expiration till October 21 is

T =10=21=2011� 5=6=2011

365=168

365

Calculating the days between 10=21=2011 and 5=6=2011 isn�t easy. Fortu-nately,we can use a calculator. BA II Plus and BA II Plus Professional have"Date" Worksheet. When using Date Worksheet, use the ACT mode. ACTmode calculates the actual days between two dates. If you use the 360 daymode, you are assuming that there are 360 days between two dates.When using the date worksheet, set DT1 (i.e. Date 1) as 5=6=2011 by

entering 5.06111; set DT2 (i.e. Date 2) as 10=21=2011 by entering 10.2111. Thecalculator should tell you that DBD=168 (i.e. the days between two days is 168days).

If you have trouble using the date worksheet, refer to the guidebook of BAII Plus or BA II Plus Professional.For the expiration date 6=17=2011, dividend is paid after the expiration date;

the dividend is irrelevant. Hence PVrL (Div) = 0:

For the expiration date 10=21=2011, the dividend time is tD=8=8=2011� 5=6=2011

365=


http://www.cboe.com/learncenter/concepts/basics/expiration.aspx

http://www.cboe.com/learncenter/concepts/basics/expiration.aspx

9.17 CONTENTS

94

365

PVrL (Div) = 0:75e�(0:003)94=365 = 0:749 4 PVrL (K) = Ke

�0:003T

K Ca P b�P b + Sb0

�� [Ca + PVrL (K) + PVrL (Div)]

160 10:15 1:16 1:16 + 168:89��10:15 + 160e�0:003�42=365 + 0

�= �4: 478� 10�2

165 6:25 2:26 2:26 + 168:89��6:25 + 165e�0:003�42=365 + 0

�= �4: 305 � 10�2

170 3:30 4:25 4:25 + 168:89��3:30 + 170e�0:003�42=365 + 0

�= �0:101 33

175 1:43 7:40 7:40 + 168:89��1:43 + 175e�0:003�42=365 + 0

�= �7: 960� 10�2

160 14:2 5:70 5:70 + 168:89��14:2 + 160e�0:003�168=365 + 0:749 4

�= �0:138 6

165 11:0 7:45 7:45 + 168:89��11 + 165e�0:003�168=365 + 0:749 4

�= �0:181 7

170 8:20 9:70 9:70 + 168:89��8:20 + 170e�0:003�168=365 + 0:749 4

�= �0:124 8

175 5:90 12:4 12:4 + 168:89��5:90 + 175e�0:003�168=365 + 0:749 4

�= �0:117 9

b. According to the put-call parity, the payo¤ of the following position isalways zero:

1. Sell the call

2. Borrow the present value of the strike price plus dividend

3. Buy the put

4. Buy one stock

If there is transaction cost such as the bid-ask spread, then to avoid arbitrage,the initial gain of the above position is zero. However, if there is a bid-ask spread,the initial gain of the above position can be zero or negative.The initial gain of the position is:Cb + PVrB (K) + PVrB (Div)� (P a + Sa0 )

There�s no arbitrage ifCb + PVrB (K) + PVrB (Div)� (P a + Sa0 ) � 0

For the expiration date 6=17=2011, dividend is paid after the expiration date;the dividend is irrelevant. Hence PVrB (Div) = 0:

For the expiration date 10=21=2011, the dividend time is tD=8=8=2011� 5=6=2011

365=

94

365

PVrB (Div) = 0:75e�(0:004)94=365 = 0:749 2


9.18 CONTENTS

K Cb P a Cb + PVrB (K) + PVrB (Div)� (P a + Sa0 )160 10:05 1:2 10:05 + 160e�0:004�42=365 + 0� (1:2 + 168:89) = �0:113 6165 6:15 2:31 6:15 + 165e�0:004�42=365 + 0� (2:31 + 168:89) = �0:125 9170 3:20 4:35 3:2 + 170e�0:004�42=365 + 0� (4:35 + 168:89) = �0:118 2175 1:38 7:55 1:38 + 175e�0:004�42=365 + 0� (7:55 + 168:89) = �0:140 5160 14:1 5:80 14:1 + 160e�0:004�168=365 + 0:749 2� (5:80 + 168:89) = �0:135 1165 10:85 7:60 10:85 + 165e�0:004�168=365 + 0:749 2� (7:60 + 168:89) = �0:194 3170 8:10 9:85 8:10 + 170e�0:004�168=365 + 0:749 2� (9:85 + 168:89) = �0:203 5175 5:80 12:55 5:80 + 175e�0:004�168=365 + 0:749 2� (12:55 + 168:89) = �0:212 7

9.18

Suppose there are 3 options otherwise identical but with di¤erent strike priceK1 < K2 < K3 where K2 = �K1 + (1� �)K2 and 0 < � < 1.Then the price of the middle strike price K2 must not exceed the price of a

diversi�ed portfolio consisting of � units of K1-strike option and (1� �) unitsof K2-strike option:

C [�K1 + (1� �)K3] � �C (K1) + (1� �)C (K3)P [�K1 + (1� �)K3] � �P (K1) + (1� �)P (K3)

The above conditions are called the convexity of the option price with respectto the strike price. They are equivalent to the textbook Equation 9.19 and 9.20.If the above conditions are violated, arbitrage opportunities exist.

K T Cb Ca

80 0:271 2 6:5 6:785 0:271 2 3:2 3:490 0:271 2 1:2 1:35

K Cb Ca

165 6:15 6:25170 3:2 3:3175 1:38 1:43

170 = � (165) + (1� �) 175 ! � = 0:5

a:We buy a 165-strike call and a 175-strike call form a diversi�ed portfolio

of calls; simultaneously, we sell two 170-strike calls. The net receipt should benon-positive, since the payo¤ of the diversi�ed portfolio of calls is always atleast as good as the payo¤ of two 170-strike calls. Otherwise, arbitrage exists.The net receipt is 2 (3:2)� (6:25 + 1:43) = �1: 28, which is non-positive. So

the convexity condition is met.

b.We sell a 165-strike call and a 175-strike call; simultaneously, we buy two

170-strike calls. The net receipt should be non-negative, since the payo¤ of thediversi�ed portfolio of calls is always at least as good as the payo¤ of two 170-strike calls. Otherwise, arbitrage exists.


10.1 CONTENTS

The net receipt is (6:15 + 1:38) � 2 (3:3) = 0:93, which is non-positive. Sothe convexity condition is met.

c. To avoid arbitrage, the following two conditions must be met:C [�K1 + (1� �)K3] � �C (K1) + (1� �)C (K3)P [�K1 + (1� �)K3] � �P (K1) + (1� �)P (K3)These conditions must be met no matter you are a market-maker or anyone

else buying or selling options, no matter you pay a bid-ask spread or not.

10.1

The stock price today is S = 100. The stock at T is either

� Su = uS = 1:3� 100 = 130

� Sd = dS = 0:8� 100 = 80

a. For a call, the payo¤ at T is

� Vu = max (0; Su �K) = max (0; 130� 105) = 25

� Vd = max (0; Sd �K) = max (0; 80� 105) = 0

We hold a replicating portfolio (4; B) at t = 0. This portfolio will havevalue Vu if the stock goes up to Su or Vd if the stock goes down to Sd. We setup the following equations:�

4Su +BerT = Vu4Sd +BerT = Vd

!�4130 +Be0:08(0:5) = 25480 +Be0:08(0:5) = 0

! B = �38: 431 6 4 = 0:5So the option premium is:V = 4S +B = 0:5� 100 + (�38: 431 6) = 11: 568 4

b. For a put, the payo¤ at T is either

� Vu = max (0;K � Su) = max (0; 105� 130) = 0

� Vd = max (0;K � Sd) = max (0; 105� 80) = 25

We hold a replicating portfolio (4; B) at t = 0. This portfolio will havevalue Vu if the stock goes up to Su or Vd if the stock goes down to Sd. We setup the following equations:�

4Su +BerT = Vu4Sd +BerT = Vd

�4130 +Be0:08(0:5) = 0480 +Be0:08(0:5) = 25

B = 62: 451 3 4 = �0:5So the option premium is:V = 4S +B = �0:5� 100 + 62: 451 3 = 12: 451 3


solutions to derivative markets 3ed by mcdonald

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