Solution to Homework One

Solution to Homework One - MAFS 5030 Quantitative Modeling...

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Unformatted text preview: MAFS 5030 Quantitative Modeling of Derivatives Securities Solution to Homework One Course Instructor: Prof. Y.K. Kwok 1. There will be two coupons before delivery: one in 6 months and one just prior to delivery. Using the present value formula: $94 . 6 = F + $4 e . 1 + $4 e . 05 , where F is the forward price, $4 is the coupon amount and interest is compounded continuously at the rate of 10%. We obtain F = $94 . 6 e . 1- $4 e . 05- $4 = $96 . 34 . Alternatively, we can interpret coupons as negative cost of carry and apply the formula: F = S- D B ( τ ) , where D is the sum of present value of the coupons. We then have F = $94 . 6- $4 e- . 05- $4 e- . 1 e- . 1 = $96 . 34 . 2. Consider the fixed-leg and floating-leg of a swap of unit notional. Suppose the current time is indexed by 0 (i.e. t = 0), and t = 1 means one year away from now. Fixed-leg At t = 1 4 , receives 10% 2 = 0 . 05 of interest. At t = 3 4 , receives 10% 2 = 0 . 05 of interest. Floating-leg At t = 1 4 , receives L 1 2 parenleftbigg- 1 4 parenrightbiggparenleftbigg 3 4- 1 4 parenrightbigg = 1 2 L 1 2 parenleftbigg- 1 4 parenrightbigg . Here, L 1 2 parenleftbigg- 1 4 parenrightbigg denotes half-year LIBOR reset at an earlier time t =- 1 4 . At t = 3 4 , receives L 1 2 parenleftbigg 1 4 parenrightbiggparenleftbigg 3 4- 1 4 parenrightbigg = 1 2 L 1 2 parenleftbigg 1 4 parenrightbigg . 1 Discount bond prices observed at t = 0: B parenleftbigg , 1 4 parenrightbigg = 0 . 972 , B parenleftbigg , 3 4 parenrightbigg = 0 . 918. The floating rate bond that receives 1 + 1 2 L 1 2 parenleftbigg- 1 4 parenrightbigg at time 1 4 is now priced at 0 . 992. This gives the present value of 1 2 L 1 2 parenleftbigg- 1 4 parenrightbigg = 0 . 992- B parenleftbigg , 1 4 parenrightbigg = 0 . 992- . 972 = 0 . 02 . Also, the implied present value of 1 2 L 1 2 parenleftbigg 1 4 parenrightbigg = B parenleftbigg , 1 4 parenrightbigg- B parenleftbigg , 3 4 parenrightbigg = 0 . 972- . 918 = 0 . 054 . Therefore, the present value of the floating-leg payments is $0 . 02 + $0 . 054 = $0 . 074 per unit notional. The present value of the fixed-leg payments is . 05 bracketleftbigg B parenleftbigg , 1 4 parenrightbigg + B parenleftbigg , 3 4 parenrightbiggbracketrightbigg = ($0 . 05)(0 . 972 + 0 . 918) = $0 . 0945 per unit notional....
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This note was uploaded on 02/01/2012 for the course MATH A taught by Professor A during the Spring '11 term at HKU.

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Solution to Homework One - MAFS 5030 Quantitative Modeling...

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