673_hw2_solution - NBA673 Introduction to Derivatives I...

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1 of 5 NBA673 Introduction to Derivatives I Tibor Jánosi Spring 2006 (1 st half) Homework 2: Arbitrage, Forwards, Futures Solutions Problem 1: (a) 987153 . 365 100 0475 . 1 1 = + = B(t, T) 59 . 45 $ 987153 . 45 $ ) , ( ) ( = = = T t B t S F(t, T) (b) At maturity, the value of the original forward contract will be S(T) -$50.25, while the value of the new forward contract will be $45.59- S(T) . The value of the net position will be V(T )=-$4.66. This example helps us understand the true meaning of “offsetting.” The second forward contract offsets the first one in the sense that it eliminates future uncertainty about payoffs; in effect (in this case) it “freezes in” the losses. The upside is that one is no longer exposed to the possibility that the stock price will fall further, thus increasing losses even more. Of course, the stock price might also rise, case in which the offsetting transaction might not appear to be such a great idea in hindsight (but hindsight is always 20/20). (c) We can determine the present value of the offsetting position by discounting the future (known) net cash flow on the two contracts. 60 . 4 $ 987153 . 66 . 4 $ ) , ( ) ( - = - = = T t B T V V(t) Problem 2: The formula linking simple discount rates and the prices of zeros is 360 1 ) , 0 ( T i T B d - = . From here, we obtain immediately the prices for US zero-coupon bonds: 0 = “3 months ago” t = “now” T = “expiration” “100 days”
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2 of 5 Maturity (Days) US Discount Rate Price of US Zeros 30 5.46 .995450 90 5.77 .985575 180 5.96 .970200 We write the equation giving the forward price in generic form (but we use subscript Y to denote yen- denominated zero-coupon bonds, as opposed to F, for a generic foreign zero-coupon bond): ) S( ,T) ( B ,T) F( ,T) B( Y 0 0 0 0 = . From here, we immediately get the prices of yen- denominated zero-coupon bonds. Using the same 360-day year, we then compute the implied simple discount rate for Japanese zeros. For this, we use the formula that expresses the simple discount rate as a function of the corresponding zero-coupon bond: [ ] T T J i j 360 ) , 0 ( 1 - = (we have used J(0,T) to denote the price of Japanese zeros, and i j to denote the Japanese simple discount rate, but this is still the usual formula). Maturity (Days) Price of US Zeros Price of Japanese Zeros Japanese Discount Rate 30 .995450 .998886 1.34 90 .985575 .995780 1.69 180 .970200 .991350 1.73 Problem 3 : This is a simple application of the formula given in class: ] [ ) 0 ( ) , 0 ( ) , 0 ( 0 contract the of life the over dividends PV S T B T F - = . We can immediately write the following: 9741 . 2 9967 . 50 . 1 37 . 63 9512 . ) , 0 ( - - = T F From this equation we get that the forward price is 00 . 63 $ ) , 0 ( = T F . Problem 4:
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673_hw2_solution - NBA673 Introduction to Derivatives I...

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