8. Optional Bonus Problem Arbitrage in Montral. As an arbitrageur with Banque de 80 Montral in...

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8. Optional Bonus Problem Arbitrage in Montral. As an arbitrageur with Banque de 80 Montral in Montreal, Province Qubec, Canada you see the following CAD/USD information on your customized trading system as inputs for algorithmic execution: Spot rate 6M forward rate 6M CAD money market rate 6M USD money market rate CAD 1.1520/USD CAD 1.1635/USD 10.00% p.a. 7.50% p.a. Having only recently started in your position fresh out of school your position limit is CAD 10m or its USD equivalent. Transaction costs for trading in these markets total around USD 1,700 and would be paid at the end of the 6 months. Being based in Canada any arbitrage profit should accrue in CAD. Assuming that you can trade, borrow and invest at the above quoted rates. (a) What principle links the four prices? What is the relevant mathematical formulation? (b) Identify any arbitrage opportunities and explain how they might have come about. (c) How would you build up an arbitrage position? Describe your strategy and its mechanics. (d) Calculate your profit resulting from the preceding arbitrage strategy. PART 2 Clearly show all work. Give explanations in complete sentences. Any justification, mathematical basis, or argument must be accompanied by mathematical reasons supporting each statement given. 7. A student starts the following computation using the standard algorithm. a) Use base ten blocks below to depict the trade the student initially made. Write a sentence that explains the physical actions in your picture. - 13 203 98 5 b) Explain why the student's work is incorrect. Use your work in part a) and make sure your explanation uses mathematical reasons for why the work is incorrect. c) Now illustrate, using base 10 blocks how to correctly carry out the subtraction step by step. Be sure to use a take from approach in your use of blocks. Also, match up each move with the blocks with the appropriate symbolic move in the algorithm. 2 PROBLEM STATEMENT There are n trading posts numbered 1 to n, as you travel downstream. At any trading post i, you can rent a canoe to be returned at any of the downstream trading posts j> i. You are given a cost array R(i, j) giving the cost of these rentals for 1 i j n. We will have to assume that R(i, i) = 0 and R(i, j) = if i > j. For example, with n = 4, the cost array may look as follows: The rows are the sources (i-s) and the columns are the destinations (j's). 1 2 3 4 1 0 2 3 7 2 0 2 4 3 4 0 2 0 The problem is to find a solution that computes the cheapest sequence of rentals taking you from post 1 all the way down to post n. In this example, the cheapest sequence is to rent from post 1 to post 3 (cost 3), then from post 3 to post 4 (cost 2), with a total cost of 5 (less than the direct rental from post 1 to post 7, which would cost 7). 3.2 DIVIDE AND CONQUER (7 points): Express the problem with a purely divide-and-conquer approach. Implement a recursive algorithm for the problem. Be sure to consider all sub-instances needed to compute a solution to the full input instance in the self-reduction, especially if it contains overlaps. As before, you need to print the solution, as well as the sequence. What is the asymptotic complexity of this algorithm? 8. Optional Bonus Problem Arbitrage in Montral. As an arbitrageur with Banque de 80 Montral in Montreal, Province Qubec, Canada you see the following CAD/USD information on your customized trading system as inputs for algorithmic execution: Spot rate 6M forward rate 6M CAD money market rate 6M USD money market rate CAD 1.1520/USD CAD 1.1635/USD 10.00% p.a. 7.50% p.a. Having only recently started in your position fresh out of school your position limit is CAD 10m or its USD equivalent. Transaction costs for trading in these markets total around USD 1,700 and would be paid at the end of the 6 months. Being based in Canada any arbitrage profit should accrue in CAD. Assuming that you can trade, borrow and invest at the above quoted rates. (a) What principle links the four prices? What is the relevant mathematical formulation? (b) Identify any arbitrage opportunities and explain how they might have come about. (c) How would you build up an arbitrage position? Describe your strategy and its mechanics. (d) Calculate your profit resulting from the preceding arbitrage strategy. PART 2 Clearly show all work. Give explanations in complete sentences. Any justification, mathematical basis, or argument must be accompanied by mathematical reasons supporting each statement given. 7. A student starts the following computation using the standard algorithm. a) Use base ten blocks below to depict the trade the student initially made. Write a sentence that explains the physical actions in your picture. - 13 203 98 5 b) Explain why the student's work is incorrect. Use your work in part a) and make sure your explanation uses mathematical reasons for why the work is incorrect. c) Now illustrate, using base 10 blocks how to correctly carry out the subtraction step by step. Be sure to use a take from approach in your use of blocks. Also, match up each move with the blocks with the appropriate symbolic move in the algorithm. 2 PROBLEM STATEMENT There are n trading posts numbered 1 to n, as you travel downstream. At any trading post i, you can rent a canoe to be returned at any of the downstream trading posts j> i. You are given a cost array R(i, j) giving the cost of these rentals for 1 i j n. We will have to assume that R(i, i) = 0 and R(i, j) = if i > j. For example, with n = 4, the cost array may look as follows: The rows are the sources (i-s) and the columns are the destinations (j's). 1 2 3 4 1 0 2 3 7 2 0 2 4 3 4 0 2 0 The problem is to find a solution that computes the cheapest sequence of rentals taking you from post 1 all the way down to post n. In this example, the cheapest sequence is to rent from post 1 to post 3 (cost 3), then from post 3 to post 4 (cost 2), with a total cost of 5 (less than the direct rental from post 1 to post 7, which would cost 7). 3.2 DIVIDE AND CONQUER (7 points): Express the problem with a purely divide-and-conquer approach. Implement a recursive algorithm for the problem. Be sure to consider all sub-instances needed to compute a solution to the full input instance in the self-reduction, especially if it contains overlaps. As before, you need to print the solution, as well as the sequence. What is the asymptotic complexity of this algorithm?