United Kingdom Pound Canadian Dollar Euro Hong Kong Dollar Japanese Yen Swiss Fr
ID: 365449 • Letter: U
Question
United Kingdom Pound Canadian Dollar Euro Hong Kong Dollar Japanese Yen Swiss Franc US Dollar Mexican Peso United Kingdom Pound 1 1.9677 1.4391 16.1286 239.798 2.4119 2.08 22.1669 Canadian Dollar 0.5079 1 0.7311 8.1937 121.825 1.2254 1.0567 11.2613 Euro 0.6947 1.3672 1 11.174 166.623 1.6759 1.4452 15.4019 Hong Kong Dollar 0.06199 0.122 0.08947 1 14.8671 0.1495 0.1285 1.3743 Japanese Yen 0.004169 0.008204 0.006001 0.06725 1 0.010056 0.008673 0.09242 Swiss Franc 0.4145 0.8156 0.5966 6.6852 99.4095 1 0.8622 9.188 US Dollar 0.4807 0.946 0.6919 7.754 115.288 1.1596 1 10.657 Mexican Peso 0.04508 0.08871 0.06488 0.7271 10.8112 0.1087 0.093775 1 Problem 2 (a) Suppose that we have N different currencies (e.g., US Dollars, Euros, etc.). For each pair of currencies, (i,j), one unit of currency i can be exchanged into rij units of currency j. We wish to find an arbitrage opportunity if it exists, i.e., a sequential series of currency trades that allows us to make a profit with no risk. Formulate this as a min-cost flow problem and provide an LP with a bounded feasible polytope which solves it. Assume no transaction costs. (b) Can you find an arbitrage opportunity given the currency exchange rates in the spreadsheet "currn cies.xlsx"? If so, describe the corresponding trades. What is the maximum amount of profit that can be made for each $1.00 invested in this procedure? (c) Suppose now that the exchange charges a p% transaction fee on each unit of currency trade in other words, if we wish to trade currency i into currency J, p% of our initial amount ofí is taken away before doing the exchange. Modify your LP from (a) to take these fees into account. Assuming p 0.50, how does this change your answer for (b)?Explanation / Answer
a. We introduce variables xi j for each i = 1,...,N and j = 1,...,N which model the amount of currency i exchanged for currency j. Since the amount of currency i that can be exchanged is limited by ui , we get the constraints X N j=1 xi j ui , i = 1,...,N. For each currency i 6= 1, we cannot exchange more currency from i to other currencies than we changed to i before. This is modelled by the constraints: X N j=1 r j i xj i X N k=1 xj k, i = 2,...,N. Since by assumption there are no profitable cycles, there is no reason to exchange currency from N to any other currency, or to change from any currency to 1. Thus we can set xi1 = 0 and xN i = 0 for each i = 1,...,N. The amount of currency we can change from 1 is then upper bounded by the initial value of B: X N j=1 x1j B. Finally the amount of currency N is given by X N j=1 r j N xj N
which is our objective to maximize. Summarizing, we get the following LP: max PN j=1 r j N xj N subject to PN j=1 xi j ui , i = 1,...,N PN j=1 x1j B PN j=1 r j i xj i PN k=1 xj k, i = 2,...,N xi1 = 0, xN i = 0 i = 1,...,N xi j 0 i = 1,...,N, j = 1,...,N
b.
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