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Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2010


Problem Statement

Consider the hypothetical country of Elbonia, where the government has declared a “currency band” policy, in which the exchange rate between the domestic currency, the Elbonian Bongo Buck, denoted by EBB, and the US Dollar is guaranteed to fluctuate in a prescribed band, namely:

0.95USD < EBB < 1.05USD

for at least one year. Suppose also that the government has issued 1-year notes denominated in the EBB that pay a simply compounded interest rate of 30%. Assuming that the corresponding interest rate for US deposits is 6%, show that there is an arbitrage opportunity. (Adapted from Quantitative Modeling of Derivative Securities, by M. Avellaneda and P. Laurence, Chapman and Hall, Boca Raton, 2000, Exercises 1.7.1, page 18).

Solution

We start by shorting (borrowing) X USD and buying the EBB. We assume that this occurs in the worst case, which is when the EBB is worth most in USD. Then we can purchase

 X USD ----------------. 1.05 USD/EBB

Now we invest these EBB in the 30% bonds, so at the end of a year, our total of EBB have grown to

 X ---- × 1.30. 1.05

Now we exchange these EBB back to USD. Again, we assume that this happens at the worst possible rate, when the EBB is least dear in USD, so that we receive

1.30X-- 0.95USD-- 1.05 EBB EBB .

Then we pay the loan, which was at the 6% rate for a total of 1.06X, so the amount left is

1.30X ------× 0.95 - 1.06X = 0.1162X 1.05

so we make a profit of at about $0.12 per dollar borrowed, exchanged, and invested.


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