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Math 489/Math 889

Stochastic Processes and

Advanced Mathematical Finance

Dunbar, Fall 2010

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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:

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).

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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

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

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

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

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

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