Stochastic Processes and

Advanced Mathematical Finance

Homework 2

Due Monday, Sept 13, 2010

- Consider the hypothetical country of Elbonia, where the government has
declared a “currency band” policy. This means 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:
$$0.95\text{USD}\le \text{EBB}\le 1.05\text{USD}$$
for at least one year. Suppose also that the government has issued 1-year bonds denominated in the EBB. The government is so shaky that it must pay a continuously compounded interest rate of 20%. Assuming that the corresponding continuously compounded interest rate for US lending and borrowing is 4%, show that there is an arbitrage opportunity. In a sentence explain the risk associated with this transaction.

- Consider a market that has
- a stock (also called a security or asset), current price $S$
- a loan market so that money (also called a bond) can be borrowed or loaned at an annual interest rate of $r$ compounded continuously.

At the end of a time period $T$, the security will either increase in value by a factor $U$ to $SU$, or decrease in value by a factor $D$ to value $SD$. Show that a forward contract with strike price $k$ that, is, a contract to buy the security at time $T$ with potential values $SU-k$ and $SD-k$ should have the strike price set at $Sexp\left(rT\right)$ to avoid an arbitrage opportunity.