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Stochastic Processes and

Advanced Mathematical Finance

Dunbar, Fall 2010

Consider a market that has a security and a bond so that money 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 have increased in value by a factor U to SU, or decreased 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 which has potential payoffs SU - k and SD - k should have the strike price set at S exp(rT) to avoid an arbitrage opportunity.

Suppose instead that the strike price k is set so that k
> S_{0}
exp(rT). Then the strategy is to
borrow S_{0} with bonds, buy the security at
current price S_{0} and enter into that forward contract
to sell the security at strike price k. Then at time T, sell the security at price k, deliver it, and pay back the bond loan in
the amount S_{0} exp(rT),
making a risk free profit of k
- S_{0}
exp(rT) on the deal.

Suppose instead that the strike price k is set so that k
< S_{0}
exp(rT). Then the strategy is to
loan S_{0} in bonds, and enter into that forward
contract to buy the security at strike price k. Then at time T, cash in the bonds, get S_{0}
exp(rT), and use it to buy the
security at price k, thereby
making a risk free profit of S_{0}
exp(rT) - k on
the deal.

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