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

### Problem Statement

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.

### Solution

Suppose instead that the strike price k is set so that k > S0 exp(rT). Then the strategy is to borrow S0 with bonds, buy the security at current price S0 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 S0 exp(rT), making a risk free profit of k - S0 exp(rT) on the deal.

Suppose instead that the strike price k is set so that k < S0 exp(rT). Then the strategy is to loan S0 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 S0 exp(rT), and use it to buy the security at price k, thereby making a risk free profit of S0 exp(rT) - k on the deal.

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