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

Problem Statement

Consider a stock whose price today is \$50. Suppose that over the next year, the stock price can either go up by 10%, or down by 3%, so the stock price at the end of the year is either \$55 or \$48.50. The interest rate on a \$1 bond is 6%. If there also exists a call on the stock with an exercise price of \$50, then what is the price of the call option? Also, what is the replicating portfolio?

Solution

The payoffs are f(SU) = max(SU - 50, 0) = 5 and F(SD) = max(SD - 50, 0) = 0.

A replicating portfolio must satisfy:

Solving this system we obtain that = 0.769 and = -35.14, so we own 0.769 shares of stock, and short (borrow) 35.14 bonds. The value of the portfolio and therefore the option is then S + = 3.32.

We can also figure the value from the risk-neutral measure, namely

and

and so

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