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#
Math 489/Math 889

Stochastic Processes and

Advanced Mathematical Finance

Dunbar, Fall 2010

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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?

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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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problem statement

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