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#
Math 489/Math 889

Stochastic Processes and

Advanced Mathematical Finance

Dunbar, Fall 2010

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Problem Statement

A stock price is currently $50. it is known that at the end
of 6 months, it will either be $60 or $42. The risk-free rate
of interest with continuous compounding on a $1 bond is 12%
per annum. Calculate the value of a 6-month European call
option on the stock with strike price $48 and find the
replicating portfolio.

This problem is adapted from Hull, page 240, problem 10.16

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Solution

The payoffs are f(SU) = max(SU
- 50, 0) = 10 and F(SD) =
max(SD - 50, 0)
= 0.

A replicating portfolio must satisfy:

Solving this system we obtain that = 0.556 and = -21.97, so we own 0.556 shares of stock, and short (borrow)
21.97 bonds. The value of the
portfolio and therefore the option is then phiS + = 5.80.

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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