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

### 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

### 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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