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Problem Statement

A stock price is currently $40. It is known that at the end of 3 months, it will either $45 or $34. The risk-free rate of interest with quarterly compounding on a $1 bond is 8% per annum. Calculate the value of a 3-month European put option on the stock with a strike price of $40, and find the replicating portfolio.

Solution

The payoffs are f(SU) = max(40 -SU, 0) = 0 and f(SD) = max(40 -SD, 0) = 6.

A replicating portfolio must satisfy:

45f + 1.02y = 0 34f + 1.02y = 6.

Solving this system we obtain that f = -0.54545 and y = 24.06, so we short -0.54545 shares of stock, and own 24.06 worth of bonds. The value of the portfolio and therefore the option is then phiS + y = 2.25.

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

 1.02 - 0.85 p = ------------ = 0.61818 1.125 - 0.85

and

1 - p = 0.38182

and so

 --1- V = 1.02(p0 + (1 - p)6) = 2.25.

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