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A stock price is currently $100
A stock price is currently $100. It is known that at the end of three months it will be either $110 or $90. The risk-free interest rate is 12% per annum with quarterly compounding. Suppose that Sy is the stock price at the end of three months. Then, the current price of a derivative that pays off S at maturity should be $ .(* Assume that short-selling is possible and that there are no arbitrage opportunities.)-
Expert Solution
If stock price becomes $110 after 3 months , derivative pays off $110^2 =$12100
If stock price becomes $90 after 3 months , derivative pays off $90^2 =$8100
Let a riskfree portfolio be constructed with long position in X units of Stock and short position in One unit of derivative
This riskless portfolio has the same value at both values i.e. when the stock price is $110 or $90
Thus,
value of portfolio If stock price becomes $110 =value of portfolio If stock price becomes $90
=> X*110- 12100 = X*90 - 8100
=> X = 200
So, the value of portfolio after 3 months = X*110- 12100 = X *90 - 8100 = $9900
Value of portfolio today = Value of portfolio after 3 months discounted at riskfree rate (as portfolio is riskless)
=9900/(+0.12*3/12) =$9611.65
Also
Value of portfolio today = X*100 - V where V is the value of derivative
200*100 - V = 9611.65
V = $10388.35
Current price of the derivative is $10388.35
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