# Describing trading strategy with an arbitrage opportunity. Tut 4, Q4

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Question: (Taken from tutorial 4, question 4)

You can assume that a non-dividend paying share that has a current price of R125 will have a price of either R110 or R140 6 months from now, depending on the state of the economy. The continuously compounded risk free rate of interest is 10% per annum.

If a call on this share with 6 months to maturity and a strike price of R130 was currently trading at R7,50 in the market, show that an arbitrage opportunity would exist by describing a trading strategy that has zero cost at time 0 and a positive value at maturity (i.e. in 6 months from now). Calculate the value at maturity.

I am not sure how calculating the theoretical call price indicates the exact strategy that is needed to create an arbitrage (especially the part about selling 3 calls).

Also are there many possible trading strategies that can be chosen, if so how do you go about finding them.

First calculate $$\phi = (10 − 0)/(140 − 110) = 1 /3$$. Then $$\psi = e ^{−0.05}[10 − 140/3] = −34.8784$$ Giving a theoretical option price of 125/3 − 34.8784 = 6.7883. Our intrinsic price is less than the mark price so we sell the market call and long our replicating portfolio. Here the memo assumes we sell 3 calls so we can long one share (in total). So, we can sell 3 calls for 22.5, borrow another 102.5 and buy one share. That cost 0 initially. After 6 months we must repay 107.7553. If the share is at 110, the calls will not be exercised and we can sell our one share for 110 and make 2.2447 profit. If the share is at 140, the three calls will be exercised. We will buy two shares for 280, sell our three shares for 390 at the same payment date and repay the original loan. This will leave us with 2.2477 profit.