A stock trades at $30. Over the first six months it could go up to $35 or down t

Question

A stock trades at $30. Over the first six months it could go up to $35 or down to $25. In the subsequent six-month period, if the stock starts at $35 it could go to $40 or it could fall to $30. If the stock starts at $25, it could go up to $30 or it might fall to $20. Value a call option on this stock with X = $28. Assume an annual risk-free rate of 1%. Suppose now that the actual market value of the call with X = $28 is $5. Demonstrate in detail how you could engage in arbitrage to take advantage of this mispricing.

Explanation / Answer

Expected price of stock after 1 year = 0.5 * (0.5*40 + 0.5*30) + 0.5 * (0.5*30 + 0.5*20) = 30

Expected Profit from Call option after 1 year = Actual price - Call price = 30 - 28 = 2

Present value of profit = 2 / (1+0.01) = $ 1.98. This is the expected worth of the option.

Hence value of 5$ implies that the option is overpriced. Hence, selling this call option will generate Arbitrage Profits.

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