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Consider a stock that does not pay dividend. A one-year European put option with strike $40...

Consider a stock that does not pay dividend. A one-year European put option with strike $40 is trading at $2.40 and a one-year European put option with strike $50 is trading at $12.30. The risk-free interest rate is 5% per annum with continuous compounding. Construct an arbitrage strategy.

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Expert Solution

Premium for One Year Put option with strike price $ 40 = $2.40

Premium for One Year Put option with strike price $ 50 = $12.30

Assume a strategy where the investor buys a put option with strike $ 40 and sells a put option with strike $50

Premium Paid (for purchasing put option with strike $ 40) = $ 2.40

Premium Received (for selling put option with strike $ 50) = $ 12.30

Net receipt of premium = $12.30-2.40=$9.90

Assuming the amount received is invested at risk free rate for 1 year, the net cash flow at end of 1 year is

Net Amount = Initail Amount * (1+risk free rate)^ years = 9.90*(1+.05) = $10.3950

Below are the option secnarios and final payoff after 1 year based on the then prevailing stock price

Stock Price Buy Put @ 40 Sell Put @ 50 Net Loss Cash flow from premium Net Gain
<=40 Excercised Excercised 10 10.395 0.395
41 Not Excercised Excercised 9 10.395 1.395
42 Not Excercised Excercised 8 10.395 2.395
43 Not Excercised Excercised 7 10.395 3.395
44 Not Excercised Excercised 6 10.395 4.395
45 Not Excercised Excercised 5 10.395 5.395
46 Not Excercised Excercised 4 10.395 6.395
47 Not Excercised Excercised 3 10.395 7.395
48 Not Excercised Excercised 2 10.395 8.395
49 Not Excercised Excercised 1 10.395 9.395
>=50 Not Excercised Not Excercised 0 10.395 10.395

In case the share price is less than 40, the investor would excercise the option to sell @ 40 and would have to buy @ 50 also due to the put option sold, resulting in a loss of $ 10, the net gain would be 0.3950

Between 41-50, the investor would sell at the market rate but would have to purchase @ $50 due to the put option sold, resulting in a loss of $50-Market Rate

At a price more than 50, bith the options would be worthless resulting in a net gain of the premium received i.e 10.3950.

In any given scenario, there would always be a positive cash flow


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