Question

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.

Answer 1

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