In: Finance
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.
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