Question

A call that is $2 in-the-money trades for $4 today. A put written on the same stock that is $2 out-of-the-money trades for $2.25 today. Both options expire in 6 months and the risk free rate is 10% p.a. continuously compounded while the stock trades for $40. Are there any arbitrage opportunities available? If yes state the arbitrage strategy and prove

that it works.

Answer 1

Solution:

It is given that Current share price is S= $40

Strike price is $38 because call option is in the money by $2 and put option is out the money by $2 .

X= $38

Call premium = C = $4

Put premium = P = $2.25

Interest rate r = 10% and t = 6 months = 0.5 years

In order to check for arbitrage opportunity we need to use put call parity formula

Here left hand side and right hand side is not equal hence arbitrage opportunity exists.

Startegy is to buy the lower side and sell the higher side and profit will be the difference of both the sides

Profit = 42.25 - 40.14672 = 2.10328

How to achieve the arbitrage

• Sell the stock at current price and earn $40
• Sell the put option and earn $2.25
• Buy the call option and pay $4
• Invest $36.14672 in the security that pays 10% return

Now you will left with 2.10328 as profit . Now irrespective of share price movement this profit is fixed.

Let's take an example to see this.

Case 1 : suppose at the expiry share price is $50

Value of call option = $50 -$38= $12

Value of investment (bond) = 38

Since we sold the share at $40 but currently we have to buy it from the market for $50 and this amount will be recovered by investment + gain from the call option $38 + 12 = $50.

Case 2

When share price is $30

Call option is worthless

Since we have sold the put option so we have to pay $38 - $30 = $8

Now we are getting $38 from the investment , this will paid towards purchasing share at $30 and paying to the put option buyer $8.