Question

Consider a European call option and a put option on a stock each with a strike price of K = $22 and each expires in six months. The price of call is C = $3 and the price of put is P = $4. The risk free interest rate is 10% per annum and current stock price is S0 = $20. Show how to create an arbitrage strategy and calculate the arbitrage traders profit.

Answer 1

Let’s verify that put–call parity holds or not

Call-put parity equation can be used in following manner

C + K* e^ (-r*t) = P + S0

Where,

C = price of the call option = $3

P= price of the put option =$4

S0 = spot price = $20

Strike price K = $22

The risk-free rate r= 10%

Time period t= 6 months or 0.5 year

Now putting all the values in the put-call parity equation

$3 + $22 * e^ (-0.10*0.5) = $4 + $20

But, $23.93 ≠ $24

As the value of both sides of equation is not same therefore the put–call parity does not hold and there is an arbitrage opportunity

We can see that left side of equation is underpriced therefore it should be brought and right side of equation is overpriced therefore it should be sold. Therefore, buy the market call, short the stock, short the put and invest the present value of the exercise price at the risk-free rate.

This arbitrage opportunity involves

Buying a call option = -$3

Selling a put option = +$4

Selling a share = +$20

Net amount received = -$3 + $4 +$20 = $21

And invests the proceeds at the risk-free rate for six months = $21 *(1+10%/2)

= $22.05

Net profit = proceeds - exercise price

= $22.05 -$22

= $0.05

Therefore arbitrage trader’s profit is $0.05