Question

# One stock is selling for \$54 per share. Calls and puts with a \$55 strike and...

One stock is selling for \$54 per share. Calls and puts with a \$55 strike and 360 days until expiration are selling for \$8 and \$4, respectively. What is the arbitrage profit, if we trade on one call and one put? Suppose risk-free rate is 10%.

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 = \$8

P= price of the put option =\$4

S0 = spot price = \$54

Strike price K = \$55

The risk-free rate r= 10%

Time period t= 360 days or 1 year

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

\$8 + \$55 * e^ (-0.10*1) = \$4 + \$54

But price of the put option is \$4; therefore

\$57.78 ≠ \$58.00

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 under-priced therefore it should be brought and right side of equation is overpriced therefore it should be sold.

This arbitrage opportunity involves buying a call option and selling a put option and a share of the company.

-\$8 + \$4 +\$55 = + \$51

After one year, if share price is more than the strike price, call option will be exercised and if it is below the strike price then put option will be exercised

Therefore, Net profit = +\$51 * (1+10%) - \$54 = + \$2.10