Question

A European call and put option on the same security both expire in three months, both have a strike price of 20, and both sell for the price 3. If the nominal continuously compounded interest rate is 10% and the stock price is currently 25 , identify an arbitrage.

Answer 1

Call payoff = Max [0, Stock price - Strike price]

Call payoff = Max[0,25-20]

Call payoff = 5

Put payoff = Max[0, strike price - stock price]

Put payoff = max[0,20-25]

Put payoff = 0

Strike price = K = 20

Stock price​ = S = 25

interest rate = 10% = 0.1

Time to expiry = T = 3 months = 3/12 = 0.25

Put call parity: C + K*Exp(-r*T) = P + S

C = P + S​ - K*Exp(-r*T)

Call = 3 + 25 - 20*exp(-0.1*0.25)

Call = 28 - 19.5062 =

Call = 8.4938 > 3

So, yes there is an arbitrage . Implied value is 8.4938 but trades at 3.00; Call option is trading cheap hence we should buy more call options.