Question

Consider a one-year call option and a one-year put option on the same stock, both with an exercise price $100. If the risk-free rate is 5%, the current stock price is $103, and the put option sells for $7.50.

1. According to the put-call parity, what should be the price of the call option?

2. To your amazement, the call option is actually traded at $15. If the call option fairly priced, overvalued, or undervalued? What would you do to exploit this mispricing? Formulate your strategy and work out the payoff worksheet. What would be your arbitrage profit when the future stock price is equal to $0, $100, and $200?

Answer 1

As per Put Call Parity, the prices of options with same strike price & expiry date are as follows:

Price of Call + PV of Exercise Price = Spot Price (Current Stock Price) + Price of Put

Interest Rate is assumed as continuous compounding

C + [100*(e^-0.05)] = 103 + 7.5

C + [100*0.9512(from table)] = 110.5

Therefore, Price of Call = C = 110.5 – 95.12 = $15.38

Actual Price of Call < Theoretical Price. Therefore, Call is Undervalued.

Arbitrage Strategy:

As Call is Undervalued, Buy Call Option, Sell Stock Now, Sell Put Option and Invest

Steps for Arbitrage:

Now,

(1) Buy a Call Option @ $15. Outflow of $15

(2) Sell Stock @ $103. Inflow of $103

(3) Sell Put Option @ $7.5. Inflow of $7.5

(4) Invest [Price of Stock+Premium of Put-Premium of Call] = 103+7.5-15 = $95.5. Outflow of $95.5

Balance = -15+103+7.5-95.5 = 0

After 1 year,

Case 1: If Stock Price is less than $100, then Exercise Put and Lapse Call. Stock will be bought for $100 under Put contract.

Case 2: If Stock Price is greater than $100, then Exercise Call and Lapse Put. Stock will be bought for $100 under Call contract.

Case 3: If Stock Price is equal to $100, then Both Lapse. Stock will be bought for $100 from Market

Therefore, In any Case, we will be able to buy the stock for $100

(5) Realize Investment 95.5*e^0.05 = 95.5*1.0513 = $100.399. Inflow of $100.399

(6) Buy Stock @ $100. Outflow of $100.

Balance = Arbitrage Gain = 100.399-100 = $0.399