Question

1-month call and put price for European options at strike 108 are 0.29 and 1.70, respectively. The prevailing short-term interest rate is 2% per year.

1. Find the current price of the stock using the put-call parity.
2. Suppose another set of call and put options on the same stock at the strike price of 106.5 is selling for 0.71 and 0.23, respectively. Is there any arbitrage opportunity at 106.5 strike price? Answer this by finding the amount of arbitrage profit available at the strike price of 106.5. (Hint: Use the price found in part (a) to find the value of both sides of the put-call parity. If the two sides are not equal, then there is some arbitrage opportunity.)
3. What would be the strategy to take advantage of arbitrage opportunity at 106.5, if there is any? (Hint: State whether you would have to be long/short in all the 4 instruments (put, call, stock, bond) that are used in the put-call parity)

Answer 1

(a) Call Premium = C = $ 0.29 and Put Premium = P = $ 1.7, Strike Price = K = $ 108, Risk-Free Interest = Rf = 2%, Maturity = 1 months

Let the current price of the stock be $ S

As per put-call parity, we have:

Call Premium + PV of Strike Price = P + Current Stock Price

0.29 + 108 / e^[(1/12) x 0.02] = 1.7 + S

S = $ 106.4

(b) Call Premium = $ 0.71 and Put Premium = $ 0.23, Strike Price = $ 106.5, Asset Price = $ 106.4

Risk-Free Rate = 2% and Expiry = 1 month

Put-Call Parity: Call Premium + PV of Strike Price = Put Premium + Stock Price

Left-Hands Side (LHS) : Call Premium + PV of Strike Price

LHS = 0.71 + 106.5 / e^[(1/12) x 0.02] = $ 107.03

RHS: Put Premium + Current Stock Price = 0.23 + 106.4 = $ 106.63

As the LHS > RHS, the put-call parity is not adhered to and an arbitrage opportunity indeed exists.

(c) In order to advantage from the put-call parity arbitrage opportunity, one needs to buy the cheaper side (i.e RHS) and sell the costlier side (i.e LHS). In other words, one should buy the put option and the stock, while selling the call option and bond (PV of strike price)