Question

A European call and put both have a strike price of $20 and expire in 3 months. Both sell for $1.5. Assume the annual interest rate is 12%, the current stock price is $19 and the dividend yield is 5%. What opportunities are available to an arbitrageur? Show the cash flows associated with your arbitrage strategy.

Answer 1

The put-call parity formula is given as-

Call + Xexp(-rt) = Put + Share

Putting the given values, we get the LHS = 1.5 + 20 x exp(-0.12 x 0.25) = 20.908

And the RHS = 1.5 + 19 = 20.5.

Since the put-call parity does not hold, there is an arbitrage opportunity here.

We can go long on the portfolio in the RHS, i.e. we can lend the present value of the strike price i.e. $20 x exp(-0.12 x 0.25) = $19.408, buy a put at $1.5 and the underlying stock at $19 and short the call option.

Now, since we have bought the share and the put, our upside and downside are balanced. After 3 months, we will get $20 (from the lending) and will have to either sell the stock at the strike price (if the stock price is higher than the strike in which case the put will expire worthless and we will sell the stock we currently own) or the call will expire unexercised (in which case we can use the put to sell the stock at the strike price). Hence, we see that one way or another our stock will be sold at the strike price.

So, total amount spent initially on this strategy = 1.5 (for buying put) - 1.5 (for selling call) + 19.408 (lending) + 19 (for stock) = $38.408.

And total amount received = 20 (from selling stock at the strike price) + 20 (from the amount lent) = $40

Hence, if we see the total profit made on this strategy, we can calculate the PV of this $40.

Total Profit = PV of $40 - $38.408 = 40 x exp(-0.15 x 0.25) - 38.408 = $0.119 (in present value terms)