Question 21.6: You are an options dealer who deals in non-publicly traded o...
You are an options dealer who deals in non-publicly traded options. One of your clients wants to purchase a one-year European call option on HAL Computer Systems stock with a strike price of $20. Another dealer is willing to write a one-year European put option on HAL stock with a strike price of $20, and sell you the put option for a price of $2.50 per share. If HAL pays no dividends and is currently trading for $18 per share, and if the risk-free interest rate is 6%, what is the lowest price you can charge for the call option and still guarantee yourself a profit?
PLAN
We can use put-call parity to determine the price of the option:
Call Price = Put Price + Stock Price – PV(Strike Price ).
In order to price a one-year European call with a strike price of $20, we need to know the price of a one-year European put with the same strike price, the current stock price, and the risk-free interest rate. We have all of that information, so we’re ready.
EXECUTE
Call Price = Put Price + Stock Price – PV( Strike Price )
= $2.50 + $18 – $20/1.06 = $1.632.
EVALUATE
Put-call parity means that we can replicate the payoff of the one-year call option with a strike price of $20 by holding the following portfolio: Buy the one-year put option with a strike price of $20 from the dealer, buy the stock, and sell a one-year, risk-free zero-coupon bond with a face value of $20. With this combination, we have the following final payoff, depending on the final price of HAL stock in one year, S_1:
Note that the final payoff of the portfolio of the three securities matches the payoff of a call option. Therefore, we can sell the call option to our client and have future payoff of zero no matter what happens. Doing so is worthwhile as long as we can sell the call option for more than the cost of the portfolio, which we found to be $1,632.
| Payoff | |||
| Final HAL Stock Price | |||
| S1 < $20 | S1 ≥ $20 | ||
| Buy Put Option | 20-S_1 | 0 | |
| + | Buy Stock | S_1 | S_1 |
| + | Sell Bond | -20 | -20 |
| = | Portfolio | 0 | S_1-20 |
| + | Sell Call Option | 0 | -(S_1-20) |
| = | Total Payoff | 0 | 0 |