Question

The put option of Joe Inc. is currently trading at $2.50 while the call option premium is $7.50. Both the put and the call have an exercise price of $25. Joe Inc. stock is currently trading at $32.25 and the risk free rate is 3%. The options will expire in one month.

I. An investor applies a protective put strategy by buying the put option of Joe Inc. to protect his holding of the company’s stock. This strategy creates a portfolio of long stock and long put, which provides a payoff of unlimited upside potential with a limited loss. This is similar to buying a call option (i.e., a long call position) of Joe Inc. that also provides unlimited upside profit potential with a limited loss (the maximum loss will be the option premium paid).

Investigate the profit/loss possibilities at options expiration for the protective put portfolio vs. the long call position. For example, if the stock price is $14 when options expire, stock’s profit/loss will be $l4 - $32.25 = - $18.25 (i.e., a loss), long put option’s profit/loss will be $25 - $14 - $2.5 = $8.5 (i.e., a profit), and long call option’s profit/loss will be $0 - $7.5 = - $7.5 (i.e., a loss). Thus, a protective-put portfolio of long stock and long put will incur portfolio’s profit/loss of

- $18.25 + $8.5 = - $9.75, while the long call position will suffer - $7.5 loss. These profits/losses are entered in the following table for the Stock Price = 14.00. Complete the following table for different stock prices other than $14. (In this exercise, ignore the interest costs on capital.)

Question:

If the call premium is $7.5, instead of its parity price of $9.81, and the put premium is $2.5, show how to take an arbitrage opportunity by trading 1,000 shares of stocks (hint: long or sell short?) with 10 contracts of puts (hint: long or short position?) and 10 contracts of calls (hint: long or short position?).

Answer 1

As per the put call parity equation: C + PV(Strike Price) = P + S where C is the price of call option, P is the price of put option, S is the spot price and PV (Strike Price) is the present value of strike price at risk free rate from expiry period to current time (1 month in this case). If there is a deviation, then there is a possible arbitrage where in we can purchase the lower priced combination and short the higher priced combination. Since on expiry the pay off for both the combinations should be same, we will make risk free arbitrage profit. We can do this as below:

• Given C = 7.5, P = 2.5, S = 32.25 and PV(25) = 25/(1+3%/12) = 24.94
• C+PV(Strike Price) = 7.5 + 24.94 = 32.44
• P + S = 2.5 + 32.25 = 34.75
• Since C + PV(Strike Price) is mispriced lower, we will go long on 10 contracts of calls (each contract with 100 units), short 10 contracts of puts (each contract with 100 units), short 1000 stock and place the residual funds in a deposit at risk free rate of 3% for 1 month. The cash flow at the time of initiating the trade will be:
  • Short Stock inflow = 1000*32.25 = 32250
  • Short Put inflow = 2.5 * 10 * 100 = 2500
  • Long Call outflow = 7.5 * 10 * 100 = 7500
  • Residual placed in deposity at 3% = 27250. After 1 month, this deposit will pay us (27250 * (1+3%/12)) = 27318.13
• Now after 1 month the pay off on expiry will be - let us denote expiry price by EP:
• If the expiry price (EP) is 25 or above:
  • Long Call pay off : (EP - Strike Price) * 1000 or (EP - 25) * 1000 = 1000 EP - 25000
  • Short Put pay off : zero - it will expire worthless
  • Short Stock pay off : - (1000 * EP)
  • Receivable from deposit = 27318.13
  • Net Profit = 1000 EP - 25000 + 0 -1000 EP + 27813.13 = 2318.13
• ??If the expiry price (EP) is less than 25:
  • Long Call pay off : zero - expires worthless
  • Short Put pay off (loss) : - (Strike Price - EP) * 1000 or - (25-EP)* 1000 = - (25000 - 1000 EP)
  • Short Stock pay off : - (1000 * EP)
  • Receivable from deposit = 27318.13
  • Net Profit = 0 - 25000 + 1000 EP - 1000 EP + 27813.13 = 2318.13
• Thus we see that the expiry stock price (EP) can close anywhere but we will make arbitrage profit and this profit is going to be there till such time that the value of C + PV(strike price) is less than P + S. We can also prove the correct price is 9.81 - the arbitrage profit is 2318.13 over 1000 units or per unit the mispricing is (2318.13/1000) = 2.31. Hence the call price should be increased by 2.31 which will make is (7.50 + 2.31) = 9.81