Question

Put-Call Parity Case Study

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.)

Stock Price		Without Considering Interest Costs
When Options Expire		Protective-Put Portfolio			Long Call
			(Profit/Loss)		Call Option Profit/Loss
	$	Stock Profit/Loss (1 )	Put Option Profit/Loss (2)	Portfolio Profit/Loss    (3) = (1) + (2)	(4)
	14.00	-18.25	8.5	-18.25+8.5 = -9.75	-9.75
	17.10
	20.30
	23.50
	25.00 (strike)
	26.70
	29.90
	32.50
	35.70
	38.90
	42.00

II. In the previous exercise, the interest costs for capital are ignored.   Now consider the interest costs for capital.

1. How much capital is needed to form a protective-put portfolio with 1,000 shares of stocks and ten long put contracts?

2. How much capital is needed to buy/long ten call contracts (assume the call premium is $9.75)?

3. Calculate the difference of capital requirements between 1. (i.e., the protective-put portfolio) and 2. (i.e., the long-call position).   If the interest rate on capital is 3% a year, what will be the difference in interest costs between the protective-put portfolio and the long call position? (Hint: the options will expire in one month. Options expiration time will be used for any comparison of different investment strategies that involve options.)

4. From the difference in interest costs between the protective-put portfolio and the long call position, explain why the call premium should be $9.81 (assume the put premium is correctly priced at $2.50).

5. Verify that the correct premium for the call is $9.81 with the put-call parity relationship.

6. 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?). (To do this question, you have to study and fully understand the Put-Call Parity Relationship and

Answer 1

The completed table is as below:

The pay off profile is as below:

Long Call: Max (Expiry Price - Strike Price - Premium, Premium)

Long Put: Max (Strike Price - Expiry Price - Premium, Premium)

As we can see that the Protective Put strategy pay off is similar to the Long Call pay off.

Part II:

(i) Capital Required for Protective Put Strategy is as below:

Long Stock : 1000 * 32.25 = 32250

Long Put : 100 * 10 * 2.50 (10 contracts of 100 each) = 2500

Total Outlay : 34750

(ii) Capital outlayLong Call : 100 * 10 * 9.75 = 9750

(iii) Difference in capital outlay = (34750 - 9750) = 25000 and interest cost difference = 25000 * (3%/12) = 62.5

(iv) Given the difference in the interest of 62.5 of per unit difference of (62.5/1000) - 10 contracts of 100 each - to make things equal the call price should be higher by per unit difference which os 0.0625 which is (9.75+0.6) = 9.81

(v) As put call parity equation : C + PV (Strike Price) = P + S where C is the call premium, P is the put premium, and S is the current stock price. Plugging in the values we have:

C=9.81, P = 2.5, S = 32.25 and PV(25) = 25/(1+3%/12) = 24.94 Thus we get

9.81 + 24.94 = 34.75 and 32.25 + 2.5 = 34.75 . Thus the put call parity is satisfied at call premium of 9.81

(vi) If the C = 7.5, then in the put call equation, the C+PV(Strike Price) combination will be priced lower than P + S. Hence we should long C + PV(Strike Price) and short P+S, as below:

• Short 10 contracts (1000 units) of put at 2.50. Cash inflow = 2500
• Short 1000 stock at 32.25. Cash inflow = 32250
• Buy 10 contracts (1000 units) of call at 7.50. Cash outlfow = 7500
• Place residual cash at 3% in a deposit for 1 month. (32250+2500-7500) * (1+3%/12) = 27318.13
• On expiry, the pay off will be:
  • Expiry stock price (S₁) is above 25 (strike price of call & put): The pay off will be:
  • Long Call: (S₁ - 25)*1000
  • Short Put: Expires worthless
  • Short stock: - S₁ * 1000
  • Cash flow from deposit = 27318.13
  • Net Pay off: (S₁ - 25) * 1000 - S₁ * 1000 + 27318.13 = 2318.13
• Expiry price (S₁) is below 25, then the pay off will be:
  • Long Call: expires worthless
  • Short Put : - (25 - S₁) * 1000
  • Short Stock : - S₁
  • Cash flow from deposit = 27318.13
  • Net Pay off: - (25 - S₁) * 1000 - S₁ + 27318.13 = 2318.13
• Thus we see that in either case there is an arbitrage profit of 2318.14 which on per unit basis (10 contracts of call total to 1000 units) is 2.31 difference. Hence for there to be no arbitrage the call price should be higher by 2.31 which is going to be 9.81