6. Options Strategies

6.1. What is an Option Strategy?

An option strategy involves combining an option with other assets and/or other options together. Types of strategies that we can consider:

• Option plus bond

• Option plus stock

• Two or more options of the same type

• Two or more options of different types

6.2. Simple Options Strategies

6.2.1. Covered Call

A covered call consists in a long position in the stock and a short position in a European call option with strike \(K\) and maturity \(T\). The figure below depicts the payoffs of the long stock, the short call and the covered call.

As can be seen from the figure, only the long stock position pays off when the stock price is less than \(K.\) In that case the call is out-of-the-money and will not be exercised. Otherwise the short call becomes active, and for every additional dollar that the stock gains in value the short call loses the same amount. The resulting payoff is therefore flat at \(K\) for stock prices greater than the strike.

This might be an interesting strategy if you think that the stock has limited upside potential. Indeed, by selling the call you give up all the upside potential if the stock price is higher than the strike price, but you can purchase the stock for less. Also, if you are planning to hold the stock for a long time, by selling the call you can generate additional income if the stock does not appreciate too much in value during the life of the option.

More formally, the payoff of a covered call on a non-dividend paying stock as a function of the stock price \(S\) can be described as follows:

	\(S \leq K\)	\(S > K\)
Long Stock	\(S\)	\(S\)
Short Call	0	\(-(S - K)\)
Covered Call	\(S\)	\(K\)

The table confirms the payoff function pictured previously. The covered call pays like the stock if \(S \leq K\) and caps the payoff at \(K\) otherwise.

Note

We will use this strategy extensively to prove many interesting facts about European options such as put-call parity.

Example 6.1

A non-dividend paying stock currently trades at $50. A call option with strike $60 and maturity 3 months sells for $3.45. A covered call with the same characteristics would then cost 50 - 3.45 = $46.55.

Below are some possible covered call payoffs and profits for different stock prices at maturity.

Stock Price	40	50	60	70	80
Payoff	40	50	60	60	60
Profit	-6.55	3.45	13.45	13.45	13.45

If the stock price is $40, then the call is OTM and pays nothing, leaving only the stock that pays $40. The profit is then \(40 - 46.55 = -\$6.55.\) If the stock price climbs to $80, then the call is ITM and the payoff of the short call is \(-(80 - 60) = -\$20.\) The payoff of the covered call is then \(80 - 20 = \$60,\) and its profit is \(60 - 46.55 = \$13.45.\)

The payoff diagram reveals that the covered call payoff is the same as the long position in the stock if \(S \leq 60,\) and is capped at $60 otherwise.

Comparing the profit diagram of the covered call and a long position in the stock, we see that the covered call profit is higher than the stock alone whenever \(S < 63.45.\) This is because the cost of the covered call, compared to the stock alone, is cheaper. For example, if you only buy the stock and the price at maturity is $60, then your profit would be $10 instead of $13.45. The covered call is then a good strategy when the stock price finishes close to the call strike. Indeed, if the stock goes up to $80, then the covered call profit is still the same, whereas a pure stock position would yield a profit of $30.

6.2.2. Protective Put

A protective put consists in a long position in the stock and a long position in a European put option with strike \(K\) and maturity \(T\). The figure below depicts the payoffs of the long stock, the long put and the protective put.

As the figure shows, the objective of the long put is to protect the stock by keeping it from falling below \(K.\) This might be an interesting strategy if you want to hedge your portfolio from potential losses during the life of the option, although the hedge comes at a cost.

For example, you might be concerned that over the next three-months your stocks might fall significantly because of current global conditions. You could then buy 3-month puts to protect your portfolio from heavy loses. Buying puts in this case amounts to buying portfolio insurance.

More formally, the payoff of a protective put on a non-dividend paying stock as a function of the stock price \(S\) can be described as follows:

	\(S \leq K\)	\(S > K\)
Long Stock	\(S\)	\(S\)
Long Put	\(K - S\)	0
Protective Put	\(K\)	\(S\)

The table confirms the payoff function pictured previously. The protective put pays like the stock if \(S > K\) and caps the payoff at \(K\) otherwise.

Example 6.2

A non-dividend paying stock currently trades at $50. A put option with strike $40 and maturity 3 months sells for $1.28. A protective put with the same characteristics would then cost 50 + 1.28 = $51.28. Below are some possible protective put payoffs and profits for different stock prices at maturity:

Stock Price	30	40	50	60	70
Payoff	40	40	50	60	70
Profit	-11.28	-11.28	-1.28	8.72	18.72

The protective put caps the losses at 40 - 51.28 = -$11.28 no matter how low is the stock price at maturity. This comes at a cost if the stock price goes up, though.

6.2.3. Straddle

A straddle is a two-leg option strategy that consists in buying a call and a put with the same strike \(K\).

The payoff of a straddle can then be described as follows:

	\(S \leq K\)	\(K < S\)
Long Put	\(K - S\)	0
Long Call	0	\(S - K\)
Straddle	\(K - S\)	\(S - K\)

The straddle pays off when the stock price moves a lot.

Example 6.3

A non-dividend paying stock currently trades at $50. A put and a call with strike \(K = \$50\) cost $4.68 and $7.12, respectively. A straddle with the same strike then costs 4.68 + 7.12 = $11.80. Below are some possible straddle payoffs and profits for different stock prices at maturity:

Stock Price	30	40	50	60	70
Payoff	20	10	0	10	20
Profit	8.20	-1.80	-11.80	-1.80	8.20

The straddle makes a profit if the stock moves below $38.20 or above $61.80.

6.2.4. Strangle

A strangle is a two-leg option strategy that consists in a long call with strike \(K_{2}\) and a long put with strike \(K_{1}\) where \(K_{1} < K_{2}\). The payoff of a strangle can then be described as follows:

	\(S \leq K_{1}\)	\(K_{1} < S \leq K_{2}\)	\(K_{2} < S\)
Long Put	\(K_{1} - S\)	0	0
Long Call	0	0	\(S - K_{2}\)
Strangle	\(K_{1} - S\)	0	\(S - K_{2}\)

Compared to the straddle, the strangle requires the stock price to move even more in order to make a profit.

Example 6.4

A non-dividend paying stock currently trades at $50. A put with strike \(K_{1} = \$45\) trades for $2.65 whereas a call with strike \(K_{2} = \$55\) costs $5.01. A strangle with strikes \(K_{1}\) and \(K_{2}\) then costs 2.65 + 5.01 = $7.66. Below are some possible strangle payoffs and profits for different stock prices at maturity:

Stock Price	35	40	45	50	55	60	65
Payoff	10	5	0	0	0	5	10
Profit	2.34	-2.66	-7.66	-7.66	-7.66	-2.66	2.34

The strangle makes a profit if the stock moves below $37.34 or above $62.66.

6.3. Options Spreads

6.3.1. Bull Spread

A bull spread is a two-leg option strategy that consists in a long position in a call with strike \(K_{1}\) and a short position in a call with strike \(K_{2}\), where \(K_{1} < K_{2}\). The payoff of a bull spread can then be described as follows:

	\(S \leq K_{1}\)	\(K_{1} < S \leq K_{2}\)	\(S > K_{2}\)
Long Call	0	\(S - K_{1}\)	\(S - K_{1}\)
Short Call	0	0	\(-(S - K_{2})\)
Bull Spread	0	\(S - K_{1}\)	\(K_{2} - K_{1}\)

If \(K_{2} - K_{1}\) is small, the bull spread is like an all-or-nothing bet on the stock going above \(K_{2}\).

Example 6.5

A non-dividend paying stock currently trades at $50. Call options with strikes \(K_{1} = \$40\) and \(K_{2} = \$60\) trade for $13.23 and $3.45, respectively. A bull spread that goes long the call with strike \(K_{1}\) and shorts the call with strike \(K_{2}\) costs 13.23 - 3.45 = $9.78. Below are some possible bull spread payoffs and profits for different stock prices at maturity:

Stock Price	30	40	50	60	70
Payoff	0	0	10	20	20
Profit	-9.78	-9.78	0.22	10.22	10.22

The bull spread caps the maximum gains and losses at 20 - 9.78 = $10.22 and 0 - 9.78 = -$9.78, respectively.

6.3.2. Bear Spread

A bear spread is a two-leg option strategy that consists in a long position in a put with strike \(K_{2}\) and a short position in a put with strike \(K_{1}\), where \(K_{1} < K_{2}\). The payoff of a bear spread can then be described as follows:

	\(S \leq K_{1}\)	\(K_{1} < S \leq K_{2}\)	\(S > K_{2}\)
Long Put	\(K_{2} - S\)	\(K_{2} - S\)	0
Short Put	\(-(K_{1} - S)\)	0	0
Bear Spread	\(K_{2} - K_{1}\)	\(K_{2} - S\)	0

If \(K_{2} - K_{1}\) is small, the bear spread is like an all-or-nothing bet on the stock going below \(K_{1}\).

Example 6.6

A non-dividend paying stock currently trades at $50. Put options with strikes \(K_{1} = \$40\) and \(K_{2} = \$60\) trade for $1.28 and $10.53, respectively. A bear spread that goes long the put with strike \(K_{2}\) and shorts the put with strike \(K_{1}\) costs 10.53 - 1.28 = $9.25. Below are some possible bear spread payoffs and profits for different stock prices at maturity:

Stock Price	30	40	50	60	70
Payoff	20	20	10	0	0
Profit	10.75	10.75	0.75	-9.25	-9.25

The bear spread caps the maximum gains and losses at 20 - 9.25 = $10.75 and 0 - 9.25 = -$9.25, respectively.

6.3.3. Butterfly

A butterfly is a three-leg option strategy that consists in a long call with strike \(K_{1}\), short two calls with strike \(K_{2}\) and a long call with strike \(K_{3}\) where \(K_{1} < K_{2} < K_{3}\) and \(K_{2} = (K_{1} + K_{3}) / 2\). The payoff of a butterfly can then be described as follows:

	\(S \leq K_{1}\)	\(K_{1} < S \leq K_{2}\)	\(K_{2} < S \leq K_{3}\)	\(S > K_{3}\)
Long Call	0	\(S - K_{1}\)	\(S - K_{1}\)	\(S - K_{1}\)
2 x Short Call	0	0	\(2 (K_{2} - S)\)	\(2 (K_{2} - S)\)
Long Call	0	0	0	\(S - K_{3}\)
Butterfly	0	\(S - K_{1}\)	\(K_{3} - S\)	0

The butterfly is a bull’s eye bet on the stock price around \(K_{2}\)!

Example 6.7

A non-dividend paying stock currently trades at $50. Call options with strikes \(K_{1} = \$45\), \(K_{2} = \$50\) and \(K_{3} = \$55\) trade for $9.85, $7.12 and $5.01, respectively. A butterfly with strikes \(K_{1}\), \(K_{2}\) and \(K_{3}\) then costs 9.85 - 2(7.12) + 5.01 = $0.63. Below are some possible straddle payoffs and profits for different stock prices at maturity:

Stock Price	35	40	45	50	55	60	65
Payoff	0	0	0	5	0	0	0
Profit	-0.62	-0.62	-0.62	4.38	-0.62	-0.62	-0.62

The butterfly makes a profit if the stock stays very close to $50. The return of getting the bet right is big. If the stock price ends up at $50 at maturity then you would make 4.38/0.62 = 706% on your investment!

Note that the butterfly can also be obtained by buying puts with strikes \(K_{1}\) and \(K_{3}\), and shorting two puts with strikes \(K_{2} = (K_{1} + K_{3})/2\).

	\(S \leq K_{1}\)	\(K_{1} < S \leq K_{2}\)	\(K_{2} < S \leq K_{3}\)	\(S > K_{3}\)
Long Put	\(K_{1} - S\)	0	0	0
2 x Short Put	\(2 (S - K_{2})\)	\(2 (S - K_{2})\)	0	0
Long Put	\(K_{3} - S\)	\(K_{3} - S\)	\(K_{3} - S\)	0
Butterfly	0	\(S - K_{1}\)	\(K_{3} - S\)	0

Using the data of example Example 6.7 yields the following payoff diagram:

No-arbitrage then implies that:

\[ P_{1} - 2 P_{2} + P_{3} = C_{1} - 2 C_{2} + C_{3} \geq 0, \]

which in turn implies that \(P_{2} \leq \dfrac{P_{1} + P_{3}}{2}\) and \(C_{2} \leq \dfrac{C_{1} + C_{3}}{2}\).

6.3.4. Condor

Similar to the butterfly, a condor is a four-leg option strategy that consists in a long call with strike \(K_{1}\), a short call with strike \(K_{2}\), a short call with strike \(K_{3}\) and a long call with strike \(K_{4}\) where \(K_{1} < K_{2} < K_{3} < K_{4}\) with \(K_{2} - K_{1} = K_{4} - K_{3}\). The payoff of a condor can then be described as follows:

	\(S \leq K_{1}\)	\(K_{1} < S \leq K_{2}\)	\(K_{2} < S \leq K_{3}\)	\(K_{3} < S \leq K_{4}\)	\(S > K_{4}\)
Long Call	0	\(S - K_{1}\)	\(S - K_{1}\)	\(S - K_{1}\)	\(S - K_{1}\)
Short Call	0	0	\(K_{2} - S\)	\(K_{2} - S\)	\(K_{2} - S\)
Short Call	0	0	0	\(K_{3} - S\)	\(K_{3} - S\)
Long Call	0	0	0	0	\(S - K_{4}\)
Condor	0	\(S - K_{1}\)	\(K_{2} - K_{1}\)	\(K_{4} - S\)	0

Example 6.8

A non-dividend paying stock currently trades at $50. Call options with strikes \(K_{1} = \$40\), \(K_{2} = \$45\), \(K_{3} = \$55\) and \(K_{4} = \$60\) trade for $13.23, $9.85, $5.01 and $3.45, respectively. A condor with strikes \(K_{1},\) \(K_{2},\) \(K_{3},\) and \(K_{4}\) then costs 13.23 - 9.85 - 5.01 + 3.45 = $1.82. Below are some possible straddle payoffs and profits for different stock prices at maturity:

\(S\)	35	40	45	50	55	60	35
Payoff	0	0	5	5	5	0	0
Profit	-1.82	-1.82	3.18	3.18	3.18	-1.82	-1.82

The condor makes a profit if the stock stays between $41.82 and $58.18.

6.4. Practice Problems

Exercise 6.1

Suppose you think FedEx stock is going to appreciate substantially in value in the next 6 months. Say the stock’s current price is $100, and the call option expiring in 6 months has an exercise price of $100 and is selling at a price of $10. With $10,000 to invest, you are considering three alternatives:

• Invest all $10,000 in the stock, buying 100 shares.

• Invest all $10,000 in 1,000 options (10 contracts).

• Buy 100 options (one contract) for $1,000 and invest the remaining $9,000 in a money market fund paying 8% per year compounded semi-annually, i.e., 4% every six months.

The total value of your portfolio in six months for each of the following stock prices is:

Price of Stock	80	100	110	120
All stocks (100 shares)
All options (1,000 options)
Bills + 100 options

The percentage return of your portfolio in six months for each of the following stock prices is:

Price of Stock	80	100	110	120
All stocks (100 shares)
All options (1,000 options)
Bills + 100 options

Solution to Exercise 6.1

The total value of your portfolio in six months for each of the following stock prices is:

Stock Price	80	100	110	120
All stocks (100 shares)	8,000	10,000	11,000	12,000
All options (1,000 options)	0	0	10,000	20,000
Bills + 100 options	9,360	9,360	10,360	11,360

The percentage return of your portfolio in six months for each of the following stock prices is:

Stock Price	80	100	110	120
All stocks (100 shares)	-20%	0%	10%	20%
All options (1,000 options)	-100%	-100%	0%	100%
Bills + 100 options	-6.40%	-6.40%	3.60%	13.60%

Exercise 6.2

Imagine that you are holding 5,000 shares of stock, currently selling at $40 per share. You are ready to sell the shares but would prefer to put off the sale until next year for tax reasons. If you continue to hold the shares until January, however, you face the risk that the stock will drop in value before year-end. You decide to use a collar to limit downside risk without laying out a good deal of additional funds. January call options with a strike price of $45 are selling at $2, and January put options with a strike price of $35 are selling at $3. Assume that you hedge the entire 5,000 shares of stock.

1. What will be the value of your portfolio in January (net of the proceeds from the options) if the stock price ends up at $30.

2. What will be the value of your portfolio in January (net of the proceeds from the options) if the stock price ends up at $40?

3. What will be the value of your portfolio in January (net of the proceeds from the options) if the stock price ends up at $50?

Solution to Exercise 6.2

To build this collar you need to buy 5,000 puts and write 5,000 calls. This will cost \(5\,000 \times (3 - 2) = \$5\,000.\) To compute the final portfolio value, we account for the value of the shares, the payoffs of the call and the put and the cost of the options.

Position	\(S = \$30\)	\(S = \$40\)	\(S = \$50\)
Initial stock portfolio	150,000	200,000	250,000
Payoff Short Call	0	0	-25,000
Payoff long put	25,000	0	0
Cost of options	-5,000	-5,000	-5,000
Final portfolio value	170,000	195,000	220,000

Exercise 6.3

Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create a bull spread and a bear spread? Construct a table that shows the profit and payoff for both spreads as a function of \(S\) when \(S \leq 30\), \(30 < S \leq 35\) and \(S > 35\).

Solution to Exercise 6.3

The bull spread is synthesized by going long the put with strike $30 and short the put with strike $35. The cost of the strategy is 4 - 7 = -$3. The payoff and profit as a function of \(S\) is:

	\(S \leq 30\)	\(30 < S \leq 35\)	\(S > 35\)
Payoff	\(-5\)	\(S - 35\)	\(0\)
Profit	\(-2\)	\(S - 32\)	\(3\)

The bear spread is synthesized by going short the put with strike $30 and long the put with strike $35. The cost of the strategy is -4 + 7 = $3. The payoff and profit as a function of \(S\) is:

	\(S \leq 30\)	\(30 < S \leq 35\)	\(S > 35\)
Payoff	\(5\)	\(35 - S\)	\(0\)
Profit	\(2\)	\(32 - S\)	\(-3\)

Exercise 6.4

A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle as a function of \(S\) when \(S \leq 60\) and \(S > 60\). For what range of stock prices would the straddle lead to a loss?

Solution to Exercise 6.4

The straddle costs 6 + 4 = $10. The payoff and profit as a function of \(S\) is:

	\(S \leq 60\)	\(S > 60\)
Payoff	\(60 - S\)	\(S - 60\)
Profit	\(50 - S\)	\(S - 70\)

The straddle is at a loss if \(50 < S < 70\).

Exercise 6.5

Three put options on a stock have the same expiration date and strike prices of $55, $60, and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy as a function of \(S\). For what range of stock prices would the butterfly spread lead to a loss?

Solution to Exercise 6.5

The butterfly spread is synthesized by going long the put with strike $55, short two puts with strike $60 and long the put with strike $65. The cost of the strategy is \(3 - 2 \cdot 5 + 8 = \$1\).

	\(S \leq 55\)	\(55 < S \leq 60\)	\(60 < S \leq 65\)	\(S > 65\)
Payoff	\(0\)	\(S - 55\)	\(65 - S\)	\(0\)
Profit	\(-1\)	\(S - 56\)	\(64 - S\)	\(-1\)

The butterfly spread is at a loss if \(S < 56\) or \(S > 64\).

Exercise 6.6

The price of a stock is $40. The price of a one-year European put option on the stock with a strike price of $30 is quoted as $7 and the price of a one-year European call option on the stock with a strike price of $50 is quoted as $5. Suppose that an investor buys 100 shares, shorts 100 call options, and buys 100 put options. Complete the following table for the strategy:

Stock Price	35	45	55	65
Payoff
Profit

Solution to Exercise 6.6

The payoff of the strategy is computed as follows:

Stock Price	35	45	55	65
Long 100 Shares	3,500	4,500	5,500	6,500
Short 100 Call	0	0	-500	-1,500
Long 100 Put	0	0	0	0
Payoff	3,500	4,500	5,000	5,000

The profit is computed by subtracting the cost of the strategy, which is 100 (40 + 7 - 5) = $4,200, to the payoff:

Stock Price	35	45	55	65
Payoff	3,500	4,500	5,000	5,000
Profit	-700	300	800	800

Exercise 6.7

Stock XYZ trades now for $75. You have the following information on different call and put options prices written on stock XYZ and expiring in 1 year:

Strike	Call	Put
$70	$10.76	$2.42
$80	$5.44	$6.22

Consider a strangle in which you purchase a put with strike $70 and purchase a call with strike $80. What is the profit of the strategy if the stock price at maturity is $60?

Solution to Exercise 6.7

The cost of the strategy is \(2.42 + 5.44 = \$7.86.\) If the stock price at maturity is $60, then only the put is in the money and its payoff is \(70 - 60 = \$10.\) Therefore, the profit is \(10 - 7.86 = \$2.14.\)

Exercise 6.8

Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create a bear spread?

Solution to Exercise 6.8

You need to go short the put with strike $30 and long the put with strike $35.

Exercise 6.9

Suppose you purchase one call option written on stock WFM, expiring in May, with strike price $100, for $5. At the same time, you write one call on WFM, expiring in May, with strike $105, for $2. If at expiration the price of a share of WFM stock is $103, compute the profit per share.

Solution to Exercise 6.9

The cost of the strategy is \(5 - 2 = \$3.\) If at expiration the price of a share of WFM stock is $103, then only the call with strike $100 is in the money. The payoff of the strategy is then \(103 - 100 = \$3.\) Thus, the profit is $0.

Exercise 6.10

Suppose you think that there is a small possibility that XYZ stock might depreciate substantially in value in the next 3 months. Say the stock’s current price is $200, and a put option expiring in 3 months has an exercise price of $180 and is selling at a premium of $5. With $10,000 to invest, you are considering investing $6,000 in the stock (30 shares) and $4,000 in puts (800 options). Compute the profit of your portfolio 3 months from now if the price of XYZ stock is $160.

Solution to Exercise 6.10

The payoff of the portfolio is \(30 \times 160 + 800 \times (180 - 160) = \$20{,}800.\) Since the portfolio has an initial cost of $10,000, your profit is $10,800.