Conventions And Assumptions

• For convenience, it will be assumed that the asset underlying the options considered is a stock. (Similar trading strategies can be developed for other underlying assets.)
• In figures throughout this chapter, the dashed line shows the relationship between profit and the stock price for the individual securities constituting the portfolio, whereas the solid line shows the relationship between profit and the stock price for the whole portfolio.

Covered Call

• A covered call consists of a long position in a stock plus a short position in a European call option. This is known as writing a covered call. The long stock position “covers” or protects the investor from the payoff on the short call that becomes necessary if there is a sharp rise in the stock price.
• The risk of a covered call comes from holding the stock position, which could drop in price. The maximum loss occurs if the stock goes to zero.
• The main goal of the covered call is to collect income via option premiums by selling calls against a stock that you already own. Assuming the stock doesn’t move above the strike price, the premium is collected, and stock position is maintained. The overall outlook is neutral to slightly bullish.

Resemblance to Short Put Profit Diagram

• The put-call parity relationship is

where

p is the price of a European put,

S₀ is the stock price,

c is the price of a European call,

K is the strike price of both call and put,

r is the risk-free interest rate,

T is the time to maturity of both call and put,

D is the present value of the dividends anticipated during the life of the options.

• The above equation can be rearranged to become

Protective Put

• This investment strategy involves buying a European put option on a stock and the stock itself. Protective put strategy is also called portfolio insurance or a hedged portfolio.
• The risk of a protective put comes from paying the premium for the put option. The overall profit is reduced by the amount of premium paid.
• The main goal of the protective put is to protect against steep decline in stock price.

Resemblance to Long Call Profit Diagram

• The put-call parity relationship is

Spreads

• A spread trading strategy involves taking a position in two or more options of the same type (i.e., two or more calls or two or more puts).
• The three main classes of spreads are the horizontal spread, the vertical spread, and the diagonal spread. They are grouped by the relationships between the strike price and expiration dates of the options involved.

1. Vertical spreads, or money spreads, are spreads involving options of the same underlying security, same expiration month, but at different strike prices.
2. Horizontal, calendar spreads, or time spreads are created using options of the same underlying security, same strike prices but with different expiration dates.
3. Diagonal spreads are constructed using options of the same underlying security but different strike prices and expiration dates. They are called diagonal spreads because they are a combination of vertical and horizontal spreads.

Bull Call Spreads

• This can be created by buying a European call option on a stock with a certain strike price and selling a European call option on the same stock with a higher strike price. Both options have the same expiration date. Because a call price always decreases as the strike price increases, the value of the option sold is always less than the value of the option bought. A bull spread, when created from calls, therefore requires an initial investment.
• Suppose that

K₁ is the strike price of the call option bought,

K₂ is the strike price of the call option sold,

S_T is the stock price on the expiration date of the options.

• This table shows the total payoff that will be realized from a bull spread in different circumstances. If the stock price does well and is greater than the higher strike price, the payoff is the difference between the two strike prices, or K₂ – K₁. If the stock price on the expiration date lies between the two strike prices, the payoff is S_T – K₁. If the stock price on the expiration date is below the lower strike price, the payoff is zero.

Stock price range	Payoff from long call option	Payoff from short call option	Total payoff
ST ≤ K1	0	0	0
K1 < ST < K2	ST – K1	0	ST – K1
ST ≥ K2	ST – K1	-(ST – K2)	K2 – K1

• The profit in this figure is calculated by subtracting the initial investment from the payoff.

• An investor buys for $3 a 3-month European call with a strike price of $30 and sells for $1 a 3-month European call with a strike price of $35. The payoff from this bull spread strategy is 0 if the stock price is above $35, and 5 if it is below $30. If the stock price is between $30 and $35, the payoff is 35 – S_T. The cost of the strategy is 3-1 = $2.

Stock Price Range	Payoff	Profit
ST ≤ 30	0	0 – 2 = -2
30 < ST < 35	ST – 30	ST – 30 – 2 = ST – 32
ST ≥ 35	5	5 – 2 = 3

• A bull spread strategy limits the investor’s upside as well as downside risk. The strategy can be described by saying that the investor has a call option with a strike price equal to K, and has chosen to give up some upside potential by selling a call option with strike price K₂ (K₂ > K₁). In return for giving up the upside potential, the investor gets the price of the option with strike price K₂.
• Three types of bull spreads can be distinguished:

1. Both calls are initially out of the money.
2. One call is initially in the money; the other call is initially out of the money.
3. Both calls are initially in the money.

• The most aggressive bull spreads are those of type 1. They cost very little to set up and have a small probability of giving a relatively high payoff (= K₂ – K₁). As we move from type 1 to type 2 and from type 2 to type 3, the spreads become more conservative.

Bear Put Spreads

• Bull spreads can also be created by buying a European put with a low strike price and selling a European put with a high strike price, as illustrated in this figure. Unlike bull spreads created from calls, those created from puts involve a positive up-front cash flow to the investor, but have margin requirements and a payoff that is either negative or zero.

• An investor who enters into a bull spread is hoping that the stock price will increase. By contrast, an investor who enters into a bear spread is hoping that the stock price will decline. Bear spreads can be created by buying a European put with one strike price and selling a European put with another strike price. The strike price of the option purchased is greater than the strike price of the option sold. (This is in contrast to a bull spread, where the strike price of the option purchased is always less than the strike price of the option sold.) A bear spread created from puts involves an initial cash outflow because the price of the put sold is less than the price of the put purchased. In essence, the investor has bought a put with a certain strike price and chosen to give up some of the profit potential by selling a put with a lower strike price. In return for the profit given up, the investor gets the price of the option sold.
• Assume that the strike prices are K₁ and K₂ with K₁ < K₂.

• This table shows the payoff that will be realized from a bear spread in different circumstances. If the stock price is greater than K₂, the payoff is zero. If the stock price is less than K₁, the payoff is K₂ – K₁. If the stock price is between K₁ and K₂, the payoff is K₂ – S_T. The profit is calculated by subtracting the initial cost from the payoff.

Stock Price Range	Payoff from Long Call Option	Payoff from Short Call Option	Total Payoff
ST ≤ K1	K2 – ST	-(K1 – ST)	K2 – K1
K1 < ST < K2	K2 – ST	0	K2 – ST
ST ≥ K2	0	0	0

• In this figure, the profit from the spread is shown by the solid line.

Bear Put Spreads – Example

• An investor buys for $3 a 3-month European put with a strike price of $35 and sells for $1 a 3-month European put with a strike price of $30. The payoff from this bear spread strategy is 0 if the stock price is above $35, and 5 if it is below $30. If the stock price is between $30 and $35, the payoff is 35 – S_T. The cost of the strategy is 3 – 1 = $2.

Stock Price Range	Payoff	Profit
ST ≤ 30	5	5 – 2 = 3
30 < ST < 35	35 – ST	35 – ST – 2 = 33 – ST
ST ≥ 35	0	0 – 2 = -2

Bear Call Spreads

• Like bull spreads, bear spreads limit both the upside profit potential and the downside risk.
• Bear spreads can be created using calls instead of puts. The investor buys a call with a high strike price and sells a call with a low strike price, as illustrated in this figure. Bear spreads created with calls involve an initial cash inflow (ignoring margin requirements).

Box Spread

• A box spread is a combination of a bull call spread with strike prices K₁ and K₂ and a bear put spread with the same two strike prices. As shown in this table, the payoff from a box spread is always K₂ – K₁. The value of a box spread is therefore always the present value of this payoff or . If it has a different value there is an arbitrage opportunity.

Stock Price Range	Payoff	Profit
ST ≤ 30	5	5 – 2 = 3
30 < ST < 35	35 – ST	35 – ST – 2 = 33 – ST
ST ≥ 35	0	0 – 2 = -2

• If the market price of the box spread is too low, it is profitable to buy the box. This involves
  • buying a call with strike price K₁,
  • buying a put with strike price K₂,
  • selling a call with strike price K₂, and
  • selling a put with strike price K₁.
• If the market price of the box spread is too high, it is profitable to sell the box. This involves
  • buying a call with strike price K₂,
  • buying a put with strike price K₁,
  • selling a call with strike price K₁, and
  • selling a put with strike price K₂.

Butterfly Spreads

• A butterfly spread involves positions in options with three different strike prices. It can be created by buying a European call option with a relatively low strike price K₁, buying a European call option with a relatively high strike price K₃, and selling two European call options with a strike price K₂ that is halfway between K₁ and K₃. Generally, K₂ is close to the current stock price. The pattern of profits from the strategy is shown in this figure. A butterfly spread leads to a profit if the stock price stays close to K₂, but gives rise to a small loss if there is a significant stock price move in either direction. It is therefore an appropriate strategy for an investor who feels that large stock price moves are unlikely. The strategy requires a small investment initially.

• Suppose that a certain stock is currently worth $61. Consider an investor who feels that a significant price move in the next 6 months is unlikely. Suppose that the market prices of 6-month European calls are as follows:

• The investor could create a butterfly spread by buying one call with a $55 strike price, buying one call with a $65 strike price, and selling two calls with a $60 strike price.

• It costs $10 + $5 – (2 x $7) = 15 – 14 = $1 to create the spread.
• If the stock price in 6 months is greater than $65 or less than $55, the total payoff is 0, and the investor incurs a net loss of $1 (cost).
• If the stock price is between $56 and $64, a profit is made.
• The maximum profit, $4, occurs when the stock price in 6 months is $60.

STRIKE PRICE($)	CALL PRICE($)
55	10
60	7
65	5

• Suppose that a certain stock is currently worth $61. Consider an investor who feels that a significant price move in the next 6 months is unlikely. Suppose that the market prices of 6-month European calls are as follows:
  • Butterfly spreads can be created using put options. The investor buys two European puts, one with a low strike price and one with a high strike price, and sells two European puts with an intermediate strike price, as illustrated in this figure.

• • If a butterfly spread has to be created with put options where the strike prices are the same as the previous example which used call options, then it would be created by buying one put with a strike price of $55, another with a strike price of $65, and selling two puts with a strike price of $60. The use of put options results in exactly the same spread as the use of call options. Put-call parity can be used to show that the initial investment is the same in both cases.
  • A butterfly spread can be sold or shorted by following the reverse strategy. Options are sold with strike prices of K₁ and K₃, and two options with the middle strike price K₂ are purchased. This strategy produces a modest profit if there is a significant movement in the stock price.

Calendar Spreads

• A calendar spread can be created by selling a European call option with a certain strike price and buying a longer-maturity European call option with the same strike price. The longer the maturity of an option, the more expensive it usually is. A calendar spread therefore usually requires an initial investment.
• Profit diagrams for calendar spreads are usually produced so that they show the profit when the short-maturity option expires on the assumption that the long-maturity option is closed out at that time. The profit pattern for a calendar spread produced from call options is shown in this figure. The pattern is similar to the profit from the butterfly spread. The investor makes a profit if the stock price at the expiration of the short-maturity option is close to the strike price of the short-maturity option. However, a loss is incurred when the stock price is significantly above or significantly below this strike price.

• Calendar spreads can be created with put options as well as call options. The investor buys a long-maturity put option and sells a short-maturity put option. As shown in this figure, the profit pattern is similar to that obtained from using calls.

Diagonal Spreads

• In a diagonal spread both the expiration date and the strike price of the calls are different. This increases the range of profit patterns that are possible.

Combinations

• A combination is an option trading strategy that involves taking a position in both calls and puts on the same stock. The following combination strategies will be considered:
  • Straddles
  • Strips
  • Straps
  • Strangles

Straddles

• A straddle involves buying a European call and put with the same strike price and expiration date. The profit pattern is shown in this figure. The strike price is denoted by K. If the stock price is close to this strike price at expiration of the options, the straddle leads to a loss. However, if there is a sufficiently large move in either direction, a significant profit will result.
• A straddle is appropriate when an investor is expecting a large move in a stock price but does not know in which direction the move will be.
• The straddle in this figure is sometimes referred to as a bottom straddle or straddle purchase. A top straddle or straddle write is the reverse position. It is created by selling a call and a put with the same exercise price and expiration date. It is a highly risky strategy. If the stock price on the expiration date is close to the strike price, a profit results. However, the loss arising from a large move is unlimited.

Strips

• A strip consists of a long position in one European call and two European puts with the same strike price and expiration date. The profit patterns from strips are shown in this figure. In a strip the investor is betting that there will be a big stock price move and considers a decrease in the stock price to be more likely than an increase.

Straps

• A strap consists of a long position in two European calls and one European put with the same strike price and expiration date. The profit patterns from straps are shown in this figure. In a strap the investor is also betting that there will be a big stock price move. However, in this case, an increase in the stock price is considered to be more likely than a decrease.

Strangles

• In a strangle, sometimes called a bottom vertical combination, an investor buys a European put and a European call with the same expiration date and different strike prices. The profit pattern is shown in this figure. The call strike price, K₂ is higher than the put strike price, K₁. A strangle is a similar strategy to a straddle. The investor is betting that there will be a large price move, but is uncertain whether it will be an increase or a decrease.

• The stock price has to move farther in a strangle than in a straddle for the investor to make a profit. However, the downside risk if the stock price ends up at a central value is less with a strangle. The profit pattern obtained with a strangle depends on how close together the strike prices are. The farther they are apart, the less the downside risk and the farther the stock price has to move for a profit to be realized.
• The sale of a strangle is sometimes referred to as a top vertical combination. It can be appropriate for an investor who feels that large stock price moves are unlikely. However, as with the sale of a straddle, it is a risky strategy involving unlimited potential loss to the investor.