Spreads

Option spreads and combinations can be useful option strategies. We first consider money spreads, in which the two options differ only by exercise price. The investor buys an option with a given expiration and exercise price and sells an option with the same expiration but a different exercise price. Of course, the options are on the same underlying asset. The term spread is used here because the payoff is based on the difference, or spread, between option exercise prices. For a bull or bear spread, the investor buys one call and writes another call option with a different exercise price, or the investor buys one put and writes another put with a different exercise price.

Bull Call Spread

Summary – Buy low, sell high. Buy a lower strike call and sell a higher strike call. Capture the spread between the two as price moves towards higher call. For a call earn the right to gains above low strike, but give up gains above higher strike in order to save on the upfront premium. Money made by price reaching target. Using calls creates a debit spread (net outflow.) Using puts, buy low put, sell higher put. Selling a the higher put will generate higher premium than buying the lower strike put. Thus profit you make is the premium from selling an option, minus the cost of the downside protection. As price approaches higher put price, you will get closer and closer to that option expiring worthless, leaving you with full premium (minus lower put price to cap your loss). Money made by price staying at target.

Regardless of whether someone constructs a bull spread with puts or with calls, the strategy requires buying one option and writing another with a higher exercise price. Because the higher exercise price call is less expensive than the lower strike, a call bull spread involves an initial cash outflow (debit spread).

A bull spread created from puts also requires the investor to write the higher-strike option and buy the lower-strike one. Because the higher-strike put is more expensive, a put bull spread involves an initial cash inflow (credit spread).

The value of the spread at expiration (V_T) depends on the stock price at expiration S_T. For a bull spread, the investor buys the low strike option (struck at X_L) and sells the high strike option (struck at X_H), so that:

Equation (5) 

• V_T = Max(0,S_T – X_L) – Max(0,S_T – X_H).

Therefore, the value depends on the terminal stock price S_T:

• V_T = 0 − 0 = 0 if S_T ≤ X_L
• V_T = S_T − X_L − 0 = S_T − X_L if X_L < S_T < X_H
• V_T = S_T − X_L− (S_T − X_H) = X_H − X_L if S_T ≥ X_H

The profit is obtained by subtracting the initial outlay for the spread from the foregoing value of the spread at expiration. To determine the initial outlay, recall that a call option with a lower exercise price will be more expensive than a call option with a higher exercise price. Because we are buying the call with the lower exercise price (for c_L) and selling the call with the higher exercise price (for c_H), the call we buy will cost more than the call we sell (c_L > c_H). Hence, the spread will require a net outlay of funds. This net outlay is the initial value of the position, V₀ = c_L – c_H, which we call the net premium. The profit is:

Equation (6) 

• Π = Max(0,S_T – X_L) – Max(0,S_T – X_H) – (c_L – c_H).

In this manner, we see that the profit is the profit from the long call, Max(0,S_T – X_L) – c_L, plus the profit from the short call, –Max(0,S_T – X_H) + c_H. Broken down into ranges, the profit is as follows:

• Π = −c_L + c_H if S_T ≤ X_L
• Π = S_T − X_L − c_L + c_H if X_L < S_T < X_H
• Π = X_H − X_L − c_L + c_H if S_T ≥ X_H

If S_T is below X_L, the strategy will lose a limited amount of money. When both options expire out of the money, the investor loses the net premium, c_L – c_H. The profit on the upside, if S_T is at least X_H, is also limited to the difference in strike prices minus the net premium.

In general, for a bull call spread:

• Maximum loss = net premium paid
• Breakeven = lower strike + net premium paid
• Maximum profit = difference between strikes – net premium paid

The bull call spread is an example of a debit spread since it entails a net outlay: the bought call, with a lower strike, is more valuable than the sold call.

The other debit spread is the bear put spread.

Bear Put Spread

Summary – Buy high, sell low. Buy a higher strike put and sell a lower strike put. As price approaches lower put, higher strike is more profitable. Below low put, further gains have been sold. Net cash outflow as higher put will be more expensive. Money is made if price moves to target. Using calls, buy high call, sell low call. The lower call will be more expensive (ie. more in the money or closer to it) and will create a net inflow. As price moves down past the bought call, it will expire worthless, allow full premium to be captured. So money is made at start, and hope to keep it.

To construct a bear put spread, we sell a lower strike put and buy a higher strike put. Because puts with higher exercise prices are (all else equal) more expensive, a put bear spread will result in an initial cash outflow (be a debit spread). For a call bear spread, the investor buys a higher exercise price call and sells the lower exercise price call. Because the higher exercise price call being purchased is less expensive than the lower strike being sold, a call bear spread will result in an initial cash inflow (credit spread).

Mathematically, the value of this bear spread position at expiration is:

• V_T = Max(0,X_H – S_T) – Max(0,X_L – S_T).

Broken down into ranges, we have the following relations:

• V_T = X_H − S_T − (X_L − S_T) = X_H − X_L if S_T ≤ X_L
• V_T = X_H − S_T − 0 = X_H − S_T if X_L < S_T < X_H
• V_T = 0 − 0 = 0 if S_T ≥ X_H

To obtain the profit, we subtract the initial outlay. Because we are buying the put with the higher exercise price and selling the put with the lower exercise price, the put we are buying is more expensive than the put we are selling. The initial value of the bear spread is V₀ = p_H – p_L. The profit is, therefore, V_T – V₀, which is:

• Π = Max(0,X_H – S_T) – Max(0,X_L – S_T) – (p_H – p_L).

We see that the profit is that on the long put, Max(0,X_H – S_T) – p_H, plus the profit from the short put, –Max(0,X_L – S_T) + p_L. Broken down into ranges, the profit is as follows:

• Π = X_H − X_L − p_H + p_L if S_T ≤ X_L
• Π = X_H − S_T − p_H + p_L if X_L < S_T < X_H
• Π = −p_H + p_L if S_T ≥ X_H

The breakeven point, S_T* = X_H – p_H + p_L, sets the profit equal to zero between the strike prices. 

Generalized At-Expiration Formulas for Spreads

In these formulas, net premium means the absolute value of the difference between the premiums.

For debit spreads (bull call and bear put):

• Maximum loss = net premium paid
• Maximum profit = difference between strikes – net premium paid

For credit spreads (bear call and bull put):

• Maximum profit = net premium received
• Maximum loss = difference between strikes – net premium received

For call spreads, breakeven = lower strike + net premium

For put spreads, breakeven = higher strike – net premium

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