Box Spread (Long Box)

The box spread, or long box, is a common arbitrage strategy that involves buying a bull call spread together with the corresponding bear put spread, with both vertical spreads having the same strike prices and expiration dates. The long box is used when the spreads are underpriced in relation to their expiration values.

Limited Risk-free Profit

Essentially, the arbitrager is simply buying and selling equivalent spreads and as long as the price paid for the box is significantly below the combined expiration value of the spreads, a riskless profit can be locked in immediately. 

Expiration Value of Box = Higher Strike Price - Lower Strike Price

Risk-free Profit = Expiration Value of Box - Net Premium Paid

Example

Suppose XYZ stock is trading at $45 in June and the following prices are available:

• JUL 40 put - $1.50
• JUL 50 put - $6
• JUL 40 call - $6
• JUL 50 call - $1

Buying the bull call spread involves purchasing the JUL 40 call for $600 and selling the JUL 50 call for $100. The bull call spread costs: $600 - $100 = $500

Buying the bear put spread involves purchasing the JUL 50 put for $600 and selling the JUL 40 put for $150. The bear put spread costs: $600 - $150 = $450

The total cost of the box spread is: $500 + $450 = $950

The expiration value of the box is computed to be: ($50 - $40) x 100 = $1000. 

Since the total cost of the box spread is less than its expiration value, a riskfree arbitrage is possible with the long box strategy. It can be observed that the expiration value of the box spread is indeed the difference between the strike prices of the options involved.

If XYZ remain unchanged at $45, then the JUL 40 put and the JUL 50 call expire worthless while both the JUL 40 call and the JUL 50 put expires in-the-money with $500 intrinsic value each. So the total value of the box at expiration is: $500 + $500 = $1000.

Suppose, on expiration in July, XYZ stock rallies to $50, then only the JUL 40 call expires in-the-money with $1000 in intrinsic value. So the box is still worth $1000 at expiration.

What happens when XYZ stock plummets to $40? A similar situation happens but this time it is the JUL 50 put that expires in-the-money with $1000 in intrinsic value while all the other options expire worthless. Still, the box is worth $1000.

As the trader had paid only $950 for the entire box, his profit comes to $50.

Commissions

These examples exclude commissions and fees. Include all transaction costs when evaluating the profit or loss of a position.

Short Box

The box spread is profitable when the component spreads are underpriced. Conversely, when the box is overpriced, you can sell the box for a profit. This strategy is known as a short box.