Options Arbitrage

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Short Box

The short box is an arbitrage strategy that involves selling a bull call spread together with the corresponding bear put spread with the same strike prices and expiration dates. The short box is a strategy that is used when the spreads are overpriced with respect to their combined expiration value.

Limited Risk-free Profit

Basically, with the short box, the arbitrager is just buying and selling equivalent spreads and as long as the net premium obtained for the selling the two spreads is significantly higher than the combined expiration value of the spreads, a risk-free profit can be captured upon entering the trade. 

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

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

Example

Suppose XYZ stock is trading at $55 in July and the following prices are available:

• AUG 50 put - $2
• AUG 60 put - $7
• AUG 50 call - $7
• AUG 60 call - $1.50

Selling the bull call spread involves shorting the AUG 50 call for $700 while buying the AUG 60 call for $150. The premiums collected from the sale of the bull call spread is: $700 - $150 = $550

Selling the bear put spread involves shorting the AUG 60 put for $700 while buying the AUG 50 put for $200. The premiums collected from the sale of the bear put spread comes to: $700 - $200 = $500

Together, the net premium received for shorting the box is: $550 + $500 = $1050

Since the total price of the box spread is more than its expiration value, a riskfree arbitrage is possible using the short box strategy. Selling the box will result in a net premium received of $1050. It can be observed that the expiration value of the box spread is indeed the difference between the strike prices of the options involved. The expiration value of the box is computed to be: ($50 - $40) x 100 = $1000.

If XYZ remain unchanged at $55, then the AUG 50 put and the AUG 60 call expire worthless while both the AUG 50 call and the AUG 60 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 August, XYZ stock rallies to $60, then only the AUG 50 call expires in-the-money with $1000 in intrinsic value. So the box is still worth $1000 at expiration.

So what happens when XYZ stock plunges to $50? A similar situation occurs but this time it is the AUG 60 put that expires in-the-money with $1000 in intrinsic value while all the other options expire worthless. Hence, the box is still worth $1000.

As the trader had collected $1050 for shorting the box, his profit comes to $50 after buying it back for $1000 on expiration date.

Note: While we have covered the use of this strategy with reference to stock options, the short box is equally applicable using ETF options, index options as well as options on futures.

Commissions

As the gains from the short box is very minimal, the commissions payable when implementing this strategy can often wipe out all of the profits. Thus, one have to take into careful consideration the commissions involved when contemplating the use of this strategy.

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Long Box

The short box is profitable when the component spreads are overpriced. When the spreads are underpriced, the converse strategy known as the long box, or simply box spread, is used instead.