## What Is a Box Spread?

A box spread, or long box, is an options arbitrage strategy that combines buying a bull call spread with a matching bear put spread. A box spread can be thought of as two vertical spreads that each has the same strike prices and expiration dates.

Box spreads are used for borrowing or lending at implied rates that are more favorable than a trader going to their prime broker, clearing firm, or bank. Because the price of a box at its expiration will always be the distance between the strikes involved (e.g., a 100-pt box might utilize the 25 and 125 strikes and would be worth 100 at expiration), the price paid for today can be thought of as that of a zero-coupon bond. The lower the initial cost of the box, the higher its implied interest rate. This concept is known as a synthetic loan. ### Key Takeaways • A box spread is an options arbitrage strategy that combines buying a bull call spread with a matching bear put spread. • A box spread's ultimate payoff will always be the difference between the two strike prices. • The longer the time to expiration, the lower the market price of the box spread today. • The cost to implement a box spread—specifically, the commissions charged—can be a significant factor in its potential profitability. • Traders use box spreads to synthetically borrow or lend for cash management purposes. ## Understanding a Box Spread A box spread is optimally used when the spreads themselves are underpriced with respect to their expiration values. When the trader believes the spreads are overpriced, they may employ a short box, which uses the opposite options pairs, instead. The concept of a box comes to light when one considers the purpose of the two vertical, bull call and bear put, spreads involved. A bullish vertical spread maximizes its profit when the underlying asset closes at the higher strike price at expiration. The bearish vertical spread maximizes its profit when the underlying asset closes at the lower strike price at expiration. By combining both a bull call spread and a bear put spread, the trader eliminates the unknown, namely where the underlying asset closes at expiration. This is so because the payoff is always going to be the difference between the two strike prices at expiration. If the cost of the spread, after commissions, is less than the difference between the two strike prices, then the trader locks in a riskless profit, making it a delta-neutral strategy. Otherwise, the trader has realized a loss comprised solely of the cost to execute this strategy. Box spreads effectively establish synthetic loans. Like a zero-coupon bond, they are initially bought at a discount and the price steadily rises over time until expiration where it equals the distance between strikes. ## Box Spread Construction \begin{aligned} &\text{BVE}=\text{ HSP }-\text{ LSP}\\ &\text{MP}=\text{ BVE }-\text{ (NPP} + \text{ Commissions)}\\ &\text{ML }= \text{ NPP }+ \text{ Commissions}\\ &\textbf{where:}\\ &\text{BVE}=\text{ Box value at expiration}\\ &\text{HSP}=\text{ Higher strike price}\\ &\text{LSP}=\text{ Lower strike price}\\ &\text{MP}=\text{ Max profit}\\ &\text{NPP}=\text{ Net premium paid}\\ &\text{ML}=\text{ Max Loss} \end{aligned} To construct a box spread, a trader buys an in-the-money (ITM) call, sells an out-of-the-money(OTM) call, buys an ITM put, and sells an OTM put. In other words, buy an ITM call and put and then sell an OTM call and put. Given that there are four options in this combination, the cost to implement this strategy—specifically, the commissions charged—can be a significant factor in its potential profitability. Complex option strategies, such as these, are sometimes referred to as alligator spreads. There will be times when the box costs more than the spread between the strikes. Should this be the case, the long box would not work but a short box might. This strategy reverses the plan, selling the ITM options and buying the OTM options. ## Box Spread Example Company A stock trades for51.00. Each options contract in the four legs of the box controls 100 shares of stock. The plan is to:

• Buy the 49 call for 3.29 (ITM) for $329 debit per options contract • Sell the 53 call for 1.23 (OTM) for$123 credit
• Buy the 53 put for 2.69 (ITM) for $269 debit • Sell the 49 put for 0.97 (OTM) for$97 credit

The total cost of the trade before commissions would be $329 -$123 + $269 -$97 = $378. The spread between the strike prices is 53 - 49 = 4. Multiply by 100 shares per contract =$400 for the box spread.

In this case, the trade can lock in a profit of $22 before commissions. The commission cost for all four legs of the deal must be less than$22 to make this profitable. That is a razor-thin margin, and this is only when the net cost of the box is less than the expiration value of the spreads, or the difference between the strikes.

## Hidden Risks in Box Spreads

While box spreads are commonly used for cash management and are seen as a way to arbitrage interest rates with low risk, there are some hidden risks. The first is that interest rates may move strongly against you, causing losses like they would on any other fixed-income investments that are sensitive to rates.

A second potential danger, which is perhaps less obvious, is the risk of early exercise. American style options, such as those options listed on most U.S. stocks may be exercised early (i.e., before expiration), and so it is possible that a short option that becomes deep in-the-money can be assigned. In the normal construction of a box, this is unlikely, since you would own the deep call and put, but the stock price can move significantly and then find yourself in a situation where you might be assigned.

This risk increases for short boxes written on single stock options, as was the infamous case of a Robinhood trader who lost more than 2,000% on a short box when the deep puts that were sold were subsequently assigned, causing Robinhood to exercise the long calls in an effort to come up with the shares needed to satisfy the assignment. This debacle was posted online including on various subreddits, where it has become a cautionary tale (especially after said trader boasted that it was a virtually riskless strategy).

The lesson here is to avoid short boxes, or to only write short boxes on indexes (or similar) that instead use European options, which do not allow for early exercise.